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What you are saying might be correct for discrete time processes. In continuous time the process $$ dX_t = X_t^2 dW_t,\quad X_0 > 0 $$ is stationary but not mean reverting.


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As you have guessed correctly, these type of questions can be answered using Ito's Lemma.We have: \begin{equation} d(M_t)= d(Z_t e^{\int_0^tF(Z_u)du})=d(Z_t) e^{\int_0^tF(Z_u)du}+Z_t d(e^{\int_0^tF(Z_u)du})+d(Z_t)d(e^{\int_0^tF(Z_u)du}) \end{equation} For the first two terms on R.H.S, we have: \begin{equation} d(Z_t) e^{\int_0^tF(Z_u)du} = (f(W_t)dW_t + ...


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I am not absolutely sure what you mean by diffusion-jump but if you mean jump-diffusion. Here are some references: Chapter 15, Concepts and Practice of Mathematical Finance, Joshi Cont and Tankov, Financial Modelling with Jump Processes Using Monte Carlo Simulation and Importance Sampling to Rapidly Obtain Jump-Diffusion Prices of Continuous Barrier ...



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