Girsanov theorem

Girsanov theorem   (Redirected from Girsanov's theorem) Jump to navigation Jump to search Visualisation of the Girsanov theorem — The left side shows a Wiener process with negative drift under a canonical measure P; on the right side each path of the process is colored according to its likelihood under the martingale measure Q. The density transformation from P to Q is given by the Girsanov theorem.

In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying.

Contents 1 History 2 Significance 3 Statement of theorem 4 Corollary 4.1 Comments 5 Application to finance 6 Application to Langevin Equations 7 References 8 External links History Results of this type were first proved by Cameron-Martin in the 1940s and by Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977).

Significance Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure that is absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale.

Statement of theorem We state the theorem first for the special case when the underlying stochastic process is a Wiener process. This special case is sufficient for risk-neutral pricing in the Black-Scholes model.

Let {displaystyle {W_{t}}} be a Wiener process on the Wiener probability space {displaystyle {Omega ,{mathcal {F}},P}} . Let {displaystyle X_{t}} be a measurable process adapted to the natural filtration of the Wiener process {displaystyle {{mathcal {F}}_{t}^{W}}} ; we assume that the usual conditions have been satisfied.

Given an adapted process {displaystyle X_{t}} define {displaystyle Z_{t}={mathcal {E}}(X)_{t},,} where {displaystyle {mathcal {E}}(X)} is the stochastic exponential of X with respect to W, i.e.

{displaystyle {mathcal {E}}(X)_{t}=exp left(X_{t}-{frac {1}{2}}[X]_{t}right),} and {displaystyle [X]_{t}} denotes the quadratic variation of the process X.

If {displaystyle Z_{t}} is a martingale then a probability measure Q can be defined on {displaystyle {sigma ,F}} such that Radon-Nikodym derivative {displaystyle left.{frac {dQ}{dP}}right|_{{mathcal {F}}_{t}}=Z_{t}={mathcal {E}}(X)_{t}} Then for each t the measure Q restricted to the unaugmented sigma fields {displaystyle {mathcal {F}}_{t}^{o}} is equivalent to P restricted to {displaystyle {mathcal {F}}_{t}^{o}.,} Furthermore if {displaystyle Y_{t}} is a local martingale under P then the process {displaystyle {tilde {Y}}_{t}=Y_{t}-left[Y,Xright]_{t}} is a Q local martingale on the filtered probability space {displaystyle {Omega ,F,Q,{{mathcal {F}}_{t}^{W}}}} .

Corollary If X is a continuous process and W is Brownian Motion under measure P then {displaystyle {tilde {W}}_{t}=W_{t}-left[W,Xright]_{t}} is Brownian motion under Q.

The fact that {displaystyle {tilde {W}}_{t}} is continuous is trivial; by Girsanov's theorem it is a Q local martingale, and by computing {displaystyle left[{tilde {W}}right]_{t}=left[Wright]_{t}=t} it follows by Levy's characterization of Brownian Motion that this is a Q Brownian Motion.

Comments In many common applications, the process X is defined by {displaystyle X_{t}=int _{0}^{t}Y_{s},dW_{s}.} For X of this form then a necessary and sufficient condition for X to be a martingale is Novikov's condition which requires that {displaystyle E_{P}left[exp left({frac {1}{2}}int _{0}^{T}Y_{s}^{2},dsright)right]

Si quieres conocer otros artículos parecidos a Girsanov theorem puedes visitar la categoría Mathematical finance.