An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach

Abstract

This paper proposes a new analytical approximation scheme for the representation of the forward–backward stochastic differential equations (FBSDEs) of Ma and Zhang (Ann Appl Probab, 2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment is taken into account.

Appendices

Appendix 1: Proof of Lemma 5.1

We prove the assertion by induction. First,

$$\begin{aligned} \frac{\partial }{\partial \varepsilon }X_{s}^{\varepsilon ,t,x}= & {} \sum _{i=1}^d \int _t^s \partial _x X_s^{\varepsilon ,t,x} (\partial _x X_u^{\varepsilon ,t,x})^{-1} \sigma _i(u, {X}_u^{\varepsilon ,t,x}) dW_u^i\end{aligned}$$

(82)

$$\begin{aligned}&+\,\varepsilon \sum _{i=1}^d \int _t^s \partial _x X_s^{\varepsilon ,t,x} (\partial _x X_u^{\varepsilon ,t,x})^{-1} \partial _x \sigma _i(u, {X}_u^{\varepsilon ,t,x})\sigma _i(u, {X}_u^{\varepsilon }) du.\quad \end{aligned}$$

(83)

Since $\partial _x X_s^{\varepsilon ,t,x}, (\partial _x X_s^{\varepsilon ,t,x})^{-1} \in \mathcal{K}_0^T$, we have $\frac{\partial }{\partial \varepsilon }X_{s}^{\varepsilon ,t,x} \in \mathcal{K}_1^T$.

For $k\ge 2$, $\frac{1}{k!}\frac{\partial ^k}{\partial \varepsilon ^k}X_{s}^{\varepsilon ,t,x}=\left( \frac{1}{k!}\frac{\partial ^k}{\partial \varepsilon ^k}X_{s}^{\varepsilon ,t,x,1},\ldots , \frac{1}{k!}\frac{\partial ^k}{\partial \varepsilon ^k}X_{s}^{\varepsilon ,t,x,d} \right)$ is recursively determined by the following:

$$\begin{aligned} \frac{1}{k!}\frac{\partial ^k}{\partial \varepsilon ^k}X_{s}^{\varepsilon ,t,x,j}= & {} \sum _{ \mathbf{l}_\beta ,\mathbf{d}_\beta }^{(k)} \int _t^s \left( \prod _{j=1}^{\beta } \frac{1}{l_j!} \frac{\partial ^{l_j}}{\partial \varepsilon ^{l_j}}X_{u}^{\varepsilon ,t,x, d_j} \right) \partial _{{d}_\beta }^{\beta } b^j(u,{X}_u^{\varepsilon ,t,x})du\end{aligned}$$

(84)

$$\begin{aligned}&+ \sum _{\mathbf{l}_\beta , \mathbf{d}_\beta }^{(k-1)} \int _t^s \left( \prod _{j=1}^{\beta } \frac{1}{l_j!} \frac{\partial ^{l_j}}{\partial \varepsilon ^{l_j}}X_{u}^{\varepsilon ,t,x, d_j} \right) \sum _{i=1}^d \partial _{\mathbf{d}_\beta }^{\beta } \sigma _i^j(u,{X}_u^{\varepsilon ,t,x})dW_u^i\end{aligned}$$

(85)

$$\begin{aligned}&+\,\varepsilon \sum _{ \mathbf{l}_\beta ,\mathbf{d}_\beta }^{(k)} \int _t^u \left( \prod _{j=1}^{k} \frac{1}{l_j!} \frac{\partial ^{l_j}}{\partial \varepsilon ^{l_j}}X_{u}^{\varepsilon ,t,x, d_j} \right) \sum _{i=1}^d \partial _{\mathbf{d}_k}^{k} \sigma _i^j(u,{X}_u^{\varepsilon ,t,x})dW_s^i\nonumber \\ \end{aligned}$$

