what is the formula for Merton Jump Diffusion model

The Merton Jump Diffusion model is a mathematical model used to describe the behavior of stock prices over time. The formula for the model is:

dS(t) = (r-q)S(t)dt + σS(t)dW(t) + J(S,t)dN(t)

where:

dS(t) = the change in the stock price over time r = the risk-free interest rate q = the dividend yield σ = the stock price volatility dW(t) = the change in the Wiener process (a mathematical model used to describe the random movement of particles) J(S,t) = the jump component of the model, which represents sudden, discontinuous changes in the stock price due to news or other unexpected events dN(t) = the change in the Poisson process (a mathematical model used to describe the occurrence of rare events)