Generalization of the Girsanov theorem. (English) Zbl 0648.60068

Skokan, Václav

Commentat. Math. Univ. Carol. 29, No. 1, 127-141 (1988).

The paper gives a generalization of the Cameron-Martin-Girsanov formula. Let \(X=X(t))_{0\leq t\leq T}\) be an \({\mathbb{R}}^ d\)-valued process on a probability space \((\Omega,{\mathcal F},P_ 0)\) of the form \(X(t)=W(t)+Y(t)- \int^{t}_{0}A(s)ds\), where W denotes a d-dimensional Wiener process, Y is an independent increment process without Gaussian part, and A is square integrable. Let the process \[ Z(t)=\exp (\int^{t}_{0}A(s)dW(s)-2^{-1}\int^{t}_{0}| A(s)|^ s ds) \] fulfill the condition \(E_{P_ 0} Z(T)=1\) and define the probability measure \(P_ 1\) on \(\Omega\) by \(P_ 1(B)=\int_{B}Z(T)dP_ 0\). Then it is proved that on \((\Omega,{\mathcal F},P_ 1)\) one has \[ X(t)=\bar W(t)+\bar Y(t), \] where \(\bar W\) is a Wiener process relative to \(P_ 1\) and \(\bar Y\) is a square integrable local martingale with \(\bar Y(t)=Y(t)\) \(P_ 1\)-a.s.
Some consequences and generalizations of this main result are also given.

Reviewer: E.Dettweiler

MSC:

60H20	Stochastic integral equations
60G57	Random measures
60H05	Stochastic integrals

Keywords:

generalization of the Cameron-Martin-Girsanov formula; independent increment process; square integrable local martingale