(86)

where $\partial _{{d}_\beta }^\beta = {\partial ^\beta \over \partial {x}_{d_1} \cdots \partial {x}_{d_\beta }}$,

$$\begin{aligned} \sum _{\mathbf{l}_\beta ,\mathbf{d}_\beta }^{(l)} := \sum _{\beta =1}^l \sum _{\mathbf{l}_\beta \in L_{l,\beta }} \sum _{\mathbf{d}_\beta \in \{1,\cdots ,d\}^\beta } \frac{1}{\beta !}, \end{aligned}$$

(87)

and $L_{l,\beta } := \left\{ \mathbf{l}_\beta = (l_1,\ldots ,l_\beta );\ \sum _{j=1}^\beta l_j = l;\ (l, l_j, \beta \in \mathbf {N}) \right\}$. The above SDE is linear and the order of the Kusuoka-Stroock function $\frac{1}{i!}\frac{\partial ^i}{\partial \varepsilon ^i}X_{s}^{\varepsilon ,t,x}$ is determined inductively by the term

$$\begin{aligned}&\sum _{\mathbf{l}_\beta , \mathbf{d}_\beta }^{(i-1)} \frac{1}{\beta !} \int _t^s \partial X_s^{\varepsilon ,t,x} \left( \partial X_u^{\varepsilon ,t,x}\right) ^{-1} \left( \prod _{j=1}^{\beta } \frac{1}{l_j!} \frac{\partial ^{l_j}}{\partial \varepsilon ^{l_j}}X_{u}^{\varepsilon ,t,x, d_j} \right) \sum _{i=1}^d \partial _{\mathbf{d}_\beta }^{\beta } \sigma _i(u,{X}_u^{\varepsilon ,t,x})dW_u^i \nonumber \\&\quad \in \mathcal{K}_i^T. \end{aligned}$$

(88)

Then, $\frac{1}{i!}\frac{\partial ^i}{\partial \varepsilon ^i}X_{s}^{\varepsilon ,t,x} \in \mathcal{K}_i^T$. $\Box$

Appendix 2: Proof of Proposition 5.1

Let $(\varphi _n)_{n \in \mathbf{N}} \subset C_b^{\infty }(\mathbf{R}^d)$ be a mollifier converging to $\varphi$. The following Taylor formula

$$\begin{aligned} \varphi _n(F_{T}^{\varepsilon ,t,x})= & {} \varphi _n(F_{T}^{0,t,x})+\sum _{i=1}^{N} \frac{\varepsilon ^i}{i!} \frac{\partial ^i}{\partial \varepsilon ^i} \varphi _n(F_{T}^{\varepsilon ,t,x})|_{\varepsilon =0}\\&+\,\varepsilon ^{N+1} \int _{0}^1 \frac{(1-u)^N}{N!} \frac{\partial ^{N+1}}{\partial \nu ^{N+1}}\varphi _n(F_{T}^{\nu ,t,x})|_{\nu =\varepsilon u}du, \end{aligned}$$

and the integration by parts on the Wiener space, we have

$$\begin{aligned} E[\varphi _n(F_{T}^{\varepsilon ,t,x})]= & {} E[\varphi _n(F_{T}^{0,t,x})] +\sum _{i=1}^{N} \varepsilon ^{i} \sum _{k}^{(i)} E[ \partial _{\alpha ^{(k)}} \varphi _n(F_{T}^{0,t,x}) \prod _{l=1}^k F_{\beta _{l},T}^{0,t,x,\alpha _l} ] \\&+\,\varepsilon ^{N+1} \int _{0}^1{(1-u)^N} (N+1)\sum _{k}^{(N+1)}E \left[ \partial _{\alpha ^{(k)}} \varphi _n(F_{T}^{\varepsilon u,t,x}) \prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} \right] du\\= & {} E[\varphi _n(F_{T}^{0,t,x})] +\sum _{i=1}^{N}{\varepsilon ^i} E[ \varphi _n(F_{T}^{0,t,x}) \pi _{i,T}^{t,x} ]\\&+\,\varepsilon ^{N+1} \int _{0}^1 {(1-u)^N} (N+1)\sum _{k}^{(N+1)} E[\partial _{\alpha ^{(1)}} \varphi _n(F_{T}^{\varepsilon u,t,x})H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\\&\times \quad \prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} )]du, \end{aligned}$$

where, $\pi _{i,T}^{t,x} =\sum _{k}^{(i)} H_{\alpha ^{(k)}}(F_{T}^{0,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{0,t,x,\alpha _l} )=\sum _{k}^{(i)} H_{\alpha ^{(k)}}(X_{1,T}^{0,t,x},\prod _{l=1}^k X_{\beta _{l}+1,T}^{0,t,x,\alpha _l} )$.

