1 INTRODUCTION

In this paper, as a robust utility maximization problem with a penalty function, we mean the problem of maximizing the functional

$$\xi\rightsquigarrow\inf_{{\textrm{{Q}}}\in\mathcal{Q}}\bigl{(}{\textrm{{E}}}_{{\textrm{{Q}}}}U(\xi)+\gamma({\textrm{{Q}}})\bigr{)},\quad\xi\in\mathcal{A},$$

over some convex set \(\mathcal{A}\) of random variables defined on a probability space \((\Omega,\mathcal{F},{\textrm{{P}}})\).

Assumption 1 (on a utility function):  \(U:\,\mathbb{R}\to[-\infty,+\infty)\) is a monotonically nondecreasing concave function such that \(U(x)=-\infty\) for \(x<0\) and \(U(x)\in\mathbb{R}\) for \(x>0\).

Let \(\mathcal{Q}\) be some convex set of probability measures on \((\Omega,\mathcal{F})\), and let the penalty function \(\gamma\) be convex (see [1]).

We introduce the function \(V\) conjugate to \(U\) by the relation

$$V(y)=\sup_{x>0}\bigl{(}U(x)-xy\bigr{)},\quad y\in\mathbb{R}.$$

For a function \(f\colon X\to\mathbb{R}\cup\{+\infty\}\), the effective set \(\text{dom }f\) is defined as

$$\text{dom }f:=\{x\in X\colon f(x)<+\infty\}.$$

By Assumption 1, \(\text{dom }V\subseteq\mathbb{R}_{+}\), the function \(V\) is not monotonically increasing, and

$$\lim_{y\to+\infty}\frac{V(y)}{y}=0.$$

By the standard utility maximization problem we mean the case where \(\mathcal{Q}=\{{\textrm{{P}}}\}\).

Denote by \(ba\) the space of bounded finitely additive set functions \(\mu\colon\mathcal{F}\to\mathbb{R}\) such that

$$A\in\mathcal{F},\quad{\textrm{{P}}}(A)=0\,\Rightarrow\,\mu(A)=0,$$

with the total variation norm. It is well known that \(ba\) is dual to the space \(L^{\infty}\), and the duality is given by the relation

$$\langle\xi,\mu\rangle:=\mu(\xi):=\int\limits_{\Omega}\xi d\mu,\quad\mu\in ba,\quad\xi\in L^{\infty}.$$

A subspace of the space \(ba\) consisting of countably additive measures is denoted by ca. For \(\mu\in ba\), there exists a unique decomposition \(\mu=\mu^{r}+\mu^{s}\) into a countably additive measure \(\mu^{r}\in ca\) and a purely finitely additive set function \(\mu^{s}\in ba\). The space \(ca\) is naturally identified with \(L^{1}\) by the relations \(\xi\in L^{1}\) and \(\xi\rightsquigarrow\xi\cdot{\textrm{{P}}}\in ca\), where \(\xi\cdot{\textrm{{P}}}\) is a measure with density \(\xi\) in P.

The proof of the results of this paper uses the notion of \(f\)-divergence. Let us give its formal definition. Let \(f:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}\) be a proper, lower semicontinuous and convex function with \(\text{dom }f\subseteq\mathbb{R}_{+}\). In [2], Gushchin gave a definition of the \(f\)-divergence \(\mathcal{J}_{f}(\mu,\nu)\) of finitely additive functions \(\mu\) and \(\nu\) given on \((\Omega,\mathcal{F})\). For \(\mu,\nu\in ba\), this definition is equivalent to the following:

$$\mathcal{J}_{f}(\mu,\nu)=\sup_{\xi,\,\eta\,\in\,L^{\infty}\colon\eta+f^{*}(\xi)\leqslant 0}\bigl{(}\mu(\xi)+\nu(\eta)\bigr{)},$$

where \(f^{*}\) is the Fenchel transform of the function \(f\). It follows from the definition that the function \(\mathcal{J}_{f}(\mu,\nu)\) on \(ba\times ba\) takes values in \(\mathbb{R}\cup\{+\infty\}\) and is convex and lower semicontinuous in the topology \(\sigma(ba\times ba,L^{\infty}\times L^{\infty})\). The properties used in this paper were proved in [2, Theorem 1].

