Abstract
In this paper we derive error estimates of the backward Euler–Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler–Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525–589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.
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1 Introduction
In this paper, we investigate the numerical approximation of multi-valued stochastic differential equations (MSDE). An important example of such equations is provided by stochastic gradient flows with a convex potential. More precisely, let \(T \in (0,\infty )\) and \((\varOmega ,{\mathcal {F}}, ({\mathcal {F}}_t)_{t \in [0,T]}, {\mathbf {P}})\) be a filtered probability space satisfying the usual conditions. By \(W :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^m\), \(m \in {\mathbf {N}}\), we denote a standard \(({\mathcal {F}}_t)_{t \in [0,T]}\)-adapted Wiener process whose increments are independent of the filtration. As a motivating example, let us consider the numerical treatment of nonlinear, overdamped Langevin-type equations of the form
where \(X_0 \in L^p(\varOmega ,{\mathcal {F}}_0,{\mathbf {P}};{\mathbf {R}}^d)\), \(p \in [2,\infty )\), \(g_0 \in {\mathbf {R}}^{d,m}\), and \(\varPhi :{\mathbf {R}}^d \rightarrow {\mathbf {R}}\) are given. The space \({\mathbf {R}}^{d,m}\) consists of the real matrices of type \(d \times m\). These equations have many important applications, for example, in Bayesian statistics and molecular dynamics. We refer to [10, 22, 23, 45, 50], and the references therein.
We recall that if the gradient \(\nabla \varPhi \) is of superlinear growth, then the classical forward Euler–Maruyama method is known to be divergent in the strong and weak sense, see [18]. This problem can be circumvented by using modified versions of the explicit Euler–Maruyama method based on techniques such as taming, truncating, stopping, projecting, or adaptive strategies, cf. [4, 6, 17, 19, 29, 49].
In this paper we take an alternative approach by considering the backward Euler–Maruyama method. Our main motivation for considering this method lies in its good stability properties, which allow its application to stiff problems arising, for instance, from the spatial semi-discretization of stochastic partial differential equations. Implicit methods have also been studied extensively in the context of stochastic differential equations with superlinearly growing coefficients. For example, see [1, 15, 16, 30, 31].
The error analysis in the above mentioned papers on explicit and implicit methods typically requires a certain degree of smoothness of \(\nabla \varPhi \) such as local Lipschitz continuity. The purpose of this paper is to derive error estimates of the backward Euler–Maruyama method for equations of the form (1.1), where the associated potential \(\varPhi :{\mathbf {R}}^d \rightarrow {\mathbf {R}}\) is not necessarily continuously differentiable, but assumed to be convex.
For the formulation of the numerical scheme, let \(N \in {\mathbf {N}}\) be the number of temporal steps, let \(k = \frac{T}{N}\) be the step size, and let
be an equidistant partition of the interval [0, T], where \(t_n = n k\) for \(n \in \{0,\dots ,N\}\). The backward Euler–Maruyama method for the Langevin equation (1.1) is then given by the recursion
where \(\varDelta W^n = W(t_n) - W(t_{n-1})\).
An example of a non-smooth potential is found by setting \(d = m = 1\) and \(\varPhi (x) = |x|^{p}\), \(x \in {\mathbf {R}}\), for \(p \in [1,2)\). Evidently, the gradient of \(\varPhi \) is not locally Lipschitz continuous at \(0 \in {\mathbf {R}}\) for \(p \in (1,2)\). Moreover, if \(p = 1\), then the gradient \(\nabla \varPhi \) has a jump discontinuity of the form
Here, the value \(c \in {\mathbf {R}}\) at \(x =0\) is not canonically determined. We have to solve a nonlinear equation of the form \(x + k \nabla \varPhi (x) = y\) in each step of the backward Euler method (1.3). However, if \(y \in (-k,k)\), then the sole candidate for a solution is \(x = 0\), since otherwise \(|x + k \nabla \varPhi (x)| \ge k\). But \(x=0\) is only a solution if \(kc = y\). Therefore, the mapping \({\mathbf {R}}\ni x \mapsto x + k \nabla \varPhi (x) \in {\mathbf {R}}\) is not surjective for any single-valued choice of c.
This problem can be bypassed by considering the multi-valued subdifferential \(\partial \varPhi :{\mathbf {R}}^d \rightarrow 2^{{\mathbf {R}}^d}\) of a convex potential \(\varPhi :{\mathbf {R}}^d \rightarrow {\mathbf {R}}\), which is given by
Recall that \(\partial \varPhi (x) = \{\nabla \varPhi (x)\}\) if the gradient exists at \(x \in {\mathbf {R}}^d\) in the classical sense. See [46, Section 23] for further details.
In the above example, one easily verifies that
This allows us to solve the nonlinear inclusion where we want to find \(x \in {\mathbf {R}}\) with \(x + k \partial \varPhi (x) \ni y\) for any \(y \in {\mathbf {R}}\).
For this reason we study the more general problem of the numerical approximation of multi-valued stochastic differential equations (MSDE) of the form
Here, we assume that the mappings \(b :{\mathbf {R}}^d \rightarrow {\mathbf {R}}^d\) and \(g :{\mathbf {R}}^d \rightarrow {\mathbf {R}}^{d,m}\) are globally Lipschitz continuous. Moreover, the multi-valued drift coefficient function \(f :{\mathbf {R}}^d \rightarrow 2^{{\mathbf {R}}^d}\) is assumed to be a maximal monotone operator, cf. Definition 2.1 below. We refer to Sect. 4 for a complete list of all imposed assumptions on the MSDE (1.5). Let us emphasize that the subdifferential of a proper, lower semi-continuous, and convex potential is an important example of a possibly multi-valued and maximal monotone mapping f, cf. [46, Corollary 31.5.2].
We use the backward Euler–Maruyama method for the approximation of the MSDE (1.5) on the partition \(\pi \), which is given by the recursion
We discuss the well-posedness of this method (1.6) under our assumptions on f, b, and g in Sect. 5. In particular, it will turn out that both problems, (1.5) and (1.6), admit single-valued solutions \((X(t))_{t \in [0,T]}\) and \((X^n)_{n = 0}^N\), respectively.
The main result of this paper, Theorem 6.4, then states that the backward Euler–Maruyama method is convergent of order at least 1/4 with respect to the norm in \(L^2(\varOmega ;{\mathbf {R}}^d)\). For the error analysis we rely on techniques for deterministic problems developed in [38]. An important ingredient is the additional condition on f that there exists \(\gamma \in (0,\infty )\) with
for all \(v, w, z \in D(f) \subset {\mathbf {R}}^d\) and \(f_v \in f(v)\), \(f_w \in f(w)\), \(f_z \in f(z)\). This assumption is easily verified for a subdifferential of a convex potential, cf. Lemma 3.2. As already noted in [38] for deterministic problems, this inequality allows us to avoid Gronwall-type arguments in the error analysis for terms involving the multi-valued mapping f.
Before we give a more detailed outline of the content of this paper let us mention that multi-valued stochastic differential equations have been studied in the literature before. The existence of a uniquely determined solution to the MSDE (1.5) has been investigated, e.g., in [7, 21, 42]. We also refer to the more recent monograph [41] and the references therein. In [14, 52] related results have been derived for multi-valued stochastic evolution equations in infinite dimensions. The numerical analysis for MSDEs has also been considered in [3, 26, 43, 54, 56]. However, these papers differ from the present paper in terms of the considered numerical methods, the imposed conditions, or the obtained order of convergence.
Further, we also mention that several authors have developed explicit numerical methods for SDEs with discontinuous drifts in recent years. For instance, we refer to [9, 24, 25, 33, 35,36,37]. While these results often apply to more irregular drift coefficients, which are beyond the framework of maximal monotone operators, the authors have to employ more restrictive conditions such as the global boundedness or piecewise Lipschitz continuity of the drift, which is not required in our framework. This allows for more general growth conditions. Moreover, none of these papers allows for a multi-valued drift coefficient.
The main motivation for this paper is to present a novel approach to analyze MSDEs. As the numerical method as such is not new, we do not provide any numerical tests in this paper. Numerical experiments for implicit methods can be found, e.g., in [1, 3], and [30].
