Many problems in finance can be posed as non-linear Hamilton Jacobi Bellman (HJB) Partial Integro Differential Equations (PIDEs). Examples of such problems include: dynamic asset allocation for pension plans, optimal operation of natural gas storage facilities, optimal execution of trades, and pricing of variable annuity products (e.g. Guaranteed Minimum Withdrawal Benefit).
This course will discuss general numerical methods for solving the HJB PDEs which arise from these types of problems. After an introductory lecture, we will give an example where seemingly reasonable methods do not converge to the correct (viscosity) solution of a nonlinear HJB equation. A set of general guidelines is then established which will ensure convergence of the numerical method to the viscosity solution. Emphasis will be placed on methods which are straightforward to implement. We then illustrate these techniques on some of the problems mentioned above.
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