. Use the two-dimensional Lévy Theorem, Theorem 4.6.5, to prove the two-dimensional Girsanov The

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. Use the two-dimensional Lévy Theorem, Theorem 4.6.5, to...

Key Concepts

Two-Dimensional Lévy Theorem

This theorem characterizes the two-dimensional Brownian motion by specifying that if a continuous process has independent and normally distributed increments together with the appropriate quadratic covariation structure, then it is a two-dimensional Brownian motion. It is used to identify or verify that a given process is indeed a Brownian motion in two dimensions, using conditions on its components and their interactions via quadratic variations.

Girsanov Theorem (in Two Dimensions)

Girsanov’s theorem is a fundamental result in stochastic calculus that describes how the dynamics of a stochastic process, particularly its drift term, change when moving from one probability measure to another. In the two-dimensional setting, it asserts that, under an absolutely continuous change of measure via an exponential martingale, a two-dimensional Brownian motion with no drift can transform into a process with a new drift component. The process remains a Brownian motion under the new measure when adjusted by the prescribed shift.

Change of Measure

The concept of changing measure is central to applying Girsanov’s theorem. It involves the use of a Radon-Nikodym derivative to define a new probability measure that is absolutely continuous with respect to the original one. This new measure alters the drift of the underlying stochastic processes, allowing the conversion of a driftless Brownian motion into one with drift, while preserving the essential properties of the process under the altered measure.

Exponential Martingales

Exponential martingales arise naturally when constructing the Radon-Nikodym derivative used in a change of measure. They are essential for ensuring that the new measure is well-defined and that the stochastic integrals retain their martingale properties. In the context of Girsanov’s theorem, they serve as the vehicle for transforming the original Brownian motion’s drift characteristics after the measure change.

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