2.1. The SDE and its approximation scheme

We introduce

\begin{align*}X_t = x + \int_0^t b(s,X_s) ds + \int_0^t \sigma(s,X_s) dB_s, \quad \quad 0 \le t \le T,\end{align*}

and its discretized counterpart

(4)\begin{align} X^n_{t_k} = x + h\sum_{j=1}^k b(t_j,X^n_{t_{j-1}}) + \sqrt{h} \sum_{j=1}^k \sigma(t_j,X^n_{t_{j-1}}) {\varepsilon}_j, \quad \quad t_j\coloneqq j\tfrac{T}{n}, \,\, j=0,\ldots,n,\end{align}

where $(\varepsilon_i)_{i=1,2, \dots}$ is a sequence of i.i.d. Rademacher random variables. Letting

(5)\begin{align} \mathcal{G}_k \coloneqq \sigma(\varepsilon_i: 1 \leq i \leq k) \quad \text{ with } \quad \mathcal{G}_0 \coloneqq \{\emptyset, \Omega\},\end{align}

it follows that the associated discrete-time random walk $(B^n_{t_k})_{k=0}^n$ is $(\mathcal{G}_k)_{k=0}^n$-adapted. Recall (2) and $h= \tfrac{T}{n}.$ If we extend the sequence $(X^n_{t_k})_{k\ge0}$ to a process in continuous time by defining $X^n_t \coloneqq X^n_{t_k} $ for $t \in [t_k, t_{k+1}),$ it is the solution of the forward SDE (3).

We formulate our first assumptions. Assumption 2.1(ii) will not be used explicitly for our estimates, but it is required for Theorem 4.1 below.

Notice that (6) implies

(8)\begin{eqnarray} |g(x)| \le K(1+|x|^{p_0+1}) \eqqcolon \Psi(x), \quad x \in {\mathbb{R}},\end{eqnarray}

for some $K>0.$ From the continuity of f we conclude that

\begin{equation*} K_f\coloneqq \sup_{0\le t\le T} |f(t,0,0,0)| < \infty.\end{equation*}

Notation:

1. • $\|\cdot\|_p\coloneqq \|\cdot\|_{L_p({\mathbb{P}})}$ for $p\ge 1$. For $p=2$ we write simply $\|\cdot\|$.

2. • If a is a function, C(a) represents a generic constant which depends on a and possibly also on its derivatives.

3. • ${\mathbb{E}}_{0,x}\coloneqq {\mathbb{E}}(\cdot|X_0=x)$.

4. • Let $\phi$ be a $C^{0,1}([0,T]\times {\mathbb{R}})$ function. Then $\phi_x$ denotes $\partial_x \phi$, the partial derivative of $\phi$ with respect to x.

2.2. The FBSDE and its approximation scheme

Recall the FBSDE (1) and its approximation (3). The backward equation in (3) can equivalently be written in the form

(9)\begin{align}Y^n_{t_k} &= g(X^n_T) + h\sum_{m=k}^{n-1} f(t_{m+1},X^n_{t_m},Y^n_{t_m},Z^n_{t_m}) - \sqrt{h}\sum_{m=k}^{n-1}Z^n_{t_m} {\varepsilon}_{m+1}, \quad 0 \leq k \leq n,\end{align}

if one puts $X^n_r\coloneqq X^n_{t_m}$, $Y^n_r\coloneqq Y^n_{t_m}$, and $Z_r^n \coloneqq {Z}_{t_m}^n$ for $r \in [t_m, t_{m+1})$.

Remark 2.1. Equations (3) and (9) do not contain any martingale orthogonal to the random walk $B^n$, since we are in a special case where the orthogonal martingale is zero (see [Reference Briand, Delyon and Mémin7, p. 3] or [Reference Privault34, Proposition 1.7.5]). Indeed, for the symmetric simple random walk $B^n$ the predictable representation property holds; i.e. for any $\mathcal{G}_n$-measurable (see (5)) random variable $\xi= F(\varepsilon_1, \dots, \varepsilon_n)$ there exists a representation

\begin{equation*}F(\varepsilon_1, \dots, \varepsilon_n) = c + \sum_{m=1}^n h_m \varepsilon_m,\end{equation*}

where $c \in {\mathbb{R}}$ and $h_m$ is $\mathcal{G}_{m-1}$-measurable for $m=1,\ldots,n$. To see this, consider

\begin{align*}F(\varepsilon_1, \ldots, \varepsilon_n) &={\mathbb{E}} [F(\varepsilon_1, \ldots, \varepsilon_n)] + \sum_{m=1}^n \Big( {\mathbb{E}}[ F(\varepsilon_1, \ldots, \varepsilon_n) |\mathcal{G}_m]- {\mathbb{E}}[F(\varepsilon_1, \ldots, \varepsilon_n) | \mathcal{G}_{m-1}] \Big).\end{align*}

Put $c= {\mathbb{E}} [F(\varepsilon_1, \ldots, \varepsilon_n)].$ Our aim is to determine a $\mathcal{G}_{m-1}$-measurable $h_m$ such that

\begin{equation*}{\mathbb{E}}[ F(\varepsilon_1, \ldots, \varepsilon_n) |\mathcal{G}_m]- {\mathbb{E}}[F(\varepsilon_1, \ldots, \varepsilon_n) | \mathcal{G}_{m-1}] = h_m \varepsilon_m.\end{equation*}

We define

\begin{equation*} F_m(\varepsilon_1, \ldots, \varepsilon_m )\coloneqq {\mathbb{E}}[ F(\varepsilon_1, \ldots, \varepsilon_n) |\mathcal{G}_m] . \end{equation*}

By the tower property it holds that

\begin{align*}&F_m(\varepsilon_1, \ldots, \varepsilon_m) - F_{m-1}(\varepsilon_1, \ldots, \varepsilon_{m-1}) \\[2pt] &\quad= F_m(\varepsilon_1,\ldots, \varepsilon_m) - {\mathbb{E}}[F_m(\varepsilon_1, \ldots, \varepsilon_m)| \mathcal{G}_{m-1}] \\[2pt] &\quad= F_m(\varepsilon_1,\ldots, \varepsilon_m) -\frac{F_m(\varepsilon_1,\ldots, \varepsilon_{m-1},1) +F_m(\varepsilon_1,\ldots, \varepsilon_{m-1},-1) }{2} \\[2pt] &\quad=\frac{F_m(\varepsilon_1,\ldots, \varepsilon_{m-1},1) -F_m(\varepsilon_1,\ldots, \varepsilon_{m-1},-1) }{2} \varepsilon_m;\end{align*}

hence

\begin{equation*}h_m = \frac{F_m(\varepsilon_1,\ldots, \varepsilon_{m-1},1) -F_m(\varepsilon_1,\ldots, \varepsilon_{m-1},-1) }{2}.\end{equation*}

One can derive an equation for $Z^n = (Z^n_{t_k})_{k=0}^{n-1}$ if one multiplies (9) by ${\varepsilon}_{k+1}$ and takes the conditional expectation with respect to ${\mathcal{G}}_k$, so that

(10)\begin{eqnarray} Z^n_{t_k} &=& \frac{{\mathbb{E}}^{{\mathcal{G}}}_{k}\big(g(X^n_T) {\varepsilon}_{k+1}\big)}{\sqrt{h}} +{\mathbb{E}}^{{\mathcal{G}}}_{k}\Bigg( \sqrt{ h}\sum_{m=k+1}^{n-1}f( t_{m+1},X^n_{t_m}, Y^n_{t_m} , Z^n_{t_m}) {\varepsilon}_{k+1}\Bigg ), \, 0\le k \le n-1, \end{eqnarray}

where ${\mathbb{E}}^{{\mathcal{G}}}_k\coloneqq {\mathbb{E}}(\cdot|{\mathcal{G}}_k)$.

2.2.1. Representation for Z

We will use the following representation for Z, due to Ma and Zhang (see [Reference Ma and Zhang30, Theorem 4.2]):

(11)\begin{align} Z_t &= {\mathbb{E}}_{t} \bigg(g(X_T) N^t_T + \int_t^T f(s,X_s,Y_s,Z_s) N^t_s ds\bigg) \sigma(t,X_t), \quad 0 \leq t \leq T,\end{align}

where ${\mathbb{E}}_t \coloneqq {\mathbb{E}}(\cdot|{\mathcal{F}}_t),$ and for all $s \in (t,T]$, we have (cf. Lemma 4.1)

(12)\begin{eqnarray} N^t_s=\frac{1}{s-t}\int_t^s\frac{\nabla X_r}{\sigma(r,X_r)\nabla X_t}dB_r, \end{eqnarray}

where $\nabla X = (\nabla X_s)_{s \in [0,T]}$ is the variational process; i.e., it solves

(13)\begin{align} \nabla X_s = 1+\int_0^s b_x(r,X_r) \nabla X_r dr + \int_0^s \sigma_x(r,X_r) \nabla X_r dB_r, \end{align}

with $(X_s)_{s \in [0,T]}$ given in (1).

Remark 2.3. In the following we will assume that gʹʹ exists. In such a case we have the following representation for Z:

(14)\begin{align} Z_t &= {\mathbb{E}}_{t} \bigg(g'(X_T) \nabla X_T + \int_t^T f(s,X_s,Y_s,Z_s) N^t_s ds\bigg) \sigma(t,X_t), \quad 0 \leq t \leq T.\end{align}

2.2.2. Approximation for $Z^n$

In this section we state the discrete counterpart to (11), which, in the general case of a forward process X, does not coincide with $Z^n$ (given by (10)). In contrast to the continuous-time case, where the variational process and the Malliavin derivative are connected by $\tfrac{\nabla X_t}{\nabla X_s} = \tfrac{D_sX_t }{\sigma(s,X_s)}$ ($s \le t$), we cannot expect equality for the corresponding expressions if we use the discretized versions of the processes $(\nabla X_t)_t$ and $(D_s X_t)_{s\le t}$ introduced in (16). This counterpart $\hat{Z}^n$ to Z is a key tool in the proof of the convergence of $Z^n$ to Z. As we will see in the proof of Theorem 3.1, the study of $\|Z^n_{t_k}-Z_{t_k}\|$ goes through the study of $\|Z^n_{t_k}-\hat{Z}^n_{t_k}\|$ and $\|\hat{Z}^n_{t_k}-Z_{t_k}\|$.

Before defining the discretized versions of $(\nabla X_t)_t$ and $(D_s X_t)_{s \le t}$, we briefly introduce the discretized Malliavin derivative. We refer the reader to [Reference Bender and Parczewski4] for more information on this topic.

Definition 2.1. (Definition of $T_{_{m,+}}$, $T_{_{m,-}}$ and $\mathcal{D}^n_m$) For any function $F\,:\,\{-1,1 \}^n \to {\mathbb{R}}$, the mappings $T_{_{m,+}}$ and $T_{_{m,-}}$ are defined by

\begin{eqnarray*}T_{_{m, \pm}} F(\varepsilon_1, \dots, \varepsilon_n) \coloneqq F(\varepsilon_1, \dots, \varepsilon_{m-1}, \pm 1, \varepsilon_{m+1}, \dots, \varepsilon_n), \quad \quad 1 \leq m \leq n.\end{eqnarray*}

For any $\xi= F(\varepsilon_1, \dots, \varepsilon_n)$, the discretized Malliavin derivative is defined by

(15)\begin{eqnarray} \mathcal{D}^n_m \xi \coloneqq \frac{ {\mathbb{E}} [\xi {\varepsilon}_{m} |\sigma( ( {\varepsilon}_l)_{l \in \{1,\ldots,n\} \setminus \{m\}} ) ]}{\sqrt{h}} = \frac{T_{_{m,+}} \xi - T_{_{m,-}} \xi}{2\sqrt{h}}, \quad 1 \leq m \leq n. \end{eqnarray}

Definition 2.2. (Definition of $\phi_x^{(k,l)}$) Let $\phi$ be a $C^{0,1}([0,T]\times {\mathbb{R}})$ function. We define

\begin{align*}\phi_x^{(k,l)} \coloneqq \frac{\mathcal{D}^n_k \phi(t_l,X^n_{t_{l-1}})}{\mathcal{D}^n_k X^n_{t_{l-1}}} \coloneqq \int_{0}^1 \phi_x(t_l, \vartheta T_{_{k,+}} \, X^n_{t_{l-1}} + (1-\vartheta) T_{_{k,-}} \, X^n_{t_{l-1}}) d\vartheta.\end{align*}

If $\mathcal{D}^n_{k} X^n_{t_{\ell-1}}\neq 0$, the second ‘$\coloneqq $’ holds as an identity.

We are now able to define the discretized versions of $(\nabla X_t)_t$ and $(D_s X_t)_{s \le t}$.

Definition 2.3. (Discretized processes $(\nabla X^{n,t_k,x}_{t_m})_{m \in \{k,\dots,n\}}$ and $(\mathcal{D}^n_k X^{n}_{t_m})_{m \in \{k,\dots,n\}}$) For all m in $\{k,\dots,n\}$ we define

(16)\begin{align} \nabla X^{n,t_k,x}_{t_m} &= 1 + h\sum_{l=k+1}^m b_x(t_l,X^{n,t_k,x}_{t_{l-1}})\nabla X^{n,t_k,x}_{t_{l-1}} \notag \\[2pt] &\quad + \sqrt{h}\!\! \sum_{l=k+1}^m \sigma_x(t_l,X^{n,t_k,x}_{t_{l-1}})\nabla X^{n,t_k,x}_{t_{l-1}} {\varepsilon}_l, \,\, \quad 0\le k \le n, \notag \\[2pt] \mathcal{D}^n_k X^n_{t_m} & = \sigma(t_{k},X^n_{t_{k-1}}) + h\sum_{l=k+1}^m b_x^{(k,l)} \mathcal{D}^n_k X^n_{t_{l-1}} \notag \\[2pt] & \quad+ \sqrt{h} \sum_{l=k+1}^m \sigma_x^{(k,l)} (\mathcal{D}^n_k X^n_{t_{l-1}}) {\varepsilon}_l, \quad 0<k \le n. \end{align}

Remark 2.4.

1. (i) Although $\nabla X^{n,t_k,X^n_{t_k}}_{t_m}$ is not equal to

   \begin{equation*}\frac{\mathcal{D}^n_{k+1}X^n_{t_m} }{\sigma(t_{k+1},X^n_{t_k})},\end{equation*}
   we can show that the difference of these terms converges in $L_p$ (see Lemma 5.4).

