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 ...
, the martingale representation theorem states that a random variable that is
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
with respect to the
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
generated by a
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
can be written in terms of an
Itô integral with respect to this Brownian motion.
The theorem only asserts the existence of the representation and does not help to find it explicitly; it is possible in many cases to determine the form of the representation using
Malliavin calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...
.
Similar theorems also exist for
martingales on filtrations induced by
jump processes, for example, by
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
s.
Statement
Let
be a
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
on a standard
filtered probability space
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
and let
be the
augmented filtration
Augment or augmentation may refer to:
Language
*Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages
*Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns i ...
generated by
. If ''X'' is a
square integrable random variable measurable with respect to
, then there exists a
predictable process ''C'' which is
adapted with respect to
, such that
:
Consequently,
:
Application in finance
The martingale representation theorem can be used to establish the existence
of a
hedging strategy.
Suppose that
is a Q-martingale process, whose
volatility is always non-zero.
Then, if
is any other Q-martingale, there exists an
-previsible process
, unique up to sets of measure 0, such that
with probability one, and ''N'' can be written as:
:
The replicating strategy is defined to be:
* hold
units of the stock at the time ''t'', and
* hold
units of the bond.
where
is the stock price discounted by the bond price to time
and
is the expected payoff of the option at time
.
At the expiration day ''T'', the value of the portfolio is:
:
and it's easy to check that the strategy is self-financing: the change in the value of the portfolio only depends on the change of the asset prices
.
See also
*
Backward stochastic differential equation
References
*Montin, Benoît. (2002) "Stochastic Processes Applied in Finance" {{full citation needed, date=November 2012
*
Elliott, Robert (1976) "Stochastic Integrals for Martingales of a Jump Process with Partially Accessible Jump Times", ''Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete'', 36, 213–226
Martingale theory
Probability theorems