Using kernel-based collocation methods to solve a delay partial differential equation with application to finance

We consider a delay partial differential equation arising in a jump diffusion model of option pricing. Under the mean-reverting jump-diffusion model, the price of options on electricity satisfies a second order partial differential equation. In this paper, we use positive definite kernels to discretise the PDE in spatial direction and achieve a linear system of first order differential equation with respect to time. We impose homogenous boundary conditions of the PDE by using a manipulated version of kernels called `recursive kernels'. The proposed methods are fast and accurate with low computational complexity. No integrations are necessary and the time dependent system of differential equations can be solving analytically. Illustrative example is included to demonstrate the validity and applicability of the new techniques.