2. (ii) With the notation introduced above, (10) can be rewritten as

   (17)\begin{eqnarray} Z^n_{t_k} &=& {\mathbb{E}}^{{\mathcal{G}}}_{k} \big ( \mathcal{D}^n_{k+1} g(X^n_T) \big) +{\mathbb{E}}^{{\mathcal{G}}}_{k}\Bigg(h\sum_{m=k+1}^{n-1} \mathcal{D}^n_{k+1} f( t_{m+1},X^n_{t_m}, Y^n_{t_m} , Z^n_{t_m}) \Bigg). \end{eqnarray}

In order to define the discrete counterpart to (11), we first define the discrete counterpart to $(N^t_s)_{s \in [t,T]}$ given in (12):

(18)\begin{eqnarray} N^{n,t_k}_{t_{\ell}}\coloneqq \sqrt{h} \sum_{m=k+1}^\ell \frac{\nabla X^{n,t_k,X^n_{t_k}}_{t_{m-1}}}{\sigma(t_m, X^n_{t_{m-1}})} \frac{{\varepsilon}_m}{t_\ell - t_k}, \quad k < \ell \leq n.\end{eqnarray}

Notice that there is some constant $\widehat \kappa_2>0$ depending on $b,\sigma,T,\delta$ such that

(19)\begin{eqnarray} \left ( {\mathbb{E}}^{{\mathcal{G}}}_{k} | N^{n,t_k}_{t_{\ell}} |^2\right )^{\frac{1}{2}} \le \frac{ \widehat \kappa_2}{(t_{\ell}-t_k)^{\frac{1}{2}}}, \quad 0\le k < \ell \le n. \end{eqnarray}

Definition 2.4. (Discrete counterpart to (14).) Let the process $\hat{Z}^n = (\hat{Z}^n_{t_k})_{k=0}^{n-1}$ be defined by

(20)\begin{align} \hat{Z}^n_{t_k} \coloneqq {\mathbb{E}}^{{\mathcal{G}}}_{k} \big( \mathcal{D}^n_{k+1} g(X^n_T) \big) + {\mathbb{E}}^{{\mathcal{G}}}_{k} \Bigg( h\sum_{m= k+1}^{n-1}f( t_{m+1}, X^n_{t_m}, Y^n_{t_m}, Z^n_{t_m}) N^{n,t_k}_{t_m}\Bigg) \sigma(t_{k+1},X^n_{t_k}).\end{align}

Remark 2.5. In (20) the approximate expression ${\mathbb{E}}^{{\mathcal{G}}}_{k} ({ g(X^n_T) N^{n,t_k}_{t_n} \sigma(t_{k+1},X^n_{t_k}) })$ also could have been used, but since we will assume that gʹʹ exists, we work with the correct term.

The study of the convergence ${\mathbb{E}}^{{\mathcal{G}}}_{0,x} | Z^n_{t_k} - \hat{Z}^n_{t_k}|^2$ requires stronger assumptions on the coefficients b, $\sigma$, f, and g.

Assumption 2.3. Assumptions 2.1 and 2.2 hold. Additionally, we assume that all first and second derivatives with respect to the variables x, y, z of b(t, x), $\sigma(t,x)$, and f(t, x, y, z) exist and are bounded Lipschitz functions with respect to these variables, uniformly in time. Moreover, gʹʹ satisfies (6).

Proposition 2.1. If Assumption 2.3 holds, then

\begin{eqnarray*} {\mathbb{E}}^{{\mathcal{G}}}_{0,x} | Z^n_{t_k} - \hat{Z}^n_{t_k} |^2 \le C_{\protect\linktarget{Pro2.1}{\textcolor{blue}{2.1}}} \hat \Psi^2(x) h^{\alpha}, \end{eqnarray*}

where ${\mathbb{E}}^{{\mathcal{G}}}_{0,x}\coloneqq {\mathbb{E}}^{{\mathcal{G}}}(\cdot|X_0=x)$, the function $\hat \Psi$ is defined in (62) below, and $C_{\hyperref[Pro2.1]{\scriptstyle\textcolor{blue}{2.1}}}$ depends on b, $\sigma$, f, g, T, $p_0$, and $\delta$.

Proof. According to [Reference Briand, Delyon and Mémin7, Proposition 5.1] one has the representations

(21)\begin{eqnarray} Y^n_{t_m} = u^n(t_m, X^n_{t_m}) \quad \text{and} \quad Z^n_{t_m} =\mathcal{D}^n_{m+1} u^n(t_{m+1},X^{n}_{t_{m+1}}),\end{eqnarray}

where $u^n$ is the solution of the finite difference equation (44) with terminal condition $u^n(t_n,x)= g(x).$ Notice that by the definition of $\mathcal{D}^n_{m+1}$ in (15) the expression $\mathcal{D}^n_{m+1} u^n(t_{m+1},X^{n}_{t_{m+1}})$ depends in fact on $X^{n}_{t_m}.$ Hence we can put

\begin{align*} f(t_{m+1}, X^n_{t_m}, Y^n_{t_m}, Z^n_{t_m})&= f(t_{m+1}, X^n_{t_m}, u^n(t_m, X^n_{t_m}), \mathcal{D}^n_{m+1} u^n(t_{m+1},X^{n}_{t_{m+1}}))\\[2pt] &\eqqcolon F^n(t_{m+1}, X^n_{t_{m}}).\end{align*}

From (20) and (17) we conclude the following (we use ${\mathbb{E}}\coloneqq {\mathbb{E}}^{{\mathcal{G}}}_{0,x}$ for $\|\cdot \|$):

\begin{align*} & \| Z^n_{t_k} - \hat{Z}^n_{t_k} \| \\[2pt] &\quad =\Bigg \| {\mathbb{E}}^{{\mathcal{G}}}_k \Bigg ( h\sum_{m=k+1}^{n-1} \mathcal{D}^n_{k+1} f(t_{m+1},X^n_{t_m}, Y^n_{t_m}, Z^n_{t_m}) \Bigg) \\[2pt] & \quad \quad - {\mathbb{E}}^{{\mathcal{G}}}_k \Bigg(h\sum_{m=k+1}^{n-1}f( t_{m+1}, X^n_{t_m}, Y^n_{t_m}, Z^n_{t_m}) N^{n,t_k}_{t_m} \sigma(t_{k+1},X^n_{t_k}) \Bigg) \notag\Bigg \| \\[2pt] &\quad\le\sum_{m=k+1}^{n-1}\!\! \frac{h}{m - k} \! \sum_{\ell=k+1}^m \!\left\| {\mathbb{E}}^{{\mathcal{G}}}_k \! \left[\! \mathcal{D}^n_{k+1} F^n(t_{m+1}, X^n_{t_m})- \mathcal{D}^n_{ \ell } F^n(t_{m+1}, X^n_{t_m}) \frac{\sigma(t_{k+1},X^n_{t_k})\nabla X^{n,t_k,X^n_{t_k}}_{t_{\ell -1 }}}{\sigma(t_\ell, X^n_{t_{\ell - 1}})} \right] \right \|\!.\end{align*}

With the notation introduced in Definition 2.2 applied to $F^n$,

\begin{align*}& \left \| \mathcal{D}^n_{k+1}F^n(t_{m+1}, X^n_{t_m})-\mathcal{D}^n_\ell F^n(t_{m+1}, X^n_{t_m}) \frac{\sigma(t_{k+1},X^n_{t_k})\nabla X^{n,t_k,X^n_{t_k}}_{t_{\ell-1}}}{\sigma(t_\ell,X^n_{t_{\ell-1}})} \right \| \\[2pt] & \quad \le \| (\mathcal{D}^n_{k+1}X^n_{t_m} )( F^{n,(k+1,m+1)}_x -F^{n,(\ell,m+1)}_x) \| \\[2pt] &\qquad+ \,\left \| F^{n,(\ell,m+1)}_x \left( (\mathcal{D}^n_{k+1} X^n_{t_m}) -(\mathcal{D}^n_\ell X^n_{t_m}) \frac{\sigma(t_{k+1},X^n_{t_k})\nabla X^{n,t_k,X^n_{t_k}}_{t_{\ell-1}}}{\sigma(t_\ell,X^n_{t_{\ell-1}})} \right)\right \| \\[2pt] &\quad\eqqcolon A_1 +A_2.\end{align*}

For $A_1$ we use Definition 2.2 again and exploit the fact that

\begin{equation*}x \mapsto F^n_x(t_{m+1},x)\coloneqq \partial_xf\big(t_{m+1}, x, u^n(t_m, x), \mathcal{D}^n_{m+1} u^n\big(t_{m+1},X^{n,t_m,x}_{t_{m+1}}\big)\big)\end{equation*}

is locally $\alpha$-Hölder continuous according to (63). By Hölder’s inequality and Lemma 5.4 Parts (i) and (iii),

\begin{align*}A_1 & \leq \| {\mathcal{D}}^n_{k+1} X^n_{t_m}\|_4\int_{0}^1 \| F^n_x(t_{m+1}, \vartheta T_{_{k+1,+}} X^n_{t_m} + (1-\vartheta) T_{_{k+1,-}}X^n_{t_m}) \\[2pt] &\quad \quad \quad \quad\quad \quad- F^n_x(t_{m+1}, \vartheta T_{_{\ell,+}} X^n_{t_m} + (1-\vartheta) T_{_{\ell,-}} X^n_{t_m})\|_4 d\vartheta \\[2pt] & \leq C(b,\sigma,f,g,T,p_0) \hat \Psi(x) h^{\frac{\alpha}{2}}.\end{align*}

For the estimate of $A_2$ we notice that by our assumptions the $L_4$-norm of $F^{n,(\ell,m+1)}_x $ is bounded by $C \Psi^2(x),$ so that it suffices to estimate

(22)\begin{eqnarray} && \left \| (\mathcal{D}^n_{k+1} X^n_{t_m}) - (\mathcal{D}^n_\ell X^n_{t_m}) \frac{\sigma(t_{k+1},X^n_{t_k})\nabla X^{n,t_k,X^n_{t_k}}_{t_{\ell-1}}}{\sigma(t_\ell, X^n_{t_{\ell-1}})}\right \|_4\notag\\[2pt] && \quad\le \left\| (\mathcal{D}^n_{k+1} X^n_{t_m}) - \frac{\sigma( t_{k+1},X^n_{t_k}) \, \mathcal{D}^n_\ell X^n_{t_m}}{ \sigma(t_\ell ,X^n_{t_{\ell-1}}) } \frac{ \mathcal{D}^n_{k+1}X^n_{t_{\ell -1}}}{ \sigma( t_{k+1},X^n_{t_k})}\right\|_4 \notag\\[2pt] && \quad \quad + \left\| \frac{\sigma( t_{k+1},X^n_{t_k}) \, \mathcal{D}^n_\ell X^n_{t_m}}{ \sigma(t_\ell,X^n_{t_{\ell-1}} )}\Bigg (\nabla X^{n,t_k,X^n_{t_k}}_{t_{\ell-1}} - \frac{ \mathcal{D}^n_{k+1} X^n_{t_{\ell-1}}}{\sigma( t_{k+1},X^n_{t_k})} \Bigg )\right\|_4. \end{eqnarray}

The second expression on the right-hand side of (22) is bounded by $C(b,\sigma,T,\delta)h^{\frac{1}{2}}$ as a consequence of Lemma 5.4 Parts (ii) and (iii). To show that the first expression is also bounded by $C(b,\sigma,T,\delta)h^{\frac{1}{2}}$, we rewrite it using (16) and get

(23)\begin{align}& \left |\frac{ \mathcal{D}^n_{\ell} X^n_{t_m}}{ \sigma(t_\ell,X^n_{t_{\ell-1}})} \mathcal{D}^n_{k+1} X^n_{t_{\ell-1}} - \mathcal{D}^n_{k+1} X^n_{t_m}\right | \nonumber\\[2pt] & = \Bigg | \Bigg (1+ \sum_{l=\ell+1}^{m} \frac{ \mathcal{D}^n_{\ell} X^n_{t_{l-1}}}{ \sigma(t_\ell,X^n_{t_{\ell-1}}) } ( b_x^{(\ell,l)} h + \sigma_x^{(\ell,l)} \sqrt{h} \varepsilon_l) \Bigg ) \nonumber \\[2pt] & \quad \quad \times \Bigg (\sigma(t_{k+1},X^n_{t_k})+ \sum_{l=k+2}^{\ell-1}\mathcal{D}^n_{k+1} X^n_{t_{l-1}}( b^{(k+1,l)}_x h + \sigma^{(k+1,l)}_x \sqrt{h} \varepsilon_l) \Bigg ) \nonumber\\[2pt] & \quad \quad - \Bigg (\sigma(t_{k+1},X^n_{t_k})+ \Bigg (\sum_{l=k+2}^{\ell-1} + \sum_{l=\ell}^{m} \Bigg )\mathcal{D}^n_{k+1} X^n_{t_{l-1}}( b^{(k+1,l)}_x h + \sigma^{(k+1,l)}_x \sqrt{h} \varepsilon_l) \Bigg ) \Bigg | \nonumber\\[2pt] &\le \big| \mathcal{D}^n_{k+1} X^n_{t_{\ell-1}}( b^{(k+1,\ell)}_x h + \sigma^{(k+1,\ell)}_x \sqrt{h} \varepsilon_\ell)\big| \nonumber\\& \quad + \Bigg | \sum_{l=\ell+1}^{m} \bigg [\frac{ \mathcal{D}^n_{\ell} X^n_{t_{l-1}}}{ \sigma(t_\ell,X^n_{t_{\ell-1}}) }\mathcal{D}^n_{k+1} X^n_{t_{\ell-1}} - \mathcal{D}^n_{k+1} X^n_{t_{l-1}} \bigg] \Big( b^{(\ell,l)}_x h + \sigma^{(\ell,l)}_x \sqrt{h} \varepsilon_l\Big) \Bigg | \nonumber\\ & \quad + \! \Bigg | \sum_{l=\ell+1}^{m} \mathcal{D}^n_{k+1} X^n_{t_{l-1}} \bigg[ b^{(\ell,l)}_x h + \sigma^{(\ell,l)}_x \sqrt{h} \varepsilon_l - \Big(b^{(k+1,l)}_x h + \sigma^{(k+1,l)}_x\sqrt{h} \varepsilon_l\Big) \bigg] \Bigg |.\end{align}

We take the $L_4$-norm of (23) and apply the Burkholder–Davis–Gundy (BDG) inequality and Hölder’s inequality. The second term on the right-hand side of (23) will be used for Gronwall’s lemma, while the first and last terms can be bounded by $ C(b,\sigma,T)h^{\frac{1}{2}},$ using Lemma 5.4(iii). For the last term we also use the Lipschitz continuity of $b_x$ and $\sigma_x$ in space and Lemma 5.4(i).

