Penalty and penalty-like methods for nonlinear HJB PDEs

Highlights：

• We present novel techniques of handling nonlinearities involving first and second spatial derivatives of the unknown function, and the use of multiple penalty and penalty-like matrices, similar to Wilmott et al. [6]. However, in this work, we handle nonlinearities associated not just with the unknown value function as is done in Wilmott et al. [6] but also with its first and second spatial derivatives.

• We discuss the advantages that these new penalty and penalty-like methods have over the policy iteration [11]. algorithms.

• We prove the monotonic convergence of our numerical methods where applicable.

• We introduce a new type of policy iteration algorithms for problems with Americanstyle exercise rights, that have improved order of convergence and allow the use of variable time stepsizes, leading to more accurate and efficient solutions, when compared to the methods in Reisinger and Zhang [11].

• In our penalty and policy iteration algorithms, we introduce stopping criteria that avoid oscillatory iterate behaviour in certain degenerate cases.

• We provide numerical results on a variety of problems, including transaction cost models developed in Pooley et al. [21] and give results for the model introduced in Zhao and Ziemba [18].

• We extend the penalty iteration algorithm to the passport option pricing problem, which is a continuous control problem.