Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

Abstract

A formal mean square error expansion (MSE) is derived for Euler–Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise, a posteriori, adaptive time-stepping Euler–Maruyama algorithm for numerical solutions of SDE, and the resulting algorithm is incorporated into a multilevel Monte Carlo (MLMC) algorithm for weak approximations of SDE. This gives an efficient MSE adaptive MLMC algorithm for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC algorithm is shown to outperform the uniform time-stepping MLMC algorithm by orders of magnitude, producing output whose error with high probability is bounded by ${\mathrm {TOL}}>0$ at the near-optimal MLMC cost rate ${\mathscr {O}}\left( {\mathrm {TOL}}^{-2} \log ({\mathrm {TOL}})^4\right)$ that is achieved when the cost of sample generation is ${\mathscr {O}}\left( 1\right)$.

Appendices

Theoretical Results

1.1 Error Expansion for the MSE in 1D

In this section, we derive a leading-order error expansion for the MSE (12) in the 1D setting when the drift and diffusion coefficients are respectively mappings of the form $a: [0,T] \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ and $b: [0,T] \times {\mathbb {R}}\rightarrow {\mathbb {R}}$. We begin by deriving a representation of the MSE in terms of products of local errors and weights.

Recalling the definition of the flow map, $\varphi (x,t):= g(X_T^{x,t})$, and the first variation of the flow map and the path itself given in Sect. 2.1.1, we use the Mean Value Theorem to deduce that

$$\begin{aligned} \begin{aligned} g \left( X_T \right) - g \left( \overline{X}_{T} \right)&= \varphi (0,x_0) - \varphi (0,\overline{X}_{T} ) \\&= \sum _{n=0}^{N-1} \varphi (t_{n},\overline{X}_{t_n}) - \varphi (t_{n+1},\overline{X}_{t_{n+1}}) \\&= \sum _{n=0}^{N-1} \varphi \left( t_{n+1},X_{t_{n+1}}^{\overline{X}_{t_n},t_{n}} \right) - \varphi (t_{n+1},\overline{X}_{t_{n+1}}) \\&= \sum _{n=0}^{N-1} \varphi _{x}\left( t_{n+1},\overline{X}_{t_{n+1}} + s_n \varDelta e_n \right) \varDelta e_n, \end{aligned} \end{aligned}$$

(48)

where the local error is given by $\varDelta e_n := X_{t_{n+1}}^{\overline{X}_{t_n},t_{n}} -\overline{X}_{t_{n+1}}$ and $s_n \in [0,1]$. Itô expansion of the local error gives the following representation:

(49)

By Eq. (48) we may express the MSE by the following squared sum

$$\begin{aligned} \begin{aligned}&{\mathrm {E}\left[ \left( g(X_T) -g \left( \overline{X}_{T} \right) \right) ^2\right] } = {\mathrm {E}\left[ \left( \sum _{n=0}^{\check{N}-1} \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} + s_n \varDelta e_n\right) \varDelta e_n \right) ^2 \right] }\\&\qquad \quad = \sum _{n,k=0}^{\check{N}-1} {\mathrm {E}\left[ \varphi _{x}\left( t_{k+1},\overline{X}_{t_{k+1}} + s_k \varDelta e_k\right) \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} + s_n \varDelta e_n\right) \varDelta e_k \varDelta e_n \right] }. \end{aligned} \end{aligned}$$

This is the first step in deriving the error expansion in Theorem 2. The remaining steps follow in the proof below.

Proof of Theorem 2. The main tools used in proving this theorem are Taylor and Itô–Taylor expansions, Itô isometry, and truncation of higher order terms. For errors attributed to the leading-order local error term, $\widetilde{\varDelta b} _n$, (cf. Eq. (49)), we do detailed calculations, and the remainder is bounded by stated higher order terms.

We begin by noting that under the assumptions in Theorem 2 Lemmas 1 and 2 respectively verify then the existence and uniqueness of the solution of the SDE X and the numerical solution $\overline{X}$, and provide higher order moment bounds for both. Furthermore, due to the assumption of the mesh points being stopping times for which $t_n$ is $\mathscr {F}_{t_{n-1}}$-measurable for all n, it follows also that the numerical solution is adapted to the filtration, i.e., $\overline{X}_{t_n}$ is $\mathscr {F}_{t_n}$-measurable for all n.

We further need to extend the flow map and the first variation notation from Sect. 2.1.1. Let $\overline{X}^{x, t_k}_{t_n}$ for $n\ge k$ denote the numerical solution of the Euler–Maruyama scheme

$$\begin{aligned} \overline{X}_{t_{j+1}}^{x,t_k} = \overline{X}_{t_{j}}^{x,t_k} + a(t_j, \overline{X}_{t_j}^{x,t_k}) \varDelta t_j + b(t_j, \overline{X}_{t_j}^{x,t_k}) \varDelta W_j, \quad j \ge k, \end{aligned}$$

(50)

with initial condition $X_{t_k} = x$. The first variation of $\overline{X}^{x, t_k}_{t_n}$ is defined by $\partial _x \overline{X}_{t_n}^{x,t_k}$. Provided that ${\mathrm {E}\left[ |x|^{2p}\right] }< \infty$ for all $p \in {\mathbb {N}}$, x is $\mathscr {F}_{t_k}$-measurable and provided the assumptions of Lemma 2 hold, it is straightforward to extend the proof of the lemma to verify that $(\overline{X}^{x, t_k}, \partial _x \overline{X}^{x,t_k})$ converges strongly to $(X^{x, t_k}, \partial _x X^{x,t_k})$ for $t \in [t_k, T]$,

$$\begin{aligned} \begin{aligned} \max _{k \le n \le \check{N} } \left( \left( {\mathrm {E}\left[ \left| \overline{X}_{t_n}^{x,t_k} - X_{t_n}^{x,t_k}\right| ^{2p}\right] } \right) ^{1/2p} \right)&\le C \check{N}^{-1/2}, \quad \forall p \in {\mathbb {N}}\\ \max _{k \le n \le \check{N} } \left( \left( {\mathrm {E}\left[ \left| \partial _{x} \overline{X}_{t_n}^{x,t_k} - \partial _{x} X_{t_n}^{x,t_k}\right| ^{2p} \right] } \right) ^{1/2p} \right)&\le C \check{N}^{-1/2}, \quad \forall p \in {\mathbb {N}}\end{aligned} \end{aligned}$$

and

$$\begin{aligned} \max _{ k \le n \le \check{N} } \left( \max \left( {\mathrm {E}\left[ \left| \overline{X}_{t_n}^{x,t_k}\right| ^{2p}\right] }, {\mathrm {E}\left[ \left| \partial _x\overline{X}_{t_n}^{x,t_k}\right| ^{2p} \right] } \right) \right) < \infty , \quad \forall p \in {\mathbb {N}}. \end{aligned}$$

(51)

In addition to this, we will also make use of moment bounds for the second and third variation of the flow map in the proof, i.e., $\varphi _{xx}\left( t,x\right)$ and $\varphi _{xxx}\left( t,x\right)$. The second variation is described in Section “Variations of the flow map”, where it is shown in Lemma 3 that provided that x is $\mathscr {F}_t$-measurable and ${\mathrm {E}\left[ |x|^{2p}\right] }<\infty$ for all $p \in {\mathbb {N}}$, then

$$ \max \left( {\mathrm {E}\left[ \left| \varphi _{xx}\left( t, x\right) \right| ^{2p} \right] }, {\mathrm {E}\left[ \left| \varphi _{xxx}\left( t, x\right) \right| ^{2p} \right] }, {\mathrm {E}\left[ \left| \varphi _{xxxx}\left( t, x\right) \right| ^{2p} \right] } \right) < \infty , \quad \forall p \in {\mathbb {N}}. $$

Considering the MSE error contribution from the leading order local error terms $\widetilde{\varDelta b} _n$, i.e.,

$$\begin{aligned} {\mathrm {E}\left[ \varphi _{x}\left( t_{k+1}, \overline{X}_{t_{k+1}} + s_k \varDelta e_k \right) \varphi _{x}\left( t_{n+1},\overline{X}_{t_{n+1}} + s_n \varDelta e_n\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \right] }, \end{aligned}$$

(52)

we have for $k=n$,

$$\begin{aligned}&{\mathrm {E}\left[ \left( \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} \right) + \varphi _{xx}\left( t_{n+1}, \overline{X}_{t_{n+1}} + \hat{s}_n \varDelta e_n \right) s_n\varDelta e_n \right) ^2 \widetilde{\varDelta b} _{n}^2 \right] } \\ =&\, {\mathrm {E}\left[ \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} \right) ^2 \widetilde{\varDelta b} _{n}^2 + { o}\left( \varDelta t_n^2\right) \right] }.\end{aligned}$$