This paper is organized as follows: in Sect. 2 we fix some notation and recall the relevant terminology for multi-valued mappings. In Sect. 3 we demonstrate how to apply the techniques from [38] to the simplified setting of the Langevin equation (1.1). In addition, we also show that if the gradient \(\nabla \varPhi \) is more regular, say Hölder continuous with exponent \(\alpha \in (0,1]\), then the order of convergence increases to \(\frac{1 + \alpha }{4}\). Moreover, it turns out that the error constant does not grow exponentially with the final time T. This is an important insight if the backward Euler method is used within an unadjusted Langevin algorithm [45], which typically requires large time intervals. See Theorem 3.7 and Remark 3.8 below.
In Sect. 4 we turn to the more general multi-valued stochastic differential equation (1.5) where we introduce all the assumptions imposed on the appearing drift and diffusion coefficients and collect some properties of the exact solution. In Sect. 5 we show that the backward Euler–Maruyama method (1.6) is well-posed under the assumptions of Sect. 4. In Sect. 6 we prove the already mentioned convergence result with respect to the root-mean-square norm. Finally, in Sect. 7 we verify that the setting of Sect. 4 applies to a Langevin equation with the discontinuous gradient (1.4). Further, we also show how to apply our results to the spatial discretization of the stochastic p-Laplace equation which indicates their usability for the numerical analysis of stochastic partial differential equations. However, a complete analysis of the latter problem will be deferred to a future work.
2 Preliminaries
In this section we collect some notation and introduce some background material. First we recall some terminology for set valued mappings and (maximal) monotone operators. For a more detailed introduction we refer, for instance, to [48, Abschn. 3.3] or [40, Chapter 6].
By \({\mathbf {R}}^d\), \(d \in {\mathbf {N}}\), we denote the Euclidean space with the standard norm \(|\cdot |\) and inner product \(\langle \cdot , \cdot \rangle \). Let \(M \subset {\mathbf {R}}^d\) be a set. A set-valued mapping \(f :M \rightarrow 2^{{\mathbf {R}}^d}\) maps each \(x \in M\) to an element of the power set \(2^{{\mathbf {R}}^d}\), that is, \(f(x) \subseteq {\mathbf {R}}^d\). The domain D(f) of f is given by
Definition 2.1
Let \(M \subset {\mathbf {R}}^d\) be a non-empty set. A set-valued map \(f :M \rightarrow 2^{{\mathbf {R}}^d}\) is called monotone if
for all \(u,v \in D(f)\), \(f_u \in f(u)\), and \(f_v \in f(v)\). Moreover, a set-valued mapping \(f :M \rightarrow 2^{{\mathbf {R}}^d}\) is called maximal monotone if f is monotone and for all \(x \in M\) and \(y \in {\mathbf {R}}^d\) satisfying
it follows that \(x \in D(f)\) and \(y \in f(x)\).
Next, we recall a Burkholder–Davis–Gundy-type inequality. For a proof we refer to [28, Chapter 1, Theorem 7.1]. For its formulation we note that the Frobenius or Hilbert–Schmidt norm of a matrix \(g \in {\mathbf {R}}^{d,m}\) is also denoted by |g|.
Lemma 2.2
Let \(p \in [2,\infty )\) and \(g \in L^p(\varOmega ; L^p(0,T;{\mathbf {R}}^{d,m}))\) be stochastically integrable. Then, for every \(s,t \in [0,T]\) with \(s<t\), the inequality
holds.
Let us also recall a stochastic variant of the Gronwall inequality. A proof that can be modified to this setting can be found in [55]. Compare also with [51].
Lemma 2.3
Let \(Z,M,\xi :[0,T] \times \varOmega \!\rightarrow \! {\mathbf {R}}\) be \(({\mathcal {F}}_t)_{t\in [0,T]}\)-adapted and \(\mathbf {P}\)-almost surely continuous stochastic processes. Moreover, M is a local \(({\mathcal {F}}_t)_{t\in [0,T]}\)-martingale with \(M(0) = 0\). Suppose that Z and \(\xi \) are nonnegative. In addition, let \(\varphi :[0, T ] \rightarrow {\mathbf {R}}\) be integrable and nonnegative. If, for all \(t \in [0, T ]\), we have
then, for every \(t \in [0,T]\), the inequality
holds.
Moreover, we often make use of generic constants. More precisely, by C we denote a finite and positive quantity that may vary from occurrence to occurrence but is always independent of numerical parameters such as the step size \(k = \frac{T}{N}\) and the number of steps \(N \in {\mathbf {N}}\).
3 Application to the Langevin equation with a convex potential
In order to illustrate our approach, we first consider a more regular stochastic differential equation with single-valued (Hölder) continuous drift term. More precisely, we consider the overdamped Langevin equation [23, Section 2.2]
where \(X_0 \in L^2(\varOmega , {\mathcal {F}}_0, {\mathbf {P}};{\mathbf {R}}^d)\), \(g_0 \in {\mathbf {R}}^{d,m}\), and \(W :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^m\) is a standard \({\mathbf {R}}^m\)-valued Wiener process.
In this section we impose the following additional assumption on the potential \(\varPhi :{\mathbf {R}}^d \rightarrow {\mathbf {R}}\). It allows us to illustrate our approach in a simplified analytical setting which avoids the full technical details required for dealing with multi-valued mappings. The assumption will be dropped in later parts of the paper.
Assumption 3.1
Let \(\varPhi :{\mathbf {R}}^d \rightarrow {\mathbf {R}}\) be a convex, nonnegative, and continuously differentiable function.
In the following, we denote by \(f :{\mathbf {R}}^d \rightarrow {\mathbf {R}}^d\) the gradient of \(\varPhi \), that is \(f(x) = \nabla \varPhi (x)\). It is well-known that the convexity of \(\varPhi \) implies the variational inequality
see, for example, [46, § 23].
In the following lemma we collect some properties of f which are direct consequences of Assumption 3.1. Both inequalities are well-known. The proof of (3.4) is taken from [38].
Lemma 3.2
Under Assumption 3.1 and with \(f = \nabla \varPhi \), the inequalities
and
are fulfilled for all \(v,w,z \in {\mathbf {R}}^d\).
Proof
The first inequality follows directly from (3.2) since
for all \(v,w \in {\mathbf {R}}^d\). For the proof of the second inequality we start by rewriting its left-hand side. For arbitrary \(v,w,z \in {\mathbf {R}}^d\) we rearrange the terms to obtain
Setting \(\sigma (v,w) := \varPhi (w) - \varPhi (v) - \langle f(v),w-v\rangle _{}\) for all \(v,w \in {\mathbf {R}}^d\), we see that
But (3.2) says that \(\sigma (v,w) \ge 0\) for all \(v,w \in {\mathbf {R}}^d\), which completes the proof. \(\square \)
It follows from Assumption 3.1 and Lemma 3.2 that the drift \(f = \nabla \varPhi \) of the stochastic differential equation (3.1) is continuous and monotone. Therefore, the stochastic differential equation (3.1) has a solution in the strong (probabilistic) sense, satisfying \({\mathbf {P}}\)-a.s. for all \(t \in [0,\infty )\)
See, [44, Thm. 3.1.1] for a proof and more details on this concept of solution. Moreover, the solution is unique up to \({\mathbf {P}}\)-indistinguishability and it is square-integrable with
Next, we turn to the numerical approximation of the solution of (3.1). Recall that for a single-valued drift the backward Euler–Maruyama method is given by the recursion
where \(\varDelta W^n = W(t_{n}) - W(t_{n-1})\), \(t_n = nk\), and \(k = \frac{T}{N}\).
The next lemma contains some a priori estimates for the backward Euler–Maruyama method (3.6).
Lemma 3.3
Let \(g_0 \in {\mathbf {R}}^{d,m}\) be given and let Assumption 3.1 be satisfied. For an arbitrary step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), let \((X^n)_{n \in \{0,\dots ,N\}}\) be a family of \(({\mathcal {F}}_{t_n})_{n \in \{0,\ldots ,N\}}\)-adapted random variables satisfying (3.6). If the initial value \(X_0 \in L^2(\varOmega ,{\mathcal {F}}_0,{\mathbf {P}};{\mathbf {R}}^d)\), then
and
Proof
First, we recall the identity
Using also (3.6), we then get
for every \(n \in \{1,\ldots ,N\}\). Hence, an application of (3.2) yields
for every \(n \in \{1,\ldots ,N\}\). From applications of the Cauchy–Schwarz inequality and the weighted Young inequality we then obtain
for every \(n \in \{1,\ldots ,N\}\).
