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 , 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 (Ω,F,P) with a filtration {Ft}t≥0
Stochastic process {Xt}t≥0 defined on the probability space
Existence of an equivalent probability measure Q such that Q≪P (absolute continuity)
Radon-Nikodym derivative dPdQ is well-defined and satisfies certain conditions
Change of measure
Radon-Nikodym derivative
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Density process that relates the original measure P to the new measure Q
Defined as Zt=dPdQFt, where Zt is Ft-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∼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 is transformed
The transformed Brownian motion {W~t}t≥0 under Q is given by W~t=Wt−∫0tθsds
{θt}t≥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
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 {Mt}t≥0 is a martingale under the original measure P if EP[Mt∣Fs]=Ms for all s≤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 {M~t}t≥0 defined by M~t=Mt−∫0tθsd⟨M,W⟩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: dXt=μ(t,Xt)dt+σ(t,Xt)dWt
μ(t,Xt) is the drift coefficient and σ(t,Xt) is the diffusion coefficient
The solution {Xt}t≥0 is a diffusion process under P
SDE after measure change
Applying Girsanov's theorem, the SDE under the new measure Q becomes: dXt=(μ(t,Xt)+σ(t,Xt)θt)dt+σ(t,Xt)dW~t
The drift coefficient is adjusted by the Girsanov kernel {θt}t≥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
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 {θt}t≥0 to satisfy EP[exp(21∫0Tθt2dt)]<∞
Ensures the integrability of the 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
Key Terms to Review (17)
Absolutely Continuous Measures: Absolutely continuous measures are measures that are dominated by another measure, meaning that if a set has zero measure under the dominating measure, it also has zero measure under the absolutely continuous measure. This concept is essential in understanding changes of measure, particularly in the context of Girsanov's theorem, where it facilitates the transformation of probability measures and helps relate different stochastic processes.
Brownian motion: Brownian motion is a continuous-time stochastic process that describes the random movement of particles suspended in a fluid, ultimately serving as a fundamental model in various fields including finance and physics. It is characterized by properties such as continuous paths, stationary independent increments, and normal distributions of its increments over time, linking it to various advanced concepts in probability and stochastic calculus.
Exponential Martingale: An exponential martingale is a type of stochastic process that arises in probability theory, characterized by a specific transformation of a martingale via an exponential function. This process is particularly important in financial mathematics and risk theory, as it helps in modeling asset prices under different measures. The exponential martingale connects closely with the concept of Girsanov's theorem, which allows for changing the probability measure in a way that transforms a Brownian motion into another process.
Girsanov's Theorem: Girsanov's Theorem is a fundamental result in stochastic calculus that provides a way to change the probability measure under which a stochastic process is defined. This theorem is crucial for understanding how to transform the dynamics of a stochastic process, particularly in the context of financial modeling and the formulation of stochastic differential equations, by allowing for the adjustment of drift terms. It is key for making connections between different probabilistic frameworks, which enhances the flexibility in modeling uncertain systems.
Igor Girsanov: Igor Girsanov is a prominent mathematician known for his groundbreaking work in the field of stochastic processes, particularly Girsanov's theorem, which provides a way to change the probability measure of stochastic processes. This theorem is pivotal in finance and mathematical statistics as it allows for the transformation of a stochastic process under a new measure, enabling the analysis of different models or scenarios while maintaining consistency in expectations and probabilities.
Itô process: An Itô process is a stochastic process that represents the solution to a stochastic differential equation (SDE) and is characterized by its continuous paths and the incorporation of both deterministic and random components. It serves as a mathematical model for various phenomena in fields such as finance, physics, and engineering, allowing for the analysis of systems influenced by randomness. The Itô process is fundamentally connected to several essential concepts in stochastic calculus.