The above ${ o}\left( \varDelta t_n^2\right)$ follows from Young’s and Hölder’s inequalities,

(53)

where the last inequality is derived by applying the moment bounds for multiple Itô integrals described in [22, Lemma 5.7.5] and under the assumptions (R.1), (R.2), (M.1), (M.2) and (M.3). This yields

(54)

And by similar reasoning,

$$\begin{aligned} {\mathrm {E}\left[ \varphi _{xx}\left( \overline{X}_{t_{n+1}} + \hat{s}_n \varDelta e_n ,t_{n+1} \right) ^2s_n^2 \varDelta e_n^2 \widetilde{\varDelta b} _{n}^2 \right] } \le C {\mathrm {E}\left[ \varDelta t_n^4\right] }. \end{aligned}$$

For achieving independence between forward paths and dual solutions in the expectations, an Itô–Taylor expansion of $\varphi _x$ leads to the equality

$$\begin{aligned} {\mathrm {E}\left[ \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} \right) ^2 \widetilde{\varDelta b} _{n}^2 \right] } = {\mathrm {E}\left[ \varphi _{x}\left( t_{n+1},\overline{X}_{t_{n}} \right) ^2 \widetilde{\varDelta b} _{n}^2 + { o}\left( \varDelta t_n^{2}\right) \right] }. \end{aligned}$$

Introducing the null set completed $\sigma$-algebra

$$\begin{aligned} \widehat{\mathscr {F}}^{n} = \overline{\sigma \left( \sigma ( \{ W_{s}\}_{ 0 \le s \le t_n}) \vee \sigma ( \{ W_s-W_{t_{n+1}}\}_{ t_{n+1} \le s \le T} ) \right) \vee \sigma (X_0)}, \end{aligned}$$

we observe that $\varphi _{x}\left( t_{n+1}, \overline{X}_{t_n}\right) ^2$ is $\widehat{\mathscr {F}}^{n}$ measurable by construction, (cf. [27, Appendix B]). Moreover, by conditional expectation,

$$\begin{aligned} \begin{aligned} {\mathrm {E}\left[ \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}\right) ^2 \widetilde{\varDelta b} _{n}^2 \right] }&= {\mathrm {E}\left[ \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}} \right) ^2 {\mathrm {E}\left[ \widetilde{\varDelta b} _{n}^2 |\widehat{\mathscr {F}}^{n} \right] } \right] }\\&= {\mathrm {E}\left[ \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}} \right) ^2 (b_xb)^2(t_n,\overline{X}_{t_n}) \frac{\varDelta t_n^2}{2} + { o}\left( \varDelta t_n^{2} \right) \right] }, \end{aligned}\end{aligned}$$

where the last equality follows from using Itô’s formula,

$$ \begin{aligned} (b_x b)^2 (t,X_{t}^{\overline{X}_{t_n},t_{n}})&= (b_x b)^2 (t_n,\overline{X}_{t_n}) + \int _{t_n}^t \left( \Big (\partial _t + a \partial _x + \frac{b^2}{2} \partial _x^2 \Big )(b_x b)^2 \right) (s,X_{s}^{\overline{X}_{t_n},t_{n}}) \, ds \\&\quad + \int _{t_n}^t \left( b \partial _x (b_x b)^2 \right) (s,X_{s}^{\overline{X}_{t_n},t_{n}}) \, dW_s, \quad t\in [t_n, t_{n+1}), \end{aligned} $$

to derive that

$$\begin{aligned} \begin{aligned} {\mathrm {E}\left[ \widetilde{\varDelta b} _{n}^2 |\widehat{\mathscr {F}}^{n} \right] }&= {\mathrm {E}\left[ \left( \int _{t_{n}}^{t_{n+1}} \int _{t_{n}}^t (b_xb)(s,X_s^{\overline{X}_{t_n},t_n}) dW_s \,dW_t \right) ^2 \Big | \overline{X}_{t_n} \right] }\\&= \frac{(b_xb)^2( t_{n},\overline{X}_{t_n} )}{2} \varDelta t_n^2 + { o}\left( \varDelta t_{n}^{2}\right) . \end{aligned} \end{aligned}$$

Here, the higher order ${ o}\left( \varDelta t_{n}^{2}\right)$ terms are bounded in a similar fashion as the terms in inequality (53), by using [22, Lemma 5.7.5].

For the terms in (52) for which $k <n$, we will show that

$$\begin{aligned} \begin{aligned}&\sum _{k,n=0}^{\check{N}-1} {\mathrm {E}\left[ \varphi _{x}\left( t_{k+1}, \overline{X}_{t_{k+1}} + s_k \varDelta e_k \right) \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} + s_n \varDelta e_n\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \right] } = \sum _{n=0}^{\check{N}-1} {\mathrm {E}\left[ { o}\left( \varDelta t_n^2 \right) \right] }, \end{aligned} \end{aligned}$$

(55)

which means that the contribution to the MSE from these terms is negligible to leading order. For the use in later expansions, let us first observe by use of the chain rule that for any $\mathscr {F}_{t_n}$-measurable y with bounded second moment,

$$ \begin{aligned} \varphi _{x}\left( t_{k+1}, y \right)&= g'(X_T^{y, t_{k+1}} ) \partial _x X_{T}^{y ,t_{k+1}}\\&= g'(X_T^{\overline{X}_{t_{k+1}} + s_m \varDelta e_k, t_{k+1}} ) \partial _x X_{T}^{X_{t_{n+1}}^{y,t_{k+1}} ,t_{n+1}} \partial _x X_{t_{n+1}}^{y,t_{k+1}}\\&= \varphi _{x}\left( t_{n+1}, X_{t_{n+1}}^{y,t_{k+1}} \right) \partial _x X_{t_{n+1}}^{y,t_{k+1}}, \end{aligned} $$

and that

$$\begin{aligned}&\partial _x X_{t_{n+1}}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}} = \partial _x X_{t_{n}}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}} \\&\qquad \qquad + \int _{t_n}^{t_{n+1}} a_x(s,X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}}) \partial _x X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}} ds \\&\qquad \qquad \qquad \qquad +\int _{t_n}^{t_{n+1}} b_x(s,X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}}) \partial _x X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}} dW_s. \end{aligned}$$

We next introduce the $\sigma$-algebra

$$ \widehat{\mathscr {F}}^{k,n} := \overline{\sigma ( \{ W_{s }\}_{ 0 \le s \le t_k}) \vee \sigma ( \{ W_s-W_{t_{k+1} }\}_{ t_{k+1} \le s \le t_n} ) \vee \sigma ( \{ W_s-W_{t_{n+1} }\}_{ t_{n+1} \le s \le T} ) \vee \sigma (X_0)}, $$

and Itô–Taylor expand the $\varphi _x$ functions in (55) about center points that are $\widehat{\mathscr {F}}^{k,n}$-measurable:

$$\begin{aligned}&\varphi _{x}\left( t_{k+1}, \overline{X}_{t_{k+1}} + s_k \varDelta e_k \right) = \varphi _{x}\left( t_{n+1}, X_{t_{n+1}}^{ \overline{X}_{t_{k+1}} + s_k \varDelta e_k,t_{k+1} }\right) \partial _x X_{t_{n+1}}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}} \nonumber \\&\,\,\, = \Bigg [\varphi _{x}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) + \varphi _{xx}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \left( X_{t_{n+1}}^{ \overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1} } - X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}} \right) \nonumber \\&\qquad \qquad \,\, + \varphi _{xxx}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \frac{\left( X_{t_{n+1}}^{ \overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1} } - X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}} \right) ^2}{2} \nonumber \\&\qquad \qquad \,\quad + \varphi _{xxxx}\left( t_{n+1}, (1-\check{s}_n) X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}} + \check{s}_n X_{t_{n+1}}^{ \overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1} }\right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \times \frac{(X_{t_{n+1}}^{ \overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1} } - X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}})^2}{2}\Bigg ]\nonumber \\&\qquad \quad \times \Bigg [ \partial _x X_{t_{n}}^{\overline{X}_{t_{k}} , t_{k+1}} + \partial _{xx} X_{t_{n}}^{\overline{X}_{t_{k}} , t_{k+1}} (a(t_k,\overline{X}_{t_k}) \varDelta t_k + b(t_k,\overline{X}_{t_k}) \varDelta W_k + s_k \varDelta e_k)\nonumber \\&\qquad \qquad \qquad + \partial _{xxx} X_{t_{n}}^{\overline{X}_{t_{k}} + \grave{s}_k (a(t_k,\overline{X}_{t_k}) \varDelta t_k + (b(t_k,\overline{X}_{t_k}) \varDelta W_k + s_k \varDelta e_k) , t_{k+1}}\nonumber \\&\qquad \qquad \,\,\qquad \quad \times \frac{(a(t_k,\overline{X}_{t_k}) \varDelta t_k +b(t_k,\overline{X}_{t_k}) \varDelta W_k + s_k \varDelta e_k)^2}{2}\nonumber \\&\qquad \qquad \quad + \int _{t_n}^{t_{n+1}} a_x(s,X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}}) \partial _x X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}} ds \nonumber \\&\qquad \qquad \qquad + \int _{t_n}^{t_{n+1}} b_x(s,X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}}) \partial _x X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}} dW_s \Bigg ], \end{aligned}$$

