An implicit function theorem for semimartingales and an application.

*(English. Russian original)*Zbl 0532.60041
Russ. Math. Surv. 38, No. 2, 196-197 (1983); translation from Usp. Mat. Nauk 38, No. 2(230), 215-216 (1983).

The subject of research is a vector integral equation driven by semimartingales (not concerning unknown terms). It appears in models of estimation of parameters and filtration by the method of maximum (conditional) likelihood. The analogous equation for stochastic flows has been considered by Bismut and others. Let a probability space with filtration of \(\sigma\)-subalgebras be given; the random matrix function \(f=f(t,x)\), \(t\in R^+\), \(x\in R^ n\), be predictable, locally bounded in t and sufficiently smooth in x, and \(F=F(t,x)\) be a smooth in x modification of the vector stochastic integral \(f\cdot M\) by the vector semimartingale M in \(R^ n.\)

First implicit function theorem for semimartingales (i.f.th.s.): If in some neighbourhood of the point \(t=0\) with probability one the assumptions of the classical implicit function theorem are fulfilled, then the unique local solution \((X_ t)\) of the system of equations, relative to x, (1) \(F(t,x)=0\), is a semimartingale. The stochastic differential \(dX_ t\) generalizes the usual formula for pathwise differentials. A special generalization of Ito-Dolean-Meyer’s formula of change of variables is proved.

The functional asymptotic developments of statistics in the ”series scheme” for specific models can be obtained by means of the second i.f.th.: Let the function \(f=f^{(\epsilon)}\) be ”sufficiently smooth” with respect to the (small) parameter \(\epsilon\), and the assumptions of the first i.f.th. be fulfilled by \(f=f^{(0)}\). Then the equation (1) determines a unique, continuous and differentiable function \(X=X^{(\epsilon)}\) on some interval \([0,\epsilon_ 0[\) with values in the metrisable space of semimartingales of Emery. The differential is the solution of some linear stochastic equation. The differential for a small parameter is obtained and the solution of the ”usual” stochastic differential equation driven by semimartingales.

First implicit function theorem for semimartingales (i.f.th.s.): If in some neighbourhood of the point \(t=0\) with probability one the assumptions of the classical implicit function theorem are fulfilled, then the unique local solution \((X_ t)\) of the system of equations, relative to x, (1) \(F(t,x)=0\), is a semimartingale. The stochastic differential \(dX_ t\) generalizes the usual formula for pathwise differentials. A special generalization of Ito-Dolean-Meyer’s formula of change of variables is proved.

The functional asymptotic developments of statistics in the ”series scheme” for specific models can be obtained by means of the second i.f.th.: Let the function \(f=f^{(\epsilon)}\) be ”sufficiently smooth” with respect to the (small) parameter \(\epsilon\), and the assumptions of the first i.f.th. be fulfilled by \(f=f^{(0)}\). Then the equation (1) determines a unique, continuous and differentiable function \(X=X^{(\epsilon)}\) on some interval \([0,\epsilon_ 0[\) with values in the metrisable space of semimartingales of Emery. The differential is the solution of some linear stochastic equation. The differential for a small parameter is obtained and the solution of the ”usual” stochastic differential equation driven by semimartingales.