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

TLDR

The paper studies n‑dimensional Ito processes with continuous trajectories, defined by stochastic differential equations driven by a Wiener process and characterized by a diffusion matrix and a drift vector. It is shown that, under suitable conditions on a vector field, an absolutely continuous change of measure transforms such a process into another Ito process with the same diffusion matrix but a modified drift, proving that measures of diffusive Markov processes with identical diffusion matrices are mutually absolutely continuous.

Abstract

Let $X\{ x(t,w),{\bf P}\} $ be a stochastic process in n-dimensi onalEuclidean space $R_n $ having continuous trajectories, which satisfy the stochastic equation: $x^i (t,\omega ) = x^i (0,\omega ) + \int_0^t {\phi _j^i (s,\omega )} + \int_0^t {\Psi ^i (s,\omega )ds} ,\quad 0 \leqq t \leqq 1.$ 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 $\tilde X = \{ x(t,\omega ),\tilde {\bf P}\} $, where \[ \tilde {\bf P}(d\omega ) = \exp \left[ {\int_0^1 {\varphi ^i (t,\omega )\delta _{ij} d\xi ^i (t,\omega )} - \frac{1}{2}\int_0^1 {\mathop \Sigma \limits_1^n (\varphi ^i (t,\omega )^2 )dt} } \right]P(d\omega ) \] is also a K. Ito process (having a matrix $\Phi (t,\omega )$ and a translation vector $\tilde \Psi (t,\omega ) = \Psi (t,\omega ) + \phi (t,\omega ) \cdot \varphi (t,\omega )$, 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 $\tilde {\bf P}$ 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.