Girsanov theorem: Difference between revisions

Latest revision as of 03:56, 13 February 2025

Girsanov Theorem[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

@@ Line 1: / Line 1: @@
-'''Girsanov Theorem''' is a fundamental result in the field of [[probability theory]] and [[stochastic processes]], particularly in the study of [[Brownian motion]] and [[martingales]]. It provides a method for changing the [[probability measure]] on a space of paths in a way that transforms a Brownian motion under the original measure into a Brownian motion with drift under the new measure. This theorem has significant applications in [[mathematical finance]], especially in the pricing of [[financial derivatives]], and in [[risk management]].
+{{DISPLAYTITLE:Girsanov Theorem}}
-==Statement of the Theorem==
+== Girsanov Theorem ==
-Let \( (\Omega, \mathcal{F}, \mathbb{P}) \) be a [[probability space]], and let \( \mathbb{Q} \) be another probability measure on \( \Omega \) that is absolutely continuous with respect to \( \mathbb{P} \) (denoted \( \mathbb{Q} \ll \mathbb{P} \)). Let \( \{W_t\}_{t \geq 0} \) be a [[Brownian motion]] under \( \mathbb{P} \), and let \( \{\mathcal{F}_t\}_{t \geq 0} \) be the filtration generated by \( W_t \). Suppose there exists an \( \mathcal{F}_t \)-adapted process \( \{\theta_t\}_{t \geq 0} \) satisfying certain integrability conditions. Then, under the measure \( \mathbb{Q} \), defined by the Radon-Nikodym derivative
+[[File:Girsanov.png|thumb|right|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]].
-\[ \frac{d\mathbb{Q}}{d\mathbb{P}} \bigg|_{\mathcal{F}_t} = \exp\left( \int_0^t \theta_s dW_s - \frac{1}{2} \int_0^t \theta_s^2 ds \right), \]
+== 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.
-the process
+== 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 \(\mathbb{Q}\).
-\[ \tilde{W}_t = W_t - \int_0^t \theta_s ds \]
+== 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.
-is a Brownian motion with respect to \( \mathbb{Q} \).
+== 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:
-==Applications==
+\[
-The Girsanov Theorem is crucial in the field of [[quantitative finance]], where it is used to derive the [[Black-Scholes equation]] for option pricing. By changing the measure from the "real world" probability measure to the "risk-neutral" measure, one can simplify the problem of pricing derivatives by removing the drift of the underlying asset's price process. This allows for the valuation of derivatives to be based solely on the risk-free rate, irrespective of the asset's expected return.
+\frac{d\mathbb{Q}}{d\mathbb{P}} \bigg|_{\mathcal{F}_t} = \exp\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2} \int_0^t \theta_s^2 \, ds\right).
+\]
-==Proof==
+Under the measure \(\mathbb{Q}\), the process \(\tilde{W}_t\) is a standard Brownian motion.
-The proof of Girsanov's Theorem involves verifying that \( \tilde{W}_t \) satisfies the definition of a Brownian motion under the measure \( \mathbb{Q} \). This includes showing that \( \tilde{W}_t \) has independent and stationary increments, and that for any \( t \), \( \tilde{W}_t \) is normally distributed with mean 0 and variance \( t \). The proof also relies on the properties of the stochastic exponential and the concept of [[martingale]]s.
-==Limitations and Conditions==
+== Related Pages ==
-The application of Girsanov's Theorem requires the process \( \theta_t \) to satisfy Novikov's condition or a similar condition ensuring the exponential martingale is a true martingale. These conditions are necessary to ensure the absolute continuity of measures and the integrability of the Radon-Nikodym derivative.
+* [[Stochastic calculus]]
+* [[Brownian motion]]
-==See Also==
 * [[Martingale (probability theory)]]
-* [[Brownian motion]]
+* [[Financial mathematics]]
-* [[Radon-Nikodym theorem]]
+* [[Radon-Nikodym derivative]]
-* [[Stochastic calculus]]
-* [[Mathematical finance]]
 [[Category:Probability theory]]
 [[Category:Stochastic processes]]
-[[Category:Mathematical finance]]
+[[Category:Financial mathematics]]
-{{math-stub}}