It will be convenient for us to extend the domain of the penalty function \(\gamma\) to the space \(ba\) by setting it equal to \(+\infty\) outside \(\mathcal{Q}\). Then \(\mathcal{Q}\) is characterized as the effective domain \(\text{dom }\gamma\).

Assumption 2 (on a penalty function):  \(\gamma\colon ba\to\mathbb{R}\cup\{+\infty\}\) is a proper convex function such that \(\text{dom }\gamma=:\mathcal{Q}\) is a subset of the set of all probability measures on \((\Omega,\mathcal{F})\), \(\inf_{{\textrm{{Q}}}\in\mathcal{Q}}\gamma({\textrm{{Q}}})\geqslant 0\), and the set

$$\{d{\textrm{{Q}}}/d{\textrm{{P}}}\colon\ {\textrm{{Q}}}\in\mathcal{Q},\ \gamma({\textrm{{Q}}})\leqslant c\}$$

is closed in \(L^{1}\) and uniformly integrable with respect to P for any \(c\geqslant 0\).

Denote by \(L^{0}\) the space of P-a.s. equivalence classes of equal random variables with real values. When we speak of random variables, we mean the equivalence classes that they generate.

Assumption 3 (on the set of terminal wealths): \(\mathcal{A}\) is a convex subset \(L^{0}\) containing a random variable \(\xi_{0}\geqslant\varkappa\) for some \(\varkappa>0\).

The cone of nonnegative random variables is denoted by \(L^{0}_{+}\). We define

$$\mathcal{D}:=\{\eta\in L^{0}_{+}\colon\ {\textrm{{E}}}_{{\textrm{{P}}}}\eta\xi\leqslant 1\text{ for any }\xi\in\mathcal{A}\}.$$
(1)

It is clear that \(\mathcal{D}\subseteq L^{1}_{+}\) since \({\textrm{{E}}}_{{\textrm{{P}}}}\eta\leqslant\varkappa^{-1}\) for any \(\eta\in\mathcal{D}\). For \(x>0\) and \(y\geqslant 0\), we put

$$\mathcal{A}(x):=x\mathcal{A},\quad\mathcal{D}(y):=y\mathcal{D}.$$

We define primal and dual optimization problems:

$$u(x):=\sup_{\xi\in\mathcal{A}(x)}\inf_{{\textrm{{Q}}}\in\mathcal{L}}\bigl{(}{\textrm{{E}}}_{{\textrm{{Q}}}}U(\xi)+\gamma({\textrm{{Q}}})\bigr{)},\quad x>0;$$
(2)
$$v(y):=\inf_{\eta\in\mathcal{D}(y),\,{\textrm{{Q}}}\in\mathcal{L}}\left({\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{\eta}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right)+\gamma({\textrm{{Q}}})\right),\quad y\geqslant 0.$$
(3)

We have the equalities (see [3]):

$$u(x)=\min_{y\geqslant 0}\bigl{(}v(y)+xy\bigr{)},\quad x>0;$$
(4)
$$v(y)=\sup_{x>0}\bigl{(}u(x)-xy\bigr{)},\quad y\geqslant 0.$$
(5)

The main result of this paper is new and answers the question: when the set \(\mathcal{D}\) defined in (1) can be replaced by a convex set \(\widetilde{\mathcal{D}}\subseteq\mathcal{D}\) in the definition of the dual function \(v\) (see (3))? This situation is considered in abstract form in Lemma 1 and in a more concrete form in Theorem 1. A similar result for the nonrobust case was obtained in the joint work of Kramkov and Schachermayer (see [4]).