The third term on the right-hand side is absorbed in the third term on the left-hand side. Summation then yields
An inductive argument over \(n \in \{1,\ldots ,N\}\) then implies that \(X^n\) is square-integrable due to the assumption \(X_0 \in L^2(\varOmega ,{\mathcal {F}}_0,{\mathbf {P}};{\mathbf {R}}^d)\). Therefore, after taking expectation the last sum vanishes. Moreover, an application of the Itô isometry then gives
Since this is true for any \(n \in \{1,\ldots ,N\}\) the assertion follows. \(\square \)
As the next theorem shows, Assumption 3.1 is also sufficient to ensure the well-posedness of the backward Euler–Maruyama method. The result follows directly from the fact that f is continuous and monotone due to (3.3). For a proof we refer, for instance, to [4, Sect. 4], [39, Chap. 6.4], and [53, Theorem C.2]. The assertion also follows from the more general result in Theorem 5.3 below.
Theorem 3.4
Let \(X_0 \in L^2(\varOmega ,{\mathcal {F}}_0,{\mathbf {P}};{\mathbf {R}}^d)\) and \(g_0 \in {\mathbf {R}}^{d,m}\) be given and let Assumption 3.1 be satisfied. Then, for every equidistant step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), there exists a unique family of square-integrable and \(({\mathcal {F}}_{t_n})_{n \in \{0,\ldots ,N\}}\)-adapted random variables \((X^n)_{n\in \{0,\dots ,N\}}\) satisfying (3.6).
We now turn to an error estimate with respect to the \(L^2(\varOmega ;{\mathbf {R}}^d)\)-norm. Since we do not impose any (local) Lipschitz condition on the drift f, classical approaches based on discrete Gronwall-type inequalities are not applicable. Instead we rely on an error representation formula, which was introduced for deterministic problems in [38].
For its formulation, we introduce some additional notation: For a given equidistant partition \(\pi = \{0=t_0< t_1< \cdots < t_N = T\} \subset [0,T]\) with step size \(k = \frac{T}{N}\), we denote by \({\mathcal {X}}:[0,T] \times \varOmega \rightarrow {\mathbf {R}}^d\) the piecewise linear interpolant of the sequence \((X^n)_{n \in \{0,\ldots ,N\}}\) generated by the backward Euler method (3.6). It is defined by \({\mathcal {X}}(0) = X^0\) and for all \(t \in (t_{n-1}, t_n]\), \(n \in \{1,\ldots ,N\}\), by
In addition, we introduce the processes \({\overline{{\mathcal {X}}}}, {\underline{{\mathcal {X}}}} :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^d\), which are piecewise constant interpolants of \((X^n)_{n \in \{0,\ldots ,N\}}\) and defined by \({\overline{{\mathcal {X}}}}(0)= {\underline{{\mathcal {X}}}}(0) = X^0\) and for all \(t \in (t_{n-1}, t_n]\), \(n \in \{1,\ldots ,N\}\), by
Analogously, we define the piecewise linear interpolated process \({\mathcal {W}} :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^m\) by \({\mathcal {W}}(0) = 0\) and
for all \(t \in (t_{n-1},t_n]\), \(n \in \{1,\ldots ,N\}\).
We are now prepared to state our first preparatory result. The underlying idea was introduced in [38], where it is used to derive a posteriori error estimates for the backward Euler method. In fact, in the absence of noise, only the first term on the right-hand side of (3.12) is non-zero. In [38] this term is used as an a posteriori error estimator, since it is explicitly computable by quantities generated by the numerical method.
Lemma 3.5
Let \(X_0 \in L^2(\varOmega ,{\mathcal {F}}_0,{\mathbf {P}};{\mathbf {R}}^d)\) as well as \(g_0 \in {\mathbf {R}}^{d,m}\) be given and let Assumption 3.1 be satisfied. Let \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), be an arbitrary equidistant step size and let \(t_n = nk\), \(n \in \{0,\dots , N\}\). Then, for every \(n \in \{1,\ldots ,N\}\) the estimate
holds, where \((X(t))_{t \in [0,T]}\) and \((X^n)_{n \in \{0,\ldots ,N\}}\) are the solutions of (3.1) and (3.6), respectively.
Proof
From (3.6) we directly deduce that for every \(n \in \{1,\ldots ,N\}\)
Then, one easily verifies for all \(t \in (t_{n-1},t_n]\), \(n \in \{1,\ldots ,N\}\), that
Hence, due to (3.5), the error process \(E := X - {\mathcal {X}}\) can be written as
for all \(t \in [0,T]\). Here, we have \(E_2(t_n)= 0\), since \({\mathcal {W}}\) is an interpolant of W. Hence, for all \(n \in \{0,\ldots ,N\}\),
To estimate the norm of \(E_1(t_n)\), we first note that \(E_1\) has absolutely continuous sample paths with \(E_1(0)=0\). Hence,
holds for almost all \(t \in [0,T]\). Therefore, by integration with respect to t, we get
Next, we write
and use (3.3) and (3.4) to obtain, for almost every \(t \in (t_{n-1}, t_n]\), that
Furthermore, the expectation of the second integral on the right-hand side of (3.15) equals
Therefore,
Since \(\int _{t_{i-1}}^{t_i} (t_i - t) \,\mathrm {d}t = \frac{1}{2} k^2\) the assertion follows. \(\square \)
The next lemma concerns the difference between the Wiener process W and its piecewise linear interpolant \({\mathcal {W}}\).
Lemma 3.6
For every \(g_0 \in {\mathbf {R}}^{d,m}\) and every step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), the equality
holds.
Proof
From the definition (3.11) of \({\mathcal {W}}\) it follows that
where we used that the two increments of the Wiener process are independent for every \(t \in (t_{n-1},t_n]\), \(n \in \{1,\ldots , N\}\), and we also applied Itô’s isometry. By symmetry of the two terms it then follows that
and the proof is complete. \(\square \)
The error estimates in Lemmas 3.5 and 3.6 allow us to determine the order of convergence of the backward Euler–Maruyama method without relying on discrete Gronwall-type inequalities. The following theorem imposes the additional assumption that the drift f is Hölder continuous. We include the parameter value \(\alpha = 0\), which simply means that f is continuous and globally bounded. The case of less regular f is treated in Sect. 6.
Observe that we recover the standard rate \(\frac{1}{2}\) if \(\alpha = 1\), that is, if the drift f is assumed to be globally Lipschitz continuous. Compare also with the standard literature, for example, [20, Chap. 12] or [32, Sect. 1.3].
For processes \(X :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^d\) and exponents \(\alpha \in [0,1]\), we define the family of Hölder semi-norms by
and the corresponding Hölder spaces
Theorem 3.7
Let \(X_0 \in L^2(\varOmega ,{\mathcal {F}}_0,{\mathbf {P}};{\mathbf {R}}^d)\) as well as \(g_0 \in {\mathbf {R}}^{d,m}\) be given, let Assumption 3.1 be fulfilled and let \(f = \nabla \varPhi \) be Hölder continuous with exponent \(\alpha \in [0,1]\), i.e., there exists \(L_f \in ( 0,\infty )\) such that
Then there exists \(C \in (0,\infty )\) such that for every step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), the estimate
holds, where \((X(t))_{t \in [0,T]}\) and \((X^n)_{n \in \{0,\ldots ,N\}}\) are the solutions to (3.1) and (3.6), respectively.
Proof
Since f is assumed to be \(\alpha \)-Hölder continuous it follows that
In particular, f grows at most linearly. Therefore, as stated in [28, Chap. 2, Thm 4.3], the solution \((X(t))_{t\in [0,T]}\) of (3.1) fulfills \(X \in C^{\frac{1}{2}}([0,T];L^2(\varOmega ;{\mathbf {R}}^d))\).
