Solving some stochastic partial differential equations driven by Lévy noise using two SDEs*

ABSTRACT

The method of characteristics is a powerful tool to solve some nonlinear second-order stochastic PDEs like those satisfied by a consistent dynamic utilities, see [N. El Karoui and M. Mrad, An exact connection between two solvable SDEs and a non linear utility stochastic PDEs, SIAM J. Financ. Math. 4(1) (2013), pp. 737–783; A. Matoussi and M. Mrad, Dynamic utility and related nonlinear SPDE driven by Lévy noise, preprint (2020), submitted for publication. Available at https://hal.archives-ouvertes.fr/hal-03025475]. In this situation the solution $V(t,z)$ is theoretically of the form ${\overline{X}}_t\left(V\right(0,{\overline{\xi }}_t\left(z\right)\left)\right)$ where $\overline{X}$ and $\overline{Y}$ are solutions of a system of two SDEs, $\overline{\xi }$ is the inverse flow of $\overline{Y}$ and $V(0,.)$ is the initial condition. Unfortunately this representation is not explicit except in simple cases where $\overline{X}$ and $\overline{Y}$ are solutions of linear equations. The objective of this work is to take advantage of this representation to establish a numerical scheme approximating the solution V using Euler approximations $X^N$ and ${\xi }^N$ of X and ξ. This allows us to avoid a complicated discretization in time and space of the SPDE for which it seems really difficult to obtain error estimates. We place ourselves in the framework of SDEs driven by Lévy noise and we establish at first a strong convergence result, in ${\mathbf{L}}_p$-norms, of the compound approximation $X_t^N\left(Y_t^N\right(z\left)\right)$ to the compound variable $X_t\left(Y_t\right(z\left)\right)$, in terms of the approximations of X and Y which are solutions of two SDEs with jumps. We then apply this result to Utility-SPDEs of HJB type after inverting monotonic stochastic flows.

KEYWORDS:

• SPDE driven by Lévy noise
• utility-SPDE
• compound random maps
• stochastic flows
• stochastic characteristics method
• strong convergence rate
• Euler schemes

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 That is for any compact K of ${\mathbb{R}}^d$ there exists a finite positive random variable $C\left(K\right)$ such that for any $x,y\in K$ we have $|\mathrm{\nabla }{X}_t(x,\omega )-\mathrm{\nabla }{X}_t(y,\omega )|\le C(K,\omega )|x-y{|}^{ϵ}$ a.s., see [18, Theorem 3.3] for details.

2 It writes ${\Vert Z\Vert }_{{\mathbf{L}}_p}\le {\Vert Z\Vert }_{{\mathbf{L}}_q}$ for any $0<p\le q$.

3 The paper [5] is a survey of strong discrete time approximations of jump-diffusion processes described by SDEs.

4 ${\int }_s^t\mu (s,X_u)\widehat{du}:=\underset{|\mathrm{\Delta }|\to 0}{lim}\sum _{k=0}^{n-1}\mu (t_{k+1},X_{t_{k+1}})(t_{k+1}-t_k)$

${\int }_s^t\sigma (s,X_u)\mathrm{d}{\overleftarrow{B}}_u:=\underset{|\mathrm{\Delta }|\to 0}{lim}\sum _{k=0}^{n-1}\sigma (t_{k+1},X_{t_{k+1}})(B_{t_{k+1}}-B_{t_k})$,

where $\mathrm{\Delta }=\{s=t_0<t_2<\cdots <t_n=t\}$ and $|\mathrm{\Delta }|=ma{x}_{0\le k\le n-1}(t_{k+1}-t_k)$.

5 $\mathbb{P}(J_t=n)={\mathrm{e}}^{-\lambda t}\frac{(\lambda t{)}^n}{n!}$, for any $n\in \mathbb{N}$