  We propose a new, unified approach to solving jump-diffusion partial
integro-differential equations (PIDEs) that 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 multi-dimensional 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 which
supersets the grid that we used for the diffusion part. These approximations
make the scheme to be unconditionally stable in time and preserve positivity of
the solution which is computed either via a matrix exponential, or via P{\'a}de
approximation of the matrix exponent. Various numerical experiments are
provided to justify these results.
