Jump Diffusion

Jump-diffusion is an n-dimensional stochastic process X(t) satisfying the following stochastic differential equation (SDE):

dX(t) = A(t,X(t)) * dt + B(t,X(t)) * dW(t) + C(t,X(t)) * dN(t),

where A(t,x) is an n-dimensional vector, B(t,x) is an n-by-m matrix, C(t,x) is an n-by-r matrix, W(t) is an m-dimensional Brownian motion and N(t) is an r-dimensional counting process. In other words, jump-diffusions are solutions of stochastic differential equations containing stochastic integrals with respect Brownian motion and various jump processes.

Properties of jump-diffusions are an integral part of stochastic calculus. Several important models in empirical finance, financial engineering, engineering and physics are phrased in terms of jump-diffusions.