Abstract
The results of this chapter deal with the existence of viable solutions for stochastic functional and backward inclusions. Weak compactness of sets of all viable weak solutions of stochastic functional inclusions is also considered.
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The results of this chapter deal with the existence of viable solutions for stochastic functional and backward inclusions. Weak compactness of sets of all viable weak solutions of stochastic functional inclusions is also considered.
1 Some Properties of Set-Valued Stochastic Functional Integrals Depending on Parameters
Let \(F : [0,T] \times {\mathbb{R}}^{d} \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) and \(G : [0,T] \times {\mathbb{R}}^{d} \rightarrow \mathrm{Cl}({\mathbb{R}}^{d\times m})\) be measurable and square integrably bounded set-valued mappings. Given a set-valued stochastic process (K(t))0 ≤ t ≤ T with values in \(\mathrm{Cl}({\mathbb{R}}^{d})\), we denote by \(\overline{SFI}(F,G,K)\) the following viability problem:
associated with \(\overline{SFI}(F,G)\). Similarly, we denote by BSDI(F, K) the backward viability problem:
associated with BSDI(F, K(T)).
We precede the existence theorems for such problems by some properties of set-valued stochastic functional integrals depending on parameters. Given a Banach space \((X,\|\cdot \|)\), by Cl(X) we denote the space of all nonempty closed subsets of X. In particular, we shall consider X to be equal to \({\mathbb{R}}^{d}\), \({\mathbb{L}}^{2}(\Omega, \mathcal{F}, {\mathbb{R}}^{r})\), and \({\mathbb{L}}^{2}([0,T] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{r}\) with r = d and r = d ×m, respectively. The Hausdorff metrics on these spaces will be denoted by h, D, and H, respectively.
Let \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\) be a filtered probability space with a filtration \(\mathbb{F} = (\mathcal{F}_{t})_{0\leq t\leq T}\) satisfying the usual conditions. Similarly as above, for set-valued mappings F and G as given above and an \(\mathbb{F}\)-nonanticipative d-dimensional stochastic process \(x = (x_{t})_{0\leq t\leq T}\), we shall denote by \(S_{\mathbb{F}}(F \circ x)\) and \(S_{\mathbb{F}}(G \circ x)\) the sets of all \(\mathbb{F}\)-nonanticipative stochastic processes \(f = (f_{t})_{0\leq t\leq T}\) and \(g = (g_{t})_{0\leq t\leq T}\), respectively, such that f t ∈ F(t, x t ) and g t ∈ G(t, x t ) a.s. for a.e. t ∈ [0, T]. It is clear that \(S_{\mathbb{F}}(F \circ x)\) and \(S_{\mathbb{F}}(G \circ x)\) are decomposable closed subsets of \({\mathbb{L}}^{2}([0,T] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{d})\) and \({\mathbb{L}}^{2}([0,T] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{d\times m})\), respectively, where \(\Sigma _{\mathbb{F}}\) denotes the σ-algebra of all \(\mathbb{F}\)-nonanticipative subsets of \([0,T] \times \Omega \). Therefore, by virtue of Theorem 3.2 of Chap. 2, there exist \(\Sigma _{\mathbb{F}}\)-measurable mappings \(\Phi \) and \(\Psi \) such that \(S_{\mathbb{F}}(F \circ x) = S_{\mathbb{F}}(\Phi )\) and \(S_{\mathbb{F}}(G \circ x) = S_{\mathbb{F}}(\Psi )\), which by virtue of Corollary 3.1 of Chap. 2, implies that \(\Phi = F \circ x\) and \(\Psi = G \circ x\).
In what follows, we shall denote by | ⋅| the norm of the Banach space \({\mathcal{X}}^{r} = {\mathbb{L}}^{2}([0,T] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{r})\) with r = d or r = d ×m. Similarly as above, \(\mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d})\) denotes the space of all d-dimensional continuous \(\mathbb{F}\)-adapted stochastic processes \(x = (x_{t})_{0\leq t\leq T}\) with norm \(\|x\| = {(E[\sup _{0\leq t\leq T}\vert x_{t}{\vert }^{2}])}^{1/2}\). Given a measurable and uniformly square integrably bounded set-valued mapping \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\), we shall assume that the set \(\mathcal{K}(t) =\{ u \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{t}, {\mathbb{R}}^{d}) : u \in K(t,\cdot \,)\,\,\,a.s.\}\) is nonempty for every 0 ≤ t ≤ T. It is clear that this requirement is satisfied for a square integrably bounded multifunction \(K : [0,T] \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\). Recall that \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) is said to be uniformly square integrably bounded if there exists \(\lambda \in {\mathbb{L}}^{2}([0,T], {\mathbb{R}}^{+})\) such that \(\|K(t,\omega )\| \leq \lambda (t)\) for a.e. \((t,\omega ) \in [0,T] \times \Omega \), where \(\|K(t,\omega )\| = h(K(t,\omega ),\{0\})\). Let us observe that for the above multifunctions F and G and a d-dimensional \(\mathcal{F}_{t}\)-measurable random variable X, the set-valued processes F ∘ X and G ∘ X are \(\beta _{T} \otimes \mathcal{F}_{t}\)-measurable.
Assume that the above set-valued mappings F and G satisfy the following conditions \((\mathcal{H}_{1})\):
-
(i)
\(F : [0,T] \times {\mathbb{R}}^{d} \rightarrow \ \mathrm{ Cl}({\mathbb{R}}^{d})\) and \(G : [0,T] \times {\mathbb{R}}^{d} \rightarrow \ \mathrm{ Cl}({\mathbb{R}}^{d\times m})\) are measurable and uniformly square integrably bounded, i.e., there exists \(m \in {\mathbb{L}}^{2}([0,T], {\mathbb{R}}^{+})\) such that \(\max (\|F(t,x)\|,\|G(t,x)\|) \leq m(t)\) for a.e. t ∈ [0, T] and \(x \in {\mathbb{R}}^{d}\), where \(\|F(t,x)\| =\sup \{ \vert z\vert : z \in F(t,x)\}\) and \(\|G(t,x)\| =\sup \{ \vert z\vert : z \in G(t,x)\}\);
-
(ii)
F(t, ⋅) and G(t, ⋅) are Lipschitz continuous for a.e. fixed t ∈ [0, T], i.e., there exists \(k \in {\mathbb{L}}^{2}([0,T], {\mathbb{R}}^{+})\) such that H(F(t, x), F(t, z)) ≤ k(t) | x − z | and H(G(t, x), G(t, z)) ≤ k(t) | x − z | for a.e. t ∈ [0, T] and \(x,z \in {\mathbb{R}}^{d}\).
Lemma 1.1.
If F and G satisfy conditions \((\mathcal{H}_{1})\) , then the set-valued mappings \(\mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d}) \ni x \rightarrow S_{\mathbb{F}}(F \circ x) \in \mathrm{ Cl}({\mathcal{X}}^{d})\) and \(\mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d}) \ni x \rightarrow S_{\mathbb{F}}(G \circ x) \in \mathrm{ Cl}({\mathcal{X}}^{d\times m})\) are Lipschitz continuous with Lipschitz constant \(L = {[\int _{0}^{T}{k}^{2}(t)\mathrm{d}t]}^{1/2}\).
Proof.
The proof is quite similar to the proof of Lemma 3.7 of Chap. 2. Let \(x,z \in \mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d})\) and \({f}^{x} \in S_{\mathbb{F}}(F \circ x)\). By virtue of Theorem 3.1 of Chap. 2 applied to the \(\Sigma _{\mathbb{F}}\)-measurable set-valued mapping F ∘ z, we get
Then \(\overline{H}(S_{\mathbb{F}}(F \circ x),S_{\mathbb{F}}(F \circ z)) \leq L\|x - z\|.\) In a similar way, we also get \(\overline{H}(S_{\mathbb{F}}(F \circ z),S_{\mathbb{F}}(F \circ x)) \leq L\vert \vert x - z\vert \vert.\) Therefore, \(H(S_{\mathbb{F}}(F \circ x),S_{\mathbb{F}}(F \circ z)) \leq L\|x - z\|\). In a similar way, we obtain \(H(S_{\mathbb{F}}(G \circ x),S_{\mathbb{F}}(G \circ z)) \leq L\|x - z\|\).
Lemma 1.2.
Let \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be \(\mathbb{F}\) -adapted and square integrably bounded uniformly with respect to t ∈ [0,T]. If K( ⋅,ω) is continuous for a.e. \(\omega \in \Omega \) , then the set-valued mapping \(\mathcal{K} : [0,T] \rightarrow \mathrm{ Cl}({\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d}))\) is continuous.
Proof.
Let t 0 ∈ [0, T] be fixed and let \((t_{k})_{k=1}^{\infty }\) be a sequence of [0, T] converging to t 0. By virtue of Theorem 3.1 of Chap. 2, for every \(u \in \mathcal{K}(t_{0})\) and k ≥ 1, one has
Then \({\overline{D}}^{2}(\mathcal{K}(t_{0}),\mathcal{K}(t_{k})) \leq E\left [{h}^{2}(K(t_{k},\cdot ),K(t_{0},\cdot ))\right ].\) In a similar way, we also get \({\overline{D}}^{2}(\mathcal{K}(t_{k}),\mathcal{K}(t_{0})) \leq E\left [{h}^{2}(K(t_{k},\cdot ),K(t_{0},\cdot ))\right ]\). Therefore, for every k ≥ 1, one has \({D}^{2}(\mathcal{K}(t_{k}),\mathcal{K}(t_{0})) \leq E\left [{h}^{2}(K(t_{k},\cdot ),K(t_{0},\cdot ))\right ].\) Hence, by the continuity of K( ⋅, ω) and its uniformly square integrable boundedness, it follows that \(\lim _{k\rightarrow \infty }D(\mathcal{K}(t_{k}),\mathcal{K}(t_{0})) = 0\).
Lemma 1.3.
If F and G satisfy conditions \((\mathcal{H}_{1})\) , then the set-valued mappings \(\mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d}) \ni x \rightarrow \mathrm{ cl}_{\mathbb{L}}\{J_{st}[S_{\mathbb{F}}(F \circ x)]\} \subset {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\) and \(\mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d}) \ni x \rightarrow \mathrm{ cl}_{\mathbb{L}}\{\mathcal{J}_{st}[S_{\mathbb{F}}(G \circ x)]\} \subset {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\) are Lipschitz continuous uniformly with respect to 0 ≤ s < t ≤ T with Lipschitz constants equal to \(\sqrt{T}L\) and L, respectively, where L is as in Lemma 1.1.
Proof.
Let \(x,z \in \mathcal{C}(\mathbb{F}, {\mathbb{R}}^{d})\) and \({f}^{x} \in S_{\mathbb{F}}(F \circ x)\). For fixed 0 ≤ s < t ≤ T, we have \(\mathrm{{dist}}^{2}\left (J_{st}({f}^{x}),J_{st}[S_{\mathbb{F}}(F \circ z]\right ) =\inf \left \{\,E\vert J_{st}({f}^{x} - {f}^{z}){\vert }^{2} : {f}^{z} \in S_{\mathbb{F}}(F \circ z)\right \}.\) But for every 0 ≤ s < t ≤ T, one has
Therefore, by Lemma 3.6 of Chap. 2, it follows that
Then for every 0 ≤ s < t ≤ T, one obtains
Similarly, for every fixed 0 ≤ s < t ≤ T, we also get
Therefore, for every 0 ≤ s < t ≤ T, one has
In a similar way, for fixed 0 ≤ s < t ≤ T, we obtain
Hence it follows that
and
Lemma 1.4.
Assume that F and G satisfy (i) of \((\mathcal{H}_{1})\) and let \(x_{n},x \in \mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d})\) for n = 1,2,… be such that \(\sup _{0\leq t\leq T}\vert x_{n}(t) - x(t)\vert \rightarrow 0\) a.s. as n →∞. If F(t,⋅) and G(t,⋅) are continuous for a.e. fixed 0 ≤ t ≤ T, and \((\theta _{n})_{n=1}^{\infty }\) is a sequence of functions θ n : [0,T] → [0,T] such that θ n (t) → t as n →∞ for every t ∈ [0,T], then \(\mathrm{cl}_{\mathbb{L}}\left \{J_{st}[S_{\mathbb{F}}(F \circ (x_{n} \circ \theta _{n}))] + \mathcal{J}_{st}[S_{\mathbb{F}}(G \circ (x_{n} \circ \theta _{n}))]\right \} \rightarrow \mathrm{ cl}_{\mathbb{L}}\left \{J_{st}[S_{\mathbb{F}}(F \circ x)] + \mathcal{J}_{st}[S_{\mathbb{F}}(G \circ x)]\right \}\) in the D-metric topology of \(\mathrm{Cl}({\mathbb{L}}^{2}\) \((\Omega, \mathcal{F}, {\mathbb{R}}^{d})\) as n →∞ for every 0 ≤ s ≤ t ≤ T.
Proof.
Let 0 ≤ s ≤ t ≤ T be fixed and set \({y}^{n} = x_{n} \circ \theta _{n}\) for every n = 1, 2, …. One has
for n = 1, 2, … and 0 ≤ τ ≤ T. Then y t n(τ) → x(τ) a.s. for every 0 ≤ τ ≤ T as n → ∞. Similarly as in the proof of Lemma 1.3, we can verify that the set-valued mappings \(\mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d}) \ni x \rightarrow \mathrm{ cl}_{\mathbb{L}}\{J_{st}[S_{\mathbb{F}}(F \circ x)]\} \in \mathrm{ Cl}({\mathbb{L}}^{2}(\Omega, \mathcal{F}, {\mathbb{R}}^{d}))\) and \(\mathbf{C}(\mathbb{F}, {\mathbb{R}}^{d}) \ni x \rightarrow \mathrm{ cl}_{\mathbb{L}}\{\mathcal{J}_{st}[S_{\mathbb{F}}(G \circ x)]\} \in \mathrm{ Cl}({\mathbb{L}}^{2}(\Omega, \mathcal{F}, {\mathbb{R}}^{d}))\) are continuous. Therefore, \(\mathrm{cl}_{\mathbb{L}}\left \{J_{st}[S_{\mathbb{F}}(F \circ (x_{n} \circ \theta _{n}))] + \mathcal{J}_{st}[S_{\mathbb{F}}(G \circ (x_{n} \circ \theta _{n}))]\right \} \rightarrow \mathrm{ cl}_{\mathbb{L}}\left \{J_{st}[S_{\mathbb{F}}(F \circ x)] + \mathcal{J}_{st}[S_{\mathbb{F}}(G \circ x)]\right \}\) in the D-metric topology as n → ∞.
2 Viable Approximation Theorems
The existence of solutions of viability problems (1.1) and (1.2) will follow from some viable approximation theorems by applying the standard methods presented in the proofs of the existence of strong and weak solutions for stochastic functional inclusions. We shall now present such approximation theorems. In what follows, it will be convenient to denote by d(x, A) the distance dist(x, A) of x ∈ X to a nonempty set A ⊂ X. We shall also denote the set-valued functional integrals \(J_{st}[S_{\mathbb{F}}\Phi )]\) and \(\mathcal{J}_{st}[S_{\mathbb{F}}\Psi )]\) of \(\mathbb{F}\)-nonanticipative set-valued processes \(\Phi \in \mathcal{L}_{\mathbb{F}}^{2}(T,\Omega, {\mathbb{R}}^{d})\) and \(\Psi \in \mathcal{L}_{\mathbb{F}}^{2}(T,\Omega, {\mathbb{R}}^{d\times m})\) by \(\int _{s}^{t}\Phi _{\tau }\mathrm{d}\tau\) and \(\int _{s}^{t}\Psi _{\tau }\mathrm{d}B_{\tau },\) respectively.
