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We propose a new, unified approach to solving jump-diffusion partialintegro-differential equations (PIDEs) that often appear in mathematicalfinance. Our method consists of the following steps. First, a second-orderoperator splitting on financial processes (diffusion and jumps) is applied tothese PIDEs. To solve the diffusion equation, we use standard finite-differencemethods, which for multi-dimensional problems could also include splitting onvarious dimensions. For the jump part, we transform the jump integral into apseudo-differential operator. Then for various jump models we show how toconstruct an appropriate first and second order approximation on a grid whichsupersets the grid that we used for the diffusion part. These approximationsmake the scheme to be unconditionally stable in time and preserve positivity ofthe solution which is computed either via a matrix exponential, or via P{\'a}deapproximation of the matrix exponent. Various numerical experiments areprovided to justify these results.

Efficient Solution of Backward Jump-Diffusion PIDEs with Splitting and Matrix Exponentials