2 AUXILIARY RESULTS

Given a probability measure \({\textrm{{Q}}}\ll{\textrm{{P}}}\), we define the functions

$$v_{{\textrm{{Q}}}}(y):=\inf_{\eta\in\mathcal{D}}{\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{y\eta}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right),\quad y\geqslant 0;$$
$$\widetilde{v}_{{\textrm{{Q}}}}(y):=\inf_{\eta\in\widetilde{\mathcal{D}}}{\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{y\eta}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right),\quad y\geqslant 0.$$

It can be seen from (3) that it suffices to consider whether or not the functions \(v_{{\textrm{{Q}}}}\) and \(\widetilde{v}_{{\textrm{{Q}}}}\) coincide.

Definition 1. For a set \(\mathcal{E}\subseteq L^{0}_{+}\), we define its polar \(\mathcal{E}^{\circ}\) by

$$\mathcal{E}^{\circ}:=\{\xi\in L^{0}_{+}\colon{\textrm{{E}}}_{{\textrm{{P}}}}\eta\xi\leqslant 1\text{ for any }\eta\in\mathcal{E}\}.$$

Using these terms, the definition of the set \(\mathcal{D}\) in (1), in which \(\mathcal{A}\) can be replaced by \(\mathcal{C}_{+}:=(\mathcal{A}-L^{0}_{+})\cap L^{\infty}_{+},\) is written as \(\mathcal{D}=\mathcal{C}^{\circ}_{+}\); \(\overline{\mathcal{C}}^{0}_{+}\) denotes the closure of the set \(\mathcal{C}_{+}\) in \(L^{0}\).

Lemma 1. Suppose that the set \(\mathcal{A}\) satisfies Assumption \(3\) , \(\mathcal{A}\subseteq L^{0}_{+}\) , the set \(\mathcal{D}\) is defined in \((1)\) and \(\widetilde{\mathcal{D}}\subseteq\mathcal{D}\) , and the set \(\widetilde{\mathcal{D}}\) is convex and not empty. We introduce the following conditions:

(i) For any \(\eta\in\mathcal{D}\), there exists \(\widetilde{\eta}\in\widetilde{\mathcal{D}}\) such that \(\eta\leqslant\widetilde{\eta}\).

(ii) \(v_{{\textrm{{Q}}}}(y)=\widetilde{v}_{{\textrm{{Q}}}}(y)\) for all \({\textrm{{Q}}}\ll{\textrm{{P}}}\) and \(y\geqslant 0\) for any function \(U\) satisfying Assumption \(1\).

(iii) \(v_{{\textrm{{Q}}}}(y)=\widetilde{v}_{{\textrm{{Q}}}}(y)\) for all \({\textrm{{Q}}}\ll{\textrm{{P}}}\) and \(y\geqslant 0\) for some strictly increasing function \(U\) satisfying Assumption \(1\).

(iv) For any \(f\in L^{0}_{+}\),

$$\sup_{g\in\mathcal{D}}{\textrm{{E}}}fg=\sup_{g\in\widetilde{\mathcal{D}}}{\textrm{{E}}}fg.$$

Then \(\textrm{(i)}\Rightarrow\textrm{(ii)}\Rightarrow\textrm{(iii)}\Rightarrow\textrm{(iv)}.\) If the closure \(\overline{\widetilde{\mathcal{D}}}^{0}_{+}\) of the set \(\widetilde{\mathcal{D}}\) in \(L^{0}\) lies in \(\widetilde{\mathcal{D}}-L^{0}_{+},\) then all four conditions are equivalent.

Remark 1. We have \(\mathcal{D}^{\circ}=\bigl{(}\mathcal{C}_{+}^{\circ}\bigr{)}^{\circ}\). As is easily seen, condition (iv) of Lemma 1 is equivalent to the fact that \(\mathcal{D}^{\circ}=\widetilde{\mathcal{D}}^{\circ}\). On the other hand, since \(\mathcal{C}\) is convex and solid, the Brannath–Schachermayer bipolar theorem [5] states that \(\bigl{(}\mathcal{C}_{+}^{\circ}\bigr{)}^{\circ}\) coincides with the closure \(\overline{\mathcal{C}}^{0}_{+}\) of the set \(\mathcal{C}_{+}\) in \(L^{0}\).