We shall prove the following approximation theorems.
Theorem 2.1.
Assume that F and G satisfy condition (i) of \((\mathcal{H}_{1})\) and let \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F},P)\) be a complete filtered probability space with a filtration \(\mathbb{F} = (\mathcal{F}_{t})_{0\leq t\leq T}\) such that there exists an m-dimensional \(\mathbb{F}\) -Brownian motion \(B = (B_{t})_{0\leq t\leq T}\) defined on \(\mathcal{P}_{\mathbb{F}}\) . Let \(K : [0,T] \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be such that a set-valued process \((\mathcal{K}(t))_{0\leq t\leq T}\) is continuous. If
for every \((t,x) \in Graph({\mathcal{K}}^{\varepsilon })\) and every \(\varepsilon \in (0,1)\) , where \({\mathcal{K}}^{\varepsilon }(t) =\{ u \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{t}, {\mathbb{R}}^{d}) : d(u,\mathcal{K}(t)) \leq \varepsilon \}\) for every 0 ≤ t ≤ T, then for every \(\varepsilon \in (0,1)\) and \(x_{0} \in \mathcal{K}(0)\) , there exist a step function \(\theta _{\varepsilon } : [0,T] \rightarrow [0,T]\) and \(\mathbb{F}\) -nonanticipative stochastic processes \({f}^{\varepsilon } = (f_{t}^{\varepsilon })_{0\leq t\leq T}\) and \({g}^{\varepsilon } = (g_{t}^{\varepsilon })_{0\leq t\leq T}\) such that
-
(i)
\({f}^{\varepsilon } \in S_{\mathbb{F}}(F \circ ({x}^{\varepsilon } \circ \theta _{\varepsilon }))\) and \({g}^{\varepsilon } \in S_{\mathbb{F}}(G \circ ({x}^{\varepsilon } \circ \theta _{\varepsilon }))\) , where \({x}^{\varepsilon }(t) = x_{0} +\int _{ 0}^{t}f_{\tau }^{\varepsilon }\mathrm{d}\tau +\int _{ 0}^{t}g_{\tau }^{\varepsilon }\mathrm{d}B_{\tau }\) for 0 ≤ t ≤ T;
-
(ii)
\(E[\mathrm{dist}({x}^{\varepsilon }(\theta _{\varepsilon }(t)),K(\theta _{\varepsilon }(t))] \leq \varepsilon\) for 0 ≤ t ≤ T;
-
(iii)
\(E\left [l({x}^{\varepsilon }(s))\left (h({x}^{\varepsilon }(t)) - h({x}^{\varepsilon }(s)) -\int _{s}^{t}(\mathbb{L}_{{f}^{\varepsilon }{g}^{\varepsilon }}^{{x}^{\varepsilon } }h)_{\tau }\mathrm{d}\tau \right )\right ] = 0\) for every 0 ≤ s ≤ t ≤ T, \(l \in C_{b}({\mathbb{R}}^{d}, \mathbb{R})\) and \(h \in C_{b}^{2}({\mathbb{R}}^{d}, \mathbb{R})\).
Proof.
Let \(\varepsilon \in (0,1)\) and \(x_{0} \in \mathcal{K}(0)\) be fixed. Select \(\delta \in (0,\varepsilon )\) such that \(\int _{t}^{t+\delta }{m}^{2}(\tau )\mathrm{d}\tau {\leq \varepsilon }^{2}/{2}^{4}\) and \(D(\mathcal{K}(t+\delta ),\mathcal{K}(t)) \leq \varepsilon /{2}^{2}\) for t ∈ [0, T]. By virtue of (2.1), there exists h 0 ∈ (0, δ) such that
Then for every \(u_{0} \in x_{0} + cl_{\mathbb{L}}\left (\int _{0}^{h_{0}}F(\tau, x_{0})\mathrm{d}\tau +\int _{ 0}^{h_{0}}G(\tau, x_{0})\mathrm{d}B_{\tau }\right )\), one has \(d(u_{0},\mathcal{K}(h_{0})) \leq \varepsilon h_{0}/{2}^{2}\). Let t 0 = 0 and t 1 = h 0. Select arbitrarily \(\beta _{T} \otimes \mathcal{F}_{0}\)-measurable selectors f 0 and g 0 of F ∘ x 0 and G ∘ x 0, respectively. It is clear that \({f}^{0} \in S_{\mathbb{F}}(F \circ x_{0}))\) and \({g}^{0} \in S_{\mathbb{F}}(G \circ x_{0}))\). Let \({x}^{\varepsilon }(t) = x_{0} +\int _{ 0}^{t}f_{\tau }^{0}\mathrm{d}\tau +\int _{ 0}^{t}g_{\tau }^{0}\mathrm{d}B_{\tau }\) for 0 ≤ t ≤ t 1. Put \(\theta _{\varepsilon }(t) = 0\) for 0 ≤ t < t 1 and \(\theta _{\varepsilon }(t_{1}) = t_{1}\). We have
Therefore, \(d({x}^{\varepsilon }(h_{0}),\mathcal{K}(h_{0})) \leq \varepsilon h_{0}/{2}^{2} \leq \varepsilon /{2}^{2}\). Together with the properties of the number δ > 0, it follows that
for 0 ≤ t ≤ t 1, because
for 0 ≤ t ≤ t 1. Let \(x_{1} \in \mathcal{K}(t_{1})\) be such that \(\|{x}^{\varepsilon }(h_{0}) - x_{1}\| \leq d({x}^{\varepsilon }(h_{0}),\mathcal{K}(h_{0})) +\varepsilon /{2}^{2}\). Hence, by Theorem 3.1 of Chap. 2, it follows that
By Itô’s formula, for every \(h \in C_{b}^{2}({\mathbb{R}}^{d}, \mathbb{R})\) and 0 ≤ s ≤ t ≤ T, we have
a.s., where B = (B 1, …, B m) and \(g_{\tau }^{0} = (g_{ij}^{0}(\tau ))_{d\times m}\). But \({x}^{\varepsilon }(s)\) is \(\mathcal{F}_{s}\)-measurable. Then for 0 ≤ s ≤ t ≤ t 2, i = 1, 2, …, n, and j = 1, 2, …, m, we have
Therefore, for every \(l \in C_{b}({\mathbb{R}}^{d}, \mathbb{R})\), \(h \in C_{b}^{2}({\mathbb{R}}^{d}, \mathbb{R})\), and 0 ≤ s ≤ t ≤ t 1, we get
Suppose h 0 < T. We have \((h_{0},{x}^{\varepsilon }(h_{0})) \in Graph({\mathcal{K}}^{\varepsilon })\) because \(d({x}^{\varepsilon }(h_{0}),\mathcal{K}(h_{0})) \leq \varepsilon\). Therefore, we can repeat the above procedure and select h 1 ∈ (0, δ) such that
Similarly as above, we can select \({f}^{1} \in S_{\mathbb{F}}(F \circ {x}^{\varepsilon }(h_{0}))\) and \({g}^{1} \in S_{\mathbb{F}}(G \circ {x}^{\varepsilon }(h_{0})),\) and define \({x}^{\varepsilon }(t) = {x}^{\varepsilon }(t_{1}) +\int _{ t_{1}}^{t}f_{\tau }^{1}\mathrm{d}\tau +\int _{ t_{1}}^{t}g_{\tau }^{1}\mathrm{d}B_{\tau }\) for t 1 ≤ t ≤ t 2, where \(t_{2} = t_{1} + h_{1}\). We can also extend the function θ on [0, t 2] by taking θ(t) = t 1 for t 1 ≤ t < t 2 and \(\theta _{\varepsilon }(t_{2}) = t_{2}\). We have
Therefore, for every t 1 ≤ t ≤ t 2, one has
because similarly as above, we get \(\|{x}^{\varepsilon }(t) - {x}^{\varepsilon }(t_{2})\| \leq \varepsilon /2\) for every t 1 ≤ t ≤ t 2. Similarly as above, for every \(l \in C_{b}({\mathbb{R}}^{d}, \mathbb{R})\), \(h \in C_{b}^{2}({\mathbb{R}}^{d}, \mathbb{R})\), and t 1 ≤ s ≤ t ≤ t 2, we also get
Let \(x_{2} \in \mathcal{K}(t_{2})\) be such that \(\|{x}^{\varepsilon }(t_{2}) - x_{2}\| \leq d({x}^{\varepsilon }(t_{2}),\mathcal{K}(t_{2})) +\varepsilon /{2}^{2}\). Hence it follows that \(E[\mathrm{dist}({x}^{\varepsilon }(t_{2}),K(t_{2}))] \leq \varepsilon\). Let us observe that the above relations can be written in the form presented in (i)–(iii) above with T = t 2, where \({f}^{\varepsilon } = 1\!\!1 _{[0,t_{1})}{f}^{0} + 1\!\!1 _{(t_{1},t_{2}]}{f}^{1}\), \({g}^{\varepsilon } = 1\!\!1 _{[0,t_{1})}{g}^{0} + 1\!\!1 _{(t_{1},t_{2}]}{g}^{1}\) and \({x}^{\varepsilon }(t) = x_{0} +\int _{ 0}^{t}f_{\tau }^{\varepsilon }\mathrm{d}\tau +\int _{ 0}^{t}g_{\tau }^{\varepsilon }\mathrm{d}B_{\tau }\) for 0 ≤ t ≤ t 2.
Continuing the above procedure, we can extend the function \(\theta _{\varepsilon }\) and processes \({f}^{\varepsilon }\), \({g}^{\varepsilon }\), and \({x}^{\varepsilon }\) on the whole interval [0, T] such that the above conditions (i)–(iii) are satisfied. To see this, let us denote by \(\Lambda _{\varepsilon }\) the set of all extensions of the vector function \(\Phi _{\varepsilon } = (\theta _{\varepsilon },{f}^{\varepsilon },{g}^{\varepsilon },{x}^{\varepsilon })\) on \([0,\alpha ] \times \Omega \) with α ∈ (0, T] and \(\theta _{\varepsilon }\vert _{[0,\alpha ]}\) not depending on \(\omega \in \Omega \). We have \(\Lambda _{\varepsilon }\neq \varnothing \). Let us introduce in \(\Lambda _{\varepsilon }\) the partial order relation ≼ by setting \(\Phi _{\varepsilon }^{\alpha } \preccurlyeq \Phi _{\varepsilon }^{\beta }\) if and only if α ≤ β and \(\Phi _{\varepsilon }^{\alpha } = \Phi _{\varepsilon }^{\beta }\vert _{[0,\alpha ]}\), where \(\Phi _{\varepsilon }^{\alpha }\) and \(\Phi _{\varepsilon }^{\beta }\) denote extensions of \(\Phi _{\varepsilon }\) to [0, α] and [0, β], respectively. Let \(P_{\varepsilon }^{\alpha }\) be a set containing an extension \(\Phi _{\varepsilon }^{\alpha }\) and all its restrictions \(\Phi _{\varepsilon }^{\alpha }\vert _{[0,a]}\) for every a ∈ (0, α]. It is clear that each completely ordered subset of \(\Lambda _{\varepsilon }\) is of the form \(P_{\varepsilon }^{\alpha }\) determined by some extension \(\Phi _{\varepsilon }^{\alpha }\). It is also clear that every set \(P_{\varepsilon }^{\alpha }\) has \(\Phi _{\varepsilon }^{\alpha }\) as its upper bound. Then by the Kuratowski–Zorn lemma, there exists a maximal element \(\Psi _{\varepsilon }\) of \(\Lambda _{\varepsilon }\) defined on \([0,b] \times \Omega \) with b ≤ T. It has to be b = T. Indeed, if it were b < T, then we could repeat the above procedure and extend \(\Psi _{\varepsilon }\) to the vector function \(\Gamma _{\varepsilon }\) defined on \([0,\gamma ] \times \Omega \) with b ≤ γ. It would be \(\Psi _{\varepsilon } \preccurlyeq \Gamma _{\varepsilon }\), in contradiction to the assumption that \(\Psi _{\varepsilon }\) is a maximal element of \(\Lambda _{\varepsilon }\). Then \(\Phi _{\varepsilon }\) can be extended on \([0,T] \times \Omega \) in such a way that conditions (i)–(iii) are satisfied.
Remark 2.1.
Theorem 2.1 is also true if instead of (2.1), we assume that
for every \((t,x) \in \ \mathrm{Graph}\ ({\mathcal{K}}^{\varepsilon })\).
Theorem 2.2.
Assume that F and G satisfy conditions \((\mathcal{H}_{1})\) . Suppose \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F},P)\) is a complete filtered probability space with a filtration \(\mathbb{F} = (\mathcal{F}_{t})_{0\leq t\leq T}\) such that there exists an m-dimensional \(\mathbb{F}\) -Brownian motion \(B = (B_{t})_{0\leq t\leq T}\) defined on \(\mathcal{P}_{\mathbb{F}}\) . Let \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be \(\mathbb{F}\) -nonanticipative such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T and \((\mathcal{K}(t))_{0\leq t\leq T}\) is continuous. If (2.1) is satisfied for every \((t,x) \in Graph(\mathcal{K})\) , then for every \(\varepsilon \in (0,1)\) , a ∈ (0,T), \(x_{0} \in \mathcal{K}(0)\,\) , and \(\mathbb{F}\) -nonanticipative processes \(\phi = (\phi _{t})_{\,0\leq t\leq T}\) and \(\psi = (\psi _{t})_{0\leq t\leq T}\) with \(\phi _{t} \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\), \(\psi _{t} \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d\times m})\) for 0 ≤ t ≤ T and \((\phi _{0},\psi _{0}) \in F(0,x_{0}) \times G(0,x_{0})\) a.s., there exist a partition \(0 = t_{0} < t_{1} < \cdots < t_{p} = a\,\) of the interval [0,a], a step function \(\theta _{\varepsilon } : [0,a] \rightarrow [0,a]\), \(\mathbb{F}\) -nonanticipative stochastic processes \({f}^{\varepsilon } = (f_{t}^{\varepsilon })_{0\leq t\leq a}\) and \({g}^{\varepsilon } = (g_{t}^{\varepsilon })_{0\leq t\leq a}\) , and a step stochastic process \({z}^{\varepsilon } = ({z}^{\varepsilon }(t))_{0\leq t\leq a}\) such that
-
(i)
\(t_{j+1} - t_{j} \leq \delta\) , where \(\delta \in (0,\varepsilon )\) is such that \(\max \left (\int _{t}^{t+\delta }{k}^{2}(\tau )\mathrm{d}\tau, \int _{t}^{t+\delta }{m}^{2}(\tau )\mathrm{d}\tau \right )\) \({\leq \varepsilon }^{2}/{2}^{4}\) and \(D(\mathcal{K}(t+\delta ),\mathcal{K}(t)) \leq \varepsilon /{2}^{2}\) for t ∈ [0,T];
-
(ii)
\(\theta _{\varepsilon }(t) = t_{j}\) for t j ≤ t < t j+1 for \(j = 0,1,\ldots p - 2\) and \(\theta _{\varepsilon }(t) = t_{p-1}\) for t p−1 ≤ t ≤ a;
-
(iii)
\({f}^{\varepsilon } \in S_{\mathbb{F}}(F \circ ({x}^{\varepsilon } \circ \theta _{\varepsilon }))\), \({g}^{\varepsilon } \in S_{\mathbb{F}}(G \circ ({x}^{\varepsilon } \circ \theta _{\varepsilon }))\), \(\vert \phi _{t}(\omega ) - f_{t}^{\varepsilon }(\omega )\vert \leq \mathrm{ dist}(\phi _{t},F(t,({x}^{\varepsilon } \circ \theta _{\varepsilon })(t)))\) and \(\vert \psi _{t}(\omega ) - g_{t}^{\varepsilon }(\omega )\vert \leq \mathrm{ dist}(\psi _{t},G(t,({x}^{\varepsilon } \circ \theta _{\varepsilon })(t)))\) for \((t,\omega ) \in [0,a] \times \Omega \) , where \({x}^{\varepsilon }(t) = x_{0} +\int _{ 0}^{t}(f_{\tau }^{\varepsilon } + {z}^{\varepsilon }(\tau ))\mathrm{d}\tau +\int _{ 0}^{t}g_{\tau }^{\varepsilon }\mathrm{d}B_{\tau }\) a.s. for 0 ≤ t ≤ a;
-
(iv)
\(\|{z}^{\varepsilon }(t)\| \leq \varepsilon /{2}^{2}\) for 0 ≤ t ≤ a, where \(\|{z}^{\varepsilon }{(t)\|}^{2} = E\vert z(t){\vert }^{2}\);
-
(v)
\(d({x}^{\varepsilon }(\theta _{\varepsilon }(t)),\mathcal{K}(\theta _{\varepsilon }(t)) = 0\) for 0 ≤ t ≤ a;
-
(vi)
\(d\left ({x}^{\varepsilon }(t) - {x}^{\varepsilon }(s),\mathrm{cl}_{\mathbb{L}}\left (\int _{s}^{t}F(\tau, ({x}^{\varepsilon } \circ \theta _{\varepsilon })(\tau ))\mathrm{d}\tau \, +\,\int _{ s}^{t}G(\tau, ({x}^{\varepsilon } \circ \theta _{\varepsilon })(\tau ))\mathrm{d}B_{\tau }\right )\right )\) \(\leq \varepsilon\) for every 0 ≤ s < t ≤ a.
