 Girsanov theorem

The Girsanov theorem describes how an absolutely continuous change of measure alters the drift of a semimartingale. In a Brownian version, let $\theta$ be predictable with ${\int }_0^T{\theta }_s^{2}ds<\infty$ almost surely, and supposeis a true martingale on $[0,T]$. Its initial value is one. Under $dQ=Z_{T}dP$, the process $W_t^Q=W_t+{\int }_0^t{\theta }_{s}ds$ is standard Brownian motion through time $T$. Novikov's condition is one sufficient test for the density assumption.^[ref]

For $d{X}_t=\mu dt+\sigma d{W}_t$ with constant $\sigma >0$, choosing $\theta =\mu /\sigma$ gives $d{X}_t=\sigma d{W}_t^Q$: the same paths have different statistical weights and zero drift under $Q$. Their quadratic variation is unchanged. A positive local martingale density is insufficient unless its expectation remains one, and consistent finite-horizon changes need not give an equivalent infinite-horizon measure.

Girsanov's theorem is a fundamental result in the theory of stochastic processes, particularly in the field of stochastic calculus and quantitative finance. It provides a way to change the probability measure under which a stochastic process is defined, transforming it into another process that may have different characteristics. This is particularly useful in financial mathematics for pricing derivatives and in risk management. ### Key Concepts: 1. **Stochastic Processes**: A stochastic process is a collection of random variables indexed by time or space.