Proof of Lemma 1. The implications \(\textrm{(i)}\Rightarrow\textrm{(ii)}\Rightarrow\textrm{(iii)}\) are obvious. Suppose that condition (iii) holds, while condition (iv) is not satisfied. Then there are \(f\in L_{+}^{0}\) and \(\eta\in\mathcal{D}\) such that

$${\textrm{{E}}}f\eta>\sup_{g\in\widetilde{\mathcal{D}}}{\textrm{{E}}}fg.$$
(6)

Cutting off \(f\) and \(\eta\) from above, we can consider that \(f\) and \(\eta\) are bounded, and, adding a small constant to \(f\) (recall that \({\textrm{{E}}}g\leqslant\varkappa^{-1}\) for any \(g\in\mathcal{D}\supseteq\widetilde{\mathcal{D}}\)), we have \(f\geqslant\varepsilon>0\). Let us now set \(q=y\eta/U^{{}^{\prime}}_{+}(f)\), where \(y>0\) is chosen from the normalization condition \({\textrm{{E}}}q=1\), and \({\textrm{{Q}}}=q\cdot{\textrm{{P}}}\). Note that here \(U^{{}^{\prime}}_{+}\) is the right derivative of the utility function \(U\). It exists since, by Assumption 1, the utility function never goes to infinity on the positive semiaxis \(\mathbb{R}_{+}\). Note that P and Q are probability measures, i.e., countably additive measures: \({\textrm{{P}}}={\textrm{{P}}}^{r}\), \({\textrm{{Q}}}={\textrm{{Q}}}^{r}\), and \({\textrm{{P}}}^{s}={\textrm{{Q}}}^{s}=0\). We have [2, Theorem 1]

$${\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{yg}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right)={\textrm{{E}}}_{{\textrm{{Q}}}}V\left(yg\frac{d{\textrm{{P}}}/d{\textrm{{P}}}}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right)=\bigl{[}\mathcal{J}_{V}\bigl{(}0,0\bigr{)}=0\bigr{]}=\mathcal{J}_{V}\bigl{(}(yg)\cdot{\textrm{{P}}},{\textrm{{Q}}}\bigr{)}+\mathcal{J}_{V}\bigl{(}0,0\bigr{)}$$
$${}=\mathcal{J}_{V}\bigl{(}(yg)\cdot{\textrm{{P}}}^{r},{\textrm{{Q}}}^{r}\bigr{)}+\mathcal{J}_{V}\bigl{(}(yg)\cdot{\textrm{{P}}}^{s},{\textrm{{Q}}}^{s}\bigr{)}=\mathcal{J}_{V}\bigl{(}(yg)\cdot{\textrm{{P}}},{\textrm{{Q}}}\bigr{)}=\sup_{\xi\in L^{\infty}:\,U(\xi)\in L^{\infty}}\bigl{(}{\textrm{{E}}}_{{\textrm{{Q}}}}U(\xi)-y{\textrm{{E}}}g\xi\bigr{)}.$$

Note that here \(\mathcal{J}_{V}\) is the \(V\)-divergence. The last equality follows from the definition of the \(V\)-divergence in terms of mathematical expectation.

