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.

摘要

特征法是求解一些非线性二阶随机偏微分方程的有力工具，例如由一致动态效用满足的偏微分方程，参见 [N. El Karoui 和 M. Mrad，两个可解 SDE 和非线性效用随机 PDE 之间的精确连接，SIAM J. Financ。数学。4(1) (2013)，第 737–783 页；A. Matoussi 和 M. Mrad，由 Lévy 噪声驱动的动态效用和相关非线性 SPDE，预印本 (2020)，已提交出版。可在 https://hal.archives-ouvertes.fr/hal-03025475] 获得。在这种情况下解决方案$五(吨,z)$理论上是形式${\overline{X}}_{吨}\left(五\right(0,{\overline{\xi }}_{吨}\left(z\right)\left)\right)$在哪里$\overline{X}$和$\overline{是}$是两个 SDE 系统的解，$\overline{\xi }$是逆流$\overline{是}$和$五(0,.)$是初始条件。不幸的是，这种表示不是明确的，除非在简单的情况下$\overline{X}$和$\overline{是}$是线性方程组的解。这项工作的目的是利用这种表示来建立一个使用欧拉近似来近似解V的数值方案$X^{ñ}$和${\xi }^{ñ}$的X和ξ。这使我们能够避免 SPDE 在时间和空间上的复杂离散化，因为这种离散化似乎很难获得误差估计。我们将自己置于由 Lévy 噪声驱动的 SDE 框架中，我们首先建立了一个强大的收敛结果，在${\mathbf{大号}}_p$-范数，复合近似$X_{吨}^{ñ}\left({是}_{吨}^{ñ}\right(z\left)\right)$到复合变量$X_{吨}\left({是}_{吨}\right(z\left)\right)$，根据X和Y的近似值，它们是两个具有跳跃的 SDE 的解。然后，我们在反转单调随机流后将此结果应用于 HJB 类型的 Utility-SPDE。