Qi Feng : Cubature method and machine learning to solve Path Dependent PDE(PPDE)

Posted by Cheng Ouyang , part of the Statistics and Data Science Seminar.

At

March 4, 2020, 3 p.m.

In

636 SEO

Abstract

The classical models for asset processes in math finance are SDEs driven by Brownian motion of the following type $X_t=x+\int_0^tb(s,X_s)ds+\int_0^t\sigma(s,X_s)\circ dB_s$. Then $u(t,X_t)=\mathbb E[{g(X_T)}|\mathcal F_{t}^X]$ is a deterministic function of $X_t$ and $u(t,x)$ solves a parabolic PDE. The cubature formula is first constructed to numerically compute functionals like $\mathbb E^{\mathbb P}[g(X_T)]$, which can be seen as a discrete approximation of the infinite dimensional Wiener measure (denoted as $\mathbb P$). In this talk, we will consider that the asset process follows a rough volatility model. For example, in the rough Heston model, the process $X_t$ is the solution of Volterra type SDEs. In this case, $X$ itself is non-Markovian, then $u(t,X_t)$ will depend on the whole path of $(X_s)_{0\le s\le t}$ and $u(t,X_{[0,t]})$ solves the so-called Path Dependent PDE (PPDE). We propose a new algorithm to numerically solve PPDE by using cubature type formulas for Volterra SDEs. The cubature formula for Volterra SDEs is solved by using machine learning method. In the end, I will show some numerical examples. The talk is based on a joint work with Jianfeng Zhang.