Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations

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 iffor 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 \(y \in {\mathbf {R}}^d\) satisfyingit follows that \(x \in D(f)\) and \(y \in f(x)\).

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 inequalityholds.

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 havethen, for every \(t \in [0,T]\), the inequalityholds.

Let \(\varPhi :{\mathbf {R}}^d \rightarrow {\mathbf {R}}\) be a convex, nonnegative, and continuously differentiable function.

Under Assumption 3.1 and with \(f = \nabla \varPhi \), the inequalitiesandare fulfilled for all \(v,w,z \in {\mathbf {R}}^d\).

The first inequality follows directly from (3.2) sincefor 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 obtainSetting \(\sigma (v,w) := \varPhi (w) - \varPhi (v) - \langle f(v),w-v\rangle _{}\) for all \(v,w \in {\mathbf {R}}^d\), we see thatBut (3.2) says that \(\sigma (v,w) \ge 0\) for all \(v,w \in {\mathbf {R}}^d\), which completes the proof. \(\square \)

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 \(X_0 \in L^2(\varOmega ,{\mathcal {F}}_0,{\mathbf {P}};{\mathbf {R}}^d)\), thenand

First, we recall the identityUsing also (3.6), we then getfor every \(n \in \{1,\ldots ,N\}\). Hence, an application of (3.2) yieldsfor every \(n \in \{1,\ldots ,N\}\). From applications of the Cauchy–Schwarz inequality and the weighted Young inequality we then obtainfor 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 yieldsAn 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 givesSince this is true for any \(n \in \{1,\ldots ,N\}\) the assertion follows. \(\square \)

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).

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 estimateholds, 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.

From (3.6) we directly deduce that for every \(n \in \{1,\ldots ,N\}\)Then, one easily verifies for all \(t \in (t_{n-1},t_n]\), \(n \in \{1,\ldots ,N\}\), thatHence, due to (3.5), the error process \(E := X - {\mathcal {X}}\) can be written asfor 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 getNext, we writeand use (3.3) and (3.4) to obtain, for almost every \(t \in (t_{n-1}, t_n]\), thatFurthermore, the expectation of the second integral on the right-hand side of (3.15) equalsTherefore,Since \(\int _{t_{i-1}}^{t_i} (t_i - t) \,\mathrm {d}t = \frac{1}{2} k^2\) the assertion follows. \(\square \)

For every \(g_0 \in {\mathbf {R}}^{d,m}\) and every step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), the equalityholds.

From the definition (3.11) of \({\mathcal {W}}\) it follows thatwhere 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 thatand the proof is complete. \(\square \)

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 thatThen there exists \(C \in (0,\infty )\) such that for every step size \(k = \frac{T}{N}\), \(N \in {\mathbf {N}}\), the estimateholds, 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.

Since f is assumed to be \(\alpha \)-Hölder continuous it follows thatIn 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 thatIndeed, we make use of the Hölder continuity of f directly and obtainwhere 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 getfor \(t \in (t_{j-1},t_j]\). 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 writeTo 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 obtainFrom the Hölder continuity of f we then deduce thatwhere 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 thatTherefore, together with (3.8) we arrive at the estimatefor 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 thatTogether with the Cauchy–Schwarz inequality and (3.19), we therefore obtainInserting the estimates for \(T_1\), \(T_2\), and (3.17) into Lemma 3.5 completes the proof. \(\square \)

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].

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 thatfor every \(v \in D(f)\) and \(f_v \in f(v)\).

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 thatfor all \(v,w \in {\mathbf {R}}^d\).

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 thatfor all \(v,w \in {\mathbf {R}}^d\).

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.

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].

A tuple \((X,\eta )\) is called a solution of the multi-valued stochastic differential equation (1.5), if the following conditions hold.

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 thatFurthermore, if \(p \in (1,\infty )\) and \(\frac{1}{p}+\frac{1}{q} = 1\), then

Existence and uniqueness is shown, for instance, in [21]. Forthe equalityholds by an application of Itô’s formula (see [12, Chap. 4.7, Theorem 7.1]). From the coercivity assumption on f we obtain thatfor 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 thatSince b and g satisfie the linear growth bound (4.1), we haveas well asThus, we getWe introduceThen 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 \(t \in [0,T]\), thatInserting 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 \(f_{x} \in f(x)\), \(x \in {\mathbf {R}}^d\), thatwith \(q = \frac{p}{p-1}\). Therefore, it follows thatsince \(\eta (s) \in f(X(s))\) for almost every \(s \in [0,T]\). \(\square \)

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 definitionIn 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.