Proof.
Let \(x_{0} \in \mathcal{K}(0)\), \(\varepsilon \in (0,1)\) and a ∈ (0, T) be fixed. Without loss of generality, we can assume that T = 1. By virtue of (2.1), there exists h 0 ∈ (0, δ) such that
where δ > 0 is such that condition (i) is satisfied. By virtue of Corollary 2.3 of Chap. 2 applied to \(\Sigma _{\mathbb{F}}\)-measurable multifunctions F ∘ x 0 and G ∘ x 0, and given the above processes ϕ and ψ, there exist \({f}^{0} \in S_{\mathbb{F}}(F \circ x_{0})\) and \({g}^{0} \in S_{\mathbb{F}}(G \circ x_{0})\) such that \(\vert \phi _{t}(\omega ) - f_{t}^{0}(\omega )\vert =\mathrm{ dist}(\phi _{t},F(t,x_{0}))\) and \(\vert \psi _{t}(\omega ) - g_{t}^{0}(\omega )\vert =\mathrm{ dist}(\psi _{t},G(t,x_{0}))\) for \((t,\omega ) \in [0,a] \times \Omega \). Similarly as in the proof of Theorem 2.1, we define now the function \(\theta _{\varepsilon }\) by taking \(\theta _{\varepsilon }(t) = 0\) for 0 ≤ t < t 1 and \(\theta _{\varepsilon }(t_{1}) = t_{1},\) where t 1 = h 0. Hence it follows that \(f_{t}^{0} \in F(t,\theta _{\varepsilon }(t))\) and \(g_{t}^{0} \in G(t,\theta _{\varepsilon }(t))\) a.s. for 0 ≤ t < t 1. Let \(y_{0} = x_{0} +\int _{ 0}^{t_{1}}f_{\tau }^{0}\mathrm{d}\tau +\int _{ 0}^{t_{1}}g_{\tau }^{0}\mathrm{d}B_{\tau }\) a.s. We have
Then \(d(y_{0},\mathcal{K}(h_{0})) \leq \varepsilon h_{0}/{2}^{2}\), which by Theorem 3.1 of Chap. 2, implies that \({d}^{2}(y_{0},\mathcal{K}(h_{0})) = E[\mathrm{dist}(y_{0},K(h_{0},\cdot \,){]}^{2}\). Therefore, by Corollary 2.3 of Chap. 2, there exists an \(\mathcal{F}_{t_{1}}\)-measurable random variable x 1 such that x 1 ∈ K(h 0, ⋅) for \(\omega \in \Omega \) and
Define \(z_{t}^{\varepsilon } = (1/h_{0})(x_{1} - y_{0})\) a.s. for 0 ≤ t ≤ t 1. We get \(\vert \vert {z}^{\varepsilon }(t)\vert \vert \leq (1/h_{0})\vert \vert x_{1} - y_{0}\vert \vert \leq (1/h_{0})(\varepsilon h_{0}/{2}^{2}) =\varepsilon /4\) for 0 ≤ t ≤ t 1. We define now a process \({x}^{\varepsilon }\) on [0, t 1) by setting
We have \({x}^{\varepsilon }(0) = x_{0} \in \mathcal{K}(0)\) and \({x}^{\varepsilon }(t_{1}) = y_{0} + h_{0}(1/h_{0})(x_{1} - y_{0}) = x_{1} \in \mathcal{K}(h_{0}),\) which is equivalent to \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t),\mathcal{K}(\theta _{\varepsilon }(t))) = 0\) for t ∈ [0, t 1]. Similarly, for 0 ≤ s ≤ t < t 1, one obtains
If h 0 < a, we can repeat the above procedure. Applying (2.1) to \((t_{1},x_{1}) \in Graph(\mathcal{K})\), we can select h 1 ∈ (0, δ) such that
Similarly as above, we can select \(x_{2} \in \mathcal{K}(t_{1} + h_{1})\), \({f}^{1} \in S_{\mathbb{F}}(F \circ x_{1})\), and \({g}^{1} \in S_{\mathbb{F}}(G \circ x_{1})\) such that \(\vert \phi _{t}(\omega ) - f_{t}^{1}(\omega )\vert =\mathrm{ dist}(\phi _{t},F(t,{x}^{1}))\) and \(\vert \psi _{t}(\omega ) - g_{t}^{1}(\omega )\vert =\mathrm{ dist}(\psi _{t},G(t,{x}^{1}))\) for \((t,\omega ) \in [0,a] \times \Omega \) and \(\|y_{1} - x_{2}\| \leq \varepsilon h_{1}/{2}^{2},\) where \(y_{1} = x_{1} +\int _{ t_{1}}^{t_{1}+h_{1}}f_{\tau }^{1}\mathrm{d}\tau +\int _{ t_{ 1}}^{t_{1}+h_{1}}g_{\tau }^{1}\mathrm{d}B_{\tau }\) a.s. We can extend the function \(\theta _{\varepsilon }\) and the process \({z}^{\varepsilon }\) on the interval [0, t 2] by setting \(\theta _{\varepsilon }(t) = t_{1}\) for t 1 ≤ t < t 2, θ(t 2) = t 2, and \({z}^{\varepsilon }(t) = (1/h_{1})(x_{2} - y_{1})\) for t 1 < t ≤ t 2, where \(t_{2} = t_{1} + h_{2}\). Define on the interval [0, t 2] the process \({x}^{\varepsilon }\) by setting
where \({f}^{\varepsilon } = 1\!\!1 _{[0,t_{1})}{f}^{0} + 1\!\!1 _{[t_{1},t_{2})}{f}^{1}\) and \({g}^{\varepsilon } = 1\!\!1 _{[0,t_{1})}{g}^{0} + 1\!\!1 _{[t_{1},t_{2})}{g}^{1}\). Similarly as above, we obtain \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t),\mathcal{K}(\theta _{\varepsilon }(t))) = 0\) for 0 ≤ t < t 2 and \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t_{2}),\) \(\mathcal{K}(\theta _{\varepsilon }(t_{2}))) = 0\), because \({x}^{\varepsilon }(t_{2}) = x_{2}\). Then \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t),\mathcal{K}(\theta _{\varepsilon }(t))) = 0\) for 0 ≤ t ≤ t 2. It is clear that \(\|z_{t}^{\varepsilon }\| \leq \varepsilon /4 \leq \varepsilon\) for every 0 ≤ t ≤ t 2. Then for every 0 ≤ s ≤ t ≤ t 2, we get
Suppose that for some i ≥ 1, the inductive procedure is realized on [0, t i ) ⊂ [0, a] and the above step function \(\theta _{\varepsilon }\), and stochastic processes \({z}^{\varepsilon }\) \({f}^{\varepsilon },\) \({g}^{\varepsilon }\), and \({x}^{\varepsilon }\) are extended to [0, t i ] and [0, t i ), respectively, with the above properties on this interval. Denote by S i the set of all positive numbers h such that \(h \in (0,\min (\delta, a - t_{i}))\) and
where \(x_{i} = {x}^{\varepsilon }(t_{i})\). We have S i ≠∅ and \(\sup S_{i} > 0\). Choose h i ∈ S i such that \(\sup S_{i} - (1/2)\sup S_{i} \leq h_{i}\). Put \(t_{i+1} = t_{i} + h_{i}\) and let \({f}^{i} \in S_{\mathbb{F}}(F \circ x_{i})\) and \({g}^{i} \in S_{\mathbb{F}}(G \circ x_{i})\) be such that \(\vert \phi _{t}(\omega ) - f_{t}^{i}(\omega )\vert =\mathrm{ dist}(\phi _{t},F(t,x_{i}))\) and \(\vert \psi _{t}(\omega ) - g_{t}^{i}(\omega )\vert =\mathrm{ dist}(\psi _{t},G(t,x_{i})).\) We can now extend \(\theta _{\varepsilon }\), \({f}^{\varepsilon }\), and \({g}^{\varepsilon }\) to the interval [0, t i + 1] by taking \(\theta _{\varepsilon }(t) = t_{i}\) for t i ≤ t < t i + 1 and \(\theta _{\varepsilon }(t_{i+1}) = t_{i+1}\,\), \(f_{t}^{\varepsilon } = f_{t}^{i}\), \(g_{t}^{\varepsilon } = g_{t}^{i}\) for t i ≤ t < t i + 1. Then \(\vert \phi _{t}(\omega ) - f_{t}^{\varepsilon }(\omega )\vert \leq \mathrm{ dist}(\phi _{t},F(t,({x}^{\varepsilon } \circ \theta _{\varepsilon })(t)))\) and \(\vert \psi _{t}(\omega ) - g_{t}^{\varepsilon }(\omega )\vert \leq \mathrm{ dist}(\psi _{t},G(t,({x}^{\varepsilon } \circ \theta _{\varepsilon })(t)))\) for \((t,\omega ) \in [0,t_{i+1}) \times \Omega \), where
a.s. for 0 ≤ t ≤ t i + 1 with
a.s for t i < t ≤ t i + 1, where \(x_{i+1} \in \mathcal{K}(t_{i+1})\) is such that
Similarly as above, we obtain \(\vert \vert {z}^{\varepsilon }(t)\vert \vert \leq \varepsilon /4\) for t i < t ≤ t i + 1. Hence it follows that
for 0 ≤ s < t < t i + 1 and \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t),\mathcal{K}(\theta _{\varepsilon }(t))) = 0\) for 0 ≤ t ≤ t 2.
We can continue the above procedure up to n > 1 such that t n ∈ [a, 1]. Suppose to the contrary that such n > 1 does not exist, i.e., that for every n > 1, one has 0 < t n < a. Then we obtain a sequence \((t_{i})_{i=1}^{\infty }\) converging to t ∗ ≤ a such that for every 0 ≤ j < k ≤ i + 1 and i ≥ 0, we have
Let \(x_{j} = {x}^{\varepsilon }(t_{j})\) and \(x_{k} = {x}^{\varepsilon }(t_{k})\) for 0 ≤ j < k < ∞. For every 0 ≤ j < k < ∞, one gets
Then \((x_{i})_{i=1}^{\infty }\) is a Cauchy sequence of \({\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\). Therefore, there exists \({x}^{{\ast}}\in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\) such that \(\|x_{i} - {x}^{{\ast}}\|\rightarrow 0\) as i → ∞. By the continuity of the set-valued mapping \(\mathcal{K}\), we get \(({t}^{{\ast}},{x}^{{\ast}}) \in Graph(\mathcal{K})\). Then by (2.1), there exists \({h}^{{\ast}}\in (0,\min (\delta, 1 - a))\) such that
Let N > 1 be such that for every i ≥ N, one has \(0 < {t}^{{\ast}}- t_{i} <\min ({h}^{{\ast}},a,\eta _{\varepsilon })\), \(\|x_{i} - {x}^{{\ast}}\|\leq \varepsilon {h}^{{\ast}}/({2}^{6}A)\), and \(D(\mathcal{K}(t_{i}),\mathcal{K}({t}^{{\ast}})) \leq \varepsilon {h}^{{\ast}}/{2}^{6}\), where \(A = 1 + 2{\left (\int _{0}^{1}{k}^{2}(t)\mathrm{d}t\right )}^{1/2}\) and \(\eta _{\varepsilon } \in (0,1 - a)\) is such that \({\left (\int _{t}^{t+\eta _{\varepsilon }}{m}^{2}(\tau )\mathrm{d}\tau \right )}^{1/2} \leq \varepsilon {h}^{{\ast}}/{2}^{7}\) for every 0 ≤ t ≤ a. For every i ≥ N and arbitrarily taken \({\phi }^{i} \in S_{\mathbb{F}}(F \circ x_{i})\) and \({\psi }^{i} \in S_{\mathbb{F}}(G \circ x_{i})\), we can select \({f}^{{\ast}}\in S_{\mathbb{F}}(F \circ {x}^{{\ast}})\) and \({g}^{{\ast}}\in S_{\mathbb{F}}(G \circ {x}^{{\ast}})\) such that \(\vert \phi _{t}^{i}(\omega ) - f_{t}^{{\ast}}(\omega )\vert =\mathrm{ dist}{(\phi }^{i},F(t,{x}^{{\ast}}))\) and \(\vert \psi _{t}^{i}(\omega ) - g_{t}^{{\ast}}(\omega )\vert =\mathrm{ dist}{(\psi }^{i},G(t,{x}^{{\ast}}))\) for \((t,\omega ) \in [t_{i},{t}^{{\ast}} + {h}^{{\ast}}] \times \Omega \). In particular, this implies
and
for i ≥ 1. Therefore, for every i ≥ N, we get
Then for every i ≥ N, we have
and \({h}^{{\ast}}\in (0,\min (\delta, 1 - a))\). But t i < a for every i ≥ 1. Then \(1 - a < 1 - t_{i}\) for every i ≥ 1. Therefore, for every i ≥ N, we have h ∗ ∈ (0, min(δ, 1 − t i )). Hence it follows that h ∗ ∈ S i for every i ≥ N. Then for every i ≥ N, one has \((1/2){h}^{{\ast}}\leq (1/2)\sup S_{i} \leq h_{i} = t_{i+1} - t_{i}\), which contradicts the convergence of the sequence \((t_{i})_{i=1}^{\infty }\). Therefore, there exists p ≥ 1 such that \(0 = t_{0} < t_{1} < \cdots < t_{p} = a\).
Remark 2.2.
Theorem 2.2 is also true if instead of (2.1), we assume that (2.2) is satisfied for every \((t,x) \in Graph(\mathcal{K})\).
Theorem 2.3.