3.1. Proof of Proposition 3.1: the approximation rates for the zero generator case

To shorten the notation, we use ${\mathbb{E}} \coloneqq {\mathbb{E}}_{0,x}.$ Let us first deal with the error of Y. If v belongs to $[t_k,t_{k+1})$ we have $Y^n_v= Y^n_{t_k}$. Then

\begin{align*} {\mathbb{E}} |Y_v - Y^n_v|^2 \le 2( {\mathbb{E}} |Y_v - Y_{t_k}|^2 + {\mathbb{E}} |Y_{t_k} - Y^n_{t_k}|^2).\end{align*}

Using Theorem 4.1 we bound $\|Y_v - Y_{t_k}\|$ by

\begin{equation*}C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}} \Psi(x) (v-t_k)^\frac{1}{2} = C(C_g, b, \sigma, T,p_0, \delta) \Psi(x) (v-t_k)^\frac{1}{2} \end{equation*}

(since $\alpha=1$ can be chosen when g is locally Lipschitz continuous). It remains to bound

\begin{eqnarray*}{\mathbb{E}} |Y_{t_k} -Y^n_{t_k}|^2&=&{\mathbb{E}}|{\mathbb{E}}_{t_k} g(X_T) -{\mathbb{E}}_{\tau_k} g(X^n_T) |^2={\mathbb{E}} | \tilde {\mathbb{E}} g(\tilde X^{t_k,X_{t_k}}_{t_n})-\tilde {\mathbb{E}} g(\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n})|^2.\end{eqnarray*}

By (6) and the Cauchy–Schwarz inequality (with $\Psi_1\coloneqq C_g(1+|\tilde X^{t_k,X_{t_k}}_{t_n}|^{p_0}+| \tilde {\mathcal{X}}^{\tau_k,{{\mathcal{X}}}_{\tau_k}}_{\tau_n}|^{p_0})$),

\begin{align*} | \tilde {\mathbb{E}} g(\tilde X^{t_k,X_{t_k}}_{t_n})-\tilde {\mathbb{E}} g(\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{ \tau_n})|^2 &\le (\tilde {\mathbb{E}} (\Psi_1 |\tilde X^{t_k,X_{t_k}}_{t_n}-\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\ \tau_n}| ))^2 \\[3pt] &\le \tilde {\mathbb{E}} (\Psi_1^2) \tilde {\mathbb{E}} |\tilde X^{t_k,X_{t_k}}_{t_n}-\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{ \tau_n}|^{2}. \end{align*}

Finally, we get by Lemma 5.2(v) that

\begin{eqnarray*}{\mathbb{E}} |Y_{t_k} -Y^n_{t_k}|^2&\le& \left( {\mathbb{E}}\tilde {\mathbb{E}} (\Psi_1^4) \right)^{\frac{1}{2}}\left({\mathbb{E}}\tilde {\mathbb{E}}|\tilde X^{t_k,X_{t_k}}_{t_n}-\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}|^{4}\right)^{\frac{1}{2}}\le C(C_g, b, \sigma, T,p_0) \Psi(x)^2 h^\frac{1}{2}. \end{eqnarray*}

Let us now deal with the error of Z. We use $ \|Z_v - Z^n_v \| \le \|Z_v - Z_{t_k}\| + \|Z_{t_k} - Z^n_{t_k}\|$ and the representation

\begin{equation*} Z_{t} = \sigma(t,X_{t}) \tilde {\mathbb{E}}( g'(\tilde X^{t,X_{t}}_{T}) \nabla \tilde X^{t,X_{t}}_{T}) \end{equation*}

(see Theorem 4.2), where

(28)\begin{align} \tilde X^{t,x}_s &= x+\int_t^s b(r, \tilde X^{t,x}_r) dr + \int_t^s \sigma(r,\tilde X^{t,x}_r) d\tilde B_{r-t}, \\[3pt] \nabla \tilde X^{t,x}_s &= 1+\int_t^s b_x(r, \tilde X^{t,x}_r) \nabla \tilde X^{t,x}_r dr + \int_t^s \sigma_x(r,\tilde X^{t,x}_r) \nabla \tilde X^{t,x}_r d\tilde B_{r-t}, \quad 0\le t\le s\le T. \notag \end{align}

For the first term we get by the assumption on g and Lemma 5.2 Parts (i) and (iii) that

\begin{eqnarray*} \|Z_v - Z_{t_k}\| &\,=\,& \| \sigma(v,X_v) \tilde {\mathbb{E}} (g'(\tilde X^{v,X_v}_{T}) \nabla \tilde X^{v,X_v}_{T}) - \sigma(t_k,X_{t_k}) \tilde {\mathbb{E}}( g'(\tilde X^{t_k,X_{t_k}}_{T}) \nabla \tilde X^{t_k,X_{t_k}}_{T}) \|\\[4pt] &\,\le\,& \| \sigma(v,X_v) - \sigma(t_k,X_{t_k}) \|_4 \| \tilde {\mathbb{E}} (g'(\tilde X^{v,X_v}_{T}) \nabla \tilde X^{v,X_v}_{T}) \|_4 \\[4pt] && +\, \|\sigma\|_\infty \| \tilde {\mathbb{E}} (g'(\tilde X^{v,X_v}_{T}) \nabla \tilde X^{v,X_v}_{T}) - \tilde {\mathbb{E}}( g'(\tilde X^{t_k,X_{t_k}}_{T}) \nabla \tilde X^{v,X_v}_{T}) \| \\[4pt] && +\, \|\sigma\|_\infty \| \tilde {\mathbb{E}}( g'(\tilde X^{t_k,X_{t_k}}_{T}) \nabla \tilde X^{v,X_v}_{T}) - \tilde {\mathbb{E}}( g'(\tilde X^{t_k,X_{t_k}}_{T}) \nabla \tilde X^{t_k,X_{t_k}}_{T}) \| \\[4pt] &\,\le\, & C( C_{g'},b,\sigma,T,p_0)\Psi(x) \Big [h^{\frac{1}{2}} + \|X_v -X_{t_k} \|_4+ \left ( {\mathbb{E}}\tilde {\mathbb{E}} |\tilde X^{v,X_v}_{T}-\tilde X^{t_k,X_{t_k}}_{T}|^{4\alpha} \right)^\frac{1}{4}\\[4pt] &&+\left ({\mathbb{E}}\tilde {\mathbb{E}} | \nabla \tilde X^{v,X_v}_{T} -\nabla \tilde X^{t_k,X_{t_k}}_{T}|^4 \right)^\frac{1}{4} \Big] \\[4pt] &\,\le\,& C( C_{g'},b,\sigma,T,p_0) \Psi(x) h^\frac{\alpha}{2}.\end{eqnarray*}

We compute the second term using $Z^n_{t_k} $ as given in (17). Hence, with the notation from Definition 2.2,

\begin{eqnarray*} \|Z_{t_k} - Z^n_{t_k}\|^2&\,=\,& {\mathbb{E}} \big | \sigma(t_k,X_{t_k}) \tilde {\mathbb{E}} g'(\tilde X^{t_k,X_{t_k}}_{t_n}) \nabla \tilde X^{t_k,X_{t_k}}_{t_n} - \tilde {\mathbb{E}} \mathcal{D}^n_{k+1} g(\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}) \big |^2 \\[4pt] &\,\le\, & \| \sigma\|_\infty ^2 \, {\mathbb{E}} \left | \tilde {\mathbb{E}} ( g'(\tilde X^{t_k,X_{t_k}}_{t_n}) \nabla \tilde X^{t_k,X_{t_k}}_{t_n} ) - \frac{ \tilde {\mathbb{E}} \mathcal{D}^n_{k+1} g(\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n})}{ \sigma(t_k,X_{t_k}) } \right |^2 \\[4pt] &\,=\,& \| \sigma\|_\infty ^2 \, {\mathbb{E}} \Bigg | \tilde {\mathbb{E}} (g'(\tilde X^{t_k,X_{t_k}}_{t_n}) \nabla \tilde X^{t_k,X_{t_k}}_{t_n} ) - \tilde {\mathbb{E}} \Bigg ( g_x^{(k+1,n+1)} \frac{ \mathcal{D}^n_{k+1} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}}{ \sigma(t_k,X_{t_k}) }\Bigg ) \Bigg |^2. \\[4pt] \end{eqnarray*}

We insert $\pm \tilde {\mathbb{E}} ( g_x^{(k+1,n+1)} \nabla \tilde X^{t_k,X_{t_k}}_{t_n})$ and get by the Cauchy–Schwarz inequality that

(29)\begin{eqnarray} && {} \Bigg | \tilde {\mathbb{E}} (g'(\tilde X^{t_k,X_{t_k}}_{t_n}) \nabla \tilde X^{t_k,X_{t_k}}_{t_n}) - \tilde {\mathbb{E}} \Bigg ( g_x^{(k+1,n+1)} \frac{ \mathcal{D}^n_{k+1} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}}{ \sigma(t_k,X_{t_k}) } \Bigg ) \Bigg |^2 \notag\\[4pt] &&\quad \le\, 2 \tilde {\mathbb{E}} | g'(\tilde X^{t_k,X_{t_k}}_{t_n})- g_x^{(k+1,n+1)}|^2 \tilde {\mathbb{E}} | \nabla \tilde X^{t_k,X_{t_k}}_{t_n} |^2 + 2\tilde {\mathbb{E}} | g_x^{(k+1,n+1)}|^2 \tilde {\mathbb{E}} \Bigg | \nabla \tilde X^{t_k,X_{t_k}}_{t_n} \!- \!\frac{ \mathcal{D}^n_{k+1} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}}{ \sigma(t_k,X_{t_k}) } \Bigg|^2.\nonumber\\ \end{eqnarray}

For the estimate of $ \tilde {\mathbb{E}} | \nabla \tilde X^{t_k,X_{t_k}}_{t_n} |^2$ we use Lemma 5.2. Since gʹ satisfies (6) we proceed with

\begin{eqnarray*}&& {} \tilde {\mathbb{E}} | g'(\tilde X^{t_k,X_{t_k}}_{t_n})- g_x^{(k+1,n+1)}|^2 \\[2pt] &&\quad\le \int_0^1 \tilde {\mathbb{E}} \Big | g'(\tilde X^{t_k,X_{t_k}}_{t_n})- g'(\vartheta T_{_{k+1,+}} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}+ (1-\vartheta) T_{_{k+1,-}}\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}) \Big |^2 d \vartheta \\[2pt] &&\quad\le \int_0^1 (\tilde {\mathbb{E}} \Psi_1^4)^{\frac{1}{2}} \Big [ \tilde {\mathbb{E}} \left |\tilde X^{t_k,X_{t_k}}_{t_n}- \vartheta T_{_{k+1,+}} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n} - (1-\vartheta) T_{_{k+1,-}}\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n} \right|^{4\alpha} \Big ]^{\frac{1}{2}} d \vartheta,\end{eqnarray*}

where $\Psi_1\coloneqq C_{g'}(1+|\tilde X^{t_k,X_{t_k}}_{t_n}|^{p_0}+|\vartheta T_{_{k+1,+}} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}+ (1-\vartheta) T_{_{k+1,-}}\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}|^{p_0}). $ For $\tilde {\mathbb{E}} \Psi_1^4$ and

\begin{eqnarray*}&& \tilde {\mathbb{E}} \left |\tilde X^{t_k,X_{t_k}}_{t_n}- (\vartheta T_{_{k+1,+}} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}+ (1-\vartheta) T_{_{k+1,-}}\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n} ) \right|^{ 4\alpha} \\[2pt] &&\quad\le\, 8 \left ( \vartheta^{2\alpha} \tilde {\mathbb{E}} \left |\tilde X^{t_k,X_{t_k}}_{t_n}- T_{_{k+1,+}} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n} \right|^{4\alpha} + (1-\vartheta)^{2\alpha} \tilde {\mathbb{E}} \left |\tilde X^{t_k,X_{t_k}}_{t_n} - T_{_{k+1,-}} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n} \right|^{4\alpha} \right ) \\[2pt] &&\quad\le\, C(b,\sigma,T) h^{2 \alpha} + C(b,\sigma,T)( |X_{t_k} - {\mathcal{X}}_{\tau_k}|^{4 \alpha}+ h^{\alpha}),\end{eqnarray*}

we use Lemma 5.4 and Lemma 5.2(v). For the last term in (29) we notice that

\begin{equation*}{\mathbb{E}} \tilde {\mathbb{E}} | g_x^{(k+1,n+1)}|^4 \le C(C_{g'}, b,\sigma,T,p_0) \Psi^4(x).\end{equation*}

By Lemma 5.2 we have ${\mathbb{E}}\tilde {\mathbb{E}} | \nabla \tilde X^{t_k,X_{t_k}}_{t_n}- \nabla \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}|^p \le C(b,\sigma,T,p) h^{\frac{p}{4}},$ and by Lemma 5.4,

\begin{eqnarray*}&& {\mathbb{E}}\tilde {\mathbb{E}} \left | \nabla \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}-\frac{\mathcal{D}^n_{k+1} \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_n}}{ \sigma(t_{k}, X_{t_k})}\right |^p \\[2pt] &&\quad\le\, C(p) {\mathbb{E}}\left | \nabla X^{n,t_k, X^n_{t_k}}_{t_n} - \dfrac{\mathcal{D}^n_{k+1}X^n_{t_n} }{\sigma(t_{k+1},X^n_{t_k})} \right |^p + C(p) {\mathbb{E}}\left | \dfrac{\mathcal{D}^n_{k+1}X^n_{t_n} }{\sigma(t_{k+1},X^n_{t_k})} - \dfrac{\mathcal{D}^n_{k+1}X^n_{t_n} }{\sigma(t_k, X_{t_k})} \right |^p \\[2pt] &&\quad\le\, C(b,\sigma,T,p, \delta) h^{\frac{p}{4}}.\end{eqnarray*}

Consequently, $ \|Z_{t_k} - Z^n_{t_k}\|^2 \le C(C_{g'}, b,\sigma,T,p_0,\delta) \Psi^2(x) h^\frac{\alpha}{2}.$

3.2. Proof of Theorem 3.1: the approximation rates for the general case

Let $u \,:\, [0,T)\times {\mathbb{R}} \to {\mathbb{R}}$ be the solution of the PDE (38) associated to (1). We use the representations $ Y_s = u(s, X_s)$ and $Z_s = \sigma(s, X_s)u_x(s, X_s)$ stated in Theorem 4.2 and define

(30)\begin{eqnarray} F(s,x) \coloneqq f(s, x, u(s, x), \sigma(s, x)u_x(s, x)).\end{eqnarray}

From (1) and (3) we conclude

\begin{align*} \|Y_{t_k}- Y^n_{t_k} \| &\le \| {\mathbb{E}}_{t_k} g(X_T) - {\mathbb{E}}_{\tau_k} g(X^n_T) \| \notag \\[3pt] & \quad + \left \| {\mathbb{E}}_{t_k}\int_{t_k}^T f(s,X_s,Y_s,Z_s)ds - h {\mathbb{E}}_{\tau_k}\sum_{m=k}^{n-1} f(t_{m+1},X^n_{t_m}, Y^n_{t_m},Z^n_{t_m}) \right \|, \end{align*}

where Proposition 3.1 provides the estimate for the terminal condition. We decompose the generator term as follows:

\begin{align*}& {\mathbb{E}}_{t_k} f(s,X_s,Y_s,Z_s)-{\mathbb{E}}_{\tau_k}f(t_{m+1}, X^n_{t_m},Y^n_{t_m},Z^n_{t_m})\\[2pt] &\quad=[{\mathbb{E}}_{t_k} f(s,X_s,Y_s,Z_s)- {\mathbb{E}}_{t_k}f(t_m,X_{t_m},Y_{t_m},Z_{t_m})] + [{\mathbb{E}}_{t_k}F(t_m,X_{t_m}) -{\mathbb{E}}_{\tau_k}F(t_m, X^n_{t_m})]\\[2pt] & \qquad+ [{\mathbb{E}}_{\tau_k}F(t_m,X^n_{t_m}) \!-\!{\mathbb{E}}_{\tau_k} F(t_m,X_{t_m})] \!+[{\mathbb{E}}_{\tau_k} f(t_m, X_{t_m},Y_{t_m},Z_{t_m}) \!-\! {\mathbb{E}}_{\tau_k}f(t_{m+1}, X^n_{t_m}, Y^n_{t_m},Z^n_{t_m})] \\[2pt] &\quad\eqqcolon d_1(s,m) +d_2(m)+ d_3(m)+ d_4(m).\end{align*}

We use

\begin{eqnarray*}&& {} \left \| {\mathbb{E}}_{t_k}\int_{t_k}^T f(s,X_s,Y_s,Z_s)ds - h {\mathbb{E}}_{\tau_k}\sum_{m=k}^{n-1} f(t_{m+1}, X^n_{t_m},Y^n_{t_m},Z^n_{t_m}) \right \| \notag\\[2pt] &&\quad \le \sum_{m=k}^{n-1} \left( \left \| \int_{t_m}^{t_{m+1}} d_1(s,m) ds\right \| + h \sum_{i=2}^4 \|d_i(m) \| \right)\end{eqnarray*}

and estimate the expressions on the right-hand side. For the function F defined in (30) we use Assumption 2.3 (which implies that (6) holds for $\alpha=1$) to derive by Theorem 4.2 and the mean value theorem that for $x_1, x_2 \in {\mathbb{R}}$ there exists $\xi \in [\min\{x_1,x_2\},\max\{x_1,x_2\}] $ such that

