On the uniqueness and existence of solutions of stochastic equations with respect to semimartingales

L. I. Gal'čuk

Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 4, pp. 782-795

Résumé

Let $a=(a_t)$, $t\in[0,\infty[$, be a predictable process with locally integrable variation, $m=(m_t)$ be a continuous local martingale, $p$ be a stochastic integer-valued measure on $\mathfrak B([0,\infty[)\times\mathfrak B(R^d\setminus\{0\})$ and $\lambda$ be a dual predictable projection of $p$. The processes $a$ and $m$ take values in $R^d$, $d\ge 1$. The uniqueness and existence theorem is proved lor the solutions of a stochastic integral equation \begin{gather*} Y_t(\omega)=N_t(\omega)+\int_0^t\sum_{j=1}^df^j(\omega,s,Y_{s-}(\omega))\,da_s^j(\omega)+ \int_0^t\sum_{j=1}^dg^j(\omega,s,Y_{s-}(\omega))\,dm_s^j(\omega)+\\ \int_0^t\int_{|u|\le 1}h(\omega,s,u,Y_{s-}(\omega))(p-\lambda)(\omega,ds,du)+\\ \int_0^t\int_{|u|>1}h(\omega,s,u,Y_{s-}(\omega))p(\omega,ds,du), \end{gather*} where $N=(N_t)$ is a known process the paths of which are right-hand continuous and have left-hand limits. The functions $f(\omega,s,x)$, $g(\omega,s,x)$, $h(\omega,s,u,x)$ satisfy the Lipschitz conditions in $x$ and are predictable in other variables.