We will use Lemma 3.5 to prove the error bound. To this end, we first show that
Indeed, we make use of the Hölder continuity of f directly and obtain
where we also used Hölder’s inequality with \(p = \frac{2}{1 + \alpha } \in [1,2]\) and \(\frac{1}{p} + \frac{1}{q} = 1\) as well as Jensen’s inequality. Due to the a priori estimate (3.8) the sum \(\sum _{i=1}^{N} {\mathbf {E}}\big [ |X^i - X^{i-1}|^2 \big ]\) is bounded independently of the step size k. Hence, we arrive at (3.17).
Therefore, it remains to estimate the second error term in Lemma 3.5:
where we inserted the definition of \({\overline{{\mathcal {X}}}}\) from (3.10). Moreover, from (3.11) we get
for \(t \in (t_{j-1},t_j]\). Hence, the random variable in the second slot of the inner product on the right-hand side of (3.18) is centered and is independent of any \({\mathcal {F}}_{t_{j-1}}\)-measurable random variable. Thus, we may write
To estimate \(T_1\) we first recall the definitions of \({\underline{{\mathcal {X}}}}\) and \({\overline{{\mathcal {X}}}}\) from (3.10). Then we apply the Cauchy–Schwarz inequality and obtain
From the Hölder continuity of f we then deduce that
where the last inequality is in fact an equality if \(\alpha = 1\), \(\frac{1}{q} = 0\) or if \(\alpha = 0\), \(\frac{1}{q}=1\). Otherwise the inequality follows from Hölder’s inequality with \(p = \frac{1}{\alpha } \in (1,\infty )\) and \(\frac{1}{p} + \frac{1}{q} = 1\), followed by an application of Jensen’s inequality. Furthermore, Lemma 3.6 states that
Therefore, together with (3.8) we arrive at the estimate
for all \(n \in \{1,\ldots ,N\}\).
The estimate of \(T_2\) follows similarly by additionally making use of the Hölder continuity of the exact solution. To be more precise, we have that
Together with the Cauchy–Schwarz inequality and (3.19), we therefore obtain
Inserting the estimates for \(T_1\), \(T_2\), and (3.17) into Lemma 3.5 completes the proof. \(\square \)
Remark 3.8
The precise form of the constant C appearing in Theorem 3.7 is, after taking squares,
with \(C_0 = 2 {\mathbf {E}}[ |X_0|^2 ] + 4 T ( \varPhi (0) + |g_0|^2 )\).
Observe that, since we avoid the use of Gronwall-type inequalities, the error constant does not grow exponentially with time T. This indicates that the backward Euler–Maruyama method is particularly suited for long-time simulations as is often required in Markov-chain Monte Carlo methods, for example, in the unadjusted Langevin algorithm [45].
4 Properties of the exact solution in the multi-valued case
In this section, we turn our attention to the multi-valued stochastic differential equation (MSDE) in (1.5). We give a complete account of the assumptions imposed on the coefficient functions. In addition, we collect some results on the existence and uniqueness of a strong solution to the MSDE. We also include useful results on higher moment bounds of the exact solution.
Assumption 4.1
The set valued mapping \(f :{\mathbf {R}}^d \rightarrow 2^{{\mathbf {R}}^d}\) is maximal monotone with \({\text {int}} D(f) \ne \emptyset \). Moreover, there exist constants \(\beta , \lambda \in [0,\infty )\), \(\mu \in (0,\infty )\), and \(p \in [1,\infty )\) such that
for every \(v \in D(f)\) and \(f_v \in f(v)\).
Assumption 4.2
The function \(b :{\mathbf {R}}^d \rightarrow {\mathbf {R}}^d\) is Lipschitz continuous; i.e., there exists a constant \(L_b \in [0,\infty )\) such that
for all \(v,w \in {\mathbf {R}}^d\).
Assumption 4.3
The function \(g :{\mathbf {R}}^d \rightarrow {\mathbf {R}}^{d,m}\) is Lipschitz continuous; i.e., there exists a constant \(L_g \in [0,\infty )\) such that
for all \(v,w \in {\mathbf {R}}^d\).
Assumption 4.4
The initial value \(X_0\) is an \({\mathcal {F}}_0\)-measurable and D(f)-valued random variable. Furthermore,
where the value of p is the same as in Assumption 4.1.
Observe that Assumptions 4.2 and 4.3 directly imply that b and g grow at most linearly. More precisely, after possibly increasing the values of \(L_b\) and \(L_g\), we obtain the bounds
for all \(v \in {\mathbf {R}}^d\).
Remark 4.5
Without loss of generality we will assume that \(0 \in D(f)\). Otherwise, since the graph of f is not empty, we take \(v_0 \in D(f)\) and \(f_{v_0} \in f(v_0)\) and replace f, b, and g by suitably shifted mappings, for instance, \({\tilde{f}}(v) := f(v + v_0)\). Then \(0 \in D({\tilde{f}})\) holds. Compare further with [48, Abschn. 3.3.3].
Next, we introduce the notion of a solution of (1.5), which we use for the remainder of this paper.
Definition 4.6
A tuple \((X,\eta )\) is called a solution of the multi-valued stochastic differential equation (1.5), if the following conditions hold.
-
(i)
The mapping \(X :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^d\) is an \(({\mathcal {F}}_t)_{t \in [0,T]}\)-adapted, \(\mathbf {P}\)-almost surely continuous stochastic process such that \(X(t) \in \overline{D(f)}\) for all \(t \in (0,T]\) with probability one.
-
(ii)
The mapping \(\eta :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^d\) is an \(({\mathcal {F}}_t)_{t \in [0,T]}\)-adapted stochastic process such that
$$\begin{aligned} \int _0^T |\eta (t)| \,\mathrm {d}t < \infty , \quad {\mathbf {P}}\text {-almost surely.} \end{aligned}$$ -
(iii)
The equality
$$\begin{aligned} X(t) + \int _{0}^{t} \eta (s) \,\mathrm {d}s = X_0 + \int _{0}^{t} b(X(s)) \,\mathrm {d}s + \int _{0}^{t} g(X(s)) \,\mathrm {d}W(s) \end{aligned}$$(4.2)holds for all \(t \in [0,T]\) and \({\mathbf {P}}\)-almost surely.
-
(iv)
For \({\mathbf {P}}\)-almost all \(\omega \in \varOmega \) and \(t \in [0,T]\), it follows that \(\eta (t,\omega ) \in f(X(t,\omega ))\); in other words, for every \(y \in D(f)\) and \(f_y \in f(y)\) the inequality
$$\begin{aligned} \langle \eta (t) - f_{y},X(t) - y\rangle _{} \ge 0 \end{aligned}$$is satisfied for almost every \(t \in [0,T]\) and \({\mathbf {P}}\)-almost surely, cf. Definition 2.1.
This notion of a solution has been considered in, for example, [7, 21, 42, 52], where also the existence of a unique solution is shown. Note that in [7] the condition on \(\eta \) is slightly milder than in [21, 42, 52]. Our concept of solution corresponds to the latter sources. For the multi-valued equation it becomes necessary to consider a tuple \((X,\eta )\) as a solution. The function X plays the usual role of the solution of the equation. As f(X) is now a set in \({\mathbf {R}}^d\), we select one unique element \(\eta \) in this set such that the inclusion (1.5) becomes an equality when exchanging f(X) by \(\eta \).
Due to their importance for the error analysis we next prove certain moment estimates.
Theorem 4.7
Let Assumptions 4.1 and 4.4 be satisfied with \(p \in [1,\infty )\). Then there exists a unique solution \((X,\eta )\) of (1.5) in the sense of Definition 4.6. There is a constant \(C\in (0,\infty )\) such that
Furthermore, if \(p \in (1,\infty )\) and \(\frac{1}{p}+\frac{1}{q} = 1\), then
Proof
Existence and uniqueness is shown, for instance, in [21]. For
the equality
holds by an application of Itô’s formula (see [12, Chap. 4.7, Theorem 7.1]). From the coercivity assumption on f we obtain that
for every \(f_{X(s)} \in f(X(s))\) and almost every \(s \in [0,T]\). The fact that \(\eta (s) \in f(X(s))\) for almost every \(s \in [0,T]\) then implies that
Since b and g satisfie the linear growth bound (4.1), we have
as well as
Thus, we get
We introduce
Then Z, M, and \(\xi \) are \(({\mathcal {F}}_t)_{t\in [0,T]}\)-adapted and \(\mathbf {P}\)-almost surely continuous stochastic processes. Furthermore, M is a local \(({\mathcal {F}}_t)_{t\in [0,T]}\)-martingale satisfying \(M(0) = 0\). Thus, an application of Lemma 2.3 yields, for every \(t \in [0,T]\), that
Inserting the definition of Z then proves the first estimate.