Assume that F satisfies conditions \((\mathcal{H}_{1})\) , and let \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F},P)\) be a complete filtered probability space with a continuous filtration \(\mathbb{F} = (\mathcal{F}_{t})_{0\leq t\leq T}\) such that \(\mathcal{F}_{T} = \mathcal{F}\) . Suppose \(\,K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) is an \(\mathbb{F}\) -adapted set-valued process such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T and such that the set-valued mapping \(\mathcal{K} : [0,T] \rightarrow \mathrm{ Cl}(\mathbb{L}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\) is continuous. If
is satisfied for every \((t,x) \in Graph(\mathcal{K})\) , where \(S(E[x +\int _{ t-h}^{t}F(\tau, x)\mathrm{d}\tau \vert \mathcal{F}_{t-h}]) =\{ E[x +\int _{ t-h}^{t}f_{\tau }\mathrm{d}\tau \vert \mathcal{F}_{t-h}] : f \in S(coF \circ x)\},\) then for every \(\varepsilon \in (0,1)\), \(x_{T} \in \mathcal{K}(x_{T})\) , a ∈ (0,T) and measurable process ϕ = (ϕ) 0≤t≤T such that \(\phi _{t} \in \mathbb{L}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\) for 0 ≤ t ≤ T and ϕ T ∈ F(T,x T ) a.s., there exist a partition \(a = t_{p} < t_{p-1} < \cdots < t_{1} < t_{0} = T\) of the interval [a,T], a step function \(\theta _{\varepsilon } : [a,T] \rightarrow [a,T],\) a step stochastic process \({z}^{\varepsilon } = (z_{t}^{\varepsilon })_{a\leq t\leq T}\) , and a measurable process \({f}^{\varepsilon } = (f_{t}^{\varepsilon })_{a\leq t\leq T}\) on \(\mathcal{P}_{\mathbb{F}}\) such that
-
(i)
\(t_{j} - t_{j+1} \leq \delta\) , where \(\delta \in (0,\varepsilon )\) is such that \(\max \{\int _{t}^{t+\delta }k(\tau )\mathrm{d}\tau, \int _{t}^{t+\delta }m(\tau )\mathrm{d}\tau \} {\leq \varepsilon }^{2}/{2}^{4}\) and \(D(\mathcal{K}(t+\delta ),\mathcal{K}(t)) \leq \varepsilon /2\) for t ∈ [0,T];
-
(ii)
\(\|z_{t}^{\varepsilon }\| \leq \varepsilon /2\) for every a ≤ t ≤ T, where \(\|z_{t}^{\varepsilon }\| = E\vert z_{t}^{\varepsilon }\vert \);
-
(iii)
\(\theta _{\varepsilon }(t) = t_{j-1}\) for t j < t ≤ t j−1 and \(\theta _{\varepsilon }(t_{j}) = t_{j}\) with \(j = 1,\ldots, p - 1\) and \(\theta _{\varepsilon }(t) = t_{p-1}\) for a ≤ t ≤ t p−1;
-
(iv)
\({f}^{\varepsilon } \in S(\mathrm{co}\,F \circ ({x}^{\varepsilon } \circ \theta _{\varepsilon }))\), \(\vert \phi _{t}(\omega ) - f_{t}^{\varepsilon }(\omega )\vert =\mathrm{ dist}(\phi _{t},\mathrm{co}\,F(t,({x}^{\varepsilon } \circ \theta _{\varepsilon })(t)))\) for \((t,\omega ) \in [a,T] \times \Omega \) , where \({x}^{\varepsilon }(t) = E[x_{T} +\int _{ t}^{T}f_{\tau }^{\varepsilon }\mathrm{d}\tau \vert \mathcal{F}_{t}] +\int _{ t}^{T}z_{\tau }^{\varepsilon }\mathrm{d}\tau\) a.s. for a ≤ t ≤ T and \(S(\mathrm{co}\,F \circ ({x}^{\varepsilon } \circ \theta _{\varepsilon }))\) = \(\{f \in \mathbb{L}([a,T] \times \Omega, \beta _{T} \otimes \mathcal{F}_{T}, {\mathbb{R}}^{d}) : f_{t} \in \mathrm{ co}\,F(t,{x}^{\varepsilon }(\theta _{\varepsilon }(t)))\,\,\,\mathrm{a.s.}\,\,\,\mathrm{for}\,\,\,a.e.\,\,\,a \leq t \leq T\}\);
-
(v)
\(E[\mathrm{dist}({x}^{\varepsilon }(s),E[{x}^{\varepsilon }(t) +\int _{ s}^{t}F(\tau, ({x}^{\varepsilon } \circ \theta _{\varepsilon })(\tau )\mathrm{d}\tau \vert \mathcal{F}_{s}])] \leq \varepsilon\) for a ≤ s ≤ t ≤ T,
-
(vi)
\(d({x}^{\varepsilon }(\theta _{\varepsilon }(t)),\mathcal{K}(\theta _{\varepsilon }(t))) = 0\) for a ≤ t ≤ T.
Proof.
Let \(\varepsilon \in (0,1)\), \(a \in (0,T)\), \(x_{T} \in \mathcal{K}(T)\), and a measurable process ϕ = (ϕ)0 ≤ t ≤ T be given. By virtue of (2.3), there exists \(h_{0} \in (0,\min (\delta, T))\) such that
Let \(t_{1} = T - h_{0}\). By virtue of Corollary 2.3 of Chap. 2, there exists f 0 ∈ S(co F ∘ x T ) such that \(\vert \phi _{t}(\omega ) - f_{t}^{0}(\omega )\vert =\mathrm{ dist}(\phi _{t}(\omega ),\mathrm{co}\,F(t,x_{T}(\omega ))\) for \((t,\omega ) \in [t_{1},T] \times \Omega \). Let \(y_{0} = E[x_{T} +\int _{ t_{1}}^{T}f_{\tau }^{0}\mathrm{d}\tau \vert \mathcal{F}_{t_{1}}]\) a.s. We have \(y_{0} \in E[x_{T} +\int _{ t_{1}}^{T}F(\tau, x_{T})\mathrm{d}\tau \vert \mathcal{F}_{t_{1}}]\) a.s., i.e., \(y_{0} \in S(E[x_{T} +\int _{ t_{1}}^{T}F(\tau, x_{T})\mathrm{d}\tau \vert \mathcal{F}_{t_{1}}])\). Therefore, \(d(y_{0},\mathcal{K}(t_{1})) \leq \varepsilon h_{0}/2\). Similarly as above, we can see that there exists \(x_{1} \in \mathcal{K}(t_{1})\) such that \(E\vert y_{0} - x_{1}\vert = E[\mathrm{dist}(y_{0},K(t_{1}\cdot \,))] = d(y_{0},\mathcal{K}(t_{1})) \leq \varepsilon h_{0}/2\). Then \(\|y_{0} - x_{1}\| \leq \varepsilon h_{0}/2\). Let \(z_{t}^{\varepsilon } = 1/h_{0}(x_{1} - y_{0})\) a.s. for t 1 ≤ t ≤ T. We have \(\|z_{t}^{\varepsilon }\| \leq (1/h_{0})\|y_{0} - x_{1}\| \leq \varepsilon /2\). Furthermore, by the definition of \(z_{t}^{\varepsilon }\), it follows that \(\int _{t}^{T}z_{\tau }^{\varepsilon }\mathrm{d}\tau\) is \(\mathcal{F}_{t_{1}}\)-measurable. Define \(\theta _{\varepsilon }(t) = T\) for t 1 < t ≤ T and θ(t 1) = t 1. One has f t 0 ∈ coF(t, x T ) a.s. for t 1 ≤ t ≤ T. Let
for t 1 ≤ t ≤ T. We have \({x}^{\varepsilon }(T) = x_{T}\) and \({x}^{\varepsilon }(t_{1}) = y_{0} + h_{0}(1/h_{0})(x_{1} - y_{0}) = x_{1}\). Therefore, \(d({x}^{\varepsilon }(\theta (t)),\mathcal{K}(\theta (t))) = 0\) for t 1 ≤ t ≤ T and \(\vert \phi _{t}(\omega ) - f_{t}^{0}(\omega )\vert =\mathrm{ dist}(\phi _{t}(\omega ),\mathrm{co}\,F(t,{x}^{\varepsilon }(\theta _{\varepsilon }(t)(\omega )))\) for \((t,\omega ) \in [t_{1},T] \times \Omega.\) By the definition of \({x}^{\varepsilon }\), it follows that it is \(\mathbb{F}\)-adapted. By properties of f 0 and \({x}^{\varepsilon }\), it follows that
If t 1 > a, we can repeat the above procedure starting with \((t_{1},x_{1}) \in Graph(\mathcal{K})\). Immediately from (2.3), it follows that there exists an h 1 ∈ (0, δ) such that
Similarly as above, we can select f 1 ∈ S(coF ∘ x 1) and \(x_{2} \in \mathcal{K}(t_{1} - h_{1})\) such that \(\vert \phi _{t}(\omega ) - f_{t}^{1}(\omega ))\vert =\mathrm{ dist}(\phi _{t}(\omega ),\mathrm{co}(F \circ x_{1})(t,\omega )\) for \((t,\omega ) \in [t_{1} - h_{1},t_{1}] \times \Omega \) and \(\|y_{1} - x_{2}\| \leq \varepsilon h_{1}/{2}^{2}\), where \(y_{1} = E[x_{1} +\int _{ t_{1}-h_{1}}^{t_{1}}f_{\tau }^{1}\mathrm{d}\tau \vert \mathcal{F}_{t_{ 1}-h_{1}}]\) and \(t_{2} = t_{1} - h_{1}\). We can now extend the step function \(\theta _{\varepsilon }\) and step process \({z}^{\varepsilon }\) on the interval [t 2, T] by taking \(\theta _{\varepsilon }(t_{2}) = t_{2},\) \(\theta _{\varepsilon }(t) = t_{1}\) for t 2 < t ≤ t 1 and \(z_{t}^{\varepsilon } = (1/h_{1})(x_{2} - y_{1})\) for t 2 ≤ t ≤ t 1. We have \(f_{t}^{1} \in \mathrm{ co}F(t,\theta _{\varepsilon }(t))\) a.s. for t 2 ≤ t ≤ t 1. We can also extend the process \({x}^{\varepsilon }\) to the interval [t 2, T] by taking
a.s. for t 2 ≤ t ≤ t 1. We have \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t)),\mathcal{K}(\theta (t))) = 0\) for t 2 ≤ t ≤ T, because \({x}^{\varepsilon }(t_{2}) = x_{2}\). Let \({f}^{\varepsilon } = 1\!\!1 _{(t_{2},t_{1}]}{f}^{1} + 1\!\!1 _{(t_{1},T]}{f}^{0}\). We have \({x}^{\varepsilon }(t) = E[x_{T} +\int _{ t}^{T}f_{\tau }^{\varepsilon }\mathrm{d}\tau \vert \mathcal{F}_{t}] +\int _{ t}^{T}z_{\tau }^{\varepsilon }\mathrm{d}\tau\) a.s. for t 2 < t ≤ T. Similarly as above, we can verify that \(f_{t}^{\varepsilon } \in \mathrm{ co}F(t,{x}^{\varepsilon }(\theta _{\varepsilon }(t)))\) a.s. for t 2 < t ≤ T and \(\vert \phi _{t} - f_{t}^{\varepsilon })\vert \leq \mathrm{ dist}(\phi _{t},\mathrm{co}\,F(t,{x}^{\varepsilon }(\theta (t)))\,\) a.s. for t 2 < t ≤ T. Furthermore, \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t)),\mathcal{K}(\theta _{\varepsilon }(t))) = 0\) and \(E[\mathrm{dist}({x}^{\varepsilon }(s),E[\int _{s}^{t}F(\tau, {x}^{\varepsilon }(\theta (\tau )))\mathrm{d}\tau \vert \mathcal{F}_{s}] \leq \varepsilon /2\) for t 2 ≤ t ≤ T and t 2 ≤ s ≤ t ≤ T, respectively.
Suppose that for some i ≥ 1, the inductive procedure is realized. Then there exist t i − 1 ∈ [a, T) and \(x_{i-1} \in \mathcal{K}(t_{i-1})\) such that we can extend the step function \(\theta _{\varepsilon }\), step process \({z}^{\varepsilon }\), and process \({f}^{\varepsilon }\) to the whole interval [t i − 1, T] such that \(f_{t}^{\varepsilon } \in \mathrm{ co}\,F(t,{x}^{\varepsilon }(\theta _{\varepsilon }(t))\) and \(\vert \phi _{t} - f_{t}^{\varepsilon }\vert =\mathrm{ dist}(\phi _{t},\mathrm{co}\,F(t,{x}^{\varepsilon }(\theta _{\varepsilon }(t)))\) for t i − 1 ≤ t ≤ T. Define
a.s. for t i − 1 ≤ t ≤ T. We have \({x}^{\varepsilon }(t_{i-1}) = x_{i-1}\), \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t)),\mathcal{K}(\theta _{\varepsilon }(t))) = 0\), and
for t i − 1 ≤ s ≤ t ≤ T.
