Generalized integration and stochastic ODEs
The paper pushes forward the theory of stochastic calculus developed by \textit{F. Russo} and \textit{P. Vallois} [Stochastics Stochastics Rep. 70, 1-40 (2000; Zbl 0981.60053)], and earlier papers cited therein. The basic concepts are those of the forward integral: \[ \int_0^{.} Yd^-X= \lim_{\varepsilon\searrow 0}\int_0^. Y_s{X_{s+\varepsilon}-X_s\over\varepsilon}ds \] and the square bracket process: \[ [X,Y]= \lim_{\varepsilon\searrow 0}{1\over\varepsilon}\int_0^.(X_{s+\varepsilon}-X_s)(Y_{s+\varepsilon}-Y_s)ds, \] where the limits are taken in probability, uniformly on compacts. All processes are assumed to be continuous. Given a continuous adapted process \(A\), such that \([A,A]<\infty\) (so not necessarily a semimartingale), the authors show the existence of the forward integral \(\int X(A)d^-A\), where \((X_t(x),\;t\in[0,1], x\in{\mathbb R})\) is a sufficiently smooth Itô field driven by a semimartingale \(N=(N^1,\dots,N^m)\), that means \(X_t(x)= f(x)+\sum_{i=1}^m\int_0^ta^i(s,x)dN^i_s\), where \(f(x)\) is \({\mathcal F}_0\) measurable and \(a^i\) are adapted sufficiently regular, and all brackets \([A,N^i]\) exist. Thus the class of integrands includes semimartingales as well as functions of \(A\). This result is established with the help of an appropriate generalization of the Itô-Wentzell formula. Then the obtained integral is used to solve the SDE \[ X_t=x+ \int_0^t\sigma(X_s)d^-A_s +\int_0^tb(s,X_s)dN_s, \] where \(A\) is as above, \(N\) is a semimartingale such that \([A,N]\) exists, \(\sigma\) is non-random and smooth, and \(b\) is continuous, Lipschitz in \(x\).
scientific article; zbMATH DE number 1906052
scientific article