Furthermore, if Assumption 4.1 holds with \(p \in (1,\infty )\), then we have, for every \(f_{x} \in f(x)\), \(x \in {\mathbf {R}}^d\), that
with \(q = \frac{p}{p-1}\). Therefore, it follows that
since \(\eta (s) \in f(X(s))\) for almost every \(s \in [0,T]\). \(\square \)
Remark 4.8
Let us mention that, for instance, in [41, Chapter 4] and the references therein, a weaker notion of a solution to (1.5) is found. More precisely, if \((X,\eta )\) is a solution in the sense of Definition 4.6, then (X, H) is a solution in the sense of [41, Chapter 4] with the definition
In particular, the process H is a continuous, progressively measurable process with bounded total variation and \(H(0) = 0\) almost surely. The stronger condition of absolute continuity of the process H, which is required in Definition 4.6, is essential in the proof of Theorem 6.4 below. This explains why we work with the stronger notion of a solution in Definition 4.6.
5 Well-posedness of the backward Euler method in the multi-valued case
In this section, we show that the backward Euler–Maruyama method (1.6) for the MSDE (1.5) is well-posed under the same assumptions as in the previous section.
Lemma 5.1
Let Assumptions 4.1 and 4.2 be satisfied. Furthermore, let \(w \in {\mathbf {R}}^d\) and \(k \in (0,T]\) be given with \(L_b k \in [0,1)\). Then there exist uniquely determined \(x_0 \in D(f)\) and \(\eta _{x_0} \in f(x_0)\), which satisfy the nonlinear equation
Proof
We first show that there exists a unique \(x_0\in D(f)\) such that
To this end, notice that for all \(x,y \in {\mathbf {R}}^d\), the inequalities
hold due to the step size bound. In addition, it follows from (4.1) that
for all \(x \in {\mathbf {R}}^d\). Hence, the mapping \((\mathrm {id}+ k f - k b)\) is the sum of the maximal monotone operator kf and the mapping \((\mathrm {id}- k b)\), which is single-valued, Lipschitz continuous, monotone, and coercive.
Thus, we can apply [2, Theorem 2.1] and obtain the existence of \(x_0 \in D(f)\) such that (5.2) holds. Furthermore, there necessarily exists a corresponding unique element \(\eta _{x_0} \in f(x_0)\) with
It remains to prove the uniqueness of \(x_0\), which directly implies the uniqueness of \(\eta _{x_0}\). Assume that there exist \(x_1 \in D(f)\) and \(\eta _{x_1} \in f(x_1)\) as well as \(x_2 \in D(f)\) and \(\eta _{x_2} \in f(x_2)\) such that
By considering the difference of these equations tested with \(x_1- x_2\), we obtain
Since \(1 - k L_b > 0\) we must have \(x_1 = x_2\) and the proof is complete. \(\square \)
For later use, we note that the solution operator of (5.1) is Lipschitz continuous.
Lemma 5.2
Let Assumptions 4.1 and 4.2 be satisfied. For \(k \in (0,T]\) with \(L_b k \in [0,1)\) let \(S_k :{\mathbf {R}}^d \rightarrow D(f)\) be the solution operator that maps \(w \in {\mathbf {R}}^d\) to the unique solution \(x_0 \in D(f)\) of (5.1). Then \(S_k\) is globally Lipschitz continuous with
Proof
Let \(w_1, w_2 \in {\mathbf {R}}^d\) and \(k \in (0,T]\) with \(L_b k \in [0,1)\) be given. Let \(x_i = S_k(w_i) \in D(f)\) and \(\eta _{x_i} \in f(x_i)\), \(i\in \{1,2\}\), denote the unique solutions of the equations
By considering the difference of these equations, tested with \(x_1 - x_2\), we obtain
By using the Cauchy–Schwarz inequality for the right-hand side as well as the monotonicity and the Lipschitz continuity for the left-hand side, we get
Reinserting \(x_i = S_k(w_i)\) then shows that
as claimed. \(\square \)
Theorem 5.3
Let Assumptions 4.1 to 4.4 be satisfied. Then for every step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), with \(L_b k \in [0,1)\) there exist uniquely determined families of square-integrable, \({\mathbf {R}}^d\)-valued and \(({\mathcal {F}}_{t_n})_{n \in \{1,\ldots ,N\}}\)-adapted random variables \((X^n)_{n\in \{1,\dots ,N\}}\) and \((\eta ^n)_{n\in \{1,\dots ,N\}}\) such that \(X^n \in D(f)\), \(\eta ^n \in f(X^n)\) for every \(n \in \{1,\dots ,N\}\) and
for every \(n \in \{1,\dots ,N\}\), \({\mathbf {P}}\)-almost surely with \(X^0 = X_0\) and \(\eta ^0 \in f(X_0)\).
Proof
We prove the existence of \((X^n)_{n\in \{0,\dots ,N\}}\) and \((\eta ^n)_{n\in \{0,\dots ,N\}}\) by induction over \(n \in \{0,\ldots ,N\}\). From the assumptions on \(X_0\) and f it is clear that \(X^0=X_0\) and \(\eta ^0 \in f(X_0)\) are \({\mathcal {F}}_{t_0}\)-adapted and square-integrable. In particular, it follows from Assumptions 4.1 and 4.4 that
Next, we assume that \((X^j)_{j \in \{0,\ldots ,n-1\}}\) and \((\eta ^j)_{j \in \{0,\ldots ,n-1\}}\) are adapted to \(({\mathcal {F}}_{t_{j}})_{j \in \{0,\ldots ,n-1\}}\), square-integrable and satisfy (5.3) for all \(j \in \{1,\ldots ,n-1\}\). By Lemma 5.1 there exist uniquely determined \(X^n(\omega ) \in D(f)\) and \(\eta ^n(\omega ) \in f(X^n(\omega ))\) for every \(\omega \in \varOmega \) such that
By Lemma 5.2, the solution operator \(S_{k} :{\mathbf {R}}^d \rightarrow D(f)\) that maps \(X^{n-1}(\omega ) + g(X^{n-1}(\omega ))\varDelta W^n(\omega )\) to \(X^n(\omega ) \in D(f)\) is Lipschitz continuous. As \(S_{k}\) is Lipschitz continuous and, hence, of linear growth it follows that \(X^n\) is an \({\mathcal {F}}_{t_n}\)-measurable and square-integrable random variable. To be more precise, we have the bound
This implies, in particular, that
is also a \({\mathcal {F}}_{t_n}\)-measurable and square-integrable random variable as \(X^n\), \(X^{n-1}\), and \(g(X^{n-1})\varDelta W^n\) have these properties. This finishes the proof of the induction and hence that of the theorem. \(\square \)
Next we state an a priori estimate for the sequence of random variables satisfying recursion (1.6).
Lemma 5.4
Let Assumptions 4.1 to 4.4 be satisfied. For a step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), with \(5 L_b k \in [0,1)\), let \((X^n)_{n \in \{0,\dots ,N\}}\) and \((\eta ^n)_{n\in \{0,\dots ,N\}}\) be two families of \(({\mathcal {F}}_{t_n})_{n \in \{0,\ldots ,N\}}\)-adapted random variables as stated in Theorem 5.3. Then there exists \(K_X \in (0,\infty )\) independent of the step size \(k = \frac{T}{N}\) such that
If, in addition, \(p \in (1,\infty )\), then there exists \(K_\eta \in (0,\infty )\) independent of the step size \(k = \frac{T}{N}\) such that
where \(q \in (1,\infty )\) is given by \(\frac{1}{p} + \frac{1}{q} = 1\).
Remark 5.5
If \(p = 1\) in Assumption 4.1, then f and, hence, \((\eta ^n)_{n \in \{1,\ldots ,N\}}\) are bounded. In particular, (5.5) holds for any \(q \in (1,\infty )\) and for any step size \(k = \frac{T}{N}\) with \(L_b k \in [0,1)\).
