Arash Fahim
|
SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Arash Fahim Abstract. We present an overview of Monte Carlo methods for partial differential equations (PDE) motivated by applications to financial risk management. PDE's appear very often in financial mathematics. As a classical example, in the Black-Scholes model pricing some derivatives leads to linear PDE which does not admit a closed-form solution. Since Monte Carlo is the least sensitive method among approximation methods to dimension of the problem, it is widely popular in th presence of several sources of uncertainty. In addition, modeling frictions in the financial markets leads to models with nonlinear PDE's. We introduce some of these models to motivate the title of the talk. Then, we discuss an efficient Monte Carlo approximation scheme for semi-linear and fully nonlinear PDEs, and analyze the discretization error and the error due to approximation of expectations used in the scheme. Some implementations performed on the test problems with fully nonlinear PDEs in high dimensions will be shown. At the end, we discuss further developments and direction of future works in the subject. |