Backward Stochastic Volterra Integral Equations ---Theory and Applications
主 题: Backward Stochastic Volterra Integral Equations ---Theory and Applications
报告人: Prof.Jiongmin Yong (University of Central Florida)
时 间: 2009-05-18 下午14:00 - 15:00
地 点: 理科一号楼 1114
Ponrtryagin type Maximum principles of optimal control are well-known for various type state equations, including ordinary differential equations, partial differential equations (either stationary or evolutionary), (deterministic Volterra) integral equations, and stochastic differential equations, etc. The main feature of Pontryagin type maximum principle is the adjoint equation (besides the maximum condition). When the controlled state equation is an evolution equation given initial condition, the corresponding adjoint equation will be a similar (evolution) equation given the terminal condition. In the deterministic case, a terminal value problem for an evolution equation is equally difficult as an initial value problem for the same type equation. Whereas, in the stochastic differential equation case, the adjoint equation has to be a so-called backward stochastic differential equation (BSDE, for short), whose study leads to some quite interesting theory in the past a couple of decades. Now, if the controlled state equation is a stochastic Volterra integral equation, what will be the corresponding adjoint equation? Naively (and naturally), mimicking the stochastic differential equation case, it should be a backward stochastic Volterra integral equation (BSVIE, for short). Yes, but, what should it look like? Is the extension from BSDEs to BSVIEs routine? In this talk, we will present a general theory of BSVIEs, as well as some interesting applications, including a maximum principle for controlled stochastic Volterra integral equations, and dynamic risk measure in mathematical finance.
