Girsanov theorem

Girsanov Theorem

Girsanov Theorem illustration

The Girsanov Theorem is a fundamental result in stochastic calculus and probability theory. It provides a method for changing the measure on a probability space in such a way that a stochastic process becomes a martingale. This theorem is particularly useful in the field of financial mathematics, especially in the pricing of derivative securities.

Background

The theorem is named after Igor Vladimirovich Girsanov, a Russian mathematician who made significant contributions to the theory of stochastic processes. The Girsanov Theorem is a key result in the theory of Brownian motion and is used to transform a Brownian motion with drift into a standard Brownian motion under a new probability measure.

Statement of the Theorem

Consider a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) and a standard Brownian motion \(W_t\) on this space. Let \(\theta_t\) be an \(\mathcal{F}_t\)-adapted process satisfying certain integrability conditions. Define a new process \(\tilde{W}_t = W_t + \int_0^t \theta_s \, ds\). The Girsanov Theorem states that there exists a new probability measure \(\mathbb{Q}\) such that \(\tilde{W}_t\) is a Brownian motion with respect to \(\mathbb{Q}\).

Applications

The Girsanov Theorem is widely used in financial engineering to model the dynamics of asset prices. It allows for the transformation of the original probability measure into a risk-neutral measure, which simplifies the pricing of options and other financial derivatives. By using the Girsanov Theorem, one can eliminate the drift term in the geometric Brownian motion model, making it easier to compute the expected value of future payoffs.

Mathematical Formulation

Let \(\mathcal{F}_t\) be the natural filtration generated by \(W_t\). The Radon-Nikodym derivative \(\frac{d\mathbb{Q}}{d\mathbb{P}}\) is given by the exponential martingale:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}} \bigg|_{\mathcal{F}_t} = \exp\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2} \int_0^t \theta_s^2 \, ds\right). \]

Under the measure \(\mathbb{Q}\), the process \(\tilde{W}_t\) is a standard Brownian motion.

Related Pages

• Stochastic calculus
• Brownian motion
• Martingale (probability theory)
• Financial mathematics
• Radon-Nikodym derivative