(31)\begin{eqnarray} |F(t,x_1) -F(t,x_2)| &\,=\,& |f(t, x_1, u(t, x_1), \sigma(t,x_1)u_x(t, x_1)) - f(t, x_2, u(t, x_2), \sigma(t,x_2)u_x(t, x_2))| \notag \\[3pt] & \,\le\, & C(L_f, \sigma) \left ( 1 + c^2_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}} \Psi( \xi ) + \frac{c^3_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}} \Psi(\xi)}{(T-t)^{\frac{1}{2}}} \right )|x_1-x_2| \notag\\[3pt]&\, \le\, & C(L_f, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, \sigma, T) (1+|x_1|^{p_0+1}+|x_2|^{p_0+1}) \frac{|x_1-x_2|}{(T-t)^{\frac{1}{2}}}.\end{eqnarray}

By (7), standard estimates on $(X_s),$ Theorem 4.1(i), and Proposition 4.1 for $p=2$, we immediately get

\begin{eqnarray*} \| d_1(s,m) \|&\,\le\,& C(L_f, C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}},C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}, b,\sigma,T) \Psi(x) \,h^{\frac{1}{2}} \\&\,=\,& C(b,\sigma, f,g, T,p_0,\delta) \Psi(x) \,h^{\frac{1}{2}}.\end{eqnarray*}

For the estimate of $d_2$ one exploits

\begin{equation*}{\mathbb{E}}_{t_k}F(t_m,X_{t_m}) -{\mathbb{E}}_{\tau_k}F(t_m,X^n_{t_m})= \tilde {\mathbb{E}} F(t_m, \tilde X^{t_k,X_{t_k}}_{t_m}) - \tilde {\mathbb{E}} F(t_m,\tilde X^{n, t_k, X^n_{t_k}} _{t_m}) \end{equation*}

and then uses (31) and Lemma 5.2(v). This gives

\begin{eqnarray*} \|d_2(m) \| &\le& C(L_f, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, b,\sigma,T, p_0) \Psi(x) \frac{1}{(T-t_m)^{\frac{1}{2}} } h^{\frac{1}{4}}.\end{eqnarray*}

For $d_3$ we start with Jensen’s inequality and then continue similarly as above to get

\begin{eqnarray*} \|d_3(m) \| \le \| F(t_m, X^n_{t_m}) - F(t_m,X_{t_m}) \|\le C(L_f, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, b,\sigma,T, p_0) \Psi(x) \frac{1}{(T-t_m)^{\frac{1}{2}}} h^\frac{1}{4},\end{eqnarray*}

and for the last term we get

\begin{eqnarray*} \|d_4(m) \| &\le & L_f ( h^{\frac{1}{2}} + \| X_{t_m} - X^n_{t_m}\| + \| Y_{t_m} - Y^n_{t_m}\|+ \| Z_{t_m} -Z^n_{t_m}\| ).\end{eqnarray*}

This implies

(32)\begin{eqnarray} \| Y_{t_k} - Y^n_{t_k}\| \le C \Psi(x) h^{\frac{1}{4}}+ h L_f \sum_{m=k}^{n-1}( \| Y_{t_m} - Y^n_{t_m}\| + \| Z_{t_m} -Z^n_{t_m}\|),\end{eqnarray}

where $C= C(L_f, C^y_{\hyperref[no f]{\scriptstyle\textcolor{blue}{3.1}}}, C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}}, C_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}},b,\sigma, T,p_0) = C(b,\sigma, f,g, T,p_0,\delta).$

For $\| Z_{t_k} -Z^n_{t_k}\|$ we use the representations (14) and (17), the approximation (20), and Proposition 2.1. Instead of $N^{n,t_k}_{t_n}$ we will use here the notation $N^{n,\tau_k}_{\tau_n}$ to indicate its measurability with respect to the filtration $({\mathcal{F}}_t)$. It holds that

(33)\begin{eqnarray} \|Z^n_{t_k} - Z_{t_k}\| &\le & \| Z^n_{t_k} - \hat Z^n_{t_k}\| + \|Z_{t_k}- \hat Z^n_{t_k}\| \notag\\[3pt] &\le & C_{\hyperref[discreteZand-wrongZdifference]{\scriptstyle\textcolor{blue}{2.1}}} \hat \Psi(x) h^{\frac{\alpha}{2}} + \| \sigma(t_k,X_{t_k}) \tilde {\mathbb{E}} g'(\tilde X^{t_k,X_{t_k}}_{t_n}) \nabla \tilde X^{t_k,X_{t_k}}_{t_n} - \tilde {\mathbb{E}} \mathcal{D}^n_{k+1} g(\tilde X^{n,t_k,X^n_{t_k}}_{t_n}) \| \notag\\[3pt] &&+ \Bigg \|{\mathbb{E}}_{t_{k}}\int_{t_{k+1}}^T f(s,X_s,Y_s,Z_s) N^{t_k}_s ds \, \sigma(t_k,X_{t_k}) \notag \\[3pt] && \quad \quad -{\mathbb{E}}_{\tau_k} h\sum_{m=k+1}^{n-1}f( t_{m+1}, X^n_{t_m}, Y^n_{t_m},Z^n_{t_m}) N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},X^n_{t_k}) \Bigg \| \notag \\&& + \bigg \|{\mathbb{E}}_{t_k}\int_{t_k}^{t_{k+1}} f(s,X_s,Y_s,Z_s) N^{t_k}_s ds \, \sigma(t_k,X_{t_k}) \bigg \| .\end{eqnarray}

For the terminal condition, Proposition 3.1 provides

(34)\begin{eqnarray} \Big\| \sigma(t_k,X_{t_k}) \tilde {\mathbb{E}} g'(\tilde X^{t_k,X_{t_k}}_{t_n}) \nabla \tilde X^{t_k,X_{t_k}}_{t_n} - \tilde {\mathbb{E}} \mathcal{D}^n_{k+1} g(\tilde X^{n,t_k,X^n_{t_k}}_{t_n}) \Big\| \le (C^z_{\hyperref[no f]{\scriptstyle\textcolor{blue}{3.1}}})^{\frac{1}{2}} \Psi(x) h^\frac{1}{4}.\end{eqnarray}

We continue with the generator terms and use F defined in (30) to decompose the difference

\begin{align*}& {\mathbb{E}}_{t_k}f(s,X_s,Y_s,Z_s) N^{t_k}_s \sigma(t_k,X_{t_k})-{\mathbb{E}}_{\tau_k}f( t_{m+1}, X^n_{t_m}, Y^n_{t_m},Z^n_{t_m}) N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},X^n_{t_k}) \\[3pt]&\quad = {\mathbb{E}}_{t_k}f(s,X_s,Y_s,Z_s) N^{t_k}_s \sigma(t_k,X_{t_k}) - {\mathbb{E}}_{t_k}f(t_m, X_{t_m},Y_{t_m},Z_{t_m}) N^{t_k}_{t_m}\sigma(t_k,X_{t_k}) \\[3pt]&\qquad + {\mathbb{E}}_{t_k}F(t_m,X_{t_m}) N^{t_k}_{t_m} \sigma(t_k,X_{t_k}) - {\mathbb{E}}_{\tau_k} F(t_m,X^n_{t_m}) N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},X^n_{t_k}) \\[3pt]&\qquad + {\mathbb{E}}_{\tau_k}\left [ [ F(t_m,X^n_{t_m}) - F(t_m,X_{t_m})] N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},X^n_{t_k}) \right ] \\[3pt]&\qquad + {\mathbb{E}}_{\tau_k} \left [ [f(t_m, X_{t_m},Y_{t_m},Z_{t_m})- f(t_{m+1}, X^n_{t_m}, Y^n_{t_m},Z^n_{t_m})]N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},X^n_{t_k}) \right ] \\[3pt]&\quad \eqqcolon {\tt t}_1(s,m)+{\tt t}_2(m)+ {\tt t}_3(m) + {\tt t}_4(m),\end{align*}

where $s \in [t_m, t_{m+1})$. For ${\tt t}_1$ we use that ${\mathbb{E}}_{t_k}f(t_m, X_{t_k},Y_{t_k},Z_{t_k}) (N^{t_k}_s -N^{t_k}_{t_m}) =0,$ so that

\begin{align*} \|{\tt t}_1(s,m)\| &\le \| {\mathbb{E}}_{t_k}f(s,X_s,Y_s,Z_s) N^{t_k}_s \sigma(t_k,X_{t_k}) - {\mathbb{E}}_{t_k}f(t_m, X_{t_m},Y_{t_m},Z_{t_m}) N^{t_k}_s\sigma(t_k,X_{t_k}) \| \\[3pt] &\quad + \| {\mathbb{E}}_{t_k}(f(t_m, X_{t_m},Y_{t_m},Z_{t_m}) - f(t_m, X_{t_k},Y_{t_k},Z_{t_k}) ) (N^{t_k}_s -N^{t_k}_{t_m})\sigma(t_k,X_{t_k}) \|.\end{align*}

As before, we rewrite the conditional expectations with the help of the independent copy $\tilde B.$ Then

\begin{align*}& {\mathbb{E}}_{t_k}f(s,X_s,Y_s,Z_s) N^{t_k}_s - {\mathbb{E}}_{t_k}f(t_m, X_{t_m},Y_{t_m},Z_{t_m}) N^{t_k}_s \\[3pt] &\quad = \tilde {\mathbb{E}} [(f(s, \tilde X^{t_k,X_{t_k}}_s, \tilde Y^{t_k,X_{t_k}}_s, \tilde Z^{t_k,X_{t_k}}_s) - f(t_m, \tilde X^{t_k,X_{t_k}}_{t_m}, \tilde Y^{t_k,X_{t_k}}_{t_m},\tilde Z^{t_k,X_{t_k}}_{t_m})) \tilde N^{t_k}_s]\end{align*}

and

\begin{align*}& {\mathbb{E}}_{t_k}(f(t_m, X_{t_m},Y_{t_m},Z_{t_m}) - f(t_m, X_{t_k},Y_{t_k},Z_{t_k}) ) (N^{t_k}_s -N^{t_k}_{t_m}) \\[3pt] &\quad = \tilde {\mathbb{E}} [(f(t_m, \tilde X^{t_k,X_{t_k}}_{t_m}, \tilde Y^{t_k,X_{t_k}}_{t_m},\tilde Z^{t_k,X_{t_k}}_{t_m}) - f(t_m, X_{t_k},Y_{t_k},Z_{t_k}) ) (\tilde N^{t_k}_s - \tilde N^{t_k}_{t_m})].\end{align*}

We apply the conditional Hölder inequality, and from the estimates (37) and

\begin{equation*}\tilde {\mathbb{E}} |\tilde N^{t_k}_s - \tilde N^{t_k}_{t_m}|^2 \le C(b,\sigma,T,\delta) \frac{h}{(s-t_k)^2}\end{equation*}

we get

\begin{eqnarray*} \|{\tt t}_1(s,m)\| &\le & \frac{\kappa_2 \| \sigma \|_{\infty}}{(s-t_k)^{\frac{1}{2}}} \| f(s, X_s, Y_s,Z_s)- f(t_m, X_{t_m},Y_{t_m},Z_{t_m})\| \\ && + C(b,\sigma,T,\delta) \frac{ h^{{\frac{1}{2}}}}{s-t_k}\ \| f(t_m, X_{t_m},Y_{t_m},Z_{t_m})- f(t_k, X_{t_k},Y_{t_k},Z_{t_k}) \| \\& \le & C(L_f, C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}},C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}},\kappa_2, b,\sigma,T,p_0, \delta) \Psi(x) \frac{h^{\frac{1}{2}} }{(s-t_k)^\frac{1}{2}},\end{eqnarray*}

since for $0\le t <s \le T$ we have by Theorem 4.1 and Proposition 4.1 that

(35)\begin{align} \| f(s, X_s, Y_s,Z_s)- f(t, X_t,Y_t, Z_t)\| &\le C(L_f,C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}},C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}, b,\sigma,T,p_0) \Psi(x)(s-t)^{\frac{1}{2}}. \end{align}

For the estimate of ${\tt t}_2$, Lemma 5.2, Lemma 5.3, (31), and (37) yield

\begin{align*} \|{\tt t}_2(m)\|&= \| \tilde {\mathbb{E}} F(t_m,\tilde X^{t_k,X_{t_k}}_{t_m}) \tilde N^{t_k}_{t_m} \sigma(t_k,X_{t_k}) - \tilde {\mathbb{E}} F(t_m, \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_m}) \tilde N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},{\mathcal{X}}_{\tau_k}) \| \\[3pt] &\le \frac{ C(\kappa_2, \sigma)}{(t_m - t_k)^{\frac{1}{2}}} {{\left( {\mathbb{E}} \tilde {\mathbb{E}} (F(t_m,\tilde X^{t_k,X_{t_k}}_{t_m}) - F(t_m, \tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_m}))^2 \right)}}^{{\frac{1}{2}}}\\[2pt] &\quad + ( {\mathbb{E}} \tilde {\mathbb{E}} |F(t_m,\tilde {\mathcal{X}}^{\tau_k,{\mathcal{X}}_{\tau_k}}_{\tau_m})-F(t_m,{\mathcal{X}}_{\tau_k})|^2 \tilde {\mathbb{E}} | \tilde N^{t_k}_{t_m} \sigma(t_k,X_{t_k}) - \tilde N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},{\mathcal{X}}_{\tau_k}) |^2 )^{\frac{1}{2}} \\[3pt] &\le C(L_f, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, \kappa_2,b,\sigma,T,p_0,\delta) \frac{\Psi(x)}{(T-t_m)^{\frac{1}{2}} } \frac{h^\frac{1}{4}}{ (t_m -t_k)^\frac{1}{2}}.\end{align*}

For ${\tt t}_3$ we use the conditional Hölder inequality, (31), (19), and Lemma 5.2:

\begin{align*} \|{\tt t}_3(m)\|&= \left\|{\mathbb{E}}_{\tau_k}\left [ [F(t_m,X^n_{t_m} ) - F(t_m,X_{t_m})] N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},{\mathcal{X}}_{\tau_k}) \right ] \right\| \\[2pt] &\le \frac{ C(\widehat\kappa_2, \sigma)}{(t_m - t_k)^{\frac{1}{2}}} \left\| F(t_m, X^n_{t_m} ) - F(t_m,X_{t_m}) \right\| \\ &\le C(L_f, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}},b,\sigma,T,p_0,\delta) \frac{\Psi(x)}{(T-t_m)^{\frac{1}{2}} } \frac{h^\frac{1}{4}}{ (t_m -t_k)^\frac{1}{2}}.\end{align*}

The term ${\tt t}_4$ can be estimated as follows:

\begin{align*} \|{\tt t}_4(m)\|&= \left\| {\mathbb{E}}_{\tau_k} \big [ [f(t_m, X_{t_m},Y_{t_m},Z_{t_m})- f(t_{m+1}, X^n_{t_m}, Y^n_{t_m},Z^n_{t_m})]N^{n,\tau_k}_{\tau_m} \sigma(t_{k+1},{\mathcal{X}}_{\tau_k}) \big] \right\| \\[3pt] &\le \frac{C(L_f, b,\sigma,T, \delta)}{ (t_m -t_k)^{\frac{1}{2}}} (h^{\frac{1}{2}} + \| X_{t_m} -X^n_{t_m} \| + \| Y_{t_m} -Y^n_{t_m} \| + \| Z_{t_m} - Z^n_{t_m} \|). \end{align*}

