On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures

I. V. Girsanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 3, pp. 314-330

Résumé

Let $X\{x(t,w),\mathbf P\}$ be a stochastic process in $n$-dimensional Euclidean space $R_n$ having continuous trajectories, which satisfy the stochastic equation: $$x^i(t,\omega)=x^i(0,\omega)+\int_0^t\Phi_j^i(s,\omega)\,d\xi ^j(s,\omega)+\int_0^t\Psi^i(s,\omega)\,ds,\quad0\leq t\leq1.$$ Here $p=p(d\omega)$ is a measure in the space $\Omega$ of elementary events, $\int_0^t\Phi_j^i\,d\xi^j$ is considered to be the stochastic integral of K. Ito with respect to the Wiener process $\xi$. The process is called a Wiener process if it satisfies conditions (1.1) and (1.2) of this paper. Process $X$ is called an K. Ito process (with respect to the Wiener process $\xi$) corresponding to the diffusion matrix $\Phi(t,\omega)=||\Phi_j^i(t,\omega)||$ and to the translation vector $\Psi(t,\omega)=\{\Psi^i (t,\omega)\} $. It is proved with certain restrictions imposed on the vector $\varphi(t,\omega)=\{\varphi^i(t,\omega)\}$ that the process $\widetilde X=\{x(t,\omega),\widetilde{\mathbf P}\}$, where $$\widetilde{\mathbf P}(d\omega)=\exp\biggl[\int_0^1\varphi^i(t,\omega)\delta_{ij}\,d\xi^i(t,\omega)-\frac12\int_0^1\sum_1^n(\varphi^i(t,\omega)^2)\,dt\biggr]\mathbf P(d\omega)$$ is also a K. Ito process (having a matrix $\Phi(t,\omega)$ and a translation vector $\widetilde\Psi(t,\omega)=\Psi(t,\omega)+\Phi(t,\omega)\cdot\varphi(t,\omega)$, with respect to the Wiener process: $$\tilde\xi(t,\omega)=\xi(t,\omega)-\int_0^t\varphi(s,\omega)\,ds.$$ This is proved by deriving several relationships for conditional assembly averages with respect to the measure $\widetilde{\mathbf P}$ making use of transformation formulas for stochastic integrals. From the results obtained it follows, in particular, that the measures are absolutely continuous in the space of trajectories for diffusive Markov processes determined by the stochastic equations of K. Ito [3] if they have identical diffusion matrices and different translation matrices.making use of transformation formulas for stochastic integrals.