Note that, for \(g=\eta\), the upper bound is attained on \(f\):

$${\textrm{{E}}}_{{\textrm{{Q}}}}U(\xi)-y{\textrm{{E}}}\eta\xi=y{\textrm{{E}}}\eta\left(\frac{U(\xi)}{U^{{}^{\prime}}_{+}(f)}-\xi\right)\leqslant y{\textrm{{E}}}\eta\left(\frac{U(f)}{U^{{}^{\prime}}_{+}(f)}-f\right),$$

the inequality follows from the concavity of \(U\), since the local maximum is global for a concave function. Therefore,

$${\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{y\eta}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right)={\textrm{{E}}}_{{\textrm{{Q}}}}U(f)-y{\textrm{{E}}}\eta f,$$

while

$${\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{yg}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right)\geqslant{\textrm{{E}}}_{{\textrm{{Q}}}}U(f)-y{\textrm{{E}}}gf.$$

Hence,

$$v_{{\textrm{{Q}}}}(y)\leqslant{\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{y\eta}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right)={\textrm{{E}}}_{{\textrm{{Q}}}}U(f)-y{\textrm{{E}}}\eta f<{\textrm{{E}}}_{{\textrm{{Q}}}}U(f)-y\sup_{g\in\widetilde{\mathcal{D}}}{\textrm{{E}}}gf$$
$${}\leqslant\inf_{g\in\widetilde{\mathcal{D}}}{\textrm{{E}}}_{{\textrm{{Q}}}}V\left(\frac{yg}{d{\textrm{{Q}}}/d{\textrm{{P}}}}\right)=\widetilde{v}_{{\textrm{{Q}}}}(y),$$

where the strict inequality follows from \((6)\). We come to the required contradiction.

Let now \(\overline{\widetilde{\mathcal{D}}}^{0}_{+}\subseteq\widetilde{\mathcal{D}}-L^{0}_{+}\), and let condition \((\textrm{iv})\) hold. It follows from Remark 1 that \((\textrm{iv})\) implies \(\mathcal{D}=\bigl{(}\widetilde{\mathcal{D}}^{\circ}\bigr{)}^{\circ}\). On the other hand, since the set \(\widetilde{\mathcal{D}}\) is convex and bounded in \(L^{1}\), standard arguments based on the transition to convex combinations show that the set \((\widetilde{\mathcal{D}}-L_{+}^{0})\cap L_{+}^{0}\) is closed in \(L^{0}\) and, consequently, there is the smallest subset of \(L_{+}^{0}\) containing \(\widetilde{\mathcal{D}}\) that is convex, solid and closed in \(L^{0}\). By the Brannath–Schachermayer bipolar theorem [5], \(\bigl{(}\widetilde{\mathcal{D}}^{\circ}\bigr{)}^{\circ}=(\widetilde{\mathcal{D}}-L_{+}^{0})\cap L_{+}^{0}\). Thus, condition \((\textrm{i})\) is satisfied.

3 MAIN RESULTS

Assume that the probability space \((\Omega,\mathcal{F},{\textrm{{P}}})\) is endowed with a filtration \(\mathbb{F}=(\mathcal{F}_{t})_{t\geqslant 0}\) satisfying the usual conditions. We have \(\mathcal{F}=\sigma(\cup_{t\geqslant 0}\mathcal{F}_{t})\) and \(\mathcal{F}_{0}\) contains only sets of P-measure 0 or 1. We denote by \(\mathbb{D}\) the set of real consistent random processes \(X=(X_{t})_{t\geqslant 0}\) whose trajectories are continuous on the right and have finite limits on the left; let \(\mathbb{D}_{+}=\{X\in\mathbb{D}:\ X\geqslant 0\}\) and \(\mathbb{D}_{++}=\{X\in\mathbb{D}:\ {\textrm{{P}}}(\inf_{t}X_{t}>0)=1\}.\) If \(X\in\mathbb{D}\) and there P-a.s. exists a finite limit \(\lim_{t\to\infty}X_{t},\) then the element \(L^{0}\) corresponding to this limit is denoted by \(X_{\infty}.\)

We assume that a family of processes \(\mathcal{X}\subseteq\mathbb{D}_{+}\) is given such that its elements are interpreted as wealth processes corresponding to all possible investment strategies, with a unit initial capital. If an investor has an initial capital \(x>0,\) then the wealth processes corresponding to his different strategies form the family \(\mathcal{X}(x)=x\mathcal{X}.\)