Denote by S i the set of all positive numbers \(h \in (0,\min (\delta, t_{i-1}))\) such that
By the properties of \({x}^{\varepsilon }\), we have \({x}^{\varepsilon }(t_{i-1}) = x_{i-1}\) and \((t_{i-1},{x}^{\varepsilon }(t_{i-1})) \in Graph(\mathcal{K})\). Therefore, by virtue of (2.3), we have S i ≠∅ and \(\sup S_{i} > 0\). Choose h i − 1 ∈ S i such that \((1/2)\sup S_{i} \leq h_{i-1}\). Put \(t_{i} = t_{i-1} - h_{i-1}\). We can extend again the step function \(\theta _{\varepsilon }\), step process \({z}^{\varepsilon }\), and processes \({f}^{\varepsilon }\) and \({x}^{\varepsilon }\) to the interval [t i , T] such that \(d({x}^{\varepsilon }(\theta _{\varepsilon }(t)),\mathcal{K}(\theta (t)))) = 0\) for t i ≤ t ≤ T, and \(f_{t}^{\varepsilon } \in \mathrm{ co}F(t,{x}^{\varepsilon }(\theta _{\varepsilon }(t))\) and \(\vert \phi _{t} - f_{t}^{\varepsilon }\vert =\mathrm{ dist}(\phi _{t},\mathrm{co}F(t,{x}^{\varepsilon }(\theta _{\varepsilon }(t))\) a.s. for t i ≤ t ≤ T. Furthermore,
for t i ≤ s ≤ t ≤ T. We can continue the above procedure up to n ≥ 1 such that 0 < t n ≤ a < t n − 1. Suppose to the contrary that there does not exist such n ≥ 1, i.e., that for every n ≥ 1, one has a < t n < T. Then we can extend the step function \(\theta _{\varepsilon }\), the step process \({z}^{\varepsilon }\), and the stochastic processes \({f}^{\varepsilon }\) and \({x}^{\varepsilon }\) to the interval [t n , T] for every n ≥ 1 such that \({x}^{\varepsilon }(t_{n}) \in \mathcal{K}(t_{n})\) a.s. for every n ≥ 1 and so that the above properties are satisfied on [t n , T] for every n ≥ 1. By the boundedness of the sequence \((t_{n})_{n=1}^{\infty }\), we can select a decreasing subsequence \((t_{i})_{i=1}^{\infty }\) converging to t ∗ ∈ [a, T]. Let \((x_{i})_{i=1}^{\infty }\) be a sequence defined by \(x_{i} = {x}^{\varepsilon }(t_{i})\) a.s. for every i ≥ 0. In particular, we have \(x_{i} \in \mathcal{K}(t_{i})\) a.s. for every i ≥ 1. For every j > k ≥ 0, we obtain
By the continuity of the filtration \(\mathbb{F}\), it follows that \(\lim _{j,k\rightarrow \infty }E\vert x_{k} - x_{j}\vert = 0\). Then \((x_{i})_{i=1}^{\infty }\) is a Cauchy sequence of \(\mathbb{L}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\). Therefore, there is \({x}^{{\ast}}\in \mathbb{L}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d}\) such that \(\|x_{i} - {x}^{{\ast}}\|\rightarrow 0\) as i → ∞. But \(x_{i} \in \mathcal{K}(t_{i}))\) for every i ≥ 1 and \(\mathcal{K}\) is continuous. Then \(({t}^{{\ast}},{x}^{{\ast}}) \in Graph(\mathcal{K})\), which by virtue of (3.1), implies that we can select \({h}^{{\ast}}\in (0,\min (\delta, {t}^{{\ast}}))\) such that
Similarly as above, for every i ≥ 1 and ϕ i ∈ S(coF ∘ x i ), we can select f ∗ ∈ S(coF ∘ x ∗) such that \(\vert \phi _{t}^{i} - f_{t}^{{\ast}})\vert =\mathrm{ dist}(\phi _{t}^{i},F(t,{x}^{{\ast}}))\) a.s. for every \({t}^{{\ast}}- {h}^{{\ast}}\leq t \leq {t}^{{\ast}}\). By the continuity of the filtration \(\mathbb{F}\), we obtain \(\|E[{x}^{{\ast}}\vert \mathcal{F}_{t_{i}-{h}^{{\ast}}}] - E[{x}^{{\ast}}\vert \mathcal{F}_{{t}^{{\ast}}-{h}^{{\ast}}}]\| \rightarrow 0\) and
as i → ∞. Let N ≥ 1 be such that for every i ≥ N, we have \(0 < t_{i} - {t}^{{\ast}} <\min ({h}^{{\ast}},\delta )\), \(\|x_{i} - {x}^{{\ast}}\| <\varepsilon {h}^{{\ast}}/({2}^{5} \cdot A)\), \(D(\mathcal{K}(t_{i} - {h}^{{\ast}}),\mathcal{K}({t}^{{\ast}}- {h}^{{\ast}})) \leq \varepsilon {h}^{{\ast}}/{2}^{5},\) \(\|E[{x}^{{\ast}}\vert \mathcal{F}_{t_{i}-{h}^{{\ast}}}] - E[{x}^{{\ast}}\vert \mathcal{F}_{{t}^{{\ast}}-{h}^{{\ast}}}]\| \leq \varepsilon {h}^{{\ast}}/{2}^{5},\) \(E\int _{t_{i}-{h}^{{\ast}}}^{{t}^{{\ast}}-{h}^{{\ast}} }\vert \phi _{\tau }^{i}\vert \mathrm{d}\tau \leq \varepsilon {h}^{{\ast}}/{2}^{5}\), \(E\int _{{t}^{{\ast}}}^{t_{i}}\vert \phi _{\tau }^{i}\vert \mathrm{d}t \leq \varepsilon {h}^{{\ast}}/{2}^{5}\), and \(E\vert E[\int _{{t}^{{\ast}}-{h}^{{\ast}}}^{{t}^{{\ast}} }f_{\tau }^{{\ast}}\mathrm{d}\tau \vert \mathcal{F}_{t_{i}-{h}^{{\ast}}}] - E[\int _{{t}^{{\ast}}-{h}^{{\ast}}}^{{t}^{{\ast}} }f_{\tau }^{{\ast}}\mathrm{d}\tau \vert \mathcal{F}_{{t}^{{\ast}}-h{\ast}}]\vert \leq \varepsilon {h}^{{\ast}}/{2}^{5},\) where \(A = 1 +\int _{ 0}^{T}k(t)\mathrm{d}t\). By the properties of the multifunction F(t, ⋅ ) and selector f ∗ of F ∘ x ∗, it follows that
For every i ≥ N, one gets
But for every i ≥ N, we have
Therefore, for every i ≥ N, one gets
which implies that
But t ∗ ≤ t i for i ≥ 1. Therefore, for every i ≥ N, one has h ∗ ∈ S i + 1 and \((1/2){h}^{{\ast}}\leq \sup S_{i+1} \leq h_{i} = t_{i} - t_{i+1}\), which contradicts the convergence of the sequence \((t_{i})_{i=1}^{\infty }\). Then there is a p > 1 such that \(a = t_{p} < t_{p-1},\ldots, t_{1} < t_{0} = T\). Taking \({f}^{\varepsilon } = 1\!\!1 _{[a,t_{p-1}]}{f}^{p} +\sum _{ i=p-2}^{0}1\!\!1 _{(t_{i+1},t_{i}]}{f}^{i}\), we obtain the desired selector of \(\mathrm{co}F \circ ({x}^{\varepsilon } \circ \theta _{\varepsilon })\).
Remark 2.3.
The above results are also true if instead of continuity of the set-valued mapping \(\mathcal{K}\), we assume that it is uniformly upper semicontinuous on [0, T], i.e., that \(\lim _{\delta \rightarrow 0}\sup _{0\leq t\leq T}\overline{D}(\mathcal{K}(t+\delta ),\mathcal{K}(t)) = 0\).
Conditions (2.1) and (2.3) can be expressed by certain types of stochastic tangent sets. To see this, let \((t,x) \in Graph(\mathcal{K})\) and denote by \(\mathcal{T}_{K}(t,x)\) the set of all pairs \((f,g) \in {\mathbb{L}}^{2}([t,T] \times \Omega, \Sigma _{\mathbb{F}}^{t}, {\mathbb{R}}^{d}) \times {\mathbb{L}}^{2}([t,T] \times \Omega, \Sigma _{\mathbb{F}}^{t}, {\mathbb{R}}^{d\times m})\) such that
where \(\Sigma _{\mathbb{F}}^{t}\) denotes the σ-algebra of all \(\mathbb{F}\)-nonanticipative subsets of \([t,T] \times \Omega \). In a similar way, for \((t,x) \in Graph({\mathcal{K}}^{\varepsilon })\) and \(\varepsilon \in (0,1)\), we can define a backward stochastic tangent set \(\mathcal{T}_{K}^{b}(t,x)\) with respect to a filtration \(\mathbb{F} = (\mathcal{F}_{t})_{0\leq t\leq T}\) as the set of all measurable processes \(f \in \mathbb{L}([0,T] \times \Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d})\) such that
Lemma 2.1.
Let \(\mathcal{P}_{\mathbb{F}}\) be a complete filtered probability space. Assume that F and G satisfy condition (i) of \((\mathcal{H}_{1})\) and let \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be \(\mathbb{F}\) -adapted and such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T. The condition (2.1) is satisfied for every \((t,x) \in Graph(\mathcal{K})\) if and only if \(S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x) \subset \mathcal{T}_{K}(t,x)\) for every \((t,x) \in Graph(\mathcal{K})\) , where \(S_{\mathbb{F}}^{t}(F \circ x)\) and \(S_{\mathbb{F}}^{t}(G \circ x)\) denote the sets of all restrictions of all elements of \(S_{\mathbb{F}}(F \circ x)\) and \(S_{\mathbb{F}}(G \circ x)\) , respectively, to the set \([t,T] \times \Omega \).
Proof.
It is clear that if (2.1) is satisfied for every \((t,x) \in Graph(\mathcal{K})\), then \(S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x) \subset \mathcal{T}_{K}(t,x)\) for every \((t,x) \in Graph(\mathcal{K})\). Let \(S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x) \subset \mathcal{T}_{K}(t,x)\) for fixed \((t,x) \in Graph(\mathcal{K})\). Then for every \((f,g) \in S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x)\), one has
Thus for every \((t,x) \in Graph(\mathcal{K})\) and \((f,g) \in S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x)\) and every \(\varepsilon \in (0,1)\), there exists \(h_{\varepsilon }^{f,g}(t) \in (0,\varepsilon )\) such that
Let \(h_{\varepsilon } =\sup \{ h_{\varepsilon }^{f,g}(t) : (f,g) \in S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x)),\,0 \leq t \leq T\}\). We have
for every \((t,x) \in Graph(\mathcal{K})\) and \((f,g) \in S_{\mathbb{F}}(F \circ x) \times S_{\mathbb{F}}(G \circ x)\). Then
which implies that
for every \((t,x) \in Graph(\mathcal{K}).\)
Remark 2.4.
The results of Theorems 2.1 and 2.2 also hold if instead of condition (2.1), we assume that \([S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x)] \cap \mathcal{T}_{K}(t,x)\neq \varnothing \) for every \(\varepsilon \in (0,1)\) and \((t,x) \in Graph({\mathcal{K}}^{\varepsilon })\).
There are another types of stochastic tangent sets. For a given \(\mathbb{F}\)-adapted set-valued stochastic process \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) and \((t,x) \in Graph(\mathcal{K})\), by \(\mathcal{S}_{K}(t,x)\) we denote the stochastic “tangent set” to K at (t, x) with respect to the filtration \(\mathbb{F}\) defined as the set of all pairs \((f,g) \in {\mathbb{L}}^{2}([t,T] \times \Omega, \Sigma _{\mathbb{F}}^{t}, {\mathbb{R}}^{d}) \times {\mathbb{L}}^{2}([t,T] \times \Omega, \Sigma _{\mathbb{F}}^{t}, {\mathbb{R}}^{d\times m})\) such that for every \((f,g) \in \mathcal{S}_{K}(t,x)\), there exist a sequence \((h_{n})_{n=1}^{\infty }\) of positive numbers converging to 0 and sequences \(({a}^{n})_{n=1}^{\infty }\) and \(({b}^{n})_{n=1}^{\infty }\) of d- and d ×m-dimensional \(\mathbb{F}\)-adapted stochastic processes \({a}^{n} = (a_{t}^{n})_{0\leq t\leq T}\) and \({b}^{n} = (b_{t}^{n})_{0\leq t\leq T}\), respectively, such that
and
We shall show that such stochastic tangent sets are smaller then \(\mathcal{T}_{K}(t,x)\), i.e., that \(\mathcal{S}_{K}(t,x) \subset \mathcal{T}_{K}(t,x)\) for every \((t,x) \in Graph(\mathcal{K})\).
Lemma 2.2.
Let \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be an \(\mathbb{F}\) -adapted set-valued process such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T. For every \((t,x) \in Graph(\mathcal{K})\) , one has \(\mathcal{S}_{K}(t,x) \subset \mathcal{T}_{K}(t,x)\).
Proof.
Let \((t,x) \in Graph(\mathcal{K})\) be fixed and \((f,g) \in \mathcal{S}_{K}(t,x)\). There exist a sequence \((h_{n})_{n=1}^{\infty }\) of positive numbers converging to 0 and sequences \(({a}^{n})_{n=1}^{\infty }\) and \(({b}^{n})_{n=1}^{\infty }\) of d- and d ×m-dimensional \(\mathbb{F}\)-adapted stochastic processes \({a}^{n} = (a_{t}^{n})_{0\leq t\leq T}\) and \({b}^{n} = (b_{t}^{n})_{0\leq t\leq T}\), respectively, such that the above conditions are satisfied. For every n ≥ 1, one has
Hence, by the properties of sequences \(({a}^{n})_{n=1}^{\infty }\) and \(({b}^{n})_{n=1}^{\infty },\) it follows that
which implies
Then \((f,g) \in \mathcal{T}_{K}(t,x)\) for every \((f,g) \in \mathcal{S}_{K}(t,x)\).
Denote by τ K (t, x) that stochastic “contingent set” to K at (t, x) with respect to \(\mathbb{F}\), defined as the set of all pairs \((f,g) \in {\mathbb{L}}^{2}([t,T] \times \Omega, \Sigma _{\mathbb{F}}^{t}, {\mathbb{R}}^{d}) \times {\mathbb{L}}^{2}([t,T] \times \Omega, \Sigma _{\mathbb{F}}^{t}, {\mathbb{R}}^{d\times m})\) such that for every such pair (f, g), there exist a sequence \((h_{n})_{n=1}^{\infty }\) of positive numbers converging to 0 and sequences \((a_{n})_{n=1}^{\infty }\) and \((b_{n})_{n=1}^{\infty }\) of d- and d ×m-dimensional \(\mathcal{F}_{t}\)-measurable random variables a n and b n , respectively, such that \(x +\int _{ t}^{t+h_{n}}f_{s}ds +\int _{ t}^{t+h_{n}}g_{s}\mathrm{d}B_{s} + h_{n}a_{n} + \sqrt{h_{n}}b_{n} \in \mathcal{K}(t + h_{n})\) for every n ≥ 1 and \(\max \left \{E\vert a_{n}{\vert }^{2},(1/h_{n})E\vert b_{n}{\vert }^{2}\right \} \rightarrow 0\) as n → ∞. Similarly as above, we obtain the following result.
Lemma 2.3.
Let \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be an \(\mathbb{F}\) -adapted set-valued process such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T. For every \((t,x) \in Graph(\mathcal{K})\) , one has \(\tau _{K}(t,x) \subset \mathcal{S}_{K}(t,x)\).
Proof.
Let \((t,x) \in Graph(\mathcal{K})\) be fixed and (f, g) ∈ τ K (t, x). There are a sequence \((h_{n})_{n=1}^{\infty }\) of positive numbers converging to zero and sequences \((a_{n})_{n=1}^{\infty }\,\), \((b_{n})_{n=1}^{\infty }\) of \(\mathcal{F}_{t}\)-measurable random variables \(a_{n} : \Omega \rightarrow {\mathbb{R}}^{d}\) and \(b_{n} : \Omega \rightarrow {\mathbb{R}}^{d\times m}\) such that the above conditions are satisfied. For every n ≥ 1, one gets
and
Hence, for n ≥ 1 sufficiently large, it follows that
which implies that
Then \((f,g) \in \mathcal{S}_{K}(t,x)\).
Remark 2.5.
The results of Theorems 2.1 and 2.2 are also true if instead of condition (2.1), we assume that \([S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x)] \cap \tau _{K}(t,x)\neq \varnothing \) for every \(\varepsilon \in (0,1)\) and \((t,x) \in Graph({\mathcal{K}}^{\varepsilon })\).
3 Existence of Viable Solutions
We shall prove now that if F and G satisfy conditions \((\mathcal{H}_{1})\), then for every continuous set-valued \(\mathbb{F}\)-adapted process \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\), the viability problems \(\overline{SFI}(F,G,K)\) and BSDI(F, K) possess viable strong solutions. Furthermore, the existence of viable weak solutions of \(\overline{SFI}(F,G,K)\) is considered. Similarly as above, we define \(\mathcal{K}(t)\) and \({\mathcal{K}}^{\varepsilon }(t)\) by setting \(\mathcal{K}(t) =\{ u \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{t}, {\mathbb{R}}^{d}) : d(u,\mathcal{K}(t)) = 0\}\) and \({\mathcal{K}}^{\varepsilon }(t) =\{ u \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{t}, {\mathbb{R}}^{d}) : d(u,\mathcal{K}(t)) \leq \varepsilon \}\).
Theorem 3.1.
Let \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\) be a complete filtered probability space and \(B = (B_{t})_{0\leq t\leq T}\) an m-dimensional \(\mathbb{F}\) -Brownian motion on \(\mathcal{P}_{\mathbb{F}}\) . Assume that F and G satisfy conditions \((\mathcal{H}_{1})\) and let \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be an \(\mathbb{F}\) -adapted set-valued process such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T and such that the mapping \(\mathcal{K} : [0,T] \ni t \rightarrow \mathcal{K}(t) \in \mathrm{ Cl}({\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d}))\) is continuous. If \(\mathcal{P}_{\mathbb{F}}\) , B, F, G, and K are such that (2.1) is satisfied for every \((t,x) \in Graph(\mathcal{K})\) , then the problem \(\overline{SFI}(F,G,K)\) possesses on \(\mathcal{P}_{\mathbb{F}}\) a strong viable solution.
