Girsanov theorem
In probability theory, Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under changes in measure.
Overview
In probability theory, Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under changes in measure. The theorem plays a significant role in the general theory of stochastic processes, where it provides the key result that if Q is a measure being absolutely continuous with respect to P, then every P-semimartingale is also a Q-semimartingale. It is also of great importance in the theory of financial mathematics as it explains 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 instrument.
History
Results of this type were first proved by Robert Horton Cameron and W. T. Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Érik Lenglart (1977).
Statement of theorem
Let
{ Ω ,
F
, P }
{\displaystyle \{\Omega ,{\mathcal {F}},P\}}
be a probability space,
F
= (
F
t
)
t ≥ 0
{\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\geq 0}}
be a filtration being complete and right-continuous (usual conditions) and
X
{\displaystyle X}
be a continuous local martingale adapted to
F
{\displaystyle \mathbb {F} }
. Define
Semimartingales
Any semimartingale
Y
{\displaystyle Y}
admits the decomposition
Y = M + A
{\displaystyle Y=M+A}
with a local martingale
M
{\displaystyle M}
and a process of finite variation
A
{\displaystyle A}
. By Girsanov's theorem, if
E
( X )
{\displaystyle {\mathcal {E}}(X)}
is a martingale, then the process
M ~
t
=
Brownian motion
If
X
{\displaystyle X}
is a continuous local martingale and
W
{\displaystyle W}
is a Brownian motion under the measure
P
{\displaystyle P}
, then
W ~
t
=
W
t
−
[
W , X
]
t
Finance
Consider a non-dividend-paying risky asset whose price under the physical measure
P
{\displaystyle P}
follows the stochastic differential equation
d
S
t
S
t
=
μ
t
d t +
σ
t
d
W
t
P
,
Langevin equations
Another application of this theorem, also given in the original paper of Igor Girsanov, is for stochastic differential equations. Specifically, let us consider the equation:
d
X
t
= h (
X
t
, t ) d t + d
W
t
.
{\displaystyle dX_{t}=h(X_{t},t)dt+dW_{t}.}
where
W
t
{\displaystyle W_{t}}
denotes a Brownian motion. Here
h
{\displaystyle h}
is a fixed deterministic function. We assume that this equation has a unique strong solution on
Diffusion models
Girsanov's theorem has also been used for the convergence analysis of score-based diffusion models. In this setting, one goal is to evaluate the Kullback–Leibler divergence between the law (under
P
{\displaystyle P}
) of the true data generating process
(
X
t
)
t ∈ [ 0 , T ]
{\displaystyle (X_{t})_{t\in [0,T]}}
given by
d
X
t
= ( −
X
t
− 2 ∇ log
p
t
See also
Cameron–Martin theorem – Theorem describing translation of Gaussian measures on Hilbert spaces It is a special case of the Girsanov theorem who has inspired all the theory. Girsanov theorem has fundamental applications to the Quantum Field Theory, Malliavin Calculus, stochastic partial differential equations, degree theory, calculus of variations on the classical and abstract Wiener spaces.
References
Liptser, Robert S.; Shiriaev, A. N. (2001). Statistics of Random Processes (2nd, rev. and exp. ed.). Springer. ISBN 3-540-63929-2. Dellacherie, C.; Meyer, P.-A. (1982). "Decomposition of Supermartingales, Applications". Probabilities and Potential. Vol. B. Translated by Wilson, J. P. North-Holland. pp. 183–308. ISBN 0-444-86526-8. Lenglart, E. (1977). "Transformation de martingales locales par changement absolue continu de probabilités". Zeitschrift für Wahrscheinlichkeit (in French). 39: 65–70. doi:10.1007/BF01844873. \"Ust\"unel, A. S.; Zakai, M. (2000). Transformation of Measure on Wiener space (1st ed.). Springer. ISBN 3-540-66455-6.
External links
Notes on Stochastic Calculus which contain a simple outline proof of Girsanov's theorem.
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