Therefore, we have

$$\begin{aligned}&E[\varphi _n(X_T^{\varepsilon ,t,x})]\nonumber \\&\quad =E[\varphi _n(\bar{X}_{T}^{t,x})] +\sum _{i=1}^{N}{\varepsilon ^i} E[ \varphi _n(\bar{X}_{T}^{t,x}) \pi _{i,T}^{t,x} ]\nonumber \\&\qquad +\,\varepsilon ^{N+1} \int _{0}^1 {(1-u)^N} (N+1)\sum _{k}^{(N+1)} E[\partial _{\alpha ^{(1)}} \varphi _n(\tilde{X}_T^{\varepsilon u,t,x})H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\nonumber \\&\qquad \times \prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} )]du, \end{aligned}$$

(89)

where $\tilde{X}_T^{\varepsilon u,t,x}=X_T^{0,t,x}+\varepsilon F_{T}^{\varepsilon u,t,x}$, $u\in [0,1]$. By Proposition 4.1 with Lemma 4.1 and 5.1, we have $\sum _{k}^{(N+1)}H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} ) \in \mathcal{K}_{N+2}^T$.

Then, we obtain

$$\begin{aligned}&\left| E[\varphi _n(X_T^{\varepsilon ,t,x})]-E[\varphi _n(\bar{X}_T^{t,x})]+\sum _{i=1}^{N} \varepsilon ^{i} E[\varphi _n(\bar{X}_T^{t,x}) \pi _{i,T}^{t,x}] \right| \nonumber \\&\quad \le \varepsilon ^{N+1} \Vert \nabla \varphi _n \Vert _{\infty } (T-t)^{(N+2)/2}. \end{aligned}$$

(90)

Finally, by mollifier argument, we have the assertion. $\square$

Appendix 3: Proof of Proposition 5.2

For a mollifier $(\varphi _n)_{n \in \mathbf{N}} \subset C_b^{\infty }(\mathbf{R}^d)$ converging to $\varphi$, we differentiate the expansion (89) of $E[\varphi _n(X_T^{\varepsilon ,t,x})]$ with respect to initial x as follows: for $1\le \eta \le d$,

$$\begin{aligned}&\frac{\partial }{\partial x_\eta }E[\varphi _n(X_T^{\varepsilon ,t,x})]\\&\quad =\frac{\partial }{\partial x_\eta }E[\varphi _n(\bar{X}_{T}^{t,x})] +\sum _{i=1}^{N}{\varepsilon ^i} \frac{\partial }{\partial x_\eta }E[ \varphi _n(\bar{X}_{T}^{t,x}) \pi _{i,T}^{t,x} ]\\&\qquad +\,\varepsilon ^{N+1} \int _{0}^1 {(1-u)^N} (N+1)\sum _{k}^{(N+1)} \frac{\partial }{\partial x_\eta }E\left[ \partial _{\alpha ^{(1)}} \varphi _n(\tilde{X}_T^{\varepsilon u,t,x})H_{\alpha ^{(k-1)}}\left( F_{T}^{\varepsilon u,t,x},\right. \right. \\&\qquad \prod _{l=1}^k\left. \left. { F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} }\right) \right] du. \end{aligned}$$

We have

$$\begin{aligned} \frac{\partial }{\partial x_\eta }E[\varphi _n(\bar{X}_T^{t,x})] =\sum _{j=1}^d E[\partial _{j} \varphi _n(\bar{X}_T^{t,x}) \partial _{\eta } \bar{X}_T^{t,x,j} ]=E[ \varphi _n(\bar{X}_T^{t,x}) N_{0,T}^{t,x,\eta }], \end{aligned}$$

