| Abstract: |
| In this talk, I will presents deep learning methods to solve some stochastic optimal control (SOC in short) problems. The first SOC is an application for solving initial path optimization of mean-field systems with memory where we consider the problem of finding the optimal initial investment strategy for a system modeled by a linear McKean-Vlasov (mean-field) stochastic differential equation with positive delay, driven by a Brownian motion and a pure jump Poisson random measure. The problem is to find the optimal initial values for the system in this period, before the system starts at t equal zero. Because of the delay in the dynamics, the system will after startup be influenced by these initial investment values. It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By using this equivalence, we can find implicit expression for the optimal investment. We deep machine learning algorithms to solve explicitly some examples
The second type of dynamic is a second BSDE that represent a fully nonlinear second order PDE.
As an application here, we study alpha-hypergeometric model with uncertain volatility (UV) where we derive a worst-case scenario for option pricing. The approach is based on the connection between a certain class of nonlinear partial differential equations of HJB-type (G-HJB equations), that govern the nonlinear expectation of the UV model and that provide an alternative to the difficult model calibration problem of UV models, and second-order backward stochastic differential equations (2BSDEs). Using a deep learning based approximation of the underlying 2BSDE we can find the solution of our problem. |
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