In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the Girsanov theorem tells how
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es change under changes in
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
. The theorem is especially important in the theory of
financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
as it tells how to convert from the
physical measure which describes the probability that an
underlying instrument
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
(such as a
share price or
interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
) will take a particular value or values to the
risk-neutral measure
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or ''equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price und ...
which is a very useful tool for evaluating the value of
derivatives
The derivative of a function is the rate of change of the function's output relative to its input value.
Derivative may also refer to:
In mathematics and economics
* Brzozowski derivative in the theory of formal languages
* Formal derivative, an ...
on the underlying.
History
Results of this type were first proved by Cameron-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 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
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
that is
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
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
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
. This special case is sufficient for risk-neutral pricing in the
Black-Scholes model.
Let
be a Wiener process on the Wiener probability space
. Let
be a measurable process adapted to the natural filtration of the Wiener process
; we assume that the usual conditions have been satisfied.
Given an adapted process
define
:
where
is the
stochastic exponential of ''X'' with respect to ''W'', i.e.
:
and
denotes the
quadratic variation In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
Definition
Suppose that X_t is a real-valued sto ...
of the process ''X''.
If
is a
martingale then a probability
measure ''Q'' can be defined on
such that
Radon-Nikodym derivative
:
Then for each ''t'' the measure ''Q'' restricted to the unaugmented sigma fields
is equivalent to ''P'' restricted to
:
Furthermore if
is a local martingale under ''P'' then the process
:
is a ''Q'' local martingale on the filtered probability space
.
Corollary
If ''X'' is a continuous process and ''W'' is Brownian Motion under measure ''P'' then
:
is Brownian motion under ''Q''.
The fact that
is continuous is trivial; by Girsanov's theorem it is a ''Q'' local martingale, and by computing
:
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
:
For ''X'' of this form then a necessary and sufficient condition for ''X'' to be a martingale is
Novikov's condition In probability theory, Novikov's condition is the sufficient condition for a stochastic process which takes the form of the Radon–Nikodym derivative in Girsanov's theorem to be a martingale. If satisfied together with other conditions, Girsanov ...
which requires that
:
The stochastic exponential
is the process ''Z'' which solves the stochastic differential equation
:
The measure ''Q'' constructed above is not equivalent to ''P'' on
as this would only be the case if the
Radon-Nikodym derivative were a uniformly integrable martingale, which the exponential martingale described above is not. On the other hand as long as Novikov's condition is satisfied the measures are equivalent on
.
Additionally, then combining this above observation in this case, we see that the process
for
is a Q Brownian motion. This was Igor Girsanov's original formulation of the above theorem.
Application to finance
This theorem can be used to show in the Black-Scholes model the unique risk-neutral measure, i.e. the measure in which the fair value of a derivative is the discounted expected value, Q, is specified by
:
Application to Langevin Equations
Another application of this theorem, also given in the original paper of Igor Girsanov, is for
stochastic differential equations
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock pr ...
. Specifically, let us consider the equation
where
denotes a Brownian motion. Here
and
are fixed deterministic functions. We assume that this equation has a unique strong solution on
. In this case Girsanov's theorem may be used to compute functionals of
directly in terms a related functional for Brownian motion. More specifically, we have for any bounded functional
on continuous functions
that
This follows by applying Girsanov's theorem, and the above observation, to the martingale process
In particular, we note that with the notation above, the process
is a Q Brownian motion. Rewriting this in differential form as
we see that the law of
under Q solves the equation defining
, as
is a Q Brownian motion. In particular, we see that the right-hand side may be written as
, where Q is the measure taken with respect to the process Y, so the result now is just the statement of Girsanov's Theorem.
A more general form of this application is that if both
admit unique strong solutions on
, then for any bounded functional on
, we have that
References
* Liptser, Robert S., and Shiriaev A. N. ''Statistics of Random Processes''. Springer, 2001.
* C. Dellacherie and P.-A. Meyer, "Probabilités et potentiel -- Théorie de Martingales" Chapitre VII, Hermann 1980
* E. Lenglart "Transformation de martingales locales par changement absolue continu de probabilités", Zeitschrift für Wahrscheinlichkeit 39 (1977) pp 65-70.
External links
Notes on Stochastic Calculuswhich contain a simple outline proof of Girsanov's theorem.
Stochastic processes
Mathematical theorems
Mathematical finance