Proof.
Let a ∈ (0, T) and select arbitrarily \(x_{0} \in \mathcal{K}(0)\). Let \(u_{0} \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{0}, {\mathbb{R}}^{d})\) and \(v_{0} \in {\mathbb{L}}^{2}(\Omega, \mathcal{F}_{0}, {\mathbb{R}}^{d\times m})\) be such that u 0 ∈ F(0, x 0) and v 0 ∈ G(0, x 0) a.s. By virtue of Theorem 2.2, for \(\varepsilon _{1} = 1/{2}^{3/2}\) and stochastic processes \({\phi }^{1} = (\phi _{t}^{1})_{a\leq t\leq T}\) and \({\psi }^{1} = (\psi _{t}^{1})_{a\leq t\leq T}\) defined by ϕ t 1 = u 0 and ψ t 1 = v 0 a.s. for every a ≤ t ≤ T, there exist a partition \(0 = t_{0}^{1} < t_{1}^{1} < \cdots < t_{p_{1}-1}^{1} < t_{p_{1}}^{1} = a\), a step function θ 1, and stochastic processes f 1, g 1, and z 1 such that conditions (i)–(v) of Theorem 2.2 are satisfied with
a.s. for a ≤ t ≤ T. Similarly, for \(\varepsilon _{2} = 1/2\) and ϕ 2 = f 1 and ψ 2 = g 1, we can select a partition \(0 = t_{0}^{2} < t_{1}^{2} < \cdots < t_{p_{2}-1}^{2} < t_{p_{1}}^{2} = a\), a step function θ 2, and stochastic processes f 2, g 2, and z 2 such that conditions (i)–(v) of Theorem 2.2 are satisfied with
a.s. for a ≤ t ≤ T. Continuing this procedure for \(\varepsilon _{k} = 1/{2}^{3k/2}\) and \({\phi }^{k} = {f}^{k-1}\) and \({\psi }^{k} = {g}^{k-1}\), we obtain a partition \(0 = t_{0}^{k} < t_{1}^{k} < \cdots < t_{p_{k}-1}^{k} < t_{p_{k}}^{k} = a\), a step function θ k , and stochastic processes f k, g k, and z k such that conditions (i)–(v) of Theorem 2.2 are satisfied for every k ≥ 1 with
a.s. for a ≤ t ≤ T such that \(d({x}^{k}(\theta _{k}(t)),\mathcal{K}(\theta _{k}(t)) = 0\). For every k ≥ 1, one has \({f}^{k} \in S_{\mathbb{F}}(F \circ ({x}^{k-1} \circ \theta _{k-1}))\), \({g}^{k} \in S_{\mathbb{F}}(G \circ ({x}^{k-1} \circ \theta _{k-1}))\), \(\vert {f}^{k} - {f}^{k-1}\vert \leq \mathrm{ dist}(f_{t}^{k-1},F(t,({x}^{k-1}(\theta _{k-1}(t))),\) \(\vert {g}^{k} - {g}^{k-1}\vert \leq \mathrm{ dist}(g_{t}^{k-1},G(t,({x}^{k-1}(\theta _{k-1}(t))),\) \(\|{z}^{k}(t)\| \leq \varepsilon _{k}\), and
for 0 ≤ s ≤ t ≤ a. Furthermore, one has \(\vert \theta _{k}(t) -\theta _{k-1}(t)\vert \leq \delta _{k-1},\)
for 0 ≤ t ≤ a, because by (i) of Theorem 2.2, \(\delta _{k} \in (0,\varepsilon _{k})\) is such that
We shall now evaluate \(E[\sup _{0\leq \tau \leq t}\vert {x}^{k+1}(\tau ) - {x}^{k}(\tau ){\vert }^{2}]\) for k = 1, 2, … and 0 ≤ t ≤ a. Let us observe first that \(E[\sup _{0\leq \tau \leq t}\vert {x}^{k}(\theta _{k+1}(\tau )) - {x}^{k}(\theta _{k}(\tau )){\vert }^{2}] \rightarrow 0\) as k → ∞, because \(\vert \theta _{k+1}(t) -\theta _{k}(t)\vert \leq \delta _{k}\) and
for k = 2, 3, … and 0 ≤ t ≤ a. Hence it follows that
for every k = 1, 2, … and 0 ≤ t ≤ a, where x t 0 = x 0, α = (4T)2 and \(\beta = {2}^{2}(T + 1)\).
Now, by the definition of the processes x 1 and x 0, one gets \(E[\sup _{0\leq \tau \leq t}\vert {x}^{1}(\tau ) - {x}^{0}(\tau ){\vert }^{2}] \leq \gamma\) with \(\gamma = {2}^{2}[(T + 1)\int _{0}^{T}{m}^{2}(t)\mathrm{d}t + {T}^{2}]\). Therefore,
for 0 ≤ t ≤ a. From this and (3), it follows that
Similarly, we get
for 0 ≤ t ≤ a. By the inductive procedure, we obtain
for n ≥ 1 with \(M =\max (\alpha, \gamma )\). By Chebyshev’s inequality, we obtain
Therefore, by the Borel–Cantelli lemma, one gets
Thus for a.e. \(\omega \in \Omega \), there exists n 0 = n 0(ω) such that \(\sup _{0\leq \tau \leq a}\vert {x}^{n+1}(\tau ) - {x}^{n}(\tau )\vert \leq {2}^{-n}\) for n ≥ n 0(ω). Therefore, the sequence \(\{{x}^{n}(\cdot )(\omega )\}_{n=1}^{\infty }\) is uniformly convergent on [0, a] for a.a. \(\omega \in \Omega \), because \({x}^{n}(t)(\omega ) = {x}^{1}(t)(\omega ) +\sum _{ k=1}^{n-1}[{x}^{k+1}(t)(\omega ) - {x}^{k}(t)(\omega )]\) for every 0 ≤ t ≤ T and a.a. \(\omega \in \Omega \). Denote the limit of the above sequence by x t (ω) for 0 ≤ t ≤ a and a.a. \(\omega \in \Omega \). By virtue of (3.1), it follows that \(E[\sup _{0\leq \tau \leq t}\vert {x}^{n+1}(\tau ) - {x}^{n}(\tau ){\vert }^{2}] \rightarrow 0\) as n → ∞. On the other hand, by the properties of sequences \(({f}^{k})_{k=1}^{\infty }\) and \(({f}^{k})_{k=1}^{\infty },\) we get
and
for every k = 0, 1, …. Hence it follows that \(({f}^{k})_{k=1}^{\infty }\) and \(({g}^{k})_{k=1}^{\infty }\) are Cauchy sequences of Banach spaces \(({\mathbb{L}}^{2}([0,a] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{d}),\mathbf{\vert }\cdot \mathbf{\vert })\) and \(({\mathbb{L}}^{2}([0,a] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{d\times m}),\mathbf{\vert }\cdot \mathbf{\vert })\), respectively. Then there exist \(f \in {\mathbb{L}}^{2}([0,a] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{d})\) and \(g \in {\mathbb{L}}^{2}([0,a] \times \Omega, \Sigma _{\mathbb{F}}, {\mathbb{R}}^{d\times m})\) such that \(\mathbf{\vert }{f}^{n} - f\mathbf{\vert }\rightarrow 0\) and \(\mathbf{\vert }{g}^{n} - g\mathbf{\vert }\rightarrow 0\) as n → ∞. Let \(y_{t} = x_{0} +\int _{ 0}^{t}f_{\tau }\mathrm{d}\tau +\int _{ 0}^{t}g_{\tau }\mathrm{d}B_{\tau }\) for 0 ≤ t ≤ a. For every n ≥ 1, one gets
Therefore, we have \(E[\sup _{0\leq t\leq a}\vert {x}^{n}(t) - y_{t}{\vert }^{2}] \rightarrow 0\) and \(E[\sup _{0\leq t\leq a}\vert {x}^{n}(t) - x(t){\vert }^{2}] \rightarrow 0\) as n → ∞, which implies that x(t) = y t a.s. for every 0 ≤ t ≤ a. Then \(x(t) = x_{0} +\int _{ 0}^{t}f_{\tau }\mathrm{d}\tau +\int _{ 0}^{t}g_{\tau }\mathrm{d}B_{\tau }\) a.s. for 0 ≤ t ≤ a. Now, by Lemma 1.3 and Theorem 2.2, we obtain
for every 0 ≤ s ≤ t ≤ a. But
Then \(\lim _{n\rightarrow \infty }\|{x}^{n} \circ \theta _{n} - x\| = 0\). Therefore, for every 0 ≤ s ≤ t ≤ a, we get
Thus
for every 0 ≤ s ≤ t ≤ a. In a similar way, we get \(d(x(t),\mathcal{K}(t)) \leq \| {x}^{n} - x\| + d({x}^{n}(t),\mathcal{K}(t)) \leq \| {x}^{n} - x\| +\varepsilon _{n}\) for every n ≥ 1 and 0 ≤ s ≤ t ≤ a. Therefore, \(d(x(t),\mathcal{K}(t)) = 0\) for every 0 ≤ t ≤ a, which by Theorem 3.1 of Chap. 2, implies that x(t) ∈ K(t, ⋅) a.s. for 0 ≤ t ≤ a.
We can now extend our solution to the whole interval [0, T]. Let us denote by \(\Lambda _{x}\) the set of all extensions of the viable solution x of \(\overline{SFI}(F,G,K)\) obtained above. We have \(\Lambda _{x}\neq \varnothing \), because we can repeat the above procedure for every interval [a, α] with α ∈ (a, T). Let us introduce in \(\Lambda _{x}\) the partial order relation ≼ by setting x ≼ z if and only if a x ≤ a z and \(x = z\vert _{[0,a_{x}]}\), where a z ∈ (0, T) is such that z is a strong viable solution for \(\overline{SFI}(F,G,K)\) on [0, a z ], and \(z\vert _{[0,a_{x}]}\) denotes the restriction of the solution z to the interval [0, a x ]. Let \(\psi : [0,\alpha ] \rightarrow {\mathbb{R}}^{d}\) be an extension of x to [0, α] with α ∈ (a, T) and denote by \(P_{x}^{\psi } \subset \Lambda _{x}\) the set containing ψ and all its restrictions ψ | [0, β] for every β ∈ [a, α). It is clear that each completely ordered subset of \(\Lambda _{x}\) is of the form P x ψ, determined by some extension ψ of x. It is also clear that every P x ψ has ψ as its upper bound. Then by the Kuratowski–Zorn lemma, there exists a maximal element γ of \(\Lambda _{x}\). It has to be a γ = T. Indeed, if we had a γ < T, then we could repeat the above procedure and extend γ as a viable strong solution \(\xi \in \Lambda _{x}\) of \(\overline{SFI}(F,G,K)\) to the interval [0, b] with a γ < b, which would imply that \(\gamma \preccurlyeq \xi\), a contradiction to the assumption that γ is a maximal element of \(\Lambda _{x}\). Then x can be extended to the whole interval [0, T].
In a similar way, by virtue of Remark 2.2, we can prove the following existence theorem for \(SFI(\overline{F},G)\).
Theorem 3.2.
Let \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\) be a complete filtered separable probability space and \(B = (B_{t})_{0\leq t\leq T}\) an m-dimensional \(\mathbb{F}\) -Brownian motion on \(\mathcal{P}_{\mathbb{F}}\) . Assume that F and G satisfy conditions \((\mathcal{H}_{1})\) and that \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) is \(\mathbb{F}\) -adapted such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T and such that the mapping \(\mathcal{K} : [0,T] \rightarrow \mathrm{ Cl}({\mathbb{L}}^{2}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d}))\) is continuous. If \(\mathcal{P}_{\mathbb{F}}\) , B, F, G, and K are such that (2.2) is satisfied for every \((t,x) \in Graph(\mathcal{K})\) , then the problem
possesses on \(\mathcal{P}_{\mathbb{F}}\) a strong viable solution.
We shall now prove the existence of weak viable solutions for stochastic functional inclusions. The proof of such an existence theorem is based on the first viable approximation theorem presented above.
Theorem 3.3.
Assume that \(F : [0,T] \times {\mathbb{R}}^{d} \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) and \(G : [0,T] \times {\mathbb{R}}^{d} \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d\times m})\) are measurable, bounded, convex-valued and are such that F(t,⋅ ) and G(t,⋅ ) are continuous for a.e. fixed t ∈ [0,T]. Let G be diagonally convex and let \(K : [0,T] \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be continuous. If there exist a complete filtered probability space \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\) and a d-dimensional \(\mathbb{F}\) -Brownian motion on \(\mathcal{P}_{\mathbb{F}}\) such that (2.1) is satisfied for every \(\varepsilon \in (0,1)\) and \((t,x) \in Graph({\mathcal{K}}^{\varepsilon })\) , then \(\overline{SFI}(F,G,K)\) possesses a weak viable solution.
Proof.
Let \(x_{0} \in \mathcal{K}(0)\) be fixed and let \(\varepsilon _{n} = 1/{2}^{n}\). By virtue of Theorem 2.1, we can define on [0, T] a step function \(\theta _{n} =\theta _{\varepsilon _{n}}\,\) and \(\mathbb{F}\)-nonanticipative stochastic processes \({f}^{n} = {f}^{\varepsilon _{n}}\), \({g}^{n} = {g}^{\varepsilon _{n}}\), and \(x_{t}^{n} = x_{0} +\int _{ 0}^{t}f_{\tau }^{n}\mathrm{d}\tau +\int _{ 0}^{t}g_{\tau }^{n}\mathrm{d}B_{\tau }\) for 0 ≤ t ≤ T such that conditions (i)–(iii) of Theorem 2.1 are satisfied. In particular, for every m ≥ 1, n ≥ 1, and 0 ≤ s ≤ t ≤ T, we obtain
where \(C_{m}^1\) and \(C_{m}^2\) are positive integers depending on m ≥ 1. Let us observe that
Therefore,
for every 0 ≤ s ≤ t ≤ T and n, m ≥ 1. In a similar way, we can verify that there exist positive numbers K and γ such that \(E\vert x_{0}^{n}{\vert }^{\gamma }\leq K\). Then the sequence \(({x}^{n})_{n=1}^{\infty }\) of continuous processes \({x}^{n} = (x_{t}^{n})_{0\leq t\leq T}\) satisfies on the probability space \((\Omega, \mathcal{F},P)\) the assumptions of Theorem 3.5 of Chap. 1. Furthermore, immediately from Theorem 2.1, it follows that \(E[\mathrm{dist}({x}^{n}(\theta _{n}(t)),K(\theta _{n}(t)))] \leq \varepsilon _{n}\) and
for every 0 ≤ s ≤ t ≤ T, \(l \in C_{b}({\mathbb{R}}^{d}, \mathbb{R})\), and \(h \in C_{b}^{2}({\mathbb{R}}^{d}, \mathbb{R})\).
