Drift reduction method for SDEs driven by heterogeneous singular Lévy noise

Abstract

We study SDE

$$d{X}_t=b(X_t)dt+A(X_{t-})d{Z}_t,X_0=x\in {ℝ}^d,t\ge 0,$$

where $Z={(Z^1,\dots ,Z^d)}^T$, with $Z^i,i=1,\dots ,d$ being independent one-dimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is one-dimensional symmetric ${\mathit{\alpha }}_i$-stable process, where ${\mathit{\alpha }}_i\in (0,2)$ are not necessarily equal but satisfy certain balance condition which prevents hypoelliptic effects. Under certain assumptions on b, A and Z we show that the weak solution to the SDE is uniquely defined and is a Markov process. We also provide a representation of the transition probability density and establish Hölder regularity of the corresponding transition semigroup. The method we propose is based on a reduction of an SDE with a drift term to another SDE without such a term but with coefficients depending on time variable. Such a method has the same spirit as the classic characteristic method and seems to be of independent interest.