Finally, for the remaining term of the estimate of $\| Z_{t_k} -Z^n_{t_k}\|, $ we use (35) and (37) to get

\begin{align*} \left \|{\mathbb{E}}_{t_k} f(s,X_s,Y_s,Z_s) N^{t_k}_s \, \sigma(t_k,X_{t_k}) \right \| &= \|{\mathbb{E}}_{t_k}[( f(s,X_s,Y_s,Z_s) - f(s,X_{t_k},Y_{t_k},Z_{t_k})) N^{t_k}_s ]\, \sigma(t_k,X_{t_k}) \| \\ &\le C(L_f, C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}},C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}, \kappa_2, b,\sigma,T, p_0) \Psi(x). \end{align*}

Consequently, from (33), (34), and the estimates for the remaining term and for ${\tt t}_1,\ldots,{\tt t}_4$, it follows that

\begin{align*} \| Z_{t_k} -Z^n_{t_k}\| &\le C_{\hyperref[discreteZand-wrongZdifference]{\scriptstyle\textcolor{blue}{2.1}}} \hat \Psi (x) h^{\frac{\alpha}{2}} + (C^z_{\hyperref[no f]{\scriptstyle\textcolor{blue}{3.1}}})^{\frac{1}{2}} \Psi(x) h^\frac{1}{4} + C(L_f, C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}},C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}},b,\sigma,T, p_0,\kappa_2) \Psi(x) h\\[2pt] &\quad + C(L_f, C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}},C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}},\kappa_2,b,\sigma,T,p_0,\delta) \Psi(x) h^{\frac{1}{2}} \int_{t_k}^{T} \frac{ds }{ (s-t_k)^\frac{1}{2}}\\[2pt] &\quad + C(L_f, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, \kappa_2, b,\sigma,T,p_0,\delta) h \sum_{m=k+1}^{n-1} \frac{\Psi(x)}{(T-t_m)^{\frac{1}{2}} } \frac{h^\frac{1}{4}}{ (t_m -t_k)^\frac{1}{2}} \\[2pt] &\quad + C(L_f, b,\sigma,T, \delta )h \sum_{m=k+1}^{n-1} (\| Y_{t_m} -Y^n_{t_m} \| + \| Z_{t_m} - Z^n_{t_m} \|)\frac{1}{ (t_m -t_k)^{\frac{1}{2}}} \\[2pt] &\le C(C_{\hyperref[discreteZand-wrongZdifference]{\scriptstyle\textcolor{blue}{2.1}}},C^z_{\hyperref[no f]{\scriptstyle\textcolor{blue}{3.1}}}) \hat \Psi (x) h^{\frac{\alpha}{2}\wedge\frac{1}{4}} + C(L_f, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}},C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}},\kappa_2,b,\sigma,T,p_0,\delta) \Psi(x) h^\frac{1}{4} \\[2pt] &\quad + C(L_f, b,\sigma,T, \delta ) \sum_{m=k+1}^{n-1} (\| Y_{t_m} -Y^n_{t_m} \| + \| Z_{t_m} - Z^n_{t_m} \|)\frac{1}{(t_m -t_k)^{\frac{1}{2}}} h.\end{align*}

Then we use (32) and the above estimate to get

\begin{align*}& \| Y_{t_k} - Y^n_{t_k}\| + \| Z_{t_k} -Z^n_{t_k}\| \\[2pt] &\le C(C_{\hyperref[discreteZand-wrongZdifference]{\scriptstyle\textcolor{blue}{2.1}}},C^z_{\hyperref[no f]{\scriptstyle\textcolor{blue}{3.1}}})\hat \Psi (x) h^{\frac{\alpha}{2}\wedge\frac{1}{4}} + C(L_f,C^y_{\hyperref[no f]{\scriptstyle\textcolor{blue}{3.1}}},C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}}, C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}, c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, \kappa_2, b, \sigma, T,p_0, \delta) \Psi(x) h^\frac{1}{4} \\[3pt] &\quad + C(L_f,b,\sigma,T, \delta) \sum_{m=k+1}^{n-1} (\| Y_{t_m} -Y^n_{t_m} \| + \| Z_{t_m} - Z^n_{t_m} \|)\frac{1}{ (t_m -t_k)^{\frac{1}{2}}} h.\end{align*}

Consequently, summarizing the dependencies, there is a $ C=C(b,\sigma,f,g,T,p_0,\delta)$ such that

\begin{eqnarray*} \| Y_{t_k} - Y^n_{t_k}\| + \| Z_{t_k} -Z^n_{t_k}\| &\le& { C } \hat \Psi(x) h^{\frac{\alpha}{2}\wedge\frac{1}{4}}.\end{eqnarray*}

By Theorem 4.1 (note that by Assumption 2.3 on g we have $\alpha=1$) it follows that

\begin{equation*}\| Y_v- Y^n_v\| \le \| Y_v- Y_{t_k}\|+ \| Y_{t_k}- Y^n_{t_k}\| \le C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}}\Psi(x) h^{\frac{1}{2}} + \hat \Psi(x) h^{\frac{\alpha}{2}\wedge\frac{1}{4}}, \end{equation*}

while Proposition 4.1 implies that

\begin{equation*}\| Z_v- Z_{t_k} \| \le C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}\Psi(x) h^{\frac{1}{2}}, \end{equation*}

and hence we have

\begin{align*} {\mathbb{E}}_{0,x} |Y_v - Y^n_v|^2 + {\mathbb{E}}_{0,x} |Z_v - Z^n_v|^2 \le C_{\hyperref[the-result]{\scriptstyle\textcolor{blue}{3.1}}} \hat{\Psi}(x)^2h^{ \frac{1}{2} \wedge \alpha}\end{align*}

with $C_{\hyperref[the-result]{\scriptstyle\textcolor{blue}{3.1}}} = C_{\hyperref[the-result]{\scriptstyle\textcolor{blue}{3.1}}}(b,\sigma,f,g,T,p_0,\delta).$

4.1. Malliavin weights

We use the SDE from (1) started in (t, x),

(36)\begin{eqnarray} X^{t,x}_s = x + \int_t^s b(r,X^{t,x}_r)dr + \int_t^s \sigma(r, X^{t,x}_r)dB_r, \quad 0\le t \le s \le T,\end{eqnarray}

and recall the Malliavin weight and its properties from [Reference Geiss, Geiss and Gobet20, Subsection 1.1 and Remark 3].

Lemma 4.1. Let $H\,:\, {\mathbb{R}} \to {\mathbb{R}}$ be a polynomially bounded Borel function. If Assumption 2.1 holds and $X^{t,x}$ is given by (36), then setting

\begin{equation*} G(t,x) \coloneqq {\mathbb{E}} H(X_T^{t,x}) \end{equation*}

implies that $G \in C^{1,2}([0,T)\times {\mathbb{R}} ).$ Specifically, it holds for $0 \le t \le r < T$ that

\begin{equation*} \partial_x G(r, X_r^{t,x}) = {\mathbb{E}} [ H(X_T^{t,x}) N_T^{r,(t,x)} |{\mathcal{F}}^t_r ],\end{equation*}

where $({\mathcal{F}}^t_r)_{r\in [t,T]}$ is the augmented natural filtration of $(B^{t,0}_r)_{r \in [t,T]},$

\begin{equation*}N_T^{r,(t,x)}= \frac{1}{T-r} \int_r^T\frac{\nabla X^{t,x}_s }{\sigma(s,X_s^{t,x}) \nabla X^{t,x}_r } dB_s,\end{equation*}

and $\nabla X^{t,x}_s$ is given in (13). Moreover, for $ q \in (0, \infty)$ there exists a $\kappa_q>0$ such that

(37)\begin{eqnarray} ({\mathbb{E}}[| N_T^{r,(t,x)}|^q |{\mathcal{F}}^t_r ])^\frac{1}{q} \le \frac{\kappa_q}{(T-r)^\frac{1}{2}} \quad \text{ and} \quad {\mathbb{E}}[ N_T^{r,(t,x)} |{\mathcal{F}}^t_r ]=0 \quad \text{ almost surely,} \end{eqnarray}

and we have

\begin{equation*}\| \partial_x G(r, X_r^{t,x}) \|_{L_p({\mathbb{P}})} \le \kappa_{q} \frac{\|H(X_T^{t,x}) - {\mathbb{E}}[ H(X_T^{t,x})|{\mathcal{F}}^t_r ]\|_p }{\sqrt{T-r}} \end{equation*}

for $1<q,p < \infty$ with $\frac{1}{p} + \frac{1}{q} =1.$

4.2. Regularity of solutions to BSDEs

The following result originates from [Reference Geiss, Geiss and Gobet20, Theorem 1], where path-dependent cases were also included. We formulate it only for our Markovian setting but use ${\mathbb{P}}_{t,x}$ since we are interested in an estimate for all $(t,x) \in [0,T) \times {\mathbb{R}}.$ A sketch of a proof of this formulation can be found in [Reference Geiss, Labart and Luoto22].

The constants $ C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}}$ and $C^z_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}}$ depend on $ (L_f, K_f, C_g,c^{1,2}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, \kappa_q, b,\sigma, T, p_0,p)$, and $\Psi(x)$ is defined in (8).

4.3. Properties of the associated PDE

The theorem below collects properties of the solution to the PDE associated to the FBSDE (1). For a proof see [Reference Zhang43, Theorem 3.2], [Reference Zhang41], and [Reference Geiss, Labart and Luoto22, Theorem 5.4].

Theorem 4.2. Consider the FBSDE (1) and let Assumptions 2.1 and 2.2 hold. Then for the solution u of the associated PDE

(38)\begin{eqnarray} \left\{ \begin{array}{l} u_t(t,x) + \tfrac{\sigma^2(t,x)}{2} u_{xx}(t,x) + b(t,x) u_x(t,x) + f(t,x,u(t,x), \sigma(t,x) u_x(t,x)) =0,\\[3pt] t\in [0,T), \quad x\in {\mathbb{R}}, \\[3pt] u(T,x)=g(x) , \quad x \in {\mathbb{R}}, \end{array}\right . \end{eqnarray}

we have the following:

1. (i) $Y_t=u(t,X_t)$ almost surely, where $u(t,x)={\mathbb{E}}_{t,x} \!\left (g(X_T)+\int_t^T\! f(r,X_r,Y_r,Z_r)dr \right )$, and $|u(t,x)|\le c^1_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}} \Psi(x)$ for $\Psi$ given in (8), where $c^{1}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}$ depends on $L_f$, $K_f$, $C_g$, T, and $p_0$, as well as on the bounds and Lipschitz constants of b and $\sigma$.

2. (ii) (a) $\partial_x u$ exists and is continuous in $[0,T)\times {\mathbb{R}}$.

3. (b) $Z^{t,x}_s= u_x(s,X_s^{t,x})\sigma(s,X_s^{t,x})$ almost surely.

4. (c)

   \begin{equation*}|u_x(t,x)|\le \frac{ c^2_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}} \Psi(x)}{(T-t)^{\frac{1-\alpha}{2}}},\end{equation*}
   where $c^{2}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}$ depends on $L_f$, $K_f$, $C_g$, T, $p_0$, and $\kappa_2= \kappa_2(b,\sigma,T,\delta)$, as well as on the bounds and Lipschitz constants of b and $\sigma$, and hence $c^{2}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}} = c^{2}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}(L_f, K_f, C_g,b,\sigma, T,p_0,\delta).$

5. (iii) (a) $\partial^2_x u$ exists and is continuous in $[0,T)\times {\mathbb{R}}$.

6. (b)

   \begin{equation*}|\partial^2_x u(t,x)|\le \frac{ c^3_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}} \Psi(x)}{(T-t)^{1-\frac{\alpha}{2}}},\end{equation*}
   where $c^{3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}$ depends on $L_f$, $C_g$, T, $p_0$, $\kappa_2= \kappa_2(b,\sigma,T,\delta)$, $C^y_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}}$, and $C^z_{\hyperref[The4.1]{\scriptstyle\textcolor{blue}{4.1}}}$, as well as on the bounds and Lipschitz constants of b and $\sigma$, and hence $c^{3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}} = c^{3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}(L_f, K_f, C_g,b,\sigma, T,p_0,\delta). $

Using Assumption 2.3, we are now in a position to improve the bound on $\| Z_s - Z_t\|_{L_p({\mathbb{P}}_{t,x})}$ given in Theorem 4.1.

Proposition 4.1. If Assumption 2.3 holds, then there exists a constant $C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}} >0$ such that for $0\le t < s \le T$ and $x\in {\mathbb{R}},$

\begin{equation*}\| Z_s - Z_t\|_{L_p({\mathbb{P}}_{t,x})}\leq C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}} \Psi(x)(s-t)^{\frac{1}{2}},\end{equation*}

where $C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}$ depends on $c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}$, b, $\sigma$, f, g, T, $p_0$, and p, and hence $C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}= C_{\hyperref[Pro4.1]{\scriptstyle\textcolor{blue}{4.1}}}(b,\sigma, f,g,T,p_0,p,\delta).$

Proof. From $Z^{t,x}_s= u_x(s,X_s^{t,x})\sigma(s,X_s^{t,x})$ and

\begin{equation*}\nabla Y_s^{t,x} = \partial_x u(s,X_s^{t,x}) =u_x(s,X_s^{t,x})\nabla X_s^{t,x},\end{equation*}

we conclude that

(39)\begin{eqnarray} Z^{t,x}_s = \frac{\nabla Y^{t,x}_s}{\nabla X^{t,x}_s} \sigma(s,X^{t,x}_s), \quad 0 \leq t \le s \leq T.\end{eqnarray}

It is well-known (see e.g. [Reference El Karoui, Peng and Quenez19]) that the solution $\nabla Y$ of the linear BSDE

(40)\begin{align}\nabla Y_s = & g'(X_T) \nabla X_T + \int_{s}^T (f_x(\Theta_r) \nabla X_r + f_y(\Theta_r) \nabla Y_r + f_z(\Theta_r) \nabla Z_r) dr \notag\\[2pt] & - \int_{s}^T \nabla Z_r dB_r, \quad 0 \leq s \leq T,\end{align}

can be represented as

(41)\begin{align} \frac{\nabla Y_s}{\nabla X_s} & = {\mathbb{E}}_s \bigg[g'(X_T) \nabla X_T \Gamma^s_T + \int_{s}^T f_x(\Theta_r) \nabla X_r \Gamma^s_rdr \bigg] \frac{1}{\nabla X_s} \notag\\[2pt] & = \tilde {\mathbb{E}}\bigg[g'(\tilde X^{s, X_s}_T) \nabla \tilde X^{s,X_s}_T \tilde \Gamma^{s,X_s}_T + \int_{s}^T f_x(\tilde \Theta^{s,X_s}_r) \nabla \tilde X^{s,X_s}_r \tilde \Gamma^{s,X_s}_rdr \bigg], \quad 0 \leq t \le s \leq T,\end{align}

where $\Theta_r \coloneqq (r,X_r,Y_r,Z_r)$, $\Gamma^s$ denotes the adjoint process given by

\begin{equation*}\Gamma^s_r = 1 + \int_s^r f_y(\Theta_u) \Gamma^s_u du + \int_s^r f_z(\Theta_u) \Gamma^s_u dB_u, \quad s \le r \leq T,\end{equation*}

and

\begin{equation*}\tilde \Gamma^{t,x}_s = 1 + \int_{t}^s f_y(\tilde \Theta^{t,x}_r) \tilde \Gamma^{t,x}_r dr + \int_{t}^s f_z(\tilde \Theta^{t,x}_r) \tilde \Gamma^{t,x}_r d \tilde B_r, \quad t \leq s \leq T, \,x \in {\mathbb{R}},\end{equation*}

where $\tilde B$ denotes an independent copy of B. Notice that $\nabla X^{t,x}_t=1,$ so that

\begin{align*}\frac{\nabla Y^{t,x}_t}{\nabla X^{t,x}_t} = \nabla Y^{t,x}_t & = \tilde {\mathbb{E}}\bigg[g'(\tilde X^{t, x}_T) \nabla \tilde X^{t,x}_T \tilde \Gamma^{t,x}_T + \int_{t}^T f_x(\tilde \Theta^{t,x}_r) \nabla \tilde X^{t,x}_r \tilde \Gamma^{t,x}_rdr \bigg].\end{align*}