Proof of Lemma 5.4
First, we recall the identity
As \(\eta ^n \in f(X^n)\), using Assumptions 4.1 and 4.2, it follows that
where we also applied (4.1). Hence,
for every \(n \in \{1,\ldots ,N\}\), where we also applied the Cauchy–Schwarz and weighted Young inequalities. After a kick-back argument, we sum from 1 to \(n \in \{1,\ldots ,N\}\) to obtain
After taking expectations, the last term on the right-hand side vanishes. Then, applications of Itô’s isometry and (4.1) give
Since the step size bound \(5 L_b k \in [0,1)\) ensures that
the discrete Gronwall inequality (see, for example, [8]) is applicable and completes the proof of (5.4). Finally, it follows from the polynomial growth bound on f that
6 Error estimates in the multi-valued case
In this section we derive an error estimate for the backward Euler method given by (1.6) for the MSDE (1.5).
To prove the convergence of the scheme (1.6) let us fix some notation. Throughout this section we assume that the equidistant step size \(k = \frac{T}{N}\) is small enough so that the a priori estimates in Lemma 5.4 hold. Furthermore, as in (3.9) and (3.10), we denote the piecewise linear interpolants of the discrete values by \({\mathcal {X}}(0) = X^0\), \({\mathcal {H}}(0) = \eta ^0\) for \(\eta ^0 \in f(X^0)\) and
for all \(t \in (t_{n-1},t_n]\) and \(n \in \{1,\ldots ,N\}\). Similarly, we define the piecewise constant interpolants by \({\overline{{\mathcal {X}}}}(0)={\underline{{\mathcal {X}}}}(0)=X^0\), \({\overline{{\mathcal {H}}}}(0) = {\underline{{\mathcal {H}}}}(0) = \eta ^0\), and
for all \(t \in (t_{n-1},t_n]\) and \(n \in \{1,\ldots ,N\}\). Moreover, we introduce the stochastic processes \(G :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^d\) and \({\mathcal {G}}:[0,T] \times \varOmega \rightarrow {\mathbf {R}}^d\) defined by
as well as by \({\mathcal {G}}(0) = 0\) and, for all \(n \in \{1,\ldots ,N\}\) and \(t \in (t_{n-1},t_n]\),
In view of (1.6) and the definition of \({\mathcal {G}}\) for \(t \in (t_{n-1},t_n]\), \(n \in \{1,\dots ,N\}\), we obtain the representation
We begin the derivation of our error estimate by considering the difference between the stochastic integral G and its approximation \({\mathcal {G}}\).
Lemma 6.1
Let Assumptions 4.1 to 4.4 be satisfied. Then there exists \(K_G \in (0,\infty )\) such that, for every equidistant step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\) with \(5 L_b k \in [0,1)\) and every \(t \in [0,T]\), we have
In addition, for every \(\rho \in [2,\infty )\), there exists \(K_\rho \in (0,\infty )\) such that, for every \(n \in \{1,\ldots ,N\}\) and \(t \in (t_{n-1},t_n]\), the following estimates hold:
and
Proof
Recall the definitions of G and \({\mathcal {G}}\) from (6.1) and (6.2). First, we add and subtract a term and then apply the triangle inequality. Then, for every \(n \in \{1,\ldots ,N\}\) and \(t \in (t_{n-1},t_n]\) we arrive at
by an application of Itô’s isometry. Furthermore, due to the Lipschitz continuity of g we obtain
where the last step follows from the identity
which holds for every \(s \in (t_{i-1},t_i]\), \(i \in \{1,\ldots ,N\}\). Finally, it follows from the same arguments as in the proof of Lemma 3.6 and by (4.1) for every \(t \in (t_{n-1},t_n]\) that
Together with the a priori bounds from Lemma 5.4 this shows (6.4).
It remains to prove the estimates (6.5) and (6.6). For (6.5) we first apply the Burkholder–Davis–Gundy-type inequality from Lemma 2.2 with constant \(C_\rho \) and obtain for every \(n \in \{1,\ldots ,N\}\) and \(t \in (t_{n-1},t_n]\) that
where we also made use of the linear growth bound (4.1) in the last step. This proves (6.5). The bound in (6.6) can be shown by analogous arguments. \(\square \)
The next lemma generalizes an important estimate from the proof of Theorem 3.7 to the multi-valued setting. In particular, we refer to Lemma 3.5 and (3.17).
Lemma 6.2
Let Assumptions 4.1 to 4.4 be satisfied. For every step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), with \(5 L_b k \in [0,1)\), the families of random variables \((X^n)_{n\in \{0,\dots ,N\}}\) and \((\eta ^n)_{n\in \{0,\dots ,N\}}\) are as stated in Theorem 5.3. Then there exists \(K_{\delta \eta } \in (0,\infty )\) independent of the step size k such that
Proof
The nonnegativity follows immediately from the monotonicity of f. To prove the second inequality, we insert the scheme (5.3) and obtain
For (6.7) we obtain
because of the telescopic structure. Furthermore, it follows from Assumptions 4.1 and 4.4 that
For the term in (6.8) we apply Hölder’s inequality with \(\rho = \max (2,p)\) and \(\frac{1}{\rho } + \frac{1}{\rho '} = 1\) to obtain
Then, from applications of the triangle inequality and Lemma 5.4, we get
We apply the polynomial growth bound satisfied by f and see that, for \(p \in [2,\infty )\),
is fulfilled, while for \(p \in [1,2)\) we have
In both cases the appearing terms are finite because of Assumption 4.4. Moreover, a further application of the triangle inequality yields
Due to the linear growth bound (4.1) on b and the a priori bound (5.4), it then follows that
By application of Lemma 2.2 with constant \(C_{\rho }\), we obtain
Together with the linear growth bound (4.1) on g this shows that
Putting the estimates together proves the desired bound. \(\square \)
We are now prepared to state and prove the main result of this section. While the main ingredients of the proof still consist of techniques introduced in [38, Sect. 4] for deterministic problems the proof is somewhat more technical than the proof of Theorem 3.7. In particular, due to the presence of Lipschitz perturbations in the general problem (1.5) it is no longer possible to avoid an application of a Gronwall lemma. Moreover, as in [38, Sect. 4], we impose the following additional assumption on the multi-valued mapping f.
Assumption 6.3
There exists \(\gamma \in (0,\infty )\) such that, for every \(v,w, z \in D(f)\), \(f_v \in f(v)\), \(f_w \in f(w)\), and \(f_z \in f(z)\),
In Lemma 3.2, we already proved that, if f is the subdifferential of a convex potential, then Assumption 6.3 is satisfied with \(\gamma = 1\). For a further example, we refer to Sect. 7.
Theorem 6.4
Let Assumptions 4.1–4.4 and Assumption 6.3 be satisfied. Let the step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), be such that \(8 L_b k \in [0,1)\). Then there exists a constant \(C \in (0,\infty )\) independent of k such that
Remark 6.5
The strong rate of convergence of 1/4 might not be optimal in the case of a piecewise Lipschitz drift coefficient. Under this additional assumption it is proved in [35] that the forward Euler–Maruyama scheme has a strong convergence rate of 1/2. In [34] a further scheme is introduced with the strong convergence order of 3/4. As proved in [33], the rate of 3/4 is a sharp lower error bound. In contrast to that our setting allows for a superlinearly growing and multi-valued drift coefficient at the cost of a lower convergence rate.
