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On the uniqueness and existence of solutions of stochastic equations with respect to semimartingales
L. I. Gal'čuk Moscow
Abstract:
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.
Received: 04.01.1977
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