(56)

where

$$\begin{aligned}&\quad X_{t_{n+1}}^{ \overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1} } - X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\\&\qquad \quad = \int _{t_n}^{t_{n+1}} a(s,X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}}) ds +\int _{t_n}^{t_{n+1}} b(s,X_{s}^{\overline{X}_{t_{k+1}} + s_k \varDelta e_k, t_{k+1}}) dW_s \\&+ \partial _x X_{t_{n}}^{\overline{X}_{t_{k}} + \tilde{s}_k(a(t_k,\overline{X}_{t_k}) \varDelta t_k+ b(t_k, \overline{X}_{t_k}) \varDelta W_k + s_k \varDelta e_k ) ,t_{k+1}} (a(t_k,\overline{X}_{t_k}) \varDelta t_k+ b(t_k, \overline{X}_{t_k}) \varDelta W_k + s_k \varDelta e_k ), \end{aligned}$$

and

$$\begin{aligned}&\varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+ 1}} + s_n \varDelta e_n \right) = \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}, t_{k+1} }\right) \nonumber \\&\qquad \quad + \varphi _{xx}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}, t_{k+1} } \right) \varDelta \nu _{k,n} + \varphi _{xxx}\left( t_{n+1}, \overline{X}_{n}^{\overline{X}_{k}, t_{k+1} } \right) \frac{\varDelta \nu _{k,n}^2}{2} \nonumber \\&\qquad \qquad \,\, + \varphi _{xxxx}\left( t_{n+1}, (1-\acute{s}_n)\overline{X}_{t_n}^{\overline{X}_{t_k}, t_{k+1} } + \acute{s}_n(\overline{X}_{t_{n+1}} +s_n \varDelta e_n)\right) \frac{\varDelta \nu _{k,n}^3}{6}, \end{aligned}$$

(57)

with

$$\begin{aligned}&\varDelta \nu _{k,n} := a( t_{n}, \overline{X}_{t_n}) \varDelta t_n + b( t_{n}, \overline{X}_{t_n}) \varDelta W_n +s_n \varDelta e_n\\&+ \partial _x \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}+\hat{s}_k( a(t_{k},\overline{X}_{t_{k}})\varDelta t_k + b(t_{k},\overline{X}_{t_{k}})\varDelta W_k),t_{k+1}} (a(t_k,\overline{X}_{t_k}) \varDelta t_k+ b(t_k, \overline{X}_{t_k}) \varDelta W_k + s_k \varDelta e_k ). \end{aligned}$$

Plugging the expansions (56) and (57) into the expectation

$$\begin{aligned}{\mathrm {E}\left[ \varphi _{x}\left( t_{k+1}, \overline{X}_{k+1} + s_k \varDelta e_k \right) \varphi _{x}\left( t_{n+1}, \overline{X}_{n+1} + s_n \varDelta e_n\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \right] }, \end{aligned}$$

the summands in the resulting expression that only contain products of the first variations vanishes,

$$ \begin{aligned}&{\mathrm {E}\left[ \varphi _{x}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \partial _x X_{t_{n}}^{\overline{X}_{t_{k}} , t_{k+1}} \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k+1}}, t_{k+1} }\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \right] }\\&= {\mathrm {E}\left[ {\mathrm {E}\left[ \widetilde{\varDelta b} _{n} \widetilde{\varDelta b} _k | \widehat{\mathscr {F}}^{k,n} \right] } \varphi _{x}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \partial _x X_{t_{n}}^{\overline{X}_{t_{k}} , t_{k+1}} \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}, t_{k+1} }\right) \right] } = 0. \end{aligned} $$

One can further deduce that all of the the summands in which the product of multiple Itô integrals $\widetilde{\varDelta b} _k$ and $\widetilde{\varDelta b} _{n}$ are multiplied only with one additional Itô integral of first-order vanish by using the fact that the inner product of the resulting multiple Itô integrals is zero, cf. [22, Lemma 5.7.2], and by separating the first and second variations from the Itô integrals by taking a conditional expectation with respect to the suitable filtration. We illustrate this with a couple of examples,

$$\begin{aligned}&\mathrm {E}\Bigg [ \varphi _{x}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \partial _{xx} X_{t_{n}}^{\overline{X}_{t_{k}} , t_{k+1}} b(t_k,\overline{X}_{t_k}) \varDelta W_k \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}, t_{k+1} }\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \Bigg ]\\&\quad \,\,\, = \mathrm {E}\Bigg [ \varphi _{x}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \partial _{xx} X_{t_{n}}^{\overline{X}_{t_{k}} , t_{k+1}} b(t_k,\overline{X}_{t_k}) \varDelta W_k \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}, t_{k+1} }\right) \widetilde{\varDelta b} _k \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times {\mathrm {E}\left[ \widetilde{\varDelta b} _{n} | \widehat{\mathscr {F}}^n\right] } \Bigg ]=0, \end{aligned}$$

and

$$\begin{aligned}&\,\,\, \mathrm {E}\Bigg [ \varphi _{x}\left( t_{n+1}, X_{t_{n}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \partial _{x} X_{t_{n}}^{\overline{X}_{t_{k}} , t_{k+1}} b( t_{n}, \overline{X}_{t_n}) \varDelta W_n \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}, t_{k+1} }\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \Bigg ]\\&= \mathrm {E}\Bigg [ \varphi _{x}\left( t_{n+1}, X_{t_{n+1}}^{\overline{X}_{t_{k}},t_{k+1}}\right) \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}^{\overline{X}_{t_{k}}, t_{k+1} }\right) \widetilde{\varDelta b} _k b( t_{n}, \overline{X}_{t_n}) {\mathrm {E}\left[ \widetilde{\varDelta b} _{n} \varDelta W_n | \widehat{\mathscr {F}}^n\right] } \Bigg ]=0. \end{aligned}$$

From these observations, assumption (M.3), inequality (54), and, when necessary, additional expansions of integrands to render the leading order integrand either $\widehat{\mathscr {F}}^{k}$- or $\widehat{\mathscr {F}}^{n}$-measurable and thereby sharpen the bounds (an example of such an expansion is

$$\begin{aligned} \widetilde{\varDelta b} _n&= \int _{t_{n}}^{t_{n+1}} \int _{t_{n}}^t (b_xb)(s,X_{s}^{\overline{X}_{t_n},t_{n}}) dW_s \,dW_t\\&= \int _{t_{n}}^{t_{n+1}} \int _{t_{n}}^t (b_xb)\left( s,X_{s}^{\overline{X}_{t_n}^{\overline{X}_{t_{k}},t_{k+1}},t_{n}} \right) dW_s \,dW_t + \text {h.o.t.}). \end{aligned}$$

We derive after a laborious computation which we will not include here that

$$\begin{aligned}&\left| {\mathrm {E}\left[ \varphi _{x}\left( t_{k+1}, \overline{X}_{t_{k+1}} + s_k \varDelta e_k \right) \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} + s_n \varDelta e_n\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \right] }\right| \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \le C \check{N}^{-3/2} \sqrt{{\mathrm {E}\left[ \varDelta t_{k}^2\right] } {\mathrm {E}\left[ \varDelta t_{n}^2\right] }}. \end{aligned}$$

This further implies that

$$ \begin{aligned}&\sum _{k,n=0, k{\ne }n}^{\check{N}-1} {\mathrm {E}\left[ \varphi _{x}\left( t_{k+1}, \overline{X}_{t_{k+1}} + s_k \varDelta e_k \right) \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} + s_n \varDelta e_n\right) \widetilde{\varDelta b} _k \widetilde{\varDelta b} _{n} \right] }\\&\le C \check{N}^{-3/2} \sum _{k,n=0, k{\ne }n}^{\check{N}-1} \sqrt{{\mathrm {E}\left[ \varDelta t_{k}^2\right] } {\mathrm {E}\left[ \varDelta t_{n}^2\right] }} \\&\le C \check{N}^{-3/2} \left( \sum _{n=0}^{\check{N}-1} \sqrt{{\mathrm {E}\left[ \varDelta t_{n}^2\right] }} \right) ^2 \\&\le C \check{N}^{-1/2 }\sum _{n=0}^{\check{N}-1} {\mathrm {E}\left[ \varDelta t_{n}^2\right] }, \end{aligned} $$

such that inequality (55) holds.

