"Methods of machine learning in stochastic control and high dimensional PDEs"
Apr 6, 2022 Schedule:
- Tea Time - Virtual ( Zoom)
- 03:00 to 03:30 PM Eastern Time (US and Canada)
- Colloquium - F2F ( 499 DSL) / Virtual ( Zoom)
- 03:30 to 04:30 PM Eastern Time (US and Canada)
In this talk, a review of numerical methods for stochastic control problems is presented, with emphasis on the most recent methods via machine learning. A major advantage of machine learning methods is tractability in high dimensions, e.g., d=100. An optimization problems can simply be solved by a suitable version of gradient descent algorithm. A stochastic control problem requires an adaptability of the solution to the historical information. In order to address this, we write the dynamic programing principle for the control problem through backward stochastic differential equation, which is equivalent to Hamilton-Jacobi-Bellman (HJB) PDE for the control problem. Then, we formulate loss function such that the solution to machine learning problem approximately solves the dynamic programing equation. In addition, finding gradient and Hessian of the the solution of the HJB in high dimension is costly. We adopt a Monte Carlo method to remove dependence of the loss function to the gradient and Hessian as they are present in the PDE.