By virtue of Theorems 3.5 and 2.4 of Chap. 1, there exist an increasing subsequence \((n_{k})_{k=1}^{\infty }\) of \((n)_{n=1}^{\infty }\), a probability space \((\tilde{\Omega },\tilde{\mathcal{F}},\tilde{P})\), and d-dimensional continuous stochastic processes \(\tilde{{x}}^{n_{k}}\) and \(\tilde{x}\,\) on \((\tilde{\Omega },\tilde{\mathcal{F}},\tilde{P})\) for k = 1, 2, … such that \(P{({x}^{n_{k}})}^{-1} = P{(\tilde{{x}}^{n_{k}})}^{-1}\) for k = 1, 2, … and \(\sup _{0\leq t\leq T}\vert \tilde{{x}}^{n_{k}} -\tilde{ x}\vert \rightarrow 0\) as k → ∞. Let \(\tilde{\mathcal{F}}_{t}^{n_{k}} =\bigcap _{\varepsilon >0}\sigma (\tilde{x}_{u}^{n_{k}} : u \leq t+\varepsilon )\) for 0 ≤ t ≤ T and let \(\tilde{\mathbb{F}}_{n_{k}} = (\tilde{\mathcal{F}}_{t}^{n_{k}})_{ 0\leq t\leq T}\). For every k ≥ 1, \({x}^{n_{k}}\) and \(\tilde{{x}}^{n_{k}}\) are continuous \(\mathbb{F}\)- and \(\tilde{\mathbb{F}}_{n_{k}}\)-adapted. Furthermore, immediately from (3), it follows that \(\mathcal{M}_{FG}^{{x}^{n_{k}} }\neq \varnothing \) for every k ≥ 1, which by Lemma 1.3 of Chap. 4, implies that \(\mathcal{M}_{FG}^{\tilde{x}}\neq \varnothing \). This, by Theorem 1.3 of Chap. 4, implies the existence of an \(\tilde{\mathbb{F}}\)-Brownian motion \(\hat{B}\) on the standard extension \(\hat{\mathcal{P}}_{\hat{\mathbb{F}}} = (\hat{\Omega },\hat{\mathcal{F}},\hat{ \mathbb{F}},\hat{P})\) of the filtered probability space \((\tilde{\Omega },\tilde{\mathcal{F}},\tilde{ \mathbb{F}},\tilde{P})\), with \(\tilde{\mathbb{F}} = (\tilde{\mathcal{F}}_{t})_{0\leq t\leq T}\) defined by \(\tilde{\mathcal{F}}_{t} =\bigcap _{\varepsilon >0}\sigma (\tilde{x}(u) : u \leq t+\varepsilon )\), such that \((\hat{\mathcal{P}}_{\hat{\mathbb{F}}},\hat{x},\hat{B})\) is a weak solution of \(\overline{SF}I(F,G)\) with \(\hat{x}_{t}(\hat{\omega }) =\tilde{ x}_{t}(\pi (\hat{\omega }))\) satisfying the initial condition \(P\hat{x}_{0}^{-1} = P\tilde{x}_{0}^{-1}\), where \(\pi :\hat{ \Omega } \rightarrow \tilde{ \Omega }\) is the \((\hat{\mathcal{F}},\tilde{\mathcal{F}})\)-measurable mapping described in the definition of the extension of \((\tilde{\Omega },\tilde{\mathcal{F}},\tilde{ \mathbb{F}},\tilde{P})\), because the standard extension \(\hat{\mathcal{P}}_{\hat{\mathbb{F}}}\) is also an extension. Similarly as in the proof of Corollary 3.2 of Chap. 1, we obtain \(P{(e_{s} \circ {x}^{n_{k}})}^{-1} = P{(e_{ s} \circ \tilde{ {x}}^{n_{k}})}^{-1}\) with \(s =\theta _{n_{k}}(t)\) for 0 ≤ t ≤ T. This, together with the inequality \(E[\mathrm{dist}({x}^{n_{k}}(\theta _{ n_{k}}(t)),K(\theta _{n_{k}}(t)))] \leq 1/{2}^{n_{k}}\) for k ≥ 1 and properties of the sequence \((\tilde{{x}}^{n_{k}})_{ k=1}^{\infty },\) implies that \(E[\mathrm{dist}(\tilde{x}_{t},K(t))] = 0\). Similarly as in the proof of Theorem 1.3 of Chap. 4, by the definition of the process \(\hat{x}\), it follows that \(P\hat{{x}}^{-1} = P\tilde{{x}}^{-1},\) which implies that \(P{(e_{t} \circ \hat{ x})}^{-1} = P{(e_{t} \circ \tilde{ x})}^{-1}\) for every 0 ≤ t ≤ T. Therefore, \(E[\mathrm{dist}(\hat{x}_{t},K(t))] = 0\) for every 0 ≤ t ≤ T. Thus \(\hat{x}_{t} \in K(t)\), \(\hat{P}\)-a.s. for 0 ≤ t ≤ T.
Remark 3.1.
The results of Theorem 3.3 again hold if instead of (2.1), we assume that (2.2) is satisfied. It is also true if instead of (2.1), we assume that \([S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x)] \cap \mathcal{T}_{K}(t,x)\neq \varnothing \) for every \((t,x) \in Graph({\mathcal{K}}^{\varepsilon })\) and \(\varepsilon \in (0,1)\).
In a similar way as above, we obtain immediately from Theorem 2.3 the following existence theorem.
Theorem 3.4.
Let \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\) be a complete filtered probability space with a continuous filtration \(\mathbb{F} = (\mathcal{F}_{t})_{0\leq t\leq T}\) such that \(\mathcal{F}_{T} = \mathcal{F}\) . Assume that F satisfies conditions \((\mathcal{H}_{1})\) and let \(K : [0,T] \times \Omega \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) be an \(\mathbb{F}\) -adapted set-valued process such that \(\mathcal{K}(t)\neq \varnothing \) for every 0 ≤ t ≤ T and that the mapping \(\mathcal{K} : [0,T] \ni t \rightarrow \mathcal{K}(t) \in \mathrm{ Cl}(\mathbb{L}(\Omega, \mathcal{F}_{T}, {\mathbb{R}}^{d}))\) is continuous. If \(\mathcal{P}_{\mathbb{F}}\) , F, and K are such that (2.3) is satisfied for every \((t,x) \in Graph(\mathcal{K})\) , then BSDI(F,K) possesses a strong viable solution.
Proof.
Let \(x_{T} \in \mathcal{K}(T)\) and a ∈ (0, T) be fixed. Put \(x_{t}^{0} = E[x_{T}\vert \mathcal{F}_{t}]\) a.s. for a ≤ t ≤ T and let \({f}^{0} = (f_{t}^{0})_{a\leq t\leq T}\) be a measurable process on \(\mathcal{P}_{\mathbb{F}}\) such that \(f_{t}^{0} \in \mathrm{ co}F(t,({x}^{0} \circ \theta _{0})(t))\) a.s. for a.e. a ≤ t ≤ T, where θ 0(t) = T for a ≤ t ≤ T. Let ϕ t = f t 0 a.s. for a.e. a ≤ t ≤ T. By virtue of Theorem 2.3, for \(\varepsilon _{1} = 1/{2}^{3/2}\) and the above process \(\phi = (\phi _{t})_{a\leq t\leq T}\), there exist a partition \(a = t_{p_{1}}^{1} < t_{p_{1}-1}^{1} < \cdots < t_{1}^{1} < t_{0}^{1} = T\), a step function θ 1 : [a, T] → [a, T], a step process \({z}^{1} = (z_{t}^{1})_{a\leq t\leq T}\), and a measurable process \({f}^{1} = (f_{t}^{1})_{a\leq t\leq T}\) on \(\mathcal{P}_{\mathbb{F}}\) such that conditions (i)–(vi) of Theorem 2.3 are satisfied. In particular, f t 1 ∈ coF(t, (x 1 ∘ θ 1)(t)), \(\vert f_{t}^{1} - f_{t}^{0}\vert =\mathrm{ dist}(f_{t}^{0},\mathrm{co}F(t,({x}^{1} \circ \theta _{1})(t)))\) a.s. for a.e. a ≤ t ≤ T and \(d({x}^{1}(t),\mathcal{K}(t)) \leq \varepsilon _{1}\) for a ≤ t ≤ T, because \(d({x}^{1}(t),\mathcal{K}(t)) \leq \vert {x}^{1}(t) -{x}^{1}(\theta (t))\vert + d({x}^{1}(\theta (t)),\mathcal{K}(\theta (t))) + D(\mathcal{K}(\theta (t)),\mathcal{K}(t)) \leq \varepsilon _{1}\), where \(x_{t}^{1} = E[x_{T} +\int _{ t}^{T}f_{\tau }^{0}\mathrm{d}\tau \vert \mathcal{F}_{t}] +\int _{ t}^{T}z_{\tau }^{1}\mathrm{d}\tau\) a.s. for a ≤ t ≤ T. In a similar way, for \(\phi = (f_{t}^{1})_{a\leq t\leq T}\) and \(\varepsilon _{2} = 1/{2}^{3}\), we can define a partition \(a = t_{p_{2}}^{2} < t_{p_{2}-1}^{2} < \cdots < t_{1}^{2} < t_{0}^{2} = T\,\), a step function θ 2 : [a, T] → [a, T], a step process \({z}^{2} = (z_{t}^{2})_{a\leq t\leq T}\), and a measurable process \({f}^{2} = (f_{t}^{2})_{a\leq t\leq T}\) such that \(f_{t}^{2} \in \mathrm{ co}F(t,({x}^{2} \circ \theta _{2})(t))\), \(\vert f_{t}^{2} - f_{t}^{1}\vert =\mathrm{ dist}(f_{t}^{1},\mathrm{co}F(t,({x}^{2} \circ \theta _{2})(t)))\) a.s. for a.e. a ≤ t ≤ T and \(d({x}^{2}(t),\mathcal{K}(t)) \leq \varepsilon _{2}\) for a ≤ t ≤ T, where \(x_{t}^{2} = E[x_{T} +\int _{ t}^{T}f_{\tau }^{1}\mathrm{d}\tau \vert \mathcal{F}_{t}] +\int _{ t}^{T}z_{\tau }^{2}\mathrm{d}\tau\) a.s. for a ≤ t ≤ T. Furthermore, for i = 1, 2, we have
a.s. for a ≤ s ≤ t ≤ T. By the inductive procedure, for \(\varepsilon _{k} = 1/{2}^{3k/2}\) and \({\phi }^{k} = (f_{t}^{k})_{a\leq t\leq T}\), we can select for every k ≥ 1, a partition \(a = t_{p_{k}}^{k} < t_{p_{k}-1}^{k} < \cdots < t_{1}^{k} < t_{0}^{k} = T\,\), a step function θ k : [a, T] → [a, T], a step process \({z}^{k} = (z_{t}^{k})_{a\leq t\leq T}\), and a measurable process \({f}^{k} = (f_{t}^{k})_{a\leq t\leq T}\) such that \(f_{t}^{k} \in \mathrm{ co}F(t,({x}^{k} \circ \theta _{k})(t))\), \(\vert f_{t}^{k} - f_{t}^{k-1}\vert =\mathrm{ dist}(f_{t}^{k},\mathrm{co}F(t,({x}^{k} \circ \theta _{k})(t)))\) a.s. for a.e. a ≤ t ≤ T and \(d({x}^{k}(t),\mathcal{K}(t)) \leq \varepsilon _{k}\) for a ≤ t ≤ T, where
a.s. for a ≤ t ≤ T. Furthermore,
for a ≤ s ≤ t ≤ T. Of course, \({x}^{k} \in \mathcal{S}(\mathbb{F}, {\mathbb{R}}^{m})\) for k ≥ 1. By Corollary 3.2 of Chap. 3 and the continuity of the filtration \(\mathbb{F}\), the process x k is continuous for every k ≥ 1. Furthermore, by the properties of the sequence \(({f}^{k})_{k=1}^{\infty },\) one gets
a.s. for a ≤ t ≤ T, where α = 8T 2. Therefore,
a.s. for a ≤ t ≤ T and k = 1, 2, …. By Doob’s inequality, we get
for a ≤ t ≤ T. Therefore, for every a ≤ t ≤ T and k = 1, 2, …, we have
where β = 8T. By the definitions of x 1 and x 0, we obtain \(E[\sup _{t\leq u\leq T}\vert {x}^{1}(u) - {x}^{0}(u){\vert }^{2}] \leq L,\) where \(L = T\int _{0}^{T}{m}^{2}(t)\mathrm{d}t\). Therefore,
for a ≤ t ≤ T. Hence it follows that
for a ≤ t ≤ T. By the inductive procedure, for every k = 1, 2, … and a ≤ t ≤ T, we obtain
where \(M =\max \{ {2\alpha }^{2},L\}\). Hence, by Chebyshev’s inequality and the Borel–Cantelli lemma, it follows that the sequence \(({x}^{k})_{k=1}^{\infty }\) of stochastic processes \(({x}^{k}(t))_{a\leq t\leq T}\) is for a.e. \(\omega \in \Omega \) uniformly convergent in [a, T] to a continuous process (x(t)) a ≤ t ≤ T . We can verify that the sequence \(({f}^{k})_{k=1}^{\infty }\) is a Cauchy sequence of \(\mathbb{L}([a,T] \times \Omega, \beta _{T} \otimes \mathcal{F}_{T}, {\mathbb{R}}^{m})\). Indeed, for every k = 0, 1, 2, …, one has
which by the properties of the sequence \(({x}^{k})_{k=1}^{\infty },\) implies that \(({f}^{k})_{k=1}^{\infty }\) is a Cauchy sequence. Then there is an \(f \in \mathbb{L}([a,T] \times \Omega, \beta _{T} \otimes \mathcal{F}_{T}, {\mathbb{R}}^{m})\) such that | f k − f | → 0 as k → ∞. Let \(y_{t} = E[x_{T} +\int _{ t}^{T}f_{\tau }\mathrm{d}\tau \vert \mathcal{F}_{t}]\) a.s. for a ≤ t ≤ T. For every k ≥ 1, we have
which implies that \(E[\sup _{a\leq t\leq T}\vert x(t) - y_{t}\vert ] = 0\). Then \(x(t) = E[x_{T} +\int _{ t}^{T}f_{\tau }\mathrm{d}\tau \vert \mathcal{F}_{t}]\) a.s. for a ≤ t ≤ T. Now, for every a ≤ s ≤ t ≤ T, we get
But
for every k ≥ 1 and a ≤ t ≤ T. Then
for every k ≥ 1 and a ≤ s ≤ t ≤ T. Thus
for every a ≤ s ≤ t ≤ T. In a similar way, we also get that \(d(x(t),\mathcal{K}(t)) = 0\) for every a ≤ t ≤ T. Then x is a strong solution of BSDI(F, K) on the interval [a, T] for every a ∈ (0, T).
We can now extend the above solution to the whole interval [0, T]. Let us denote by \(\Lambda _{x}\) the set of all extensions of the above-obtained viable solution x of BSDI(F, K). We have \(\Lambda _{x}\neq \varnothing \), because we can repeat the above procedure for every interval [α, T] with α ∈ (0, a] and get a solution x α of BSDI(F, K) on the interval [α, T]. The process \(z = 1\!\!1 _{[\alpha, a]}{x}^{\alpha } + 1\!\!1 _{(a,T]}x\) is an extension of x to the interval [α, T]. Let us introduce in \(\Lambda _{x}\) the partial order relation \(\preccurlyeq \) by setting \(x \preccurlyeq z\) if and only if a z ≤ a x and \(x = z\vert _{[a_{x},T]}\), where a x , a z ∈ (0, a) are such that x and z are strong viable solutions for BSDI(F, K) on [a x , T] and [a z , T], respectively, and \(z\vert _{[a_{x},T]}\) denotes the restriction of the solution z to the interval [a x , T]. Let \(\psi : [\alpha, T] \rightarrow {\mathbb{R}}^{d}\) be an extension of x to [α, T] with α ∈ (0, a] and denote by \(P_{x}^{\psi } \subset \Lambda _{x}\) the set containing ψ and all its restrictions ψ | [β, T] for every β ∈ (α, a). It is clear that each completely ordered subset of \(\Lambda _{x}\) is of the form P x ψ determined by some extension ψ of x and contains its upper bound ψ. Then by the Kuratowski–Zorn lemma, there exists a maximal element γ of \(\Lambda _{x}\). It has to be a γ = 0, where a γ ∈ [0, T) is such that γ is a strong viable solution of BSDI(F, K) on the interval [a γ , T]. Indeed, if we had a γ > 0, then we could repeat the above procedure and extend γ as a viable strong solution \(\xi \in \Lambda _{x}\) of BSFI(F, K) to the interval [b, T] with 0 ≤ b < a γ . This would imply that \(\gamma \preccurlyeq \xi\), a contradiction to the assumption that γ is a maximal element of \(\Lambda _{x}\). Then x can be extended to the whole interval [0, T].