So far, we have shown that

$$\begin{aligned}&{\mathrm {E}\left[ \left( \sum _{n=0}^{N-1} \varphi _{x}\left( t_{n+1},\overline{X}_{t_{n+1}} + s_n \varDelta e_n \right) \widetilde{\varDelta b} _{n} \right) ^2 \right] } \nonumber \\&\qquad \qquad \qquad \qquad = {\mathrm {E}\left[ \sum _{n=0}^{N-1} \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n}}\right) ^2 \frac{(b_x b)^2}{2}(t_n,\overline{X}_{t_n}) \varDelta t_n^2 + { o}\left( \varDelta t_n^2\right) \right] }. \end{aligned}$$

(58)

The MSE contribution from the other local error terms, and , can also be bounded using the above approach with Itô–Taylor expansions, $\widehat{\mathscr {F}}^{m,n}$-conditioning and Itô isometries. This yields that

(59)

$$\begin{aligned} \begin{aligned}&{\mathrm {E}\left[ \varphi _{x}\left( t_{k+1}, \overline{X}_{t_{k+1}} + s_k \varDelta e_k \right) \varphi _{x}\left( t_{n+1}, \overline{X}_{t_{n+1}} + s_n \varDelta e_n \right) \widetilde{\varDelta a} _{k} \widetilde{\varDelta a} _{n}\right] } \\&\qquad = {\left\{ \begin{array}{ll} {\mathrm {E}\left[ \varphi _{x}\left( t_{n}, \overline{X}_{t_{n}} \right) ^2 \frac{ (a_x b)^2 }{2}(t_n, \overline{X}_{t_n}) \varDelta t_n^3 + { o}\left( \varDelta t_n^3\right) \right] }, &{} \text {if } k=n,\\ {\mathscr {O}}\left( \check{N}^{-3/2}\left( {\mathrm {E}\left[ \varDelta t_k^3\right] } {\mathrm {E}\left[ \varDelta t_n^3\right] } \right) ^{1/2}\right) , &{} \text {if } k\, {\ne }\, n, \end{array}\right. } \end{aligned} \end{aligned}$$

and

Moreover, conservative bounds for error contributions involving products of different local error terms, e.g., , can be induced from the above bounds and Hölder’s inequality. For example,

The proof is completed in two replacement steps applied to $\varphi _x$ on the right-hand side of equality (58). First, we replace $\varphi _{x}\left( t_{n+1},\overline{X}_{t_{n}}\right)$ by $\varphi _{x}\left( t_{n}, \overline{X}_{t_n}\right)$. Under the regularity assumed in this theorem, the replacement is possible without introducing additional leading order error terms as

$$ \begin{aligned}&{\mathrm {E}\left[ |\varphi _{x}\left( t_{n+1},\overline{X}_{t_{n}}\right) -\varphi _{x}\left( t_{n},\overline{X}_{t_{n}}\right) | \right] } = {\mathrm {E}\left[ \left| g'(X_T^{\overline{X}_{t_{n}},t_{n+1}})\partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}} - g'(X_T^{\overline{X}_{t_{n}},t_{n}})\partial _x X_T^{\overline{X}_{t_{n}},t_{n}}\right| \right] }\\&\le {\mathrm {E}\left[ \left| (g'(X_T^{\overline{X}_{t_{n}},t_{n+1}}) - g'(X_T^{\overline{X}_{t_{n}},t_{n}}))\partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}\right| \right] } \\&\qquad + {\mathrm {E}\left[ \left| g'(X_T^{\overline{X}_{t_{n}},t_{n}})(\partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}- \partial _x X_T^{\overline{X}_{t_{n}},t_{n}})\right| \right] } \\&= {\mathscr {O}}\left( \check{N}^{-1/2}\right) . \end{aligned} $$

Here, the last equality follows from the assumptions (M.2), (M.3), (R.2), and (R.3), and Lemmas 1 and 2,

$$ \begin{aligned}&{\mathrm {E}\left[ \left| \Big (g'(X_T^{\overline{X}_{t_{n}},t_{n+1}}) - g'(X_T^{\overline{X}_{t_{n}},t_{n}})\Big )\partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}\right| \right] } \\&\le C \sqrt{{\mathrm {E}\left[ \left| X_T^{\overline{X}_{t_{n}},t_{n+1}} - X_T^{X_{t_{n+1}}^{\overline{X}_{t_{n}},t_n},t_{n+1}}\right| ^2\right] } {\mathrm {E}\left[ \left| \partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}\right| ^2\right] } }\\&\le C \left( {\mathrm {E}\left[ \left| \partial _x X_T^{(1-s_n)\overline{X}_{t_{n}}+ s_nX_{t_{n+1}}^{\overline{X}_{t_{n}},t_n} ,t_{n+1}}\right| ^4\right] } \right) ^{1/4}\\&\qquad \times \left( {\mathrm {E}\left[ \left| \int _{t_n}^{t_{n+1}} a(s,X_{s}^{\overline{X}_{t_{n}},t_n}) ds + \int _{t_n}^{t_{n+1}} b(s,X_{s}^{\overline{X}_{t_{n}},t_n})dW_s\right| ^4 \right] } \right) ^{1/4}\\&\le C \left( {\mathrm {E}\left[ \sup _{ t_n\le s \le t_{n+1} } |a(s,X_{s}^{\overline{X}_{t_{n}},t_n})|^4 \varDelta t_n^4 + \sup _{ t_n\le s \le t_{n+1} } |b(s,X_{s}^{\overline{X}_{t_{n}},t_n})|^4 \varDelta t_n^2\right] } \right) ^{1/4} \\&= {\mathscr {O}}\left( \check{N}^{-1/2}\right) , \end{aligned} $$

and that

$$ \begin{aligned}&{\mathrm {E}\left[ \left| g'(X_T^{\overline{X}_{t_{n}},t_{n}})(\partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}- \partial _x X_T^{\overline{X}_{t_{n}},t_{n}})\right| \right] } \le C \sqrt{{\mathrm {E}\left[ \left| \partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}- \partial _x X_T^{\overline{X}_{t_{n}},t_{n}}\right| ^2\right] }}\\&= C \sqrt{{\mathrm {E}\left[ \left| \partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}- \partial _x X_T^{X_{t_{n+1}}^{\overline{X}_{t_{n}},t_n},t_{n+1}}\partial _x X_{t_{n+1}}^{\overline{X}_{t_{n}},t_n}\right| ^2\right] }}\\&\le C\Bigg (\sqrt{{\mathrm {E}\left[ \left| \partial _x X_T^{\overline{X}_{t_{n}},t_{n+1}}- \partial _x X_T^{X_{t_{n+1}}^{\overline{X}_{t_{n}},t_n},t_{n+1}}\right| \right] }}\\&\qquad + \sqrt{{\mathrm {E}\left[ \left| \partial _x X_T^{X_{t_{n+1}}^{\overline{X}_{t_{n}},t_n},t_{n+1}} \left( \int _{t_n}^{t_{n+1}} a_x(s,X_{s}^{\overline{X}_{t_{n}},t_n}) ds + \int _{t_n}^{t_{n+1}} b_x(s,X_{s}^{\overline{X}_{t_{n}},t_n})dW_s \right) \right| ^2\right] }}\Bigg )\\&\le C\sqrt{{\mathrm {E}\left[ \left| \partial _{xx} X_T^{(1-\hat{s}_n) \overline{X}_{t_{n}} + \hat{s}_n X_{t_{n+1}}^{\overline{X}_{t_{n}},t_n},t_{n+1}}\left( \int _{t_n}^{t_{n+1}} a_x(s,X_{s}^{\overline{X}_{t_{n}},t_n}) ds + \int _{t_n}^{t_{n+1}} b_x(s,X_{s}^{\overline{X}_{t_{n}},t_n})dW_s \right) \right| ^2\right] }}\\&\qquad + {\mathscr {O}}\left( \check{N}^{-1/2}\right) \\&= {\mathscr {O}}\left( \check{N}^{-1/2}\right) . \end{aligned} $$

