On the uniqueness of solutions of stochastic differential equations with singular drifts
The author considers the stochastic differential equation \[ (1)\quad dx^ i(t)=\sum^{d}_{j=1}a^ i_ j(X(t))dB^ j(t)+b^ i(X(t))dt+ \] \[ \sum^{d}_{j=1}\tau^ i_ j(X(t))dM^ j(t)+B^ i(X(t))dL^ u_ t(X^ d),\quad i=1,...,d. \] Here a,\(\tau\) : \({\mathbb{R}}^ d\to {\mathbb{R}}^ d\otimes {\mathbb{R}}^ d\), b,\(\beta\) : \({\mathbb{R}}^ d\to {\mathbb{R}}^ d\) are bounded measurable functions and \(\mu\) (d\(\eta)\) is a nonnegative bounded measure on \({\mathbb{R}}^ 1\). A solution of (1) is a system of stochastic processes \(X(t)=(X^ 1(t),...,X^ d(t))\), \(B(t)=(B^ 1(t),...,B^ d(t))\), \(M(t)=(M^ 1(t),...,M^ d(t))\) defined on a filtered probability space (\(\Omega\),\({\mathcal F},P,{\mathcal F}_ t)\) such that (i) X(t) is a system of \({\mathcal F}_ t\)-semimartingales, (ii) [B(t),M(t)] is a system of \({\mathcal F}_ t\)-martingales with \(B(0)=M(0)=0\) such that \(<B^ i,B^ j>(t)=\delta^ i_ jt\), \(<B^ i,M^ j>(t)=0\) and \(<M^ i,M^ j>(t)=\delta^ i_ jL_ t^{\mu}(X^ d),\) (iii) with probability one \(X^ i(t)=X^ i(0)+\sum^{d}_{j=1}\int^{t}_{0}a^ i_ j(X(s))dB^ j(x)+\int^{t}_{0}b^ i(X(s))ds+\sum^{d}_{j=1}\int^{t}_{0}\tau^ i_ j(X(s))dM^ j(s)+\int^{t}_{0}\beta^ i(X(s))dL_ s^{\mu}(X^ d)\), \(i=1,...,d\), where \([L_ i^{\eta}(X^ d)\); \(t\geq 0\), \(\eta \in {\mathbb{R}}^ 1]\) denotes the local time of a continuous \({\mathcal F}_ t\)- semimartingale \(X^ d(t)\) and \(L_ t^{\mu}(X^ d)=\int^{+\infty}_{-\infty}L_ t^{\eta}(X^ d)\mu (d\eta).\) The existence of solutions of (1) is established when \(\mu\) belongs to a certain class of discrete measures, for example \(\mu =\delta_ 0\). If the coefficients are in addition Lipschitz continuous the solution is shown to be unique in law.
scientific article; zbMATH DE number 4013673
scientific article