## A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation.(English)Zbl 1372.91115

Summary: A fast preconditioned policy iteration method is proposed for the Hamilton-Jacobi-Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in $$\mathcal O(N\log N)$$ operations under some mild conditions, where $$N$$ is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 91G20 Derivative securities (option pricing, hedging, etc.) 60G40 Stopping times; optimal stopping problems; gambling theory

TFPDE
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### References:

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