Itô's lemma: Itô's lemma is a fundamental result in stochastic calculus that provides a formula for the differential of a function of a stochastic process, particularly Brownian motion. This lemma is crucial because it allows for the extension of classical calculus to stochastic processes, enabling the analysis of how functions evolve when their inputs are subject to randomness. It connects deeply with various concepts such as stochastic integrals, stochastic differential equations, and specific processes like the Ornstein-Uhlenbeck process.
Kiyosi Itô: Kiyosi Itô was a prominent Japanese mathematician known for his groundbreaking work in stochastic calculus, particularly the development of Itô calculus. His contributions fundamentally transformed the field of probability theory and its applications, especially in financial mathematics and the modeling of random processes.
Lévy Process: A Lévy process is a type of stochastic process that generalizes the concept of random walks and has stationary and independent increments. It plays a crucial role in various areas such as finance and insurance, as it models phenomena like stock prices and claim amounts, making it essential for understanding complex systems in probabilistic frameworks.
Martingale representation: Martingale representation refers to a property in stochastic processes where every martingale can be expressed as a stochastic integral with respect to a Brownian motion. This concept is fundamental in financial mathematics, especially for pricing options and understanding risk-neutral valuation. It connects the idea of martingales with the notion of predictable processes, providing a powerful tool for modeling and analyzing various stochastic systems.
Measure change: Measure change refers to the process of adjusting or transforming a probability measure in the context of stochastic processes, often to analyze or simulate different probabilistic scenarios. This concept is crucial when dealing with different stochastic environments, allowing for a re-weighting of probabilities and facilitating the use of various mathematical tools, such as Girsanov's theorem, which provides a method for changing the probability measure while maintaining certain properties of stochastic processes.
Novikov Condition: The Novikov Condition is a criterion used in stochastic calculus that ensures the existence of equivalent martingale measures for a given probability space, particularly in the context of Girsanov's theorem. This condition plays a crucial role in guaranteeing that the Radon-Nikodym derivative associated with a change of measure is indeed a martingale, which is essential for risk-neutral pricing in financial mathematics. In essence, it provides a set of conditions under which the transformation of stochastic processes preserves the necessary properties for pricing and hedging.
Pricing of derivatives: Pricing of derivatives refers to the process of determining the fair value of financial contracts whose value is derived from the price of an underlying asset, such as stocks, bonds, or commodities. This concept is fundamental in finance as it enables market participants to make informed decisions about trading, hedging, and investing in these complex instruments. The pricing often involves mathematical models and theories to evaluate various factors, including time, volatility, and interest rates.
Radon-Nikodym Derivative: The Radon-Nikodym derivative is a fundamental concept in measure theory that provides a way to relate two different measures on the same measurable space. It essentially describes how one measure can be expressed in terms of another, capturing the density of one measure with respect to another. This derivative plays a crucial role in the change of measure and is central to Girsanov's theorem, enabling transformations between probability measures that affect stochastic processes.
Risk-Neutral Measure: A risk-neutral measure is a probability measure under which the present value of future cash flows is equal to their expected value, discounting at the risk-free rate. This concept is essential in financial mathematics, particularly in pricing derivatives and managing financial risk. It helps simplify complex financial models by allowing analysts to focus on expected returns without considering risk preferences, facilitating the evaluation of uncertain outcomes.
Stochastic dominance: Stochastic dominance is a method used to compare the risk and return of different random variables or distributions, determining which one is preferred by a risk-averse decision-maker. It provides a framework for making choices under uncertainty by assessing how one probability distribution performs against another across all potential outcomes. This concept is vital in finance and economics as it helps identify superior investment options and supports decision-making in uncertain environments.
Stochastic integral: A stochastic integral is an extension of the traditional integral that is defined for stochastic processes, specifically in the context of integration with respect to a stochastic process, usually a Brownian motion. This type of integral allows for the analysis and modeling of systems where randomness plays a significant role, making it essential for areas such as finance, insurance, and physics. It is crucial for constructing solutions to stochastic differential equations and understanding the behavior of random systems over time.