(91)

and, for $1\le i \le N$,

$$\begin{aligned} \frac{\partial }{\partial x_\eta } E[\varphi _n(\bar{X}_T^{t,x}) \pi _{i,T}^{t,x}]= & {} \sum _{j=1}^d \{ E[ \partial _{j} \varphi _n(\bar{X}_T^{t,x}) \partial _{\eta } \bar{X}_T^{t,x,j} \pi _{i,T}^{t,x} ] +E[\varphi _n(\bar{X}_T^{t,x}) \partial _{\eta } \pi _{i,T}^{t,x}] \}\nonumber \\= & {} E[\varphi (\bar{X}_T^{t,x})N_{i,T}^{t,x,\eta }]. \end{aligned}$$

(92)

Moreover, $1\le \eta \le d$, $u \in [0,1]$,

$$\begin{aligned}&\frac{\partial }{\partial x_\eta }E[\partial _{\alpha ^{(1)}} \varphi _n(\tilde{X}_T^{\varepsilon u,t,x})H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} )]\\&\quad =\sum _{j=1}^d E[\partial _{j,\alpha ^{(1)}} \varphi _n(\tilde{X}_T^{\varepsilon u,t,x})\partial _{\eta } \tilde{X}_T^{\varepsilon u,t,x,j}H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} )]\\&\qquad +\,E[\partial _{\alpha ^{(1)}} \varphi _n(\tilde{X}_T^{\varepsilon u,t,x}) \partial _\eta H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} )]\\&\quad = E \Biggl [\partial _{\alpha ^{(1)}} \varphi _n(\tilde{X}_T^{\varepsilon u,t,x}) \Biggl \{ \sum _{j=1}^dH_j( \tilde{X}_T^{\varepsilon u,t,x}, \partial _{\eta } \tilde{X}_T^{\varepsilon u,t,x,j}H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} ))\\&\qquad +\, \partial _\eta H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l}) \Biggr \} \Biggr ] \end{aligned}$$

where $\sum _{j=1}^dH_j( \tilde{X}_T^{\varepsilon u,t,x}, \partial _{\eta } \tilde{X}_T^{\varepsilon u,t,x,j}H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} ))$ $+ \partial _\eta H_{\alpha ^{(k-1)}}(F_{T}^{\varepsilon u,t,x},\prod _{l=1}^k F_{\beta _{l},T}^{\varepsilon u,t,x,\alpha _l} ) \in \mathcal{K}_{N+1}^T$. Therefore, we have the assertion. $\square$

Appendix 4: Proof of Lemma 5.2

$u^{\varepsilon ,0,N}$ and $\partial _x u^{\varepsilon ,0,N}\sigma$ are represented as

$$\begin{aligned} u^{\varepsilon ,0,N}(t,x)= & {} E [g(\bar{X}_T^{t,x}) \vartheta _{T} ] +E \left[ \int _t^{T} f(s,\bar{X}_s^{t,x}, 0,0)\vartheta _{s} ds \right] ,\\ \partial _x u^{\varepsilon ,0,N}\sigma (t,x)= & {} \left\{ E \left[ g(\bar{X}_T^{t,x}) \gamma _{T} \right] +E \left[ \int _t^{T} f(s,\bar{X}_s^{t,x}, 0,0)\gamma _{s} ds \right] \right\} \varepsilon \sigma (t,x) , \end{aligned}$$

where $\vartheta _s=1+ \sum _{i=1}^{N} \varepsilon ^i \pi _{i,s}^{t,x}$ and $\gamma _s=\sum _{i=0}^{N} \varepsilon ^i N_{i,s}^{t,x}$. Remark that $\vartheta _s \in \mathcal{K}_{\min \{0,1,\ldots ,N \} }^T=\mathcal{K}_0^T$ and $\gamma _s \in \mathcal{K}_{\min \{-1,0,\ldots ,N-1\}}^T=\mathcal{K}_{-1}^T$. Since g is Lipschitz continuous and of linear growth, we obtain

$$\begin{aligned}&\left| E [g(\bar{X}_T^{t,x}) \vartheta _{T} ]\right| \le \Vert g(\bar{X}_T^{t,x})\Vert _{L^p} \Vert \vartheta _{T}\Vert _{L^q} \le C(T,x), \end{aligned}$$

