## Stratonovich stochastic differential equations driven by general semimartingales.(English)Zbl 0823.60046

Let $$Z$$ be a semimartingale, possibly with jumps, and $$f \in C^ 1$$. The s.d.e. $X_ t = X_ 0 + \int^ t_ 0 f(X_ s) \circ dZ_ s \tag{1}$ is understood as $\begin{split} X_ t = X_ 0 + \int^ t_ 0 f(X_{s^ -})dZ_ s + {1\over 2} \int^ t_ 0 f'f(X_ s) d[Z, Z]^ c_ s +\\ + \sum_{0 < s \leq t} \{\varphi (f \Delta Z_ s, X_{s^ -}) - X_{s^ -} - f(X_{s^ -})\Delta Z_ s\}, \end{split}\tag{2}$ where the first integral is the Itô integral, $$[Z, Z]^ c$$ is the continuous part of the quadratic variation of $$Z$$ and $$\varphi(g,x)$$ is the solution at $$u = 1$$ of the o.d.e. $${dy\over du} (u) = g(y(u))$$, $$y(0) = x$$.
There is no known definition of a stochastic integral leading to this interpretation. However, if $$X$$ satisfies (2) for some $$Z$$ and $$f$$, then $$\int^ t_ 0 g(X_ s) \circ dZ_ s$$ $$(g \in C^ 1)$$ can be consistently defined and a change of variable formula holds: $\Psi \in C^ 2 \Rightarrow \Psi(X_ t) = \Psi(X_ 0) + \int^ t_ 0 \Psi'(X_ s) f(X_ s) \circ dZ_ s.$ It is also shown that: a) under Lipschitz conditions, there exists a càdlàg solution to (1), which is unique and is a semimartingale; b) the flow $$x \mapsto X_ t(x, \omega)$$ is a diffeomorphism if $$f$$ is $$C^ \infty$$ and all derivatives of $$f$$ and $$f'f$$ are bounded; c) if $$Z$$ is a Lévy process, $$X_ 0$$ and $$Z$$ are independent and $$f$$, $$f'f$$ are Lipschitz, the solution has the strong Markov property. The main result is the following: If $$Z$$ is substituted by the continuous process $${1\over h} \int^ t_{t-h}Z_ s ds$$, $$X^ h$$ is the corresponding solution, and $$\gamma_ h$$ is a suitably defined time change, the process $$Y^ h_ t := X^ h_{\gamma^{-1}_ h(t)}$$ converges in the u.c.p. topology as $$h \to 0$$ to a process $$Y$$ such that $$X_ t = Y_{\gamma_ 0(t)}$$. And for countably many $$t > 0$$, $$X^ h_ t \to X_ t$$ in probability. This approximation result justifies the interpretation (2) from the modeling viewpoint, since the approximants of $$Z$$ are continuous processes even when $$Z$$ has jumps. This is not possible if (1) is taken as a Stratonovich equation. All the results in the paper are stated for the multidimensional case.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H20 Stochastic integral equations 60J25 Continuous-time Markov processes on general state spaces 60J60 Diffusion processes
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