Assumption 4 (on a family of wealth processes): the set \(\mathcal{X}\subseteq\mathbb{D}_{+}\) is convex, \(X_{0}=1\) for any process \(X\in\mathcal{X},\) \(1\in\mathcal{X}\), and there P-a.s. exists a finite limit \(\lim_{t\to\infty}X_{t}\) for any \(X\in\mathcal{X}.\)

We set \(\mathcal{A}=\{X_{\infty}:X\in\mathcal{X}\}.\) If \(\mathcal{X}\) satisfies Assumption 4, then \(\mathcal{A}\) satisfies Assumption 3 and \(\mathcal{A}\subseteq L^{0}_{+}.\) We define \(\mathcal{D}\) by (1).

Definition 2. A process \(Y\in\mathbb{D}_{+}\) is called the supermartingale density for the class of processes \(\mathcal{X}\) if \(Y_{0}=1\) and \(YX\) is a P-supermartingale for any \(X\in\mathcal{X}.\)

The class of all supermartingale densities is denoted by \(\mathcal{Y}\); let \(\widetilde{\mathcal{D}}:=\{Y_{\infty}:Y\in\mathcal{Y}\}.\)

The next lemma is standard.

Lemma 2. The set \(\widetilde{\mathcal{D}}\) is convex, \(\widetilde{\mathcal{D}}\subseteq\mathcal{D}\) , and \(\overline{\widetilde{\mathcal{D}}}^{0}_{+}\subseteq\widetilde{\mathcal{D}}-L^{0}_{+}.\)

Proof of Lemma 2. The convexity of \(\widetilde{\mathcal{D}}\) is obvious. If \(Y\in\mathcal{Y}\), then, for any process \(X\in\mathcal{X}\), due to Fatou’s lemma and the supermartingale property, we have

$${\textrm{{E}}}Y_{\infty}X_{\infty}\leqslant\lim_{t\to\infty}{\textrm{{E}}}Y_{t}X_{t}\leqslant{\textrm{{E}}}Y_{0}X_{0}=1;$$

therefore, \(Y_{\infty}\in\mathcal{D}\).

Let now a sequence \((Y^{n})\) of \(\mathcal{Y}\) be given, and let \(Y_{\infty}^{n}\) converge P-a.s. to \(\eta\). By Lemma 5.2 of [6], there are a sequence \(Z^{n}\in\text{conv}(Y^{n},Y^{n+1},\ldots)\) and a supermartingale \(Z\) with \(Z_{0}\leqslant 1\) such that \(Z^{n}\) are Fatou convergent on a countable everywhere dense subset of \(\mathbb{R}_{+}\) (we refer the reader to the mentioned paper [6] for the definition of Fatou convergence); in this case, we can assume that \(Z_{\infty}^{n}\to Z_{\infty}\)P-a.s.  (by a deterministic change of time, we can reduce the processes \(Y^{n}\) to \([0,1)\) and continue them to \([1,\infty)\) by \(Y_{\infty}^{n}\)). Since \(XZ^{n}\) are Fatou convergent to \(XZ\) for \(X\in\mathbb{D}_{+}\) and the Fatou convergence retains the supermartingale property, the process \(XZ\) is a supermartingale for any \(X\in\mathcal{X}\). Since it is obvious that \(\xi=Z_{\infty}\), it remains to note that, in the case \(0<Z_{0}\leqslant 1\), we have \(Z/Z_{0}\in\mathcal{Y}\) and the quantity \(Z_{\infty}/Z_{0}\) majorizes \(\xi\), and the case \(Z_{0}=0\) is trivial.

Recall that we are interested in the following question: under what assumptions the set \(\mathcal{Y}\) is nonempty and conditions (i)–(iv) of Lemma 1 are satisfied for \(\widetilde{\mathcal{D}}\), i.e., when the solution of the robust utility maximization problem (2) satisfies equalities (4) and (5) with the dual function \(v\), in the definition (3) of which the set \(\{Y_{\infty}:Y\in\mathcal{Y}\}\) stands instead of the set \(\mathcal{D}\)?