Then, by (39),

\begin{eqnarray*}\| Z_s - Z_t\|_{L_p({\mathbb{P}}_{t,x})} &\le&C(\sigma) \bigg [ \bigg\|\frac{\nabla Y_s}{\nabla X_s} - \frac{\nabla Y_t}{\nabla X_t} \bigg\|_{L_{p}({\mathbb{P}}_{t,x})} \\[3pt] &&+ \|\nabla Y_t\|_{L_{2p}({\mathbb{P}}_{t,x})} [(s-t)^{\frac{1}{2}} \!+ \| X^{t,x}_s -x\|_{L_{2p}({\mathbb{P}}_{t,x})}] \bigg ].\end{eqnarray*}

Since $(\nabla Y_s, \nabla Z_s)$ is the solution to the linear BSDE (40) with bounded $f_x, f_y, f_z,$ we have that $\|\nabla Y_t\|_{L_{2p}({\mathbb{P}}_{t,x})} \le C(b,\sigma,f,g,T,p).$ Obviously, $ \| X^{t,x}_s -x\|_{L_{2p}({\mathbb{P}}_{t,x})} \le C(b,\sigma,T,p) (s-t)^{\frac{1}{2}}.$ So it remains to show that

\begin{equation*}\bigg\|\frac{\nabla Y_s}{\nabla X_s} - \frac{\nabla Y_t}{\nabla X_t}\bigg\|_{L_{p}({\mathbb{P}}_{t,x})} \le C\Psi(x)(s-t)^{\frac{1}{2}}.\end{equation*}

We intend to use (41) in the following. There is a certain degree of freedom in how to connect B and $\tilde B$ in order to compute conditional expectations. Here, unlike in (27), we define the processes

\begin{equation*} B^{\prime}_u= B_{u\wedge s} + \tilde B_{u \vee s}- \tilde B_s \quad \text{and} \quad B^{\prime\prime}_u= B_{u\wedge t} + \tilde B_{u \vee t}- \tilde B_t, \quad u\ge 0, \end{equation*}

as driving Brownian motions for $\tfrac{\nabla Y_s}{\nabla X_s}$ and $\tfrac{\nabla Y_t}{\nabla X_t},$ respectively. This will especially simplify the estimate for $ \tilde {\mathbb{E}} |\tilde \Gamma^{s,X_s}_T -\tilde \Gamma^{t,x}_T|^ q $ below. From the above relations we get the following (with $X_s\coloneqq X^{t,x}_s $):

\begin{align*}\bigg\|\frac{\nabla Y_s}{\nabla X_s} - \frac{\nabla Y_t}{\nabla X_t} \bigg\|_{L_{p}({\mathbb{P}}_{t,x})} & \leq \left \| \tilde {\mathbb{E}} \Big[g'(\tilde X^{s, X_s}_T) \nabla \tilde X^{s,X_s}_T \tilde \Gamma^{s,X_s}_T - g'(\tilde X^{t,x}_T) \nabla \tilde X^{t,x}_T \tilde \Gamma^{t,x}_T \Big] \right \|_p\\[2pt] & \quad + \int_t^s \left\| \tilde {\mathbb{E}} \Big[ f_x(\tilde \Theta^{t,x}_r) \nabla \tilde X^{t,x}_r \tilde \Gamma^{t,x}_r\Big]\right\|_p dr \\[2pt] & \quad +\left\| \int_s^T \tilde {\mathbb{E}} \Big[ f_x(\tilde \Theta^{s,X_s}_r) \nabla \tilde X^{s,X_s}_r\tilde \Gamma^{s,X_s}_r- f_x(\tilde \Theta^{t,x}_r) \nabla \tilde X^{t,x}_r \tilde \Gamma^{t,x}_r \Big] dr \right\|_p\\[2pt] & \eqqcolon J_1 + J_2 + J_3.\end{align*}

Since gʹ is Lipschitz continuous and of polynomial growth, we have

\begin{equation*}J_1\le C(b,\sigma,g,T,p) \Psi(x) (s-t)^{\frac{1}{2}} \end{equation*}

by Hölder’s inequality and the $L_q$-boundedness for any $q >0$ of all the factors, as well as from the estimates for $ \tilde X^{s, X_s}_T - \tilde X^{t,x}_T$ and $\nabla \tilde X^{s,X_s}_T - \nabla \tilde X^{t,x}_T $ as in Lemma 5.2. For the $\Gamma$ differences we first apply the inequalities of Hölder and BDG:

\begin{align*} \tilde {\mathbb{E}} |\tilde \Gamma^{s,X_s}_T -\tilde \Gamma^{t,x}_T|^ q \le C(T,q)& \bigg [ (s-t)^{q-1} \tilde {\mathbb{E}} \int_t^s |f_y(\tilde \Theta^{s,X_s}_r) \tilde \Gamma^{s,X_s}_r|^q dr + \tilde {\mathbb{E}} \bigg(\int_t^s |f_z (\tilde \Theta^{s,X_s}_r) \tilde \Gamma^{s,X_s}_r |^2 dr \bigg)^\frac{q}{2} \\[3pt] &\quad + \tilde {\mathbb{E}} \int_s^T |f_y(\tilde \Theta^{s,X_s}_r) \tilde \Gamma^{s,X_s}_r - f_y(\tilde \Theta^{t,x}_r) \tilde \Gamma^{t,x}_r|^q dr \\[3pt] &\quad + \tilde {\mathbb{E}} \bigg( \int_s^T |f_z(\tilde \Theta^{s,X_s}_r) \tilde \Gamma^{s,X_s}_r- f_z(\tilde \Theta^{t,x}_r) \tilde \Gamma^{t,x}_r|^2 dr \bigg)^\frac{q}{2} \bigg ]. \end{align*}

Since $f_y$ and $f_z$ are bounded we have $\tilde {\mathbb{E}} | \tilde \Gamma^{s,X_s}_r|^q + \tilde {\mathbb{E}} |\tilde \Gamma^{t,x}_r |^q \le C(f,T,q). $ Similarly to (31), since $f_x, f_y,f_z$ are Lipschitz continuous with respect to the space variables,

\begin{align*} |f_x(\tilde \Theta^{s,X_s}_r) - f_x(\tilde \Theta^{t,x}_r) |&= |f_x(r, \tilde X_r^{s,X_s}, u(r, \tilde X_r^{s,X_s}), \sigma(r, \tilde X_r^{s,X_s}) u_x(r, \tilde X_r^{s,X_s}) ) \\[3pt] & \quad - f_x(r, \tilde X_r^{t,x}, u(r, \tilde X_r^{t,x}), \sigma(r, \tilde X_r^{t,x}) u_x(r, \tilde X_r^{t,x}) )| \\[3pt] &\le C(c^{2,3}_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}, \sigma, f, T) (1+|\tilde X_r^{s,X_s}|^{p_0+1}+|\tilde X_r^{t,x}|^{p_0+1}) \frac{|\tilde X_r^{s,X_s}-\tilde X_r^{t,x}|}{(T-r)^{\frac{1}{2}}}, \end{align*}

so that Lemma 5.2 yields

\begin{equation*}\tilde {\mathbb{E}} |f_x(\tilde \Theta^{s,X_s}_r) -f_x(\tilde \Theta^{t,x}_r) |^q \le C(c_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}^{2,3}, b,\sigma, f, T,p_0,q) (1+|X_s|^{p_0+1}+|x|^{p_0+1})^q \frac{ |X_s-x|^q + |s-t|^\frac{q}{2}}{ (T-r)^{\frac{1}{2}}}.\end{equation*}

The same holds for $|f_y(\tilde \Theta^{s,X_s}_r) - f_y(\tilde \Theta^{t,x}_r) |$ and $|f_z(\tilde \Theta^{s,X_s}_r) - f_z(\tilde \Theta^{t,x}_r) |.$ Applying these inequalities and Gronwall’s lemma, we arrive at

\begin{eqnarray*} \| \tilde {\mathbb{E}} [\tilde \Gamma^{s,X_s}_T -\tilde \Gamma^{t,x}_T] \|_p &\le& C(c_{\hyperref[betterZ]{\scriptstyle\textcolor{blue}{4.2}}}^{2,3}, b,\sigma,f,g, T,p_0,p) \Psi(x) |s-t|^\frac{1}{2}\end{eqnarray*}

for $p> 0.$

For $J_2\le C (t-s)$ it is enough to realise that the integrand is bounded. The estimate for $J_3$ follows similarly to that of $J_1.$

4.4. Properties of the solution to the finite difference equation

Recall the definition of $ \mathcal{D}^n_m$ given in (15). By (4),

(42)\begin{eqnarray} X_{t_{m+1}}^{n,t_m,x} = x+ h b(t_{m+1},x) + \sqrt{h} \sigma(t_{m+1},x) {\varepsilon}_{m+1},\end{eqnarray}

so that

(43)\begin{eqnarray} T_{_{m+1,\pm}} u^n(t_{m+1}, X_{t_{m+1}}^{n,t_m,x }) = u^n(t_{m+1},x+ h b(t_{m+1},x) \pm \sqrt{h} \sigma(t_{m+1},x)).\end{eqnarray}

While for the solution to the PDE (38) one can observe in Theorem 4.2 the well-known smoothing property which implies that u is differentiable on $[0,T)\times {\mathbb{R}}$ even though g is only Hölder continuous, in the following proposition, for the solution $u^n$ to the finite difference equation we have to require from g the same regularity as we want for $u^n.$

Proposition 4.2. Let Assumption 2.3 hold and assume that $u^n$ is a solution of

(44)\begin{align} & u^n(t_m,x) -h f(t_{m+1}, x,u^n(t_m,x), \mathcal{D}^n_{m+1} u^n(t_{m+1},X_{t_{m+1}}^{n,t_m,x})) \notag \\[2pt] &\quad= {\frac{1}{2}} [T_{_{m+1,+}} u^n(t_{m+1}, X_{t_{m+1}}^{n,t_m,x }) + T_{_{m+1,-}} u^n(t_{m+1}, X_{t_{m+1}}^{n,t_m,x })], \quad m = 0, \dots, n-1, \end{align}

with terminal condition $u^n(t_n,x)= g(x).$ Then, for sufficiently small h, the map $x \mapsto u^n(t_m,x)$ is $C^2,$ and it holds that

\begin{equation*}|u^n(t_m,x)| + |u_x^n(t_m,x)| \le C_{u^n\!,1}\, \Psi(x), \quad |u_{xx}^n(t_m,x)| \le C_{u^n\!,2}\, \Psi^2(x),\end{equation*}

and

(45)\begin{eqnarray} |u_{xx}^n(t_m,x) - u_{xx}^n(t_m, \bar x)| \le C_{u^n\!,3} \,(1+|x|^{6p_0+7} +|\bar x|^{6p_0+7})|x-\bar x|^\alpha,\end{eqnarray}

uniformly in $ m=0,\dots,n-1$. The constants $C_{u^n\!,1}$, $C_{u^n\!,2}$, and $C_{u^n\!,3}$ depend on the bounds of f, g, b, $\sigma$, and their derivatives, and on T and $p_0$.

Proof. Step 1. From (44), since g is $C^2$ and $f_y$ is bounded, for sufficiently small h we conclude by induction (backwards in time) that $u^n_x(t_m,x)$ exists for $m=0,\ldots,n-1,$ and that

\begin{align*}u_x^n(t_m,x)&=h f_x(t_{m+1}, x,u^n(t_m,x), \mathcal{D}^n_{m+1} u^n(t_{m+1},X_{t_{m+1}}^{n,t_m,x})) \\[3pt] &\quad +h f_y(t_{m+1}, x,u^n(t_m,x), \mathcal{D}^n_{m+1} u^n(t_{m+1},X_{t_{m+1}}^{n,t_m,x})) \, u_x^n(t_m,x) \\[3pt] &\quad +h f_z(t_{m+1}, x,u^n(t_m,x), \mathcal{D}^n_{m+1} u^n(t_{m+1},X_{t_{m+1}}^{n,t_m,x})) \, \partial_x\mathcal{D}^n_{m+1} u^n(t_{m+1},X_{t_{m+1}}^{n,t_m,x}) \\[3pt] &\quad +\tfrac{1}{2}\Big ( \partial_xT_{_{m+1,+}} u^n(t_{m+1}, X_{t_{m+1}}^{n,t_m,x }) + \partial_xT_{_{m+1,-}} u^n(t_{m+1}, X_{t_{m+1}}^{n,t_m,x} )\Big). \end{align*}

Similarly one can show that $u^n_{xx}(t_m,x)$ exists and solves the derivative of the previous equation.

Step 2. As stated in the proof of Proposition 2.1, the finite difference equation (44) is the associated equation to (9) in the sense that we have the representations (21). We will use that $u^n(t_m,x) =Y_{t_m}^{n,t_m,x}$ and exploit the BSDE

(46)\begin{align} Y_{t_m}^{n,t_m,x}&= g(X_T^{n,t_m,x}) +\int_{(t_m,T]} f(s,X^{n,t_m,x}_{s-},Y^{n,t_m,x}_{s-},Z^{n,t_m,x}_{s-})d[B^n]_s \notag \\[3pt] &\quad - \int_{(t_m,T]} Z^{n,t_m,x} _{s-} dB^n_s, \end{align}

in which we will drop the superscript $t_m,x$ from now on. For $u^n_x(t_m,x)$ we will consider

(47)\begin{align} \nabla Y^n_{t_m}\coloneqq \partial_x Y_{t_m}^{n} &= g'(X_T^{n}) \partial_x X_T^{n} +\int_{(t_m,T]} f_x \partial_x X_{s-}^{n} + f_y \partial_x Y_{s-}^{n} + f_z \partial_x Z_{s-}^{n}d[B^n]_s \notag \\[3pt] & \quad -\int_{(t_m,T]} \partial_x Z_{s-}^{n} dB^n_s.\end{align}

Similarly as in the proof of [Reference Ma and Zhang30, Theorem 3.1], the BSDE (47) can be derived from (46) as a limit of difference quotients with respect to x. Notice that the generator of (47) is random but has the same Lipschitz constant and linear growth bound as f. Assumption 2.3 allows us to find a $p_0\ge 0$ and a $K>0$ such that

\begin{equation*}|g(x)| +|g'(x)|+|g^{\prime\prime}(x)| \le K(1+|x|^{p_0+1}) =\Psi(x).\end{equation*}

In order to get estimates simultaneously for (46) and (47) we prove the following lemma.

