# A Finite Element Method for Hamilton-Jacobi-Bellman equations

**Max Jensen
**

Department of Mathematical Sciences

University of Durham

Durham, United Kingdom

### Abstract:

Hamilton-Jacobi-Bellman equations describe how the cost of an optimal control problem changes as problem parameters vary.

For example, the cost of hedging a financial derivative typically depends on the volatility of an underlying asset. If the volatility evolves as unknown function within certain bounds, then Hamilton-Jacobi-Bellman equations give the highest possible cost which the hedging of the derivative can entail.

The equations are fully non-linear and not in divergence form. In this talk I will discuss how finite element methods can be adapted to solve these equations efficiently. The main question is, how it can be ensured that the numerical solutions converge to the correct viscosity solution of the problem.