Efficient Solution of Backward Jump-Diffusion Partial Integro-Differential Equations with Splitting and Matrix Exponentials

42 Pages Posted: 15 Jun 2016

Date Written: January 27, 2016

Abstract

We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs), which often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finite-difference methods, which for multidimensional problems could also include splitting on various dimensions. For the jump part, we transform the jump integral into a pseudo-differential operator. Then, for various jump models, we show how to construct an appropriate first- and second order approximation on a grid that supersets the grid we used for the diffusion part. These approximations make the scheme unconditionally stable in time and preserve the positivity of the solution, which is computed either via a matrix exponential or via a Padé approximation of the matrix exponent. Various numerical experiments are provided to justify these results.

Keywords: jump-diffusion, partial integro-differential equation (PIDE), splitting, matrix exponential, unconditionally stable schemes

Suggested Citation

Itkin, Andrey, Efficient Solution of Backward Jump-Diffusion Partial Integro-Differential Equations with Splitting and Matrix Exponentials (January 27, 2016). Journal of Computational Finance, Vol. 19, No. 3, 2016, Available at SSRN: https://ssrn.com/abstract=2794898

Andrey Itkin (Contact Author)

New York University (NYU) ( email )

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