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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
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