Proof Theorem 6.4
Let us first introduce some additional notation. We will denote the error between the exact solution X to (1.5) and the numerical approximation \({\mathcal {X}}\) defined in (6.3) by \(E(t):= X(t) - {\mathcal {X}}(t)\), \(t \in [0,T]\). Furthermore, it will be convenient to split the error into two parts
where
\({\mathbf {P}}\)-almost surely for every \(t \in (0,T]\). We expand the square of the norm of E as
In order to estimate the terms on the right-hand side of (6.11) we first observe in (6.9) that \(E_1\) has absolutely continuous sample paths with \(E_1(0)=0\). Hence we have \(\frac{1}{2} \frac{\mathrm {d}}{\,\mathrm {d}t}|E_1(t)|^2 = \langle {\dot{E}}_1(t), E_1(t) \rangle \) for almost every \(t \in [0,T]\). Therefore, after integrating from 0 to \(t \in (0,T]\), we get
Furthermore, we also have that
Thus, after combining (6.12) and (6.13) we obtain
For the first integral on the right-hand side of (6.14) we insert the derivative of \(E_1\) and the definition of the error process E. This yields, for almost every \(s \in (0, T]\),
After recalling the definition of \({\mathcal {X}}\) we use Assumptions 4.1 and 6.3. Then, for almost every \(s \in (t_{n-1},t_n]\) and all \(n \in \{1,\ldots ,N\}\), we get
where the second term in the last step is non-positive due to the monotonicity of f (cf. Definition 2.1). Moreover, due to the Lipschitz continuity of b, it follows for almost every \(s \in (0,T]\) that
where we also made use of Young’s inequality. In addition, for every \(n \in \{1,\ldots ,N\}\) and \(s \in (t_{n-1},t_n]\), we have that
Therefore,
Altogether, for every \(t \in (t_{n-1}, t_n]\) and \(n \in \{1,\ldots ,N\}\), we have shown that
where we also inserted that \(\int _{t_{n-1}}^{t} (t_n - s) \,\mathrm {d}s \le \int _{t_{n-1}}^{t_n} (t_n - s) \,\mathrm {d}s = \frac{1}{2} k^2\) as well as \(\int _{t_{n-1}}^{t} (t_n - s)^2 \,\mathrm {d}s \le \frac{1}{3} k^3\). It follows that, for every \(n \in \{1,\ldots ,N\}\) and \(t \in (t_{n-1},t_n]\),
Hence, together with Lemmas 5.4 and 6.2 this shows that
Next, we give an estimate for the second integral on the right-hand side of (6.14). For every \(n \in \{1,\ldots ,N\}\) and \(t \in (t_{n-1},t_n]\) we decompose the integral as follows
For every \(i \in \{1,\ldots ,n-1\}\) we then add and subtract \(E_2(t_i)\) in the second slot of the inner product in the first term on the right-hand side of (6.16). This gives
After inserting the definition of \(E_2\) from (6.10) the first integral is then equal to
for all \(i, n \in \{1,\ldots ,N\}\), \(i < n\), and \(t \in (t_{n-1},t_n]\). Since \(E_1(t_i) - E_1(t_{i-1}) = E(t_i) - E(t_{i-1}) - (E_2(t_i) - E_2(t_{i-1}))\) is square-integrable and \({\mathcal {F}}_{t_i}\)-measurable it therefore follows that
for all \(n \in \{1,\ldots ,N\}\), \(t \in (t_{n-1},t_n]\), and \(t_i < t\). Hence, after taking expectations in (6.16), we arrive at
Inserting the definitions (6.9) and (6.10) of \(E_1\) and \(E_2\) and applying Hölder’s inequality with \(\rho = \max (2,p)\) and \(\frac{1}{\rho }+\frac{1}{\rho '}=1\), we get
In the following, we will estimate \(\varGamma _1\), \(\varGamma _2\), \(\varGamma _3\), and \(\varGamma _4\) separately. For \(\varGamma _1\) we obtain after an application of Hölder’s inequality for sums that
If \(p \in [2,\infty )\) then \(\rho = p\) and \(\rho ' = q\). In this case all integrals appearing are finite due to the bounds in Theorem 4.7 and Lemma 5.4. Moreover, if \(p \in (1,2)\) then \(\rho = \rho ' = 2 < q\). Then it follows from further applications of Hölder’s inequality and Jensen’s inequality that
as well as
Hence, we arrive at the same conclusion. If \(p = 1\) then the processes \((\eta (t))_{t \in [0,T]}\) and \((\eta ^n)_{n \in \{1,\ldots ,N\}}\) are globally bounded due to the bound on f in Assumption 4.1. Using Lemma 6.1 we see that
Altogether, this yields
for a suitable constant \(C_{\varGamma } \in (0,\infty )\), which is independent of k. To estimate \(\varGamma _2\), we argue analogously as in the case for \(\varGamma _1\) to obtain that
The first factor is bounded as we saw in the case for \(\varGamma _1\). Furthermore, using Lemma 6.1, we have that
Due to the a priori bound (5.4), it follows that there exists a constant \(C_{\varGamma _2} \in (0,\infty )\), which does not depend on k such that
The estimates \(\varGamma _3\) and \(\varGamma _4\) follow analogously with the only new term that appears is of the form
which is bounded due to Theorem 4.7 and the a priori bound (5.4). Therefore, there exist constants \(C_{\varGamma _3}, C_{\varGamma _4} \in (0,\infty )\) such that
Hence, we obtain
After taking expectations in (6.11) and inserting (6.14), (6.15), (6.17) as well as (6.4) from Lemma 6.1, we obtain for every \(t \in (0,T]\) that
The assertion then follows from an application of Gronwall’s lemma, see for example, [11, Appendix B]. \(\square \)
Remark 6.6
Up to this point, we only proved convergence for X but not for \(\eta \). However, from the existence of \(X^n\) we also obtain that
Analogously, we can write for the exact solution \(\eta \) that
Therefore, from the convergence of \({\mathcal {X}}\) to X and the Lipschitz continuity of b and g we also obtain the estimate
for every \(n \in \{1,\ldots ,N\}\).
7 Examples
7.1 Discontinuous drift coefficient
In this example we show that Assumption 4.1 includes overdamped Langevin-type equations with a possibly discontinuous drift f. We consider the convex, nonnegative, yet not continuously differentiable function \(\varPhi (x) := |x|\), \(x \in {\mathbf {R}}\), which has a multi-valued subdifferential \(f :{\mathbf {R}}\rightarrow 2^{{\mathbf {R}}}\) defined by
This mapping fulfills Assumption 4.1 for \(p=1\). To be more precise, f is a monotone function and there exists no proper monotone extension of its graph. In fact, the subdifferential of any proper, lower semi-continuous, and convex function is a maximal monotone mapping by a well-known theorem of Rockafellar, cf. [46, Cor. 31.5.2] or [48, Satz 3.23].
Furthermore, we notice that \(f_x x = \text {sgn} (x) x = |x|\) as well as \(|f_x | \le 1\) for every \(x \in {\mathbf {R}}\) and \(f_x \in f(x)\). This shows that f fulfills all the conditions of Assumption 4.1. It remains to verify Assumption 6.3. Since f is the subdifferential of \(\varPhi \) the variational inequality (3.2) is still satisfied in the sense that
for all \(x,y \in {\mathbf {R}}\) and \(f_x \in f(x)\). Following the same steps as in the proof of Lemma 3.2 but replacing f(v), f(w), and f(z) by arbitrary elements \(f_v \in f(v)\), \(f_w \in f(w)\), and \(f_z \in f(z)\), respectively, shows that Assumption 6.3 is fulfilled. Therefore, the backward Euler–Maruyama method (1.6) is well-defined and yields an approximation of the exact solution X of
where \(b :{\mathbf {R}}\rightarrow {\mathbf {R}}\) and \(g :{\mathbf {R}}\rightarrow {\mathbf {R}}^{1,m}\) are Lipschitz continuous and \(X_0 \in L^2(\varOmega )\). To be more precise, the piecewise linear interpolant \({\mathcal {X}}\) of the values \((X^n)_{n\in \{0,\dots ,N\}}\) defined in (6.3) fulfills
for \(C \in (0,\infty )\) that does not depend on the step size \(k = \frac{T}{N}\). However, let us mention that the strong order of convergence of 1/4 is not necessarily optimal in this particular example. We refer the reader to [9] for a corresponding result on the forward Euler–Maruyama method.
7.2 Stochastic p-Laplace equation
As a second example, we consider the discretization of the stochastic p-Laplace equation. A similar setting is studied in [5]. For a more detailed introduction to this class of problems, we refer the reader to this work and the references therein.
