## Existence and uniqueness of solutions of stochastic differential equations with non-Lipschitz diffusion and Poisson measure.(Ukrainian, English)Zbl 1224.60156

Teor. Jmovirn. Mat. Stat. 80, 43-54 (2009); translation in Theory Probab. Math. Stat. 80, 47-59 (2010).
The authors consider the following stochastic differential equation: $dX(t)=a(X(t))dt+g(X(t))dW(t)+ \int_{\mathbb R}q_1(X(t),y)\tilde{\nu}(dt, dy)+ \int_{\mathbb R}q_2(X(t),y)\mu(dt,dy),$ where $$W(t)$$ is a Wiener process, $$\nu(dt,dy)$$ is a Poisson measure, $$E\nu(dt,dy)=\Pi(dy)dt$$, $$\tilde{\nu}(dt,dy)=\nu(dt,dy)-\Pi(dy)dt$$ is a centered Poisson measure, $$\Pi(\cdot)$$ is a $$\sigma$$-finite measure on the $$\sigma$$-algebra of Borel sets of $$\mathbb R$$, $$\mu(dt, dy)$$ is a noncentered Poisson measure such that $$E\mu(dt, dy) = m(dy) dt$$, and $$m(\cdot)$$ is a finite measure on the $$\sigma$$-algebra of Borel sets of $$\mathbb R$$. The coefficients $$a(x)$$, $$g(x)$$, $$q_1(x,y)$$, and $$q_2(x, y)$$ are nonrandom measurable functions. The drift coefficient $$a(x)$$ is Lipschitzian, while the diffusion $$g(x)$$ is Hölderian. The existence and uniqueness of a solution of this stochastic differential equation is proved. It is shown that the pathwise uniqueness of a solution and the existence of a weak solution imply the existence of a strong solution for the equation.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H05 Stochastic integrals 60J65 Brownian motion

### Citations:

Zbl 0242.60003; Zbl 0557.60041; Zbl 1103.60005; Zbl 0684.60040
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