Remark 3.2.
Theorem 3.4 is also true if \(\mathcal{K}(t) =\{ u \in \mathbb{L}(\Omega, \mathcal{F}_{0}, {\mathbb{R}}^{d}) : u \in K(t)\}\). In such a case, instead of (2.3), we can assume that \(\liminf _{h\rightarrow 0+}\overline{D}(x +\int _{ t-h}^{t}F(\tau, x)\mathrm{d}\tau, \mathcal{K}(t)) = 0\) for every \((t,x) \in Graph(\mathcal{K})\).
Proof.
For every \((t,x) \in Graph(\mathcal{K})\), f ∈ S(coF ∘ x), and \(u \in \mathcal{K}(t)\), we have
Therefore, \(d(E[x +\int _{ t-h}^{t}f_{\tau }\mathrm{d}\tau \vert \mathcal{F}_{t-h}],\mathcal{K}(t)) \leq d(x +\int _{ t-h}^{t}f_{\tau }\mathrm{d}\tau, \mathcal{K}(t))\) for every f ∈ S(coF ∘ x). Then
for every \((t,x) \in Graph(\mathcal{K})\). Thus, \(\liminf _{h\rightarrow 0+}\overline{D}(x +\int _{ t-h}^{t}F(\tau, x)\mathrm{d}\tau, \) \(\mathcal{K}(t - h)) = 0\) implies that (2.3) is satisfied.
Remark 3.3.
The results of the above existence theorems are also true if instead of continuity of the set-valued mapping \(\mathcal{K}\), we assume that it is uniformly upper semicontinuous on [0, T], i.e., that \(\lim _{\delta \rightarrow 0}\sup _{0\leq t\leq T}\overline{D}(\mathcal{K}(t+\delta ),\mathcal{K}(t)) = 0\).
It can be verified that the requirement X t ∈ K(t) a.s. for 0 ≤ t ≤ T in the above viability problems is too strong to be satisfied for some stochastic differential equations. For example, the stochastic differential equation \(dX_{t} = f(X_{t}) + dB_{t}\) with Lipschitz continuous and bounded function \(f : \mathbb{R} \rightarrow \mathbb{R}\) does not have any solution \(X = (X_{t})_{0\leq t\leq T}\) with X t belonging to a compact set \(K \subset \mathbb{R}\) a.s. for every 0 ≤ t ≤ T. This is a consequences of the following theorem.
Theorem 3.5.
Let \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F},P)\) be a filtered probability space and \(B = (B_{t})_{t\geq 0}\) a real-valued \(\mathbb{F}\) -Brownian motion on \(\mathcal{P}_{\mathbb{F}}\) . Assume that \(\xi = (\xi _{t})_{0\leq t\leq T}\) is an Itô diffusion such that \(d\xi _{t} =\alpha _{t}(\xi )dt + dB_{t}\) , ξ 0 = 0 for 0 ≤ t ≤ T. Then \(P(\{\int _{0}^{T}\alpha _{t}^{2}(\xi )\mathrm{d}t < \infty \}) = 1\) and \(P(\{\int _{0}^{T}\alpha _{t}^{2}(B)\mathrm{d}t < \infty \}) = 1\) if and only if ξ and B have the same distributions as C T -random variables on \(\mathcal{P}_{\mathbb{F}}\) , where \(C_{T} = C([0,T], \mathbb{R})\).
Example 3.1.
Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be bounded and Lipschitz continuous. Let \(\mathcal{P}_{\mathbb{F}}\) and B be as in Theorem 3.5. Put α t (x) = f(e t (x)) for x ∈ C T , where \(C_{T} = C([0,T], \mathbb{R})\) and e t is the evaluation mapping on C T , i.e., e t (x) = x(t) for x ∈ C T and 0 ≤ t ≤ T. Assume that K is a nonempty compact subset of \(\mathbb{R}\) such that 0 ∈ K and consider the viable problem
Suppose there is a solution X, an Itô diffusion, of the above viability problem such that X 0 = 0. By the properties of f, we have \(\int _{0}^{T}{f}^{2}(X_{t})\mathrm{d}t < \infty \) and \(\int _{0}^{T}{f}^{2}(B_{t})\mathrm{d}t < \infty \) a.s. Therefore, by virtue of Theorem 3.4, for every A ∈ β(C T ) with \(P{X}^{-1}(A) = 1\), one has \(P{X}^{-1}(A) = P{B}^{-1}(A)\). By the properties of the process X, one has P({X t ∈ K}) = 1. But \(P(\{X_{t} \in K\}) = P(\{e_{t}(X) \in K\}) = P{X}^{-1}(e_{t}^{-1}(K))\), where e t is the evolution mapping. Hence it follows that \(1 = P{X}^{-1}(e_{t}^{-1}(K)) = P{B}^{-1}(e_{t}^{-1}(K)) = P(\{B_{t} \in K\}) < 1\), a contradiction. Then the problem (3.1) does not have any K-viable strong solution.
Remark 3.4.
We can consider viability problems with weaker viable requirements of the form \(P(\{X_{t} \in K(t)\,\}) \in (\varepsilon, 1)\) for 0 ≤ t ≤ T and \(\varepsilon \in (0,1)\) sufficiently large. Solutions to such problems can be regarded as a type of approximations to viable solutions.
4 Weak Compactness of Viable Solution Sets
Let us denote by \(\mathcal{X}(F,G,K)\) the set of (equivalence classes of) all weak viable solutions of SFI(F, G, K). We shall show that for every F, G, and K satisfying the assumptions of Theorem 3.3, the set \(\mathcal{X}(F,G,K)\) is weakly compact, i.e., the set \({\mathcal{X}}^{P}(F,G,K)\) of distributions of all weak solutions of SFI(F, G, K) is weakly compact subsets of the space \(\mathcal{M}(C_{T})\) of all probability measures on the Borel σ-algebra β(C T ), where \(C_{T} =: C([0,T], {\mathbb{R}}^{d})\).
Theorem 4.1.
Assume that F and G are measurable, bounded, and convex-valued such that F(t,⋅ ) and G(t,⋅ ) are continuous for a.e. fixed t ∈ [0,T]. Let G be diagonally convex and \(K : [0,T] \rightarrow \mathrm{ Cl}({\mathbb{R}}^{d})\) continuous. If there exist a complete filtered probability space \(\mathcal{P}_{\mathbb{F}} = (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\) with a filtration \(\mathbb{F}\) satisfying the usual conditions and an m-dimensional \(\mathbb{F}\) -Brownian motion on \(\mathcal{P}_{\mathbb{F}}\) such that (2.1) is satisfied for every \(\varepsilon \in (0,1)\) and \((t,x) \in Graph({\mathcal{K}}^{\varepsilon })\) , then the set \(\mathcal{X}(F,G,K)\) of all weak viable solutions \((\mathcal{P}_{\mathbb{F}},x,B)\) of \(\overline{SFI}(F,G,K)\) is weakly compact.
Proof.
By virtue of Theorem 3.3, the set \(\mathcal{X}(F,G,K)\) is nonempty. Similarly as in the proof of Theorem 4.1 of Chap. 4, we can verify that \(\mathcal{X}(F,G,K)\) is relatively weakly compact. We shall prove that it is a weakly closed subset of the space \(\mathcal{M}(C_{T})\). Let \(({x}^{r})_{r=1}^{\infty }\) be a sequence of \(\mathcal{X}(F,G,K)\) convergent in distribution. Then there exists a probability measure \(\mathcal{P}\) on β(C T ) such that \(P{({x}^{r})}^{-1} \Rightarrow \mathcal{P}\) as r → ∞. By virtue of Theorem 2.3 of Chap. 1, there are a probability space \((\tilde{\Omega },\tilde{\mathcal{F}},\tilde{P})\) and random variables \(\tilde{{x}}^{r} :\tilde{ \Omega } \rightarrow C_{T}\) and \(\tilde{x} :\tilde{ \Omega } \rightarrow C_{T}\) for r = 1, 2, … such that \(P{({x}^{r})}^{-1} = P{(\tilde{{x}}^{r})}^{-1}\) for \(r = 1,2,\ldots, \,\tilde{P}{(\tilde{x})}^{-1} = \mathcal{P}\) and \(\lim _{r\rightarrow \infty }\sup _{0\leq t\leq T}\vert \tilde{x}_{t}^{r} -\tilde{ x}_{t}\vert = 0\) with \((\tilde{P}.1).\) By Theorem 1.3 of Chap. 4, we have \(\mathcal{M}_{FG}^{x_{r}}\neq \varnothing \) for every r ≥ 1, which by Theorem 1.5 of Chap. 4, implies that \(\mathcal{M}_{FG}^{\tilde{x}}\neq \varnothing.\) Therefore, by Theorem 1.3 of Chap. 4, there exist a standard extension \(\hat{\mathcal{P}}_{\hat{\mathbb{F}}} = (\hat{\Omega },\hat{\mathcal{F}},\hat{ \mathbb{F}},\hat{P})\) of \((\tilde{\Omega },\tilde{\mathcal{F}},\tilde{ \mathbb{F}},\tilde{P})\) and an m-dimensional Brownian motion \(\hat{\mathcal{P}}_{\hat{\mathbb{F}}}\) such that \((\hat{\mathcal{P}}_{\hat{\mathbb{F}}},\hat{x},\hat{B})\) is a weak solution of \(\overline{SFI}(F,G,\mu )\) with an initial distribution μ equal to the probability distribution \(P\tilde{x}_{0}^{-1}\). Similarly as in the proof of Theorem 3.3, this solution is defined by \(\hat{x}(\hat{\omega }) =\tilde{ x}(\pi (\hat{x}))\) for \(\hat{\omega }\in \hat{ \Omega }\). Similarly as in the proof of Theorem 4.1 of Chap. 4, we obtain \(P{({x}^{r})}^{-1} \Rightarrow P{(\hat{x})}^{-1}\) as r → ∞, which by the properties of the sequence \((\tilde{{x}}^{r})_{n=1}^{\infty }\) implies that \(P{(\tilde{{x}}^{r})}^{-1} \Rightarrow P{(\hat{x})}^{-1}\) as r → ∞. By the properties of the sequence \((x^r)_{r=1}^\infty\), we have \({E}^{r}[\mathrm{dist}({x}^{r}(t),K(t))] = 0\) for every r ≥ 1, which implies that \(\tilde{E}[\mathrm{dist}(\tilde{{x}}^{r}(t),K(t))] = 0\) for every r ≥ 1. Hence, by the continuity of the mapping dist( ⋅, K(t)) and properties of the sequence \((\tilde{{x}}^{r})_{n=1}^{\infty }\), it follows that \(\hat{E}[\mathrm{dist}(\hat{x}_{t},K(t))] = 0\). Thus \((\hat{\mathcal{P}}_{\hat{\mathbb{F}}},\hat{x},\hat{B})\) is a weak solution of \(\overline{SFI}(F,G,\mu ),\) with an appropriately chosen initial distribution μ, such that \({x}^{r} \Rightarrow \hat{ x}\) and \(\hat{x}_{t} \in K(t)\) with (\(\hat{P}\).1) for every t ∈ [0, T]. Then \((\hat{\mathcal{P}}_{\hat{\mathbb{F}}},\hat{x},\hat{B}) \in \mathcal{X}(F,G,K)\), and \(\mathcal{X}(F,G,K)\) is weakly closed.
Remark 4.1.
The results of Theorem 4.1 continue to hold if instead of (2.1), we assume that \([S_{\mathbb{F}}^{t}(F \circ x) \times S_{\mathbb{F}}^{t}(G \circ x)] \cap \mathcal{T}_{K}(t,x)\neq \varnothing \) for every \((t,x) \in {\mathcal{K}}^{\varepsilon }\) and \(\varepsilon \in (0,1)\).
5 Notes and Remarks
The viability approach to optimal control problems is especially useful for problems with state constraints. There is a great number of papers dealing with viability problems for differential inclusions. The first results dealing with viability problems for differential inclusions were given by Aubin and Cellina in [5]. The first result extending to the stochastic case of Nagumo’s viability theorem due to Aubin and Da Prato [7]. Most of the results concerning this topic have now been collected in the excellent book by Aubin [6]. Interesting generalizations of viability and invariance problems were given by Plaskacz [88]. A new approach to viability problems for stochastic differential equations was initiated by Aubin and Da Prato in [8] and [9] and by Millian in [79]. Later on, these results were extended by Aubin, Da Prato, and Frankowska [10, 12] in the case of stochastic inclusions written in differential form. Independently, viability problems for stochastic inclusions were also considered by Kisielewicz in [54] and Motyl in [85]. Viability theory provides geometric conditions that are equivalent to viability or invariance properties. Illustrations of viability approach to some issues in control theory and dynamical games with the problem of dynamic valuation and management of a portfolio, can be found in Aubin et al. [13]. The stochastic viability condition presented in Example 3.1 was constructed by M. Michta. The results contained in the present chapter are mainly based on methods applied in Aitalioubrahim and Sajid [3], Van Benoit and Ha [18], and Aubin and Da Prato [9]. The main results of this chapter dealing with the existence of viable strong and weak solutions of stochastic and backward stochastic inclusions and weak compactness with respect to convergence in the sense of distributions of viable weak solution sets are due to the author of this book.
References
Aitalioubrahim, M., Sajid, S.: Second order viability result in Banach spaces. Discuss. Math. Disc. Math. DICO 30, 5–21 (2010)
Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, New York (1984)
Aubin, J.P.: Viability Theory. Birkhäuser, Boston (1991)
Aubin, J.P., Da Prato, G.: Stochastic Nagumo’s viability theorem. Stoch. Anal. Appl. 13(1), 1–11 (1995)
Aubin, J.P., Da Prato, G.: The viability theorem for stochastic differential inclusions. Stoch. Anal. Appl. 16, 1–15 (1998)
Aubin, J.P., Da Prato, G., Frankowska, H.: Stochastic invariance for differential inclusions. Set-valued Anal. 8, 181–201 (2000)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Aubin, J.P., Pujol, D., Saint-Pierre, P.: Dynamical Management of Portfolios Under Contingent Uncertainty. Cener Rech. Viabilité, Jeux, Contrôle (2000)
Benoit, T.V., Ha, T.X.D.: Existence of viable solutions for a nonconvex stochastic differential inclusion. Disc. Math. DI 17, 107–131 (1997)
Kisielewicz, M.: Viability theorem for stochastic differential inclusions. Discuss. Math. 16, 61–74 (1995)
Millian, A.: A note on the stochastic invariance for Itô equation. Bull. Acad. Sci. Math. 41(2), 139–150 (1993)
Motyl, J.: Viable solutions to set-valued stochastic equations. Optimization 48, 157–176 (2000)
Plaskacz, S.: Value functions in control systems and differential games: A viability method. Lecture Notes in Nonlinear Analysis. Journal of Schauder Center for Nonlinear Analysis, N.C.University (2003)
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Kisielewicz, M. (2013). Viability Theory. In: Stochastic Differential Inclusions and Applications. Springer Optimization and Its Applications, vol 80. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6756-4_5
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