We are going to talk about two independent approaches for solving certain types of stochastic control problems numerically.
The first part is about a method for solving optimal stopping problems based on the signature of the underlying path. This method targets particularly non Markovian situations. The algebraic properties of the signature and the introduction of randomized stopping times allow to approximate the optimal stopping value by solutions of deterministic maximization problems derived from the expected signature. The introduction of deep signature stopping rules yields numerical effectiveness as demonstrated by concrete examples.
In the second part we are concerned with solving a certain class of stochastic control problems with a finite set of control policies in Markovian settings. Rooted in the regression approach derived from the dynamic programming principle, this method reinforces the regression basis with the value function of the previous step in the backwards induction. A hierarchical structure of value functions is introduced in order to avoid deep recursions. This method targets the case of a high-dimensional underlying path, allowing to improve regression accuracy with little computational increase in comparison to adding, for example, one higher degree of polynomials to the basis.
Paul is a postdoctoral researcher at the Humboldt University of Berlin in the Applied Financial Mathematics & Applied Stochastic Analysis research group.
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