The last step is to replace the first variation of the exact path $\varphi _{x}\left( t_{n},\overline{X}_{t_{n}}\right)$ with the first variation of the numerical solution $\overline{\varphi }_{x, n } = g'(\overline{X}_T) \partial _x \overline{X}_T^{\overline{X}_{t_{n}},t_{n}}$. This is also possible without introducing additional leading order error terms by the same assumptions and similar bounding arguments as in the two preceding bounds as

$$\begin{aligned} \begin{aligned}&{\mathrm {E}\left[ \left| \overline{\varphi }_{x, n } -\varphi _{x}\left( t_{n},\overline{X}_{t_{n}}\right) \right| \right] } = {\mathrm {E}\left[ \left| g'(\overline{X}_T) \partial _x \overline{X}_T^{\overline{X}_{t_{n}},t_{n}} - g'(X_T^{\overline{X}_{t_{n}},t_{n}}) \partial _x X_T^{\overline{X}_{t_{n}},t_{n}} \right| \right] }\\&\le {\mathrm {E}\left[ |g'(\overline{X}_T)|\left| \partial _x \overline{X}_T^{\overline{X}_{t_{n}},t_{n}} - \partial _x X_T^{\overline{X}_{t_{n}},t_{n}} \right| \right] } + {\mathrm {E}\left[ \left| g'(\overline{X}_T) - g'(X_T^{\overline{X}_{t_{n}},t_{n}})\right| \left| \partial _x X_T^{\overline{X}_{t_{n}},t_{n}} \right| \right] }\\&= {\mathscr {O}}\left( \check{N}^{-1/2}\right) . \end{aligned} \end{aligned}$$

   $\Box$

1.2 Variations of the Flow Map

The proof of Theorem 2 relies on bounded moments of variations of order up to four of the flow map $\varphi$. Furthermore, the error density depends explicitly on the first variation. In this section, we we will verify that these variations are indeed well defined random variables with all required moments bounded. First, we present the proof of Lemma 1. Having proven Lemma 1, we proceed to present how essentially the same technique can be used in an iterative fashion to prove the existence, pathwise uniqueness and bounded moments of the higher order moments. The essentials of this procedure are presented in Lemma 3.

First, let us define the following set of coupled SDE

$$\begin{aligned} \begin{aligned} d{Y^{(1)}_{u}} =&a (u,{Y^{(1)}_{u}}) du + b(u,{Y^{(1)}_{u}}) dW_u,\\ d{Y^{(2)}_{u}} =&a_x (u,{Y^{(1)}_{u}}) {Y^{(2)}_{u}} du + b_x (u,{Y^{(1)}_{u}}) {Y^{(2)}_{u}} dW_u,\\ d{Y^{(3)}_{u}} =&\left( a_{xx} (u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^2 + a_x(u,{Y^{(1)}_{u}}) {Y^{(3)}_{u}} \right) du \\&\quad + \left( b_{xx} (u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^2 + b_x(u,{Y^{(1)}_{u}}) {Y^{(3)}_{u}} \right) dW_u,\\ d{Y^{(4)}_{u}} =&\left( a_{xxx} (u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^3 + 3a_{xx} (u,{Y^{(1)}_{u}}) {Y^{(2)}_{u}} {Y^{(3)}_{u}} + a_x(u,{Y^{(1)}_{u}}) {Y^{(4)}_{u}} \right) du \\&\quad + \left( b_{xxx} (u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^3 + 3b_{xx} (u,{Y^{(1)}_{u}}) {Y^{(2)}_{u}} {Y^{(3)}_{u}} + b_x(u,{Y^{(1)}_{u}}) {Y^{(4)}_{u}} \right) dW_u,\\ d{Y^{(5)}_{u}} =&\left( a_{xxxx}( u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^4 + 6 a_{xxx} (u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^2 {Y^{(3)}_{u}} \right) du \\&\quad + \left( a_{xx} (u,{Y^{(1)}_{u}}) \left( 3 \left( {Y^{(3)}_{u}} \right) ^2 + 4 {Y^{(2)}_{u}}{Y^{(4)}_{u}} \right) + a_x(u,{Y^{(1)}_{u}}) {Y^{(5)}_{u}} \right) du \\&\quad + \left( b_{xxxx} (u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^4 + 6 b_{xxx} (u,{Y^{(1)}_{u}}) \left( {Y^{(2)}_{u}} \right) ^2 {Y^{(3)}_{u}} \right) dW_u \\&\quad + \left( b_{xx} (u,{Y^{(1)}_{u}}) \left( 3 \left( {Y^{(3)}_{u}} \right) ^2 + 4 {Y^{(2)}_{u}} {Y^{(4)}_{u}} \right) + b_x(u,{Y^{(1)}_{u}}) {Y^{(5)}_{u}} \right) dW_u, \end{aligned} \end{aligned}$$

(60)

defined for $u \in (t,T]$ with the initial condition $Y_t = (x,1,0,0,0)$. The first component of the vector coincides with Eq. (13), whereas the second one is the first variation of the path from Eq. (16). The last three components can be understood as the second, third and fourth variations of the path, respectively.

Making use of the solution of SDE (60), we also define the second, third and fourth variations as

$$\begin{aligned} \varphi _{xx}\left( t,x\right)&= g ' (X^{x,t}_T) \partial _{xx}X^{x,t}_T + g'' (X^{x,t}_T) (\partial _{x}X^{x,t}_T)^2, \nonumber \\ \varphi _{xxx}\left( t,x\right)&= g ' (X^{x,t}_T) \partial _{xxx }X^{x,t}_T + \dots + g''' (X^{x,t}_T) (\partial _{x}X^{x,t}_T)^3, \\ \varphi _{xxxx}\left( t,x\right)&= g ' (X^{x,t}_T) \partial _{xxxx}X^{x,t}_T + \dots + g'''' (X^{x,t}_T) (\partial _{x}X^{x,t}_T)^4. \nonumber \end{aligned}$$

(61)

In the sequel, we prove that the solution to Eq. (60) when understood in the integral sense that extends (13) is a well defined random variable with bounded moments. Given sufficient differentiability of the payoff g, this results in the boundedness of the higher order variations as required in Theorem 2.

Proof of Lemma 1. By writing $(Y_s^{(1)}, Y^{(2)}_s ):= (X^{x,t}_s, \partial _x X^{x,t}_s)$, (13) and (16) together form an SDE:

$$\begin{aligned} \begin{aligned} dY_s^{(1)}&= a(s, {Y^{(1)}_{s}} ) ds + b(s, {Y^{(1)}_{s}} ) dW_s \\ dY_s^{(2)}&= a_x(s, {Y^{(1)}_{s}}) {Y^{(2)}_{s}} ds + b_x(s, {Y^{(1)}_{s}}){Y^{(2)}_{s}} dW_s \\ \end{aligned} \end{aligned}$$

(62)

for $s \in (t, T]$ and with initial condition $Y_t = (x, 1)$. As before, $a_x$ stands for the partial derivative of the drift function with respect to its spatial argument. We note that (62) has such a structure that dynamics of $Y_s^{(2)}$ depends on $Y_s^{(1)}$, that, in turn, is independent of $Y_s^{(2)}$. By the Lipschitz continuity of $a(s, {Y^{(1)}_{s}} )$ and the linear growth bound of the drift and diffusion coefficients $a(s, {Y^{(1)}_{s}} )$ and $b(s, {Y^{(1)}_{s}} )$, respectively, there exists a pathwise unique solution of ${Y^{(1)}_{s}}$ that satisfies

$$ {\mathrm {E}\left[ \sup _{s \in [t, T]}|{Y^{(1)}_{s}}|^{2p}\right] } < \infty , \quad \forall p \in {\mathbb {N}}, $$

(cf. [22, Theorems 4.5.3 and 4.5.4 and Exercise 4.5.5]). As a solution of an Itô SDE, $X^{x,t}_T$ is measurable with respect to $\mathscr {F}_T$ it generates.

Note that Theorem [20, Theorem 5.2.5] establishes that the solutions of (62) are pathwise unique. Kloeden and Platen [22, Theorems 4.5.3 and 4.5.4] note that the existence and uniqueness theorems for SDEs they present can be modified in order to account for looser regularity conditions, and the proof below is a case in point. Our approach below follows closely presentation of Kloeden and Platen, in order to prove the existence and moment bounds for ${Y^{(2)}_{s}}$.

