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Consider a stochastic differential equation (SDE) on some filtered probability space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$ : for all $t>0$

$$dX_t = u_tf(X_{t-})dt+ u_t g(X_{t-})dW_t + u_t\int_{|z|<1}h(X_{t-},z)\tilde{N}(dt,dz),\quad (\ast)$$

where $W$ is a standard Brownian motion, and $N$ is an independent Poisson measure on $\mathbb R_+\times (\mathbb R\setminus\{0\})$ with intensity measure $\nu$ and compensator $\tilde N$. Assume that $f,g :\mathbb R\to\mathbb R$ and $h:\mathbb R^2\to\mathbb R$ are Lipschitz.

  1. If $u=(u_t)_{t\ge 0}$ is $\mathbb F$-predictable and takes values in $\{0,1\}$, do we have the existence and uniqueness of the solution to $(\ast)$?
  2. If $u_t:=U(X_{t-})$ with $U:\mathbb R\to \{0,1\}$ being measurable, do we have the existence and uniqueness of the solution to $(\ast)$?

I referred to Chapter 5.3 of Protter [https://link.springer.com/book/10.1007/978-3-662-10061-5] (on SDEs driven by semimartingales) and to some lecture notes on SDEs driven by Lévy processes, while it seems not straightforward to me to obtain the desired wellposedness.

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  • $\begingroup$ When $u_t\equiv 1$ or $U$ is bounded and Lipschitz, the wellposedness clearly holds. For the above particular $u$ and $U$, do we have the same result? $\endgroup$
    – Fawen90
    Feb 12 at 11:07

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