##
**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.

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.

Reviewer: A.Alabert (Bellaterra)

### 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 |