Definition 3. A family \(\mathcal{X}\subseteq\mathbb{D}_{+}\) is called forked if, for any \(X^{i}\in\mathcal{X}\cap\mathbb{D}_{++},\) \(i=1,\,2,\,3,\) for any \(s\geqslant 0\) and every \(B\in\mathcal{F}_{s}\), the process

$$X_{t}=X^{1}_{t}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{\{t<s\}}+X^{1}_{s}\left({\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{B}\frac{X^{2}_{t}}{X^{2}_{s}}+{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{\Omega\backslash B}\frac{X^{3}_{t}}{X^{3}_{s}}\right){\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{\{t\geqslant s\}}$$

belongs to \(\mathcal{X}.\)

This definition is very close to the definition of the fork-convex family (see [7]), in which \({\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{B}\) and \({\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{\Omega\backslash B}\) are replaced by \(h\) and \(1-h\), respectively, where \(h\) is a \(\mathcal{F}_{s}\)-measurable random variable with values in \([0,1].\) Even when combined with convexity, our forking property is rather weaker than the property of fork-convexity.

Obviously, for any family \(\mathcal{X}\in\mathbb{D}_{+}\) there is the smallest forked family containing \(\mathcal{X}\), which we denote by fork(\(\mathcal{X}\)).

Theorem. Suppose that Assumption \(4\) holds true, \(\mathcal{A}=\{X_{\infty}:X\in\mathcal{X}\},\) \(\mathcal{D}\neq\{0\},\) where the set \(\mathcal{D}\) is defined in \((1)\) , and \(\widetilde{\mathcal{D}}:=\{Y_{\infty}:Y\in\mathcal{Y}\}.\) In order that the set \(\widetilde{\mathcal{D}}\) be nonempty and conditions (i)–(iv) of Lemma \(1\) hold for it, it is necessary and sufficient that

$$\{X_{\infty}:X\in\text{fork}(\mathcal{X})\}\subseteq\overline{\mathcal{C}}^{0}_{+}.$$
(7)

Proof. It is easier to prove necessity than sufficiency. Assume

$$\mathcal{X}_{0}:=\text{fork}(\mathcal{X})\cap\{X\in\mathbb{D}_{+}:\ XY\text{ is a supermartingale for any}\ Y\in\mathcal{Y}\}.$$

It is easy to verify that the set \(\mathcal{X}_{0}\) is forked. Hence, \(\mathcal{X}_{0}=\text{fork}(\mathcal{X})\) and the process \(XY\) is a supermartingale for any \(X\in\text{fork}(\mathcal{X})\) and \(Y\in\mathcal{Y}\).

We take \(X\in\text{fork}(\mathcal{X})\) and let \(\eta\in\mathcal{D}\). By condition \(\rm(i)\) of Lemma 1, there exists a process \(Y\in\mathcal{Y}\) such that \(Y_{\infty}\geqslant\eta\). Then

$${\textrm{{E}}}X_{\infty}\eta\leqslant{\textrm{{E}}}X_{\infty}Y_{\infty}\leqslant{\textrm{{E}}}X_{0}Y_{0}=1,$$

i.e., \(X_{\infty}\in\mathcal{D}^{\circ}=\overline{\mathcal{C}}^{0}_{+}\).

Let us prove the sufficiency of condition (7). Take an arbitrary variable \(\eta\in\mathcal{D},\) \(\eta\neq 0\). We have \({\textrm{{E}}}X_{\infty}\eta\leqslant 1\) for any process \(X\in\text{fork}(\mathcal{X})\). For \(t\in\mathbb{R}_{+}\), we define a random variable \(Y_{t}\) by the equality

$$Y_{t}=\mathop{\text{ess sup}\,}_{X\in\text{fork}({\mathcal{X}})\cap\mathbb{D}_{++}}\frac{{\textrm{{E}}}(\eta X_{\infty}|\mathcal{F}_{t})}{X_{t}}.$$