For \(p \in [2,\infty )\) and \(T \in (0,\infty )\) the stochastic p-Laplace equation is given by
where \({\mathcal {D}}\subset {\mathbf {R}}^n\), \(n \in {\mathbf {N}}\), is a bounded Lipschitz domain. By \(W :[0,T] \times \varOmega \rightarrow {\mathbf {R}}^m\), \(m \in {\mathbf {N}}\), we denote a standard \(({\mathcal {F}}_t)_{t \ge 0}\)-adapted Wiener process. We also assume that the initial value \(u_0 :{\mathcal {D}}\times \varOmega \rightarrow {\mathbf {R}}\) fulfills
Furthermore, let \(\varPsi :{\mathbf {R}}\rightarrow {\mathcal {L}}_2({\mathbf {R}}^m;{\mathbf {R}})\) be a Lipschitz continuous mapping, where \({\mathcal {L}}_2({\mathbf {R}}^m;{\mathbf {R}})\) denotes the space of Hilbert–Schmidt operators from \({\mathbf {R}}^m\) to \({\mathbf {R}}\). Note that the Nemytskii operator \({\tilde{\varPsi }} :L^2({\mathcal {D}}) \rightarrow {\mathcal {L}}_2({\mathbf {R}}^m;L^2({\mathcal {D}}))\), given by \([{\tilde{\varPsi }}(u)](x) = \varPsi (u(x))\) for \(u \in L^2({\mathcal {D}})\), is also Lipschitz continuous and will be of importance in the weak formulation below.
Let \(W^{1,p}_0({\mathcal {D}})\) be the Sobolev space of weakly differentiable and p-fold integrable functions on \({\mathcal {D}}\) with vanishing trace on the boundary \(\partial {\mathcal {D}}\), see [47, Section 1.2.3] or [40, Section 4.5] for a precise definition. The dual space of \(W_0^{1,p}({\mathcal {D}})\) is denoted by \(W^{-1,p}({\mathcal {D}})\) in the following. Then, the stochastic p-Laplace equation (7.1) has a unique variational solution \((u(t))_{t \in [0,T]}\) which is progressively measurable and an element of \(L^2(\varOmega ; C([0,T]; L^2({\mathcal {D}}))) \cap L^p(\varOmega ; L^p(0,T;W_0^{1,p}({\mathcal {D}})))\). For further details we refer to [27, Example 4.1.9, Theorem 4.2.4].
For a spatial discretization of (7.1), we use a family of finite element spaces \((V_h)_{h>0}\) such that \(V_h \subset W_0^{1,p}({\mathcal {D}})\) for every \(h >0\). Hereby, we interpret h as a spatial refinement parameter. In the following, we consider a fixed parameter value \(h >0\). By \(d \in {\mathbf {N}}\) we then denote the dimension of the space \(V_h\).
The spatially semi-discrete problem consists of finding a progressively measurable \(L^2(\varOmega ; C([0,T]; L^2({\mathcal {D}}))) \cap L^p(\varOmega ; L^p(0,T;V_h))\)-valued stochastic process \((u_h(t))_{t \in [0,T]}\) such that
for every \(v_h \in V_h\) and \(t\in [0,T]\). Here, \(P_h :L^2({\mathcal {D}}) \rightarrow V_h\) denotes the \(L^2({\mathcal {D}})\)-orthogonal projection onto \(V_h\).
In order to apply our results from the previous sections, we rewrite (7.3) as a problem in \({\mathbf {R}}^d\). To this end, we consider a one-to-one relation between \(V_h\) and \({\mathbf {R}}^d\) given by
for a basis \(\{\varphi _1,\dots ,\varphi _d\}\) of \(V_h\). Through (7.4) we induce additional norms on \({\mathbf {R}}^d\) which are given by
for every \(x \in {\mathbf {R}}^d\). Observe that the norm \(\Vert \cdot \Vert _0\) is also induced by the inner product
where the mass matrix \(M_h\) is symmetric and positive definite. Since all norms on \({\mathbf {R}}^d\) are equivalent, for each \(i \in \{-1,0,1\}\) there exist \(c_i, C_i \in (0,\infty )\) such that
for all \(x \in {\mathbf {R}}^d\).
The p-Laplace operator in the spatially semi-discrete problem (7.3) can be written as \(A_h :V_h \rightarrow V_h\) which is implicitly defined by
for all \(v_h, w_h \in V_h\). By the same arguments as in [27, Example 4.1.9] one can easily verify that \(A_h\) fulfills
for all \(v_h, w_h \in V_h\). Then, for \(x, y \in {\mathbf {R}}^d\) and associated \(v_x, v_y \in V_h\), we introduce mappings \({\tilde{f}} :{\mathbf {R}}^d \rightarrow {\mathbf {R}}^d\) and \({\tilde{g}} :{\mathbf {R}}^d \rightarrow {\mathbf {R}}^{d,m}\) implicitly by
for \(z \in {\mathbf {R}}^m\) and use these functions to define \(f(x) := M_h {\tilde{f}}(x)\) as well as \(g(x) := M_h^{\frac{1}{2}}{\tilde{g}}(x)\) for every \(x \in {\mathbf {R}}^d\). As we assumed that \(v_x \mapsto {\tilde{\varPsi }} (v_x )\) is Lipschitz continuous, there exists \(L_g \in (0,\infty )\) such that
for \(x,y \in {\mathbf {R}}^d\) and \(v_x,v_y \in V_h\) fulfilling (7.4) as well as an orthonormal basis \(\{e_j\}_{j \in \{1,\dots ,m\}}\) of \({\mathbf {R}}^m\). Thus, g fulfills Assumption 4.3. Due the integrability condition to (7.2) for \(u_0\), it follows that \(X_0\) fulfills Assumption 4.4.
Moreover, we see that f is monotone, coercive, and bounded as we can write
as well as
and
for all \(x,y \in {\mathbf {R}}^d\) and \(v_x,v_y \in V_h\) fulfilling (7.4). Here, \(\Vert \cdot \Vert _{{\mathcal {L}}({\mathbf {R}}^m)}\) denotes the matrix norm in \({\mathbf {R}}^m\) which is induced by \(|\cdot |\). Therefore, Assumption 4.1 is satisfied. To prove that f fulfills Assumption 6.3 we note that the mapping \(\varPhi :V_h \rightarrow [0,\infty )\) given by
is a potential of \(A_h\), compare [47, Example 4.23]. Since \(\varPhi \) is convex it follows that
where we use [13, Kapitel III, Lemma 4.10]. In the same way as in Lemma 3.2 we obtain that
for all \(v_z,v_x,v_y \in V_h\). Applying the definition of f, we then get
for \(x,y,z \in {\mathbf {R}}^d\) and \(v_x,v_y,v_z \in V_h\) fulfilling (7.4). This shows that f also fulfills Assumption 6.3.
Consequently, the results of the previous sections are applicable. More precisely, the backward Euler scheme (1.6) has a unique solution \((X^n)_{n\in \{0,\dots ,N\}}\) (cf. Theorem 5.3). Theorem 6.4 then states that the piecewise linear interpolant \({\mathcal {X}}\) of the values \((X^n)_{n\in \{1,\dots ,N\}}\) defined in (6.3) fulfills
for \(C \in (0,\infty )\) that does not depend on the step size k where X is the solution to the single-valued stochastic differential equation
Observe that our proof does not yet rule out that the constant C above depends on the dimension d of the finite element space \(V_h\). Hence, this is not a complete analysis of a full discretization of the stochastic partial differential equation (7.1) and a more detailed analysis is subject to future work. We refer to [5] for a related result in this direction.
Let us emphasize that, unlike the results in [5], we do not have to impose any temporal regularity assumption on the exact solution of (7.1) or on the solution of the semi-discrete problem (7.3). Since such regularity conditions are often not easily verified for quasi-linear stochastic partial differential equations we are confident that our approach could lead to interesting new insights in the numerical analysis of such infinite dimensional problems.
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Acknowledgements
ME would like to thank the Berlin Mathematical School for the financial support. RK also gratefully acknowledges financial support by the German Research Foundation (DFG) through the research unit FOR 2402 – Rough paths, stochastic partial differential equations and related topics – at TU Berlin. MK and SL were supported by Vetenskapsrådet (VR) through Grant No. 2017-04274.
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Eisenmann, M., Kovács, M., Kruse, R. et al. Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations. Bit Numer Math 62, 803–848 (2022). https://doi.org/10.1007/s10543-021-00893-w
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DOI: https://doi.org/10.1007/s10543-021-00893-w