(93)

$$\begin{aligned}&\left| E [g(\bar{X}_T^{t,x}) \gamma _{T} ] \varepsilon \sigma (t,x) \right| \le \varepsilon C_L C(T,x). \end{aligned}$$

(94)

Also, as f is of linear growth, we have

$$\begin{aligned}&\left| E [\int _t^T f(s,\bar{X}_s^{t,x},0,0) \vartheta _s ds ]\right| \le \int _t^T C(T,x) ds, \end{aligned}$$

(95)

$$\begin{aligned}&\left| E [\int _t^T f(s,\bar{X}_s^{t,x},0,0) \gamma _s ds ] \varepsilon \sigma (t,x) \right| \le \int _t^T C(T,x) \frac{1}{\sqrt{s-t}} ds, \end{aligned}$$

(96)

where C(T, x) denotes a non-negative, non-decreasing and finite function of at most polynomial growth in x depending on T. Then, we obtain estimates for $u^{\varepsilon ,0,N}$ and $\partial _x u^{\varepsilon ,0,N}\sigma$:

$$\begin{aligned} |u^{\varepsilon ,0,N}(t,x) |\le & {} C(T,x), \end{aligned}$$

(97)

$$\begin{aligned} | \partial _x u^{\varepsilon ,0,N}\sigma (t,x) |\le & {} C(T,x). \end{aligned}$$

(98)

Note that for $k \ge 1$,

$$\begin{aligned} u^{\varepsilon ,k,N}(t,x)= & {} E [g(\bar{X}_T^{t,x}) \vartheta _{T} ]\\&+\,E \left[ \int _t^{T} f(s,\bar{X}_s^{t,x}, u^{\varepsilon ,k-1,N}(s,\bar{X}_s^{t,x}),\partial _x u^{\varepsilon ,k-1,N}\right. \\&\left. \times \,\sigma (s,\bar{X}_s^{t,x}))\vartheta _{s} ds \right] ,\\ \partial _x u^{\varepsilon ,k,N}\sigma (t,x)= & {} E [g(\bar{X}_T^{t,x}) \gamma _{T} ]\varepsilon \sigma (t,x)\\&+\,E \left[ \int _t^{T} f(s,\bar{X}_s^{t,x}, u^{\varepsilon ,k-1,N}(s,\bar{X}_s^{t,x}),\partial _x u^{\varepsilon ,k-1,N}\right. \\&\left. \times \,\sigma (s,\bar{X}_s^{t,x}))\gamma _{s} ds \right] \varepsilon \sigma (t,x), \end{aligned}$$

with (93), (94) and

$$\begin{aligned}&\left| E\left[ \int _t^T f(s,\bar{X}_s^{0,t,x},u^{\varepsilon ,k-1,N}(s,\bar{X}_s^{t,x}),\partial _x u^{\varepsilon ,k-1,N}\sigma (s,\bar{X}_s^{t,x})) \vartheta _s ds \right] \right| \nonumber \\&\le \int _t^T C(T,x) ds, \end{aligned}$$

(99)

$$\begin{aligned}&\left| E\left[ \int _t^T f(s,\bar{X}_s^{0,t,x},u^{\varepsilon ,k-1,N}(s,\bar{X}_s^{t,x}),\partial _x u^{\varepsilon ,k-1,N}\sigma (s,\bar{X}_s^{t,x})) \gamma _s ds\right] \varepsilon \sigma (t,x)\right| \nonumber \\&\le \int _t^T C(T,x) \frac{1}{\sqrt{s-t}} ds. \end{aligned}$$

(100)

Then, recursively using (93), (94), (99) and (100) we obtain (49) and (50). $\square$