On approximation of stochastic differential equations with coefficients depending on the past
Let us consider a multidimensional stochastic differential equation \[ dX(t)=f(X(t))\circ dZ(t),\qquad X(0)=x \tag{1} \] (in matrix notation), or, equivalently (using a convention of summation over repeating indices), \[ dX_ i(t)=f_{ij}(X(t))\circ dZ_ j(t),\qquad X_ i(0)=x_ i,\qquad i=1,2,\ldots,d,\tag{2} \] where \(Z=(Z_ 1,Z_ 2,\ldots,Z_ r)\) is an \(r\)-dimensional continuous semimartingale, \(f(x)=(f_{ij}(x)\), \(i=1,\ldots,d\), \(j=1,\ldots,r)\), \(x\in{\mathbb{R}}^ d\), is a sufficiently smooth matrix function (for example, \(f_{ij}\in C^ 2_ b({\mathbb{R}}^ d))\), the stochastic differential (denoted by the sign \(\circ\)) is understood in the Stratonovich sense. Let \(Z^ \delta\), \(\delta>0\), be a family of \(r\)-dimensional continuous semimartingales, and let \(X^ \delta\), \(\delta>0\), be the family of solutions of (1) with \(Z\) replaced by \(Z^ \delta\), that is, \[ dX^ \delta_ i(t)=f_{ij}(X^ \delta(t))\circ dZ^ \delta_ j(t),\qquad X^ \delta_ i(0)=x_ i,\qquad i=1,2,\ldots,d.\tag{\(2^ \delta\)} \] When does the uniform convergence of \(Z^ \delta\) to \(Z\) (as \(\delta\to 0\)) on some time interval imply the uniform convergence of \(X^ \delta\) to \(X\)? It is known [see, e.g., the author, Lith. Math. J. 25, 343-352 (1985); translation from Litov. Mat. Sb. 25, No. 4, 72-84 (1985; Zbl 0588.60050) and Ann. Inst. Henri Poincaré, Probab. Stat. 23, 575-592 (1987; Zbl 0636.60057) and \textit{J. Picard}, Probab. Theory Relat. Fields 81, No. 3, 383-452 (1989; Zbl 0659.60088) and the bibliography therein]\ that for the positive answer the approximation \((Z^ \delta)\) of \(Z\) has to be symmetric in a certain sense. What happens if the coefficient \(f\) depends on paths of solutions, and we consider equations \[ dX_ i(t)=f_{ij}(X)(t)\circ dZ_ j(t),\qquad X_ i(0)=x_ i,\qquad i=1,2,\ldots,d,\tag{3} \] \[ dX^ \delta_ i(t)=f_{ij}(X^ \delta)(t)\circ dZ^ \delta_ j(t),\qquad X^ \delta_ i(0)=x_ i,\qquad i=1,2,\ldots,d.\tag{\(3^ \delta\)} \] We propose here a general class of coefficients depending on the paths unifying a major part of the known cases. For this class we extend the results known for simple coefficients without any additional assumptions on the approximation \((Z^ \delta)\) of the driving semimartingale \(Z\). Roughly speaking, we consider coefficients \(f\) representable in the form \(f(X)= h(X)\cdot X+ g(X)\cdot Y+ A(X)\), where \(Y\) is a fixed (multidimensional) semimartingale, \(A(X)\) is an FV process for each semimartingale \(X\) \((X\cdot Y(t)\) denotes the Itô integral \(\int_ 0^ t X(s) dY(s)\)).
scientific article; zbMATH DE number 578651
scientific article