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

We first show that there exists a unique \(x_0\in D(f)\) such thatTo this end, notice that for all \(x,y \in {\mathbf {R}}^d\), the inequalitieshold due to the step size bound. In addition, it follows from (4.1) thatfor 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 \(\eta _{x_0} \in f(x_0)\) withIt 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 thatBy considering the difference of these equations tested with \(x_1- x_2\), we obtainSince \(1 - k L_b > 0\) we must have \(x_1 = x_2\) and the proof is complete. \(\square \)

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

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 equationsBy considering the difference of these equations, tested with \(x_1 - x_2\), we obtainBy 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 getReinserting \(x_i = S_k(w_i)\) then shows thatas claimed. \(\square \)

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 \(n \in \{1,\dots ,N\}\) andfor every \(n \in \{1,\dots ,N\}\), \({\mathbf {P}}\)-almost surely with \(X^0 = X_0\) and \(\eta ^0 \in f(X_0)\).

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 thatNext, 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 thatBy 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 boundThis implies, in particular, thatis 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. \(\square \)

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 thatIf, in addition, \(p \in (1,\infty )\), then there exists \(K_\eta \in (0,\infty )\) independent of the step size \(k = \frac{T}{N}\) such thatwhere \(q \in (1,\infty )\) is given by \(\frac{1}{p} + \frac{1}{q} = 1\).

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 \(L_b k \in [0,1)\).

First, we recall the identityAs \(\eta ^n \in f(X^n)\), using Assumptions 4.1 and  4.2, it follows thatwhere 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 obtainAfter taking expectations, the last term on the right-hand side vanishes. Then, applications of Itô’s isometry and (4.1) giveSince the step size bound \(5 L_b k \in [0,1)\) ensures thatthe 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 thatand an application of (5.4) then yields (5.5). \(\square \)

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 haveIn 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

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 atby an application of Itô’s isometry. Furthermore, due to the Lipschitz continuity of g we obtainwhere the last step follows from the identitywhich 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 \(t \in (t_{n-1},t_n]\) thatTogether 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 \(t \in (t_{n-1},t_n]\) thatwhere 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. \(\square \)

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

The nonnegativity follows immediately from the monotonicity of f. To prove the second inequality, we insert the scheme (5.3) and obtainFor (6.7) we obtainbecause of the telescopic structure. Furthermore, it follows from Assumptions 4.1 and 4.4 thatFor the term in (6.8) we apply Hölder’s inequality with \(\rho = \max (2,p)\) and \(\frac{1}{\rho } + \frac{1}{\rho '} = 1\) to obtainThen, from applications of the triangle inequality and Lemma 5.4, we getWe 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 haveIn both cases the appearing terms are finite because of Assumption 4.4. Moreover, a further application of the triangle inequality yieldsDue to the linear growth bound (4.1) on b and the a priori bound (5.4), it then follows thatBy application of Lemma 2.2 with constant \(C_{\rho }\), we obtainTogether with the linear growth bound (4.1) on g this shows thatPutting the estimates together proves the desired bound. \(\square \)

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 \(f_z \in f(z)\),

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

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.

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 partswhere\({\mathbf {P}}\)-almost surely for every \(t \in (0,T]\). We expand the square of the norm of E asIn 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 getFurthermore, we also have thatThus, after combining (6.12) and (6.13) we obtainFor 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 getwhere 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 \(s \in (0,T]\) thatwhere 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 thatTherefore,Altogether, for every \(t \in (t_{n-1}, t_n]\) and \(n \in \{1,\ldots ,N\}\), we have shown thatwhere 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 thatNext, 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 followsFor 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 givesAfter inserting the definition of \(E_2\) from (6.10) the first integral is then equal tofor 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 thatfor 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 atInserting 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 getIn 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 thatIf \(p \in [2,\infty )\) 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 thatas well asHence, 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 thatAltogether, this yieldsfor 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 thatThe first factor is bounded as we saw in the case for \(\varGamma _1\). Furthermore, using Lemma 6.1, we have thatDue 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 thatThe estimates \(\varGamma _3\) and \(\varGamma _4\) follow analogously with the only new term that appears is of the formwhich 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 thatHence, we obtainAfter 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 \(t \in (0,T]\) thatThe assertion then follows from an application of Gronwall’s lemma, see for example, [11, Appendix B]. \(\square \)

Up to this point, we only proved convergence for X but not for \(\eta \). However, from the existence of \(X^n\) we also obtain thatAnalogously, we can write for the exact solution \(\eta \) thatTherefore, from the convergence of \({\mathcal {X}}\) to X and the Lipschitz continuity of b and g we also obtain the estimatefor every \(n \in \{1,\ldots ,N\}\).