GirsanovSatz

The Girsanov theorem, also known as the Girsanov change of measure, is a fundamental result in the theory of stochastic processes, particularly in the context of stochastic calculus and stochastic differential equations. It provides a method to change the probability measure under which a given stochastic process is defined, while preserving the properties of the process itself. This theorem is named after the Soviet mathematician I. V. Girsanov, who first published it in 1960.

The Girsanov theorem is particularly useful in the study of stochastic differential equations driven by Brownian motion. It allows one to transform a given stochastic differential equation under one probability measure into an equivalent equation under a different measure, often simplifying the analysis of the process. The theorem is based on the concept of equivalent measures and the Radon-Nikodym derivative, which describes the change in the probability measure.

In its simplest form, the Girsanov theorem states that if we have a Brownian motion W(t) under a probability measure P, and we define a new probability measure Q such that the Radon-Nikodym derivative dQ/dP is given by the exponential of an integral involving a predictable process, then the process W(t) under Q is a new Brownian motion with a drift. Specifically, if we define a new process W'(t) = W(t) - ∫₀ᵗ μ(s) ds, where μ(t) is the predictable process, then W'(t) is a Brownian motion under Q.

The Girsanov theorem has numerous applications in various fields, including finance, where it is used to model the dynamics of financial assets under different risk measures, and in physics, where it is employed in the study of quantum fields and stochastic processes in quantum mechanics. It is also a crucial tool in the theory of stochastic control and optimization, where it helps in transforming control problems under one measure into equivalent problems under another measure.