Lemma 4.2. We fix n and assume a BSDE

(48)\begin{eqnarray} \textsf{Y}_{t_k}&=& \xi^n +\int_{(t_k,T]} \textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-},\textsf{Z}_{s-})d[B^n]_s - \int_{(t_k,T]} \textsf{Z} _{s-} dB^n_s, \quad m\le k\le n, \end{eqnarray}

with $\xi^n = g(X_T^{n,t_m,x}) $ or $\xi^n = g'(X_T^{n,t_m,x}) \partial_x X_T^{n,t_m,x} $, and $\textsf{X}_s\coloneqq X_s^{n,t_m,x}$ or $\textsf{X}_s\coloneqq \partial_x X_s^{n,t_m,x}$, such that $\textsf{f}:\Omega \times [0,T] \times {\mathbb{R}}^3 \to {\mathbb{R}} $ is measurable and satisfies

(49)\begin{align} |\textsf{f}(\omega,t,x,y,z) - {\textsf{f}}(\omega,t,x',y',z') |& \le L_f ( |x-x'| + |y-y'| + |z-z'|), \notag \\[3pt] |\textsf{f}(\omega,t,x,y,z) |& \le (K_f+L_f) (1+ |x| + |y| + |z|). \end{align}

Then for any $p \ge 2,$

1. (i)

   \begin{equation*} {\mathbb{E}} |\textsf{Y}_{t_k}|^p + \frac{\gamma_p}{4} {\mathbb{E}} \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} |\textsf{Z}_{s-}|^2 d[B^n]_s \le C\Psi^p(x)\end{equation*}
   for $k=m,\ldots,n$ and some $\gamma_p>0,$

2. (ii) $ {\mathbb{E}} \sup_{t_m < s \le T}|\textsf{Y}_{s-}|^p \le C \Psi^p(x), $ and

3. (iii) $ {\mathbb{E}} \Big ( \int_{(t_m,T]} |\textsf{Z}_{s-}|^2 d[B^n]_s \Big)^{\frac{p}{2}} \le C\Psi^p(x),$

for some constant $ C=C(b,\sigma, f,g,T,p,p_0)$.

Proof.

1. (i) By Itô’s formula (see [Reference Jacod and Shiryaev25, Theorem 4.57]) we get for $p\ge 2$ that

   (50)\begin{align} |\textsf{Y}_{t_k}|^p& = | \xi^n|^p - p \int_{(t_k,T]} \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} \textsf{Z}_{s-} dB^n_s \notag \\[3pt] &\quad + \ p \int_{(t_k,T]} \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} \textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-}, \textsf{Z}_{s-})d[B^n]_s \notag\\[3pt] &\quad - \sum_{s \in (t_k,T]} [|\textsf{Y}_s|^p - |\textsf{Y}_{s-}|^p - p \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} (\textsf{Y}_s-\textsf{Y}_{s-})].\end{align}
   Following the proof of [Reference Kruse and Popier27, Proposition 2] (which is carried out there in the Lévy process setting but can be done also for martingales with jumps, like $B^n$) we can use the estimate
   \begin{eqnarray*} - \!\sum_{s \in (t_k,T]} [ |\textsf{Y}_s|^p \,{-}\, |\textsf{Y}_{s-}|^p \,{-}\, p \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} (\textsf{Y}_s\,{-}\,\textsf{Y}_{s-})]\le {-} \gamma_p \!\sum_{s \in (t_k,T]} |\textsf{Y}_{s-}|^{p-2} (\textsf{Y}_s{-}\textsf{Y}_{s-})^2,\end{eqnarray*}
   where $\gamma_p >0$ is computed in [Reference Yao40, Lemma A4]. Since
   \begin{equation*} \textsf{Y}_{t_{\ell+1}} - \textsf{Y}_{{t_{\ell+1}}-}=\textsf{f}(t_{\ell+1},\textsf{X}_{t_\ell},\textsf{Y}_{t_\ell}, \textsf{Z}_{t_\ell})h- \textsf{Z}_{t_\ell} \sqrt{h} {\varepsilon}_{\ell+1}\end{equation*}
   we have
   \begin{align*}& - \sum_{s \in (t_k,T]} [ |\textsf{Y}_s|^p - |\textsf{Y}_{s-}|^p - p \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} (\textsf{Y}_s-\textsf{Y}_{s-})] \\[3pt] &\quad\le - \gamma_p \,\sum_{\ell=k}^{n-1} |\textsf{Y}_{t_\ell} |^{p-2} \, \Big (\textsf{f}(t_{\ell+1},\textsf{X}_{t_\ell},\textsf{Y}_{t_\ell}, \textsf{Z}_{t_\ell})h - \textsf{Z}_{t_\ell} \sqrt{h} {\varepsilon}_{\ell+1} \Big)^2 \\[3pt] &\quad= - \gamma_p \, h \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} \, \textsf{f}^2(s,\textsf{X}_{s-},\textsf{Y}_{s-},\textsf{Z}_{s-})d[B^n]_s - \gamma_p \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} |\textsf{Z}_{s-}|^2 d[B^n]_s \\[3pt] &\qquad + 2 \gamma_p \, \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} \, \textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-}, \textsf{Z}_{s-}) \textsf{Z}_{s-}(B^n_s-B^n_{s-}) d[B^n]_s.\end{align*}
   Hence we get from (50) that
   \begin{align*} |\textsf{Y}_{t_k}|^p&\le | \xi^n|^p- p \int_{(t_k,T]} \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} \textsf{Z}_{s-} dB^n_s \\[3pt] &\quad + p\int_{(t_k,T]} \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} \, \textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-}, \textsf{Z}_{s-})d[B^n]_s \notag\\[3pt] &\quad - \gamma_p \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} \, |\textsf{Z}_{s-}|^2 d[B^n]_s \\[3pt] &\quad + 2 \gamma_p \, \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} \, \textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-}, \textsf{Z}_{s-}) \textsf{Z}_{s-}(B^n_s-B^n_{s-}) d[B^n]_s.\end{align*}
   From Young’s inequality and (49) we conclude that there is a $ c'= c'(p,K_f,L_f, \gamma_p)>0$ such that
   \begin{eqnarray*} p|\textsf{Y}_{s-} |^{p-1}\, |\textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-},\textsf{Z}_{s-})|\le \tfrac{\gamma_p}{4} |\textsf{Y}_{s-} |^{p-2} \, |\textsf{Z}_{s-}|^2 + c'(1+| \textsf{X}_{s-}|^p + |\textsf{Y}_{s-} |^p),\end{eqnarray*}
   and for $ \sqrt{h} < \tfrac{1}{8 (L_f + K_f)}$ we find a $c^{\prime\prime} =c^{\prime\prime}(p, L_f, K_f, \gamma_p )>0$ such that
   \begin{align*} & 2 \gamma_p \, \sqrt{h} |\textsf{Y}_{s-} |^{p-2} \, |\textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-},\textsf{Z}_{s-})| | \textsf{Z}_{s-}| \le \tfrac{ \gamma_p}{4} |\textsf{Y}_{s-} |^{p-2} \, | \textsf{Z}_{s-}|^2\\[3pt] &\quad + c^{\prime\prime} \,(1+|\textsf{X}_{s-}|^p +|\textsf{Y}_{s-}|^p).\end{align*}
   Then for $c=c'+c^{\prime\prime}$ we have
   (51)\begin{align} |\textsf{Y}_{t_k}|^p&\le | \xi^n|^p - p \int_{(t_k,T]} \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2}\, \textsf{Z}_{s-} dB^n_s + c\int_{(t_k,T]} 1+| \textsf{X}_{s-}|^p + |\textsf{Y}_{s-} |^p d[B^n]_s \notag\\[3pt] &\quad - \tfrac{\gamma_p}{2} \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} \, |\textsf{Z}_{s-}|^2 d[B^n]_s.\end{align}
   By standard methods, approximating the terminal condition and the generator by bounded functions, it follows that for any $a>0$,
   \begin{equation*}{\mathbb{E}} \sup_{t_k \le s\le T} |\textsf{Y}_s|^a < \infty \quad \text{ and } \quad {\mathbb{E}} \bigg (\int_{(t_k,T]} |\textsf{Z}_{s-}|^2 d[B^n]_s \bigg)^{\frac{a}{2}} < \infty.\end{equation*}
   Hence $ \int_{(t_k,T]} \textsf{Y}_{s-} |\textsf{Y}_{s-} |^{p-2} \textsf{Z}_{s-} dB^n_s$ has expectation zero. Taking the expectation in (51) yields
   (52)\begin{align} &{\mathbb{E}} |\textsf{Y}_{t_k}|^p + \tfrac{\gamma_p}{2} {\mathbb{E}} \int_{(t_k,T]} |\textsf{Y}_{s-}|^{p-2} |\textsf{Z}_{s-}|^2 d[B^n]_s\le {\mathbb{E}}| \xi^n|^p \nonumber\\[3pt] &\quad + c {\mathbb{E}} \int_{(t_k,T]} 1+| \textsf{X}_{s-}|^p + |\textsf{Y}_{s-} |^p d[B^n]_s.\end{align}
   Since ${\mathbb{E}}| \xi^n|^p$ and $ {\mathbb{E}} \int_{(t_k,T]} 1+| \textsf{X}_{s-}|^p d[B^n]_s$ are polynomially bounded in x, Gronwall’s lemma gives
   \begin{eqnarray*}&& {} \| \textsf{Y}_{t_k}\|_p \le C(b,\sigma, f,g, T,p,p_0) (1 + |x|^{ p_0+1}), \quad k=m,\ldots,n,\end{eqnarray*}
   and inserting this into (52) yields
   \begin{align*} & \bigg ({\mathbb{E}} \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} |\textsf{Z}_{s-}|^2 d[B^n]_s \bigg )^\frac{1}{p} \le C(b,\sigma,f,g,T,p,p_0) (1 + |x|^{ p_0+1}),\\[3pt] &\quad k=m,\ldots,n-1.\end{align*}

2. (ii) From (51) we derive by the inequality of BDG and Young’s inequality that for ${t_m \le t_k\,{\le}\,T}$,

   \begin{align*}& {\mathbb{E}} \sup_{t_k < s \le T}|\textsf{Y}_{s-}|^p \\[3pt] &\quad \le {\mathbb{E}}| \xi^n|^p + C(p) {\mathbb{E}} \bigg ( \int_{(t_k,T]} |\textsf{Y}_{s-} |^{2p-2} |\textsf{Z}_{s-} |^2 d[B^n]_s\bigg)^{\frac{1}{2}}\\[3pt] &\qquad +c{\mathbb{E}} \int_{(t_k,T]}1+| \textsf{X}_{s-}|^p + |\textsf{Y}_{s-} |^p d[B^n]_s \notag\\ &\quad \le {\mathbb{E}}| \xi^n|^p + c {\mathbb{E}} \int_{(t_k,T]} 1+| \textsf{X}_{s-}|^p d[B^n]_s \notag\\[2pt] &\qquad + C(p) {\mathbb{E}} \left [ \sup_{t_k < s \le T} |\textsf{Y}_{s-} |^{\frac{p}{2}} \left ( \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} |\textsf{Z}_{s-} |^2 d[B^n]_s\right )^{\frac{1}{2}} \right ]\\[3pt] &\qquad + c {\mathbb{E}} \int_{(t_k,T]} |\textsf{Y}_{s-} |^pd[B^n]_s \\[3pt] &\quad \le {\mathbb{E}}| \xi^n|^p + c {\mathbb{E}} \int_{(t_k,T]} 1+| \textsf{X}_{s-}|^p d[B^n]_s + C(p) {\mathbb{E}} \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} |\textsf{Z}_{s-} |^2 d[B^n]_s \\[3pt] &\qquad + {\mathbb{E}} \sup_{t_k < s \le T} |\textsf{Y}_{s-} |^p( \tfrac{1}{4} + c (T-t_k)).\end{align*}
   We assume that h is sufficiently small so that we find a $t_k$ with $c (T-t_k) < \tfrac{1}{4}.$ We rearrange the inequality to have $ {\mathbb{E}} \sup_{t_k < s \le T} |\textsf{Y}_{s-} |^p$ on the left-hand side, and from (i) we conclude that
   \begin{align*} {\mathbb{E}} \sup_{t_k < s \le T}|\textsf{Y}_{s-}|^p &\le 2 {\mathbb{E}} | \xi^n|^p + 2c {\mathbb{E}} \int_{(t_k,T]} 1+| \textsf{X}_{s-}|^p d[B^n]_s \\[3pt] &\quad + 2C(p) {\mathbb{E}} \int_{(t_k,T]} |\textsf{Y}_{s-} |^{p-2} |\textsf{Z}_{s-} |^2 d[B^n]_s \\[3pt] &\le C(b,\sigma, f,g, T,p,p_0) (1+|x|^{(p_0+1)p}). \end{align*}
   Now we may repeat the above step for ${\mathbb{E}} \sup_{t_\ell < s \le t_k}|\textsf{Y}_{s-}|^p$ with $c (t_k-t_\ell) < \tfrac{1}{4}$ and $\xi^n=\textsf{Y}_T$ replaced by $\textsf{Y}_{t_k},$ and continue doing so until we eventually get the assertion (ii).

3. (iii) We proceed from (48):

   \begin{align*} &\sup_{k\le \ell\le n} \Big | \int_{(t_\ell,T]} \textsf{Z}_{s-} dB^n_s \Big|^p\\[3pt] &\quad \le C(p) \bigg ( | \xi^n|^p + \sup_{k\le \ell \le n} |\textsf{Y}_{t_\ell}|^p + \Big |\int_{(t_k,T]} | \textsf{f}(s,\textsf{X}_{s-},\textsf{Y}_{s-}, \textsf{Z}_{s-}) | \,d[B^n]_s\Big |^p \bigg),\end{align*}
   so that by (49) and the inequalities of BDG and Hölder we have that
   \begin{align*}& {\mathbb{E}} \bigg ( \int_{(t_k,T]} |\textsf{Z}_{s-}|^2 d[B^n]_s \bigg)^{\frac{p}{2}} \\[2pt] &\quad\le C(p) \bigg ( {\mathbb{E}} | \xi^n|^p + {\mathbb{E}} \sup_{k\le \ell\le n} | \textsf{Y}_{t_\ell}|^p \bigg) +C(p,L_f,K_f) {\mathbb{E}} \bigg(\int_{(t_k,T]} 1+|\textsf{X}_{s-}|+| \textsf{Y}_{s-}|d[B^n]_s\bigg)^p \\[2pt] &\qquad +C(p,L_f,K_f) (T-t_k)^{\frac{p}{2}} {\mathbb{E}} \left (\int_{(t_k,T]} |\textsf{Z}_{s-}|^2d[B^n]_s\right )^{\frac{p}{2}}. \end{align*}
   Hence for $C(p,L_f,K_f) (T-t_k)^{\frac{p}{2}} <{\frac{1}{2}}$ we derive from the assertion (ii) and from the growth properties of the other terms that
   (53)\begin{align} {\mathbb{E}} \bigg ( \int_{(t_k,T]} |{\textsf{Z}_{s-}}|^2 d[B^n]_s \bigg)^{\frac{p}{2}} \le C(b,\sigma, f,g,T,p,p_0) (1+|x|^{(p_0+1)p}).\end{align}
   Repeating this procedure eventually yields (iii).