Let us define ${Y^{(2)}_{u,n}}$, $n \in \mathbb N$ by

$$\begin{aligned} {Y^{(2)}_{u,n+1}} = \int _t^u a_x(s,{Y^{(2)}_{s}}) {Y^{(2)}_{s,n}} ds + \int _t^u b_x(s,{Y^{(2)}_{s}}) {Y^{(2)}_{s,n}} dW_s, \end{aligned}$$

with ${Y^{(2)}_{u,1}} = 1$, for all $u \in [t,T]$. We then have, using Young’s inequality, that

$$\begin{aligned} {\mathrm {E}\left[ \left| {Y^{(2)}_{u,n+1}}\right| ^2\right] }&\le 2 {\mathrm {E}\left[ \left| \int _t^u a_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}} ds \right| ^2\right] } + 2 {\mathrm {E}\left[ \left| \int _t^u b_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}} dW_s \right| ^2\right] } \\&\le 2 (u-t) {\mathrm {E}\left[ \int _t^u \left| a_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}} \right| ^2 ds \right] } + 2 {\mathrm {E}\left[ \int _t^u \left| b_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}} \right| ^2 ds\right] } . \end{aligned}$$

Boundedness of the partial derivatives of the drift and diffusion terms in (62) gives

$$\begin{aligned} {\mathrm {E}\left[ \left| {Y^{(2)}_{u,n+1}}\right| ^2\right] }&\le C (u-t+1) {\mathrm {E}\left[ \int _t^u \left( 1+ \left| {Y^{(2)}_{s,n}}\right| ^2 \right) ds \right] }. \end{aligned}$$

By induction, we consequently obtain that

$$\begin{aligned} \sup _{t\le u \le T} {\mathrm {E}\left[ \left| {Y^{(2)}_{u,n}}\right| ^2\right] } < \infty , \qquad \forall n \in {\mathbb {N}}. \end{aligned}$$

Now, set ${\varDelta {Y^{(2)}_{u,n}}} = {Y^{(2)}_{u,n+1}} -{Y^{(2)}_{u,n}}$. Then

$$\begin{aligned} {\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{u,n}}}\right| ^2\right] }&\le 2 {\mathrm {E}\left[ \left| \int _t^u a_x(s,{Y^{(1)}_{s}}) {\varDelta {Y^{(2)}_{s,n-1}}} ds \right| ^2 \right] } + 2 {\mathrm {E}\left[ \left| \int _t^u b_x(s,{Y^{(1)}_{s}}) {\varDelta {Y^{(2)}_{s,n-1}}} dW_s \right| ^2 \right] } \\&\le 2(u-t) \int _t^u {\mathrm {E}\left[ \left| a_x(s,{Y^{(1)}_{s}}) {\varDelta {Y^{(2)}_{s,n-1}}} \right| ^2\right] } ds + 2\int _t^u {\mathrm {E}\left[ \left| b_x(s,{Y^{(1)}_{s}}) {\varDelta {Y^{(2)}_{s,n-1}}} \right| ^2\right] } ds \\&\le C_1 \int _t^u {\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{s,n-1}}}\right| ^2\right] } ds. \end{aligned}$$

Thus, by Grönwall’s inequality,

$$\begin{aligned} {\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{u,n}}}\right| ^2\right] } \le \frac{C_1^{n-1}}{(n-1)!} \int _t^u (u-s)^{n-1} {\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{s,1}}} \right| ^2\right] } ds . \end{aligned}$$

Next, let us show that ${\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{s,1}}} \right| ^2\right] }$ is bounded. First,

$$\begin{aligned} \begin{aligned} {\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{u,1}}} \right| ^2\right] }&= {\mathrm {E}\left[ \left| \int _t^u a_{x} (s,{Y^{(1)}_{s}}){Y^{(2)}_{s,2}} ds + \int _t^u b_{x}(s,{Y^{(1)}_{s}} ){Y^{(3)}_{u,2}} dW_s \right| ^2\right] }\\&\le C(u-t + 1) \sup _{s \in [t, u]} {\mathrm {E}\left[ \left| {Y^{(2)}_{s,2}}\right| ^2\right] }. \end{aligned} \end{aligned}$$

Consequently, there exists a $C \in {\mathbb {R}}$ such that

$$\begin{aligned} {\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{u,n}}} \right| ^2\right] } \le \frac{C^{n} (u-t)^n}{n!}, \qquad \sup _{u \in [t, T]} {\mathrm {E}\left[ \left| {\varDelta {Y^{(2)}_{u,n}}}\right| ^2\right] } \le \frac{C^{n} (T-t)^n}{n!}. \end{aligned}$$

Define

$$\begin{aligned} Z_n = \sup _{t \le u \le T} \left| {\varDelta {Y^{(2)}_{u,n}}} \right| , \end{aligned}$$

and note that

$$\begin{aligned} Z_n&\le \int _t^T \left| a_{x}(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n+1}} - a_{x}(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}} \right| ds \\&\quad + \sup _{t\le u \le T} \left| \int _t^u b_{x}(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n+1}} - b_{x}(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}} dW_s\right| . \\ \end{aligned}$$

Using Doob’s and Schwartz’s inequalities, as well as the boundedness of $a_x$ and $b_x$,

$$\begin{aligned} {\mathrm {E}\left[ \left| Z_n\right| ^2\right] }&\le 2 (T-t) \int _t^T {\mathrm {E}\left[ \left| a_{x}(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n+1}} - a_{x}(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}}\right| ^2 \right] }ds \\&\quad + 8 \int _t^T {\mathrm {E}\left[ \left| b_{x}(s,{Y^{(1)}_{s}}){Y^{(2)}_{s,n+1}} - b_{x}(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s,n}}\right| ^2 \right] } ds \\&\le \frac{ C^{n} (T-t)^{n}}{n!}, \end{aligned}$$

for some $C \in {\mathbb {R}}$. Using the Markov inequality, we get

$$\begin{aligned} \sum _{n=1}^\infty \mathrm {P}\left( Z_n > n^{-2} \right) \le \sum _{n=1}^{\infty } \frac{n^4 C^{n} (T-t)^{n}}{n!}. \end{aligned}$$

The right-hand side of the equation above converges by the ratio test, whereas the Borel–Cantelli Lemma guarantees the (almost sure) existence of $K^*\in {\mathbb {N}}$, such that $Z_k < k^2, \forall k>K^*$. We conclude that ${Y^{(2)}_{u,n}}$ converges uniformly in $L^2(\mathrm {P})$ to the limit ${Y^{(2)}_{u}} = \sum _{n=1}^{\infty } {\varDelta {Y^{(2)}_{u,n}}}$ and that since $\{{Y^{(2)}_{u,n}}\}_n$ is a sequence of continuous and $\mathscr {F}_u$-adapted processes, ${Y^{(2)}_{u}}$ is also continuous and $\mathscr {F}_u$-adapted. Furthermore, as $n\rightarrow \infty$,

$$\begin{aligned} \left| \int _t^u a_{x}(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s,n}} ds - \int _t^u a_{x}(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s}} ds \right| \le C \int _t^u \left| {Y^{(3)}_{s,n}} -{Y^{(3)}_{s}}\right| ds \rightarrow 0, \quad \text {a.s.}, \end{aligned}$$

and, similarly,

$$ \left| \int _t^u b_{x}(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s,n}} dW_s - \int _t^u b_{x}(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s}} dW_s \right| \rightarrow 0, \quad \text {a.s.} $$

This implies that $({Y^{(1)}_{u}}, {Y^{(2)}_{u}})$ is a solution to the SDE (62).

Having established that ${Y^{(2)}_{u}}$ solves the relevant SDE and that it has a finite second moment, we may follow the principles laid out in [22, Theorem 4.5.4] and show that all even moments of

$$\begin{aligned} {Y^{(2)}_{u}} = + \int _t^u a_x (t,{Y^{(1)}_{s}}) {Y^{(2)}_{s}} ds + \int _t^u b_x (t,{Y^{(1)}_{s}}) {Y^{(2)}_{s}} dW_s \end{aligned}$$

are finite. By Itô’s Lemma, we get that for any even integer l,

$$\begin{aligned} \left| {Y^{(3)}_{u}}\right| ^{l}&= \int _t^u \left| {Y^{(2)}_{s}}\right| ^{l-2} {Y^{(2)}_{s}} a_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s}} ds \\&\quad + \int _t^u \frac{l (l-1)}{2} \left| {Y^{(2)}_{s}}\right| ^{l-2} \left( b_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s}} \right) ^2 ds \\&\quad + \int _t^u \left| {Y^{(2)}_{s}}\right| ^{l-2} {Y^{(2)}_{s}} \left( b_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s}} \right) dW_s. \end{aligned}$$

Taking expectations, the Itô integral vanishes,

$$\begin{aligned}&{\mathrm {E}\left[ \left| {Y^{(2)}_{s}}\right| ^{l}\right] } = {\mathrm {E}\left[ \int _t^u \left| {Y^{(2)}_{s}}\right| ^{l-2} {Y^{(2)}_{s}} \left( a_x(s,{Y^{(1)}_{s}}){Y^{(2)}_{s}} \right) ds \right] } \\&\qquad \qquad \qquad \qquad \qquad + {\mathrm {E}\left[ \int _t^u \frac{l (l-1)\left| {Y^{(2)}_{s}}\right| ^{l-2} }{2} \left( b_x(s,{Y^{(1)}_{s}}) {Y^{(2)}_{s}} \right) ^2 ds \right] }. \end{aligned}$$

