Girsanov'sTheorem

Girsanov's theorem is a mathematical result used in the field of stochastic calculus, specifically in the context of stochastic differential equations and stochastic processes. It is named after the Russian mathematician Igor Girsanov, and the name is pronounced as "Geer-sah-nov."

The theorem is particularly useful when dealing with a change of measure in probability theory. In simple terms, it provides a way to transform one stochastic process into another by changing the underlying probability measure. This transformation is often necessary when working with processes that are not directly observable, such as in quantitative finance, where the theorem is frequently applied to model financial instruments and risk management.

Girsanov's theorem can be stated as follows:

Let $W(t)$ be a standard Wiener process (also known as a Brownian motion) defined on a probability space $(\Omega, \mathcal{F}, P)$ with a filtration $\mathcal{F}(t)$, and let $X(t)$ be an $\mathcal{F}(t)$-adapted stochastic process that satisfies the Novikov's condition:

$$ E\left[\exp\left(0.5 * \int_{0}^{T} X(s)^2 ds\right)\right] < \infty $$

Then there exists a unique probability measure $Q$, equivalent to $P$, such that the process $Z(t)$ defined by:

$$ Z(t) = W(t) - \int_{0}^{t} X(s) ds $$

is a standard Wiener process under $Q$. The relationship between $P$ and $Q$ is given by the Radon-Nikodym derivative:

$$ \frac{dQ}{dP} = \exp\left(- \int_{0}^{T} X(s) dW(s) - 0.5 * \int_{0}^{T} X(s)^2 ds\right) $$

In essence, Girsanov's theorem allows us to change the underlying probability measure in a way that simplifies the behavior of the stochastic process we are analyzing. The transformed process is often easier to work with, enabling the derivation of important results and the development of efficient computational methods.

In conclusion, Girsanov's theorem is a powerful tool in stochastic calculus that helps in analyzing and transforming stochastic processes by changing the probability measure. The name is pronounced "Geer-sah-nov," after its discoverer, the Russian mathematician Igor Girsanov.

References

There are several authoritative references on the subject of stochastic processes and Girsanov's theorem. Some of the most widely respected textbooks that cover the topic include:

• "Stochastic Calculus for Finance II: Continuous-Time Models" by Steven E. Shreve (2004)

This book, particularly Chapter 5, provides a detailed introduction to Girsanov's theorem in the context of continuous-time stochastic processes and mathematical finance. It's widely used for learning about stochastic calculus and its applications in finance.

• "Brownian Motion and Stochastic Calculus" by Ioannis Karatzas and Steven E. Shreve (1991)

This book provides a comprehensive treatment of stochastic calculus and Brownian motion, including Girsanov's theorem. It's a more advanced and mathematically rigorous text that is suitable for graduate students and researchers in the field.

• "Stochastic Differential Equations: An Introduction with Applications" by Bernt Øksendal (2003)

This is another widely used textbook that covers stochastic processes and stochastic differential equations. It offers a detailed treatment of Girsanov's theorem in Chapter 7.

• "Probability and Stochastics" by Cinlar, Erhan (2011)

This textbook offers a comprehensive introduction to probability theory, stochastic processes, and their applications. Girsanov's theorem is covered in Chapter 13.

These are just a few examples of authoritative references on the subject of Girsanov's theorem and related topics in stochastic processes. Depending on your background and level of expertise, you may find one of these books more suitable for your needs than the others.