Further, the proof of Lemma 4 from [8] is repeated almost verbatim, which requires only forking but not fork-convexity of the set \(\mathcal{X}_{>}:=\text{fork}(\mathcal{X})\cap\mathcal{D}_{++}.\) We first prove that, for each \(t\in\mathbb{R}_{+}\), there is a sequence \((X^{n})\) of \(\mathcal{X}_{>}\) such that the random variables \(\dfrac{{\textrm{{E}}}(\eta X_{\infty}^{n}|\mathcal{F}_{t})}{X_{t}^{n}}\) are monotonically increasing towards \(Y_{t}\). Then we verify the supermartingale property of the process \(YX\) for any process \(X\in\mathcal{X}_{>}\). Finally, we check that the mathematical expectation \({\textrm{{E}}}Y_{t}\) is right-continuous. This implies that the process \(Y\) has a modification from \(\mathbb{D}_{+}\), which we denote by \(Y\) from now on.

Let now \(X\in\mathcal{X}\). Then, for any positive integer \(n\), the process \(X^{n}:=(1-1/n)X+1/n\) belongs to \(\mathcal{X}\cap\mathcal{X}_{>}\); therefore, \(X^{n}Y\) is a supermartingale. Whence it follows in an elementary way that \(XY\) is a supermartingale. Obviously, \(Y_{t}\geqslant{\textrm{{E}}}(\eta|\mathcal{F}_{t})\) for every \(t\), which implies \(Y_{\infty}\geqslant\eta\). On the other hand, \(Y_{0}=\sup_{X\in\mathcal{X}_{>}}{\textrm{{E}}}\eta X_{\infty}\leqslant 1\). Therefore, \(Z:=Y/Y_{0}\in\mathcal{D}\) and \(Z_{\infty}\geqslant\eta\).

As a corollary, we obtain a slight generalization of the result of Rokhlin (see [7]), where a fork-convex family of random processes was considered.

Corollary. Let \(\mathcal{W}\subseteq\mathbb{D}_{+}\) be a convex and forked family of random processes, \(1\in\mathcal{W}\) , and let \(X_{0}=1\) for any process \(X\in\mathcal{W}.\) The following conditions are equivalent:

(i) The set \(\{X_{t}:X\in\mathcal{W},\,t\in\mathbb{R}_{+}\}\) is bounded in probability.

(ii) There exists a supermartingale density \(Y\) for the family \(\mathcal{W}\) with \({\textrm{{P}}}(Y_{\infty}>0)=1.\)

Proof. The implication \(\rm(ii)\Rightarrow\rm(i)\) is elementary and can be proved in the same way as in [7]. Let us prove \(\rm(i)\Rightarrow\rm(ii)\). To this end, we introduce a family of processes

$$\mathcal{X}:=\{X^{t}:\,X\in\mathcal{W},\,t\in\mathbb{R}_{+}\},$$

where \(X^{t}\) denotes the process stopped at time \(t:\ X_{s}^{t}=X_{s\wedge t}.\) Since the family \(\mathcal{W}\) is forked and \(1\in\mathcal{W}\), we have \(\mathcal{X}\subseteq\mathcal{W}\). It is obvious that the family \(\mathcal{X}\) satisfies Assumption 4 and is forked.

Condition \(\rm(i)\) means that the set \(\mathcal{A}=\{X_{\infty}:\,X\in\mathcal{X}\}\) is bounded in probability. By Yan’s theorem [9, Theorem 1], there exists \(\eta\in L_{+}^{\infty}\) with \({\textrm{{P}}}(\eta>0)=1\) and \(\sup_{X\in\mathcal{X}}{\textrm{{E}}}\eta X_{\infty}\leqslant 1.\) The theorem implies the existence of a supermartingale density \(Y\) for the family \(\mathcal{X}\) with \(Y_{\infty}\geqslant\eta\). But it can be easily seen that \(Y\) is a supermartingale density for the family \(\mathcal{W}\) as well.