Using Young’s inequality and exploiting the boundedness of $a_x$, we have that

$$\begin{aligned} {\mathrm {E}\left[ \left| {Y^{(2)}_{u}} \right| ^l \right] }&\le C \int _t^u {\mathrm {E}\left[ |Y_{2,u}|^{l} \right] } ds \\&\quad + {\mathrm {E}\left[ \int _t^u \frac{l(l-1) \left| {Y^{(2)}_{s}}\right| ^{l-2} }{2} \left( b_x \left( s,{Y^{(1)}_{s}} \right) {Y^{(2)}_{s}} \right) ^2 ds \right] }. \end{aligned}$$

By the same treatment for the latter integral, using that $b_x$ is bounded,

$$\begin{aligned} {\mathrm {E}\left[ \left| {Y^{(2)}_{u}}\right| ^l\right] }&\le C \int _t^u {\mathrm {E}\left[ \left| {Y^{(2)}_{u}}\right| ^{l} \right] } ds. \end{aligned}$$

Thus, by Grönwall’s inequality, ${\mathrm {E}\left[ \left| {Y^{(2)}_{u}}\right| ^{l}\right] } < \infty$.   $\Box$

Lemma 3

Assume that (R.1), (R.2), and (R.3) in Theorem 2 hold and that for any fixed $t \in [0,T]$ and x is $\mathscr {F}_t$-measurable such that ${\mathrm {E}\left[ \left| x\right| ^{2p}\right] } < \infty$ for all $p \in {\mathbb {N}}$. Then, Eq. (60) has pathwise unique solutions with finite moments. That is,

$$\begin{aligned} \max _{i \in \{1,2,\ldots ,5\}} \left( \sup _{u \in [t,T]}{\mathrm {E}\left[ \left| {Y^{(i)}_{u}}\right| ^{2p}\right] } \right) < \infty , \qquad \forall p \in \mathbb N. \end{aligned}$$

Furthermore, the higher variations as defined by Eq. (61) satisfy are $\mathscr {F}_T$-measurable and for all $p \in {\mathbb {N}}$,

$$\begin{aligned} \mathrm {max} \left\{ {\mathrm {E}\left[ \left| \varphi _{x}\left( t,x\right) \right| ^{2p}\right] }, {\mathrm {E}\left[ \left| \varphi _{xx}\left( t,x\right) \right| ^{2p}\right] },{\mathrm {E}\left[ \left| \varphi _{xxx}\left( t,x\right) \right| ^{2p}\right] },{\mathrm {E}\left[ \left| \varphi _{xxxx}\left( t,x\right) \right| ^{2p}\right] } \right\} < \infty . \end{aligned}$$

Proof

We note that (60) shares with (62) the triangular dependence structure. That is, the truncated SDE for $\{{Y^{(j)}_{u}}\}_{j=1}^{d_1}$ for $d_1 < 5$ has drift and diffusion functions $\hat{a}: [0,T] \times \mathbb R^{d_1} \rightarrow \mathbb R^{d_1}$ and $\hat{b}: [0,T] \times \mathbb R^{d_1} \rightarrow \mathbb R^{d_1\times d_2}$ that do not depend on ${Y^{(j)}_{u}}$ for $j \ge d_1$.

This enables verifying existence of solutions for the SDE in stages: first for $({Y^{(1)}_{}}, {Y^{(2)}_{}})$, thereafter for $({Y^{(1)}_{}}, {Y^{(2)}_{}}, {Y^{(3)}_{}})$, and so forth, proceeding iteratively to add the next component ${Y^{(d_1+1)}_{}}$ of the SDE. We shall also exploit this structure for proving the result of bounded moments for each component. The starting point for our proof is Lemma 1, which guarantees existence, uniqueness and the needed moment bounds for the first two components ${Y^{(1)}_{}}$, and ${Y^{(2)}_{}}$. As one proceeds to ${Y^{(i)}_{}}$, $i>2$, the relevant terms in (64) feature derivatives of a and b of increasingly high order. The boundedness of these derivatives is guaranteed by assumption (R.1).

Defining a successive set of approximations ${Y^{(3)}_{u,n}}$, $n \in \mathbb ~N$ by

$$\begin{aligned}&{Y^{(3)}_{u,n+1}} = \int _t^u a_{xx} (s,{Y^{(1)}_{s}}) \left( {Y^{(2)}_{s}} \right) ^2 + a_x(s,{Y^{(2)}_{s}}) {Y^{(3)}_{s,n}} ds \\&\qquad \qquad \qquad \qquad \qquad \qquad + \int _t^u b_{xx} (s,{Y^{(1)}_{s}}) \left( {Y^{(2)}_{s}} \right) ^2 + b_x(s,{Y^{(2)}_{s}}) {Y^{(3)}_{s,n}} dW_s, \end{aligned}$$

with the initial approximation defined by ${Y^{(3)}_{u,1}} = 0$, for all $u \in [t,T]$. Let us denote by

$$\begin{aligned} Q=\int _t^u a_{xx} (s,{Y^{(1)}_{s}}) \left( {Y^{(1)}_{s}} \right) ^2 ds + \int _t^u b_{xx} (s,{Y^{(1)}_{s}}) \left( {Y^{(2)}_{s}} \right) ^2 dW_s \end{aligned}$$

(63)

the terms that do not depend on the, highest order variation ${Y^{(3)}_{u,n}}$. We then have, using Young’s inequality, that

$$\begin{aligned} {\mathrm {E}\left[ \left| {Y^{(3)}_{u,n+1}}\right| ^2\right] }&\le 3 {\mathrm {E}\left[ \left| Q\right| ^2\right] } + 3 {\mathrm {E}\left[ \left| \int _t^u a_x(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s,n}} ds \right| ^2\right] } + 3 {\mathrm {E}\left[ \left| \int _t^u b_x(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s,n}} dW_s \right| ^2\right] } \\&\le 3 {\mathrm {E}\left[ \left| Q\right| ^2\right] } + 3 (u-t) {\mathrm {E}\left[ \int _t^u \left| a_x(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s,n}} \right| ^2 ds \right] } + 3 {\mathrm {E}\left[ \int _t^u \left| b_x(s,{Y^{(1)}_{s}}) {Y^{(3)}_{s,n}} \right| ^2 ds\right] } . \end{aligned}$$

The term Q is bounded by Lemma 1 and the remaining terms can be bounded by the same methods as in the proof of 1. Using the same essential tools: Young’s and Doob’s inequalities, Grönwall’s lemma, Markov inequality and Borel–Cantelli Lemma, we can establish the existence of a limit to which ${Y^{(3)}_{u,n}}$ converges. This limit is the solution of of ${Y^{(3)}_{u}}$, and has bounded even moments through arguments that are straightforward generalisations of those already presented in the proof of Lemma 1.

Exploiting the moment bounds of ${Y^{(3)}_{u}}$ and the boundedness of derivatives of g, we can establish the measurability of the second order variation $\varphi _{x}\left( t,x\right)$. Repeating the same arguments in an iterative fashion, we can establish the same properties for ${Y^{(4)}_{u}}$ and ${Y^{(5)}_{u}}$ as well as $\varphi _{xx}\left( t,x\right)$, $\varphi _{xxx}\left( t,x\right)$, $\varphi _{xxxx}\left( t,x\right)$.   $\Box$

1.3 Error Expansion for the MSE in Multiple Dimensions

In this section, we extend the 1D MSE error expansion presented in Theorem 2 to the multi-dimensional setting.

Consider the SDE

$$\begin{aligned} \begin{aligned} dX_t&= a\left( t, X_t \right) dt + b\left( t,X_t \right) dW_t, \qquad t \in (0,T]\\ X_0&= x_0, \end{aligned} \end{aligned}$$

(64)

where $X: [0,T] \rightarrow \mathbb {\mathbb {R}}^{d_1}$, $W : [0,T] \rightarrow {\mathbb {R}}^{d_2}$, $a: [0,T] \times {\mathbb {R}}^{d_1} \rightarrow {\mathbb {R}}^{d_1}$ and $b: [0,T] \times {\mathbb {R}}^{d_1} \rightarrow {\mathbb {R}}^{d_1 \times d_2}$. Let further $x_i$ denote the ith component of $x \in \mathbb R^{d_1}$, $a^{(i)}$, the ith component of a drift coefficient and $b^{(i,j)}$ and $b^\mathrm {T}$ denote the (i, j)th element and the transpose of the diffusion matrix b, respectively. (To avoid confusion, this derivation does not make use of any MLMC notation, particularly not the multilevel superscript ${\cdot }^{\{\ell \}}$.)

Using the Einstein summation convention to sum over repeated indices, but not over the time index n, the 1D local error terms in Eq. (49) generalize into

where all the above integrand functions in all equations implicitly depend on the state argument $X_s^{\overline{X}_{t_n},t_n}$. In flow notation, $a^{(i)}_{t}$ is shorthand for $a^{(i)}_{t}(s,X_s^{\overline{X}_{t_n},t_n})$.