Step 3. Applying Lemma 4.2 to (46) and (47) we see that for all $m =0,\ldots,n$ we have

\begin{equation*} |u^n(t_m, x)| = |Y_{t_m}^{n, t_m,x} | = ( {\mathbb{E}} (Y_{t_m}^{n, t_m,x} )^2)^\frac{1}{2} \le C(b,\sigma, f,g,T,p_0) (1+|x|^{p_0+1})\end{equation*}

and

(54)\begin{eqnarray} |u^n_x(t_m, x)| = ( {\mathbb{E}} ( \partial_xY_{t_m}^{n, t_m,x})^2 )^\frac{1}{2} \le C(b,\sigma, f,g,T,p_0) (1+|x|^{p_0+1}).\end{eqnarray}

Our next aim is to show that $u^n_{xx}(t_m,x)$ is locally Lipschitz in x. We first show that $u^n_{xx}(t_m,x)$ has polynomial growth. We introduce the BSDE which describes $u^n_{xx}(t_m,x)$, for simplicity writing

\begin{equation*}f(t,x_1,x_2,x_3)\coloneqq f(t,x,y,z) \quad \text{and } \quad D^a \coloneqq \partial_{x_1}^{i_1}\partial_{x_2}^{i_2}\partial_{x_3}^{i_3} \quad \text{with} \quad a \coloneqq (i_1,i_2,i_3),\end{equation*}

and consider

(55)\begin{align} \partial_x^2 Y_{t_m}^{n}&=g^{\prime\prime}(X_T^{n}) ( \partial_x X_T^{n})^2 + g'(X_T^{n}) \partial_x^2 X_T^{n} \notag\\[3pt] & \quad+\int_{(t_m,T]} \sum_{\substack{a \in \{0,1,2\}^3 \\ i_1+i_2+i_3=2}} (D^a f)(s,X_{s-}^{n},Y^{n}_{s-},Z^{n}_{s-}) (\partial_x X_{s-}^{n})^{i_1} (\partial_x Y^{n}_{s-})^{i_2} (\partial_x Z^{n}_{s-})^{i_3} d[B^n]_s \notag \\[3pt] &\quad + \int_{(t_m,T]} \sum_{\substack{a \in \{0, 1\}^3\\ i_1+i_2+i_3=1}} (D^a f)(s,X_{s-}^{n},Y^{n}_{s-},Z^{n}_{s-}) (\partial_x^2 X_{s-}^{n})^{i_1} (\partial_x^2 Y^{n}_{s-})^{i_2} (\partial_x^2 Z^{n}_{s-})^{i_3} d[B^n]_s \notag \\[3pt] &\quad -\int_{(t_m,T]} \partial_x^2 Z^n_{s-} dB^n_s.\end{align}

We denote the generator of this BSDE by $\hat f$ and notice that it is of the structure

\begin{equation*} \hat f(\omega, t,x,y,z) = f_0(\omega, t) + f_1(\omega, t) x+ f_2(\omega, t) y+ f_3(\omega, t) z.\end{equation*}

Here $f_0(\omega, t)$ denotes the integrand of the first integral on the right-hand side of (55), and from the previous results one concludes that ${\mathbb{E}} (\!\int_{(t_m,T]} |f_0(s-\!)| d[B^n]_s)^p < \infty.$ The functions $f_1(t) = (D^{(1,0,0)} f)(t, \cdot) = (\partial_x f)(t, \cdot)$, $f_2(t) = (\partial_y f)(t, \cdot)$, and $f_3(t) = (\partial_z f)(t, \cdot)$ are bounded by our assumptions. We put

\begin{equation*}\hat\xi^n \coloneqq g^{\prime\prime}(X_T^{n}) ( \partial_x X_T^{n})^2 + g'(X_T^{n}) \partial_x^2 X_T^{n}.\end{equation*}

Denoting the solution by $(\hat{\textsf{Y}}, \hat{\textsf{Z}})$, we get for $C(f_3 ) (T-t_m) \le \tfrac{1}{2}$ that

(56)\begin{eqnarray} && {} {\mathbb{E}} |\hat{\textsf{Y}}_{t_m}|^2 + {\frac{1}{2}} {\mathbb{E}} \int_{(t_m,T]} |\hat{\textsf{Z}}_{s-}|^2 d[B^n]_s \notag\\& \le\,& C \bigg [ {\mathbb{E}} |\hat \xi^n|^2 + {\mathbb{E}} \bigg (\int_{(t_m,T]} |f_0( s-) | d[B^n]_s \bigg )^2 + {\mathbb{E}} \int_{(t_m,T]} |\hat{\textsf{X}}_{s-}|^2 + |\hat{\textsf{Y}}_{s-} |^2 d[B^n]_s \bigg].\end{eqnarray}

Now we derive the polynomial growth ${\mathbb{E}} | \hat \xi^n|^2 \le C \Psi^2(x) $ from the properties of gʹ and gʹʹ and from the fact that ${\mathbb{E}} \sup_{t_m< s \le T} |\partial^j_x X_{s}^{n}|^p$ is bounded for $j=1,2$ under our assumptions. Then the estimate

\begin{equation*} {\mathbb{E}} \bigg (\int_{(t_m,T]} | f_0(s-) | d[B^n]_s \bigg )^2 \le C \Psi^4(x)\end{equation*}

can be derived from Lemma 4.2 Parts (ii) and (iii), so that Gronwall’s lemma implies

(57)\begin{eqnarray} |\hat{\textsf{Y}}_{t_m}^{t_m,x}|= |u_{xx}(t_m,x)| \le C \Psi^2(x).\end{eqnarray}

Finally, to show (45), one uses (55) and derives an inequality as in (56), but now for the difference $\partial_x^2 Y_{t_m}^{n,t_m,x}-\partial_x^2 Y_{t_m}^{n,t_m,\bar x}.$

Before proving (45), let us state the following lemma.

Lemma 4.3 Let Assumption 2.3 hold. We have

(58)\begin{align}\left({\mathbb{E}} \sup_s | Z^{n,t_m,x}_{s-}- Z^{n,t_m,\bar x}_{s-} |^p\right)^{1/p} \le C( \Psi(x)^2 + \Psi(\bar x)^2)|x-\bar x|,\quad p\ge 2, \end{align}

(59)\begin{align} {\mathbb{E}} \bigg ( \int_{(t_m,T]} |\partial_x Z^{n,t_m,x}_{s-} - \partial_x Z^{n,t_m,\bar x}_{s-} |^2 d[B^n]_s \bigg )^\frac{p}{2} \le C (\Psi^{4p}(x) + \Psi^{4p}(\bar x))|x-\bar x|^p, \quad p\ge 2, \end{align}

(60)\begin{align} {\mathbb{E}} \left ( \int_{(t_m,T]} |\partial^2_x Z^{n,t_m,x}_{s-}|^{2} d[B^n]_s \right )^\frac{p}{2} \le C\Psi^{4p}(x), \quad p \geq2,\qquad\qquad\qquad\qquad\qquad\end{align}

for some constant $C=C(b,\sigma,f,g,T,p,p_0)$.

Proof of Lemma 4.3. Proof of (58): Introduce $G(t_{k+1}, x) \coloneqq \mathcal{D}^n_{k+1} u^n(t_{k+1}, X^{n,t_k,x}_{t_{k+1}})$. Using the relations (42)–(43) and the bounds (54) and (57) for $u^n_x$ and $u^n_{xx}$, respectively, one obtains

\begin{align*} |G(t_{k+1}, x) - G(t_{k+1}, \bar{x})| \leq C(1+|x|^{2(p_0+1)} + |\bar{x}|^{2(p_0+1)})|x-\bar{x}|, \quad x,\bar{x} \in {\mathbb{R}}, \end{align*}

uniformly in $t_{k+1}$. Since $Z^{n,t_m,x}_{t_k} = \mathcal{D}^n_{k+1} u^n(t_{k+1}, X^{n,t_k, \eta}_{t_{k+1}}) = G(t_{k+1}, \eta)$, where $\eta = X^{n,t_m,x}_{t_k}$, the previous bound yields

\begin{align*} |Z^{n,t_m,x}_{t_k} - Z^{n,t_m, \bar{x}}_{t_k}| \leq C(1+ |X^{n,t_m,x}_{t_{k}}|^{2(p_0+1)} + |X^{n,t_m, \bar{x}}_{t_{k}}|^{2(p_0+1)})|X^{n,t_m,x}_{t_{k}}-X^{n,t_m,\bar{x}}_{t_{k}}| \end{align*}

uniformly for each $t_m \leq t_k < T$. The inequality (58) then follows by applying the Cauchy–Schwarz inequality and standard $L_p$-estimates for the process $X^n$.

Proof of (59): This can be shown similarly to Lemma 4.2(iii), by considering the BSDE for the difference $\partial_x Y_{t_m}^{n,t_m,x}-\partial_x Y_{t_m}^{n,t_m,\bar x}$ instead of (47) itself.

Proof of (60): This can again be shown by repeating the proof of Lemma 4.2(iii), but now for the BSDE (55).

We return to the main proof. By our assumptions we have

\begin{equation*} {\mathbb{E}} |\hat\xi^{n,t_m,x} - \hat\xi^{n,t_m, \bar x}|^2 \le C(\Psi^2(x) + \Psi^2(\bar x))(1+|x|^2+|\bar x|^2) |x-\bar x|^{2\alpha}, \end{equation*}

where we use $|x-\bar x|^2 \le C(1+|x|^2+|\bar x|^2) |x-\bar x|^{2\alpha}.$ (The term $|x-\bar x|^2$ appears, for example, in the estimate of $(\partial_x X_T^{n,t_m,x })^2 - ( \partial_x X_T^{n,t_m, \bar x })^2$.) To see that

\begin{equation*} {\mathbb{E}} \bigg (\int_{(t_m,T]} | f_0^{t_m,x} (s-) -f_0^{t_m,\bar x}(s-) | d[B^n]_s \bigg )^2 \le C( \Psi^{10}(x) + \Psi^{10}(\bar x) ) (1+|x|^{2} +|\bar x|^{2}) |x-\bar x|^{2\alpha},\end{equation*}

we check the terms with the highest polynomial growth. We have to deal with terms like

\begin{equation*}{\mathbb{E}} \bigg (\!\int_{(t_m,T]} | Z^{n,t_m,x}_{s-}- Z^{n,t_m,\bar x}_{s-} |\,|\partial_xZ^{n,t_m,x}_{s-}|^{2} d[B^n]_s \!\bigg )^2\!\!\end{equation*}

and

\begin{equation*}{\mathbb{E}} \bigg(\!\int_{(t_m,T]} |\partial_x Z^{n,t_m,x}_{s-}|^2- |\partial_x Z^{n,t_m, \bar x}_{s-}|^{2} d[B^n]_s \bigg )^2\!,\end{equation*}

for example. We bound the first term by using (53) and (58):

\begin{eqnarray*} && {} {\mathbb{E}} \bigg (\int_{(t_m,T]} | Z^{n,t_m,x}_{s-}- Z^{n,t_m,\bar x}_{s-} | \,|\partial_xZ^{n,t_m,x}_{s-}|^{2} d[B^n]_s \bigg )^2 \\[3pt] &&\quad\le\, \bigg( {\mathbb{E}} \sup_s | Z^{n,t_m,x}_{s-}- Z^{n,t_m,\bar x}_{s-} |^4 \bigg)^{\frac{1}{2}} \bigg ({\mathbb{E}} \bigg (\int_{(t_m,T]} |\partial_x Z^{n,t_m,x}_{s-}|^{2} d[B^n]_s \bigg )^4 \bigg )^{\frac{1}{2}}\\[3pt] &&\quad\le\, C( \Psi^4(x) + \Psi^4(\bar x) )|x-\bar x|^2 \Psi^4(x). \end{eqnarray*}

We bound the second term by using (53) and (59):

\begin{align*} & {\mathbb{E}} \bigg (\int_{(t_m,T]} |\partial_x Z^{n,t_m,x}_{s-}|^2- |\partial_x Z^{n,t_m, \bar x}_{s-}|^{2} d[B^n]_s \bigg )^2 \\[3pt] &\quad\le C {\mathbb{E}} \int_{(t_m,T]} |\partial_x Z^{n,t_m,x}_{s-}|^2+ |\partial_x Z^{n,t_m, \bar x}_{s-}|^{2} d[B^n]_s \int_{(t_m,T]} |\partial_x Z^{n,t_m,x}_{s-}- \partial_x Z^{n,t_m, \bar x}_{s-}|^{2} d[B^n]_s \\[3pt] &\quad\le C( \Psi^2(x) + \Psi^2(\bar x) ) ( \Psi^8(x) + \Psi^8(\bar x) )|x-\bar x|^2 \\ &\quad\le C( \Psi^{10}(x) + \Psi^{10}(\bar x) ) (|x|^{2-2\alpha} +|\bar x|^{2-2\alpha}) |x-\bar x|^{2\alpha} \\[3pt] &\quad\le C( \Psi^{10}(x) + \Psi^{10}(\bar x) ) (1+|x|^{2} +|\bar x|^{2}) |x-\bar x|^{2\alpha}. \end{align*}

While all the other terms can be easily estimated using the results we have obtained already, for

\begin{align*}{\mathbb{E}} \bigg ( &\int_{(t_m,T]} |(f_3^{t_m,x} (s-\!) -f_3^{t_m,\bar x}(s-\!))\partial^2_x Z^{n,t_m,x}_{s-} |d[B^n]_s \bigg)^2 \\[3pt] &\le\, C( \Psi^{12}(x) + \Psi^{12}(\bar x) ) (1+|x|^{2} +|\bar x|^{2}) |x-\bar x|^{2\alpha}\end{align*}

we need the bound (60).

The result then follows from Gronwall’s lemma.

Remark 4.1. Under Assumption 2.3 we conclude that by Proposition 4.2 there exists a constant $C = C(b,\sigma, f,g, T,p,p_0) > 0$ such that

(61)\begin{align} |u^n(t_m,x) - u^n(t_m,\bar x) | &\le C( 1+ \Psi(x) + \Psi(\bar x))|x-\bar{x}|, \notag \\[3pt] |\mathcal{D}^n_{m+1} u^n(t_{m+1},X^{n,t_m,x}_{t_{m+1}})-\mathcal{D}^n_{m+1} u^n(t_{m+1},X^{n,t_m,x}_{t_{m+1}})| &\le C( 1+ \Psi^2(x) + \Psi^2(\bar x))|x-\bar{x}|, \notag \\[3pt] |u_x^n(t_m,x) - u_x^n(t_m,\bar x) | &\le C( 1+ \Psi^2(x) + \Psi^2(\bar x))|x-\bar{x}|, \notag \\[3pt] |\partial_x \mathcal{D}^n_{m+1} u^n(t_{m+1}, X^{n,t_m,x}_{t_{m+1}}) - \partial_x \mathcal{D}^n_{m+1} u^n(t_{m+1}, X^{n,t_m,{\bar x}}_{t_{m+1}})| &\leq C(1+ \hat{\Psi}(x) + \hat{\Psi}(\bar x))|x-\bar{x}|^{\alpha}, \notag \\[3pt] |\partial_x \mathcal{D}^n_{m+1} u^n(t_{m+1}, X^{n,t_m,x}_{t_{m+1}})| &\le C(1+ \Psi^2(x)), \end{align}

uniformly in $m = 0,1, \dots, n-1$, where

(62)\begin{align} \hat{\Psi}(x) \coloneqq 1+|x|^{6p_0+8}.\end{align}

In addition, for

\begin{equation*}\partial_x F^n(t_{m+1}, x)\coloneqq \partial_xf(t_{m+1}, x, u^n(t_m, x), \mathcal{D}^n_{m+1} u^n(t_{m+1},X^{n,t_m,x}_{t_{m+1}})),\end{equation*}

we have

(63)\begin{align}|\partial_x F^n(t_{m+1}, x) - \partial_x F^n(t_{m+1}, {\bar x}) |\leq C(1+ \hat{\Psi}(x) + \hat{\Psi}(\bar x))|x-\bar{x}|^{\alpha}\end{align}

uniformly in $m = 0,1, \dots, n-1$. The latter inequality follows from the assumption that the partial derivatives of f are bounded and Lipschitz continuous with respect to the spatial variables, from estimates proved in Proposition 4.2, and from those stated in (61) above.

From the calculations it can be seen that in general Assumption 2.3 cannot be weakened if one needs $ \partial_x F^n(t_{m+1}, x)$ to be locally $\alpha$-Hölder continuous.