Girsanov's theorem is a game-changer in stochastic calculus. It lets us switch up the probability measure of a stochastic process, turning it into a martingale under a new measure. This powerful tool is key for pricing financial derivatives and solving complex problems in math finance.

The theorem hinges on the Radon-Nikodym derivative, which links the original and new measures. It preserves volatility while transforming drift, keeping the martingale property intact. This makes it invaluable for simplifying calculations in various fields, from filtering theory to stochastic control.

Definition of Girsanov's theorem

• Fundamental result in stochastic calculus that allows for changing the probability measure of a stochastic process
• Enables the transformation of a stochastic process into a martingale under a new probability measure
• Plays a crucial role in various applications within the field of Stochastic Processes (mathematical finance, filtering theory, stochastic control)

Key assumptions

• Original probability space $(\Omega, \mathcal{F}, P)$ with a filtration ${\mathcal{F}t}{t \geq 0}$
• Stochastic process ${X_t}_{t \geq 0}$ defined on the probability space
• Existence of an equivalent probability measure $Q$ such that $Q \ll P$ (absolute continuity)
• Radon-Nikodym derivative $\frac{dQ}{dP}$ is well-defined and satisfies certain conditions

Change of measure

Radon-Nikodym derivative

• Density process that relates the original measure $P$ to the new measure $Q$
• Defined as $Z_t = \frac{dQ}{dP}\bigg|_{\mathcal{F}_t}$, where $Z_t$ is $\mathcal{F}_t$-measurable
• Represents the likelihood ratio between the two probability measures at time $t$

Equivalent probability measures

• Two probability measures $P$ and $Q$ are equivalent if they assign zero probability to the same events
• Denoted as $P \sim Q$
• Ensures that the change of measure preserves the null sets of the original measure

Brownian motion under measure change

Drift transformation

• Under the new measure $Q$, the drift of the Brownian motion is transformed
• The transformed Brownian motion ${\tilde{W}t}{t \geq 0}$ under $Q$ is given by $\tilde{W}_t = W_t - \int_0^t \theta_s ds$
• ${\theta_t}_{t \geq 0}$ is the Girsanov kernel, which determines the change in drift

Volatility invariance

• Girsanov's theorem preserves the volatility of the stochastic process under the measure change
• The quadratic variation of the Brownian motion remains unchanged
• This property is crucial for maintaining the martingale property under the new measure

Martingale property

Martingale under original measure

• A stochastic process ${M_t}_{t \geq 0}$ is a martingale under the original measure $P$ if $E^P[M_t | \mathcal{F}_s] = M_s$ for all $s \leq t$
• Martingales are essential in modeling fair games and pricing financial derivatives

Martingale under new measure

• Girsanov's theorem ensures that if a process is a martingale under the original measure $P$, it remains a martingale under the new measure $Q$
• The transformed process ${\tilde{M}t}{t \geq 0}$ defined by $\tilde{M}_t = M_t - \int_0^t \theta_s d\langle M, W \rangle_s$ is a $Q$-martingale
• This property allows for simplifying calculations and deriving pricing formulas in mathematical finance

Girsanov's theorem for diffusions

SDE before measure change

• Consider a stochastic differential equation (SDE) under the original measure $P$: $dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$
• $\mu(t, X_t)$ is the drift coefficient and $\sigma(t, X_t)$ is the diffusion coefficient
• The solution ${X_t}_{t \geq 0}$ is a diffusion process under $P$

SDE after measure change

• Applying Girsanov's theorem, the SDE under the new measure $Q$ becomes: $dX_t = (\mu(t, X_t) + \sigma(t, X_t)\theta_t)dt + \sigma(t, X_t)d\tilde{W}_t$
• The drift coefficient is adjusted by the Girsanov kernel ${\theta_t}_{t \geq 0}$
• The diffusion coefficient remains unchanged, preserving the volatility structure

Applications of Girsanov's theorem

Mathematical finance

• Widely used in pricing and hedging financial derivatives
• Allows for changing the probability measure to a risk-neutral measure
• Simplifies the calculation of expected payoffs and reduces the dimensionality of the problem

Pricing derivatives

• Girsanov's theorem enables the derivation of closed-form solutions for derivative prices
• Examples include the Black-Scholes formula for European options and the Heath-Jarrow-Morton framework for interest rate derivatives

Filtering theory

• Applied in estimating the state of a system based on noisy observations
• Girsanov's theorem facilitates the derivation of filtering equations (Kalman filter, particle filter)
• Helps in obtaining optimal estimates and quantifying the associated uncertainties

Stochastic control

• Used in solving optimization problems under uncertainty
• Girsanov's theorem allows for transforming the stochastic control problem into a deterministic one
• Enables the application of dynamic programming and the derivation of optimal control strategies

Examples of Girsanov's theorem

Black-Scholes model

• Seminal model in mathematical finance for pricing European options
• Girsanov's theorem is used to change the measure from the physical measure to the risk-neutral measure
• Under the risk-neutral measure, the discounted stock price becomes a martingale, simplifying the pricing formula

Ornstein-Uhlenbeck process

• Mean-reverting stochastic process used in modeling interest rates and commodity prices
• Girsanov's theorem allows for transforming the process into a martingale under a new measure
• Facilitates the derivation of analytical solutions and the application of pricing techniques

Limitations and extensions

Novikov's condition

• Sufficient condition for the existence of the Radon-Nikodym derivative
• Requires the Girsanov kernel ${\theta_t}_{t \geq 0}$ to satisfy $E^P\left[\exp\left(\frac{1}{2}\int_0^T \theta_t^2 dt\right)\right] < \infty$
• Ensures the integrability of the exponential martingale and the well-definedness of the measure change

Multidimensional case

• Girsanov's theorem can be extended to multidimensional stochastic processes
• Involves a vector-valued Brownian motion and a matrix-valued Girsanov kernel
• Requires appropriate modifications to the drift transformation and the Radon-Nikodym derivative

Jump processes

• Girsanov's theorem can be generalized to include jump processes (Lévy processes)
• Involves a compensated Poisson random measure and a jump measure change
• Requires additional conditions on the jump intensity and the Girsanov kernel for the jump component