Under sufficient regularity, a tedious calculation similar to the proof of Theorem 2 verifies that, for a given smooth payoff, $g:{\mathbb {R}}^{d_1} \rightarrow {\mathbb {R}}$,

$$\begin{aligned} {\mathrm {E}\left[ \left( g \left( X_T \right) -g \left( \overline{X}_{T} \right) \right) ^2\right] } \le {\mathrm {E}\left[ \sum _{n=0}^{N-1} \overline{\rho }_n \varDelta t_{n}^2 + { o}\left( \varDelta t_n^2\right) \right] }, \end{aligned}$$

where

$$\begin{aligned} \overline{\rho }_n := \frac{1}{2} \overline{\varphi }_{x_i, n } \left( (bb^\mathrm {T})^{(k, \ell )} (b_{x_k} b_{x_\ell }^\mathrm {T}) \right) ^{(i,j)} (t_n,\overline{X}_{t_n}) \overline{\varphi }_{x_j, n }. \end{aligned}$$

(65)

In the multi-dimensional setting, the ith component of first variation of the flow map, $\varphi _x = (\varphi _{x_1}, \varphi _{x_2}, \ldots , \varphi _{x_{d_1}})$, is given by

$$\begin{aligned} \varphi _{x_i} \left( t, y \right) = g_{x_j}(X^{y,t}_T) \partial _{x_i} \left( X^{y,t}_T \right) ^{(j)}. \end{aligned}$$

The first variation is defined as the second component to the solution of the SDE,

$$\begin{aligned} \begin{aligned} d {Y^{(1,i)}_{s}}&= a^{(i)} \left( s, {Y^{(1)}_{s}} \right) ds + b^{(i,j)} \left( s, {Y^{(1)}_{s}} \right) d W^{(j)}_s \\ d {Y^{(2,i,j)}_{s}}&= a^{(i)}_{x_k} \left( s, {Y^{(1)}_{s}} \right) {Y^{(2,k,j)}_{s}} ds + b^{(i,\ell )}_{x_k} \left( s, {Y^{(1)}_{s}} \right) {Y^{(2,k,j)}_{s}} dW^{(\ell )}_s, \end{aligned} \end{aligned}$$

where $s \in (t,T]$ and the initial conditions are given by ${Y^{(1)}_{t}} = x \in {\mathbb {R}}^{d_1}$, ${Y^{(2)}_{t}} = I_{d_1}$, with $I_{d_1}$ denoting the $d_1\times d_1$ identity matrix. Moreover, the extension of the numerical method for solving the first variation of the 1D flow map (23) reads

$$\begin{aligned} \overline{\varphi }_{x_i,n}&= c_{x_i}^{(j)} (t_n,\overline{X}_{t_{n}}) \overline{\varphi }_{x_j,n+1} , \quad n=N-1,N-2,\ldots 0. \\ \overline{\varphi }_{x_i,N}&= g_{x_i}(\overline{X}_T), \nonumber \end{aligned}$$

(66)

with the jth component of $c: [0,T] \times {\mathbb {R}}^{d_1} \rightarrow {\mathbb {R}}^{d_1}$ defined by

$$\begin{aligned} c^{(j)} \left( t_n, \overline{X}_{t_n} \right) = \overline{X}_{t_n}^{(j)} + a^{(j)}(t_n,\overline{X}_{t_n}) \varDelta t_n + b^{(j,k)}(t_n, \overline{X}_{t_n}) \varDelta W^{(k)}_n. \end{aligned}$$

Let U and V denote subsets of Euclidean spaces and let us introduce the multi-index $\nu = (\nu _1, \nu _2, \ldots , \nu _d)$ to represent spatial partial derivatives of order $|\nu | := \sum _{j=1}^d \nu _j$ on the following short form $\partial _{x_\nu } := \prod _{j=1}^d \partial _{x_j}^\nu$. We further introduce the following function spaces.

$$ \begin{aligned}&C(U; V) := \{f:U \rightarrow V \, |\, f \text { is continuous} \},\\&C_b(U;V) := \{f:U \rightarrow V \, |\, f \text { is continuous and bounded} \},\\&C_b^k(U;V) := \Big \{f:U \rightarrow V \, |\, f \in C(U;V) \text { and } \frac{d^j}{dx^j} f \in C_b(U;V) \\&\quad \text {for all integers } 1 \le j \le k \Big \},\\&C_b^{k_1,k_2}([0,T] \times U; V):= \Big \{f:[0,T] \times U \rightarrow V \, |\, f \in C([0,T] \times U; V), \text { and } \\&\quad \partial _t^{j}\partial _\nu f \in C_b([0,T]\times U; V) \text { for all integers } j \le k_1 \text { and } 1 \le j+\left| \nu \right| \le k_2 \Big \}. \end{aligned} $$

Theorem 3

(MSE leading order error expansion in the multi-dimensional setting) Assume that drift and diffusion coefficients and input data of the SDE (64) fulfill

• (R.1) $a \in C_b^{2,4}([0,T]\times {\mathbb {R}}^{d_1}; {\mathbb {R}}^{d_1})$ and $b \in C_b^{2,4}([0,T]\times {\mathbb {R}}^{d_1}; {\mathbb {R}}^{d_1 \times d_2})$,

• (R.2) there exists a constant $C>0$ such that

  $$\begin{aligned} |a(t,x)|^2 + |b(t,x)|^2&\le C(1+|x|^2),&\forall x \in {\mathbb {R}}^{d_1} \text { and } \forall t \in [0,T],\\ \end{aligned}$$

• (R.3) $g \in C^4_b({\mathbb {R}}^{d_1})$,

• (R.4) for the initial data, $X_0$ is $\mathscr {F}_0$-measurable and ${\mathrm {E}\left[ |X_0|^p\right] } < \infty$ for all $p\ge 1$.

Assume further the mesh points $0=t_0<t_1< \cdots <t_N = T$

• (M.1) are stopping times such that $t_n$ is $\mathscr {F}_{t_{n-1}}$-measurable for $n=1,2,\ldots , N$,

• (M.2) there exists $\check{N} \in {\mathbb {N}}$, and a $c_1>0$ such that $c_1\check{N} \le \mathop {\mathrm {inf}}\nolimits _{\omega \in \varOmega } N (\omega )$ and $\mathop {\mathrm {sup}}\nolimits _{\omega \in \varOmega } N (\omega ) \le \check{N}$ holds for each realization. Furthermore, there exists a $c_2>0$ such that $\mathop {\mathrm {sup}}\nolimits _{\omega \in \varOmega } \max _{n\in \{0,1,\ldots , N-1\}} \varDelta t_n (\omega ) < c_2 \check{N}^{-1}$,

• (M.3) and there exists a $c_3>0$ such that for all $p {\in } [1,8]$ and $n {\in } \{0,1,\ldots , \check{N}-1\}$,

  $$ {\mathrm {E}\left[ \varDelta t_n^{2p}\right] } \le c_3 \left( {\mathrm {E}\left[ \varDelta t_n^2 \right] } \right) ^{p}. $$

Then, as $\check{N}$ increases,

$$\begin{aligned}&{\mathrm {E}\left[ \left( g(X_T) -g \left( \overline{X}_{T} \right) \right) ^2\right] } \\&\qquad \qquad = {\mathrm {E}\left[ \sum _{n=0}^{N-1} \frac{\left( \varphi _{x_i} \left( (bb^\mathrm {T})^{(k, \ell )} (b_{x_k} b_{x_\ell }^\mathrm {T}) \right) ^{(i,j)} \varphi _{x_j} \right) (t_n,\overline{X}_{t_n})}{2} \varDelta t_n^2 + o(\varDelta t_n^2)\right] }, \end{aligned}$$

where we have dropped the arguments of the first variation as well as the diffusion matrices for clarity.

Replacing the first variation $\varphi _{x_i}\left( t_n,\overline{X}_n\right)$ by the numerical approximation $\overline{\varphi }_{x_i, n }$, as defined in (66) and using the error density notation $\overline{\rho }$ from (65), we obtain the following to leading order all-terms-computable error expansion:

$$\begin{aligned} {\mathrm {E}\left[ \left( g(X_T) -g \left( \overline{X}_{T} \right) \right) ^2\right] } = {\mathrm {E}\left[ \sum _{n=0}^{N-1} \overline{\rho }_n \varDelta t_n^2 + o(\varDelta t_n^2)\right] }. \end{aligned}$$

(67)

A Uniform Time Step MLMC Algorithm

The uniform time step MLMC algorithm for MSE approximations of SDE was proposed in [8]. Below, we present the version of that method that we use in the numerical tests in this work for reaching the approximation goal (2).