In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion 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.
Similar theorems also exist for martingales on filtrations induced by jump processes, for example, by Markov chains.
Let
Bt
(\Omega,l{F},l{F}t,P)
l{G}t
B
l{G}infty
l{G}t,
X=E(X)+
infty | |
\int | |
0 |
CsdBs.
Consequently,
E(X|l{G}t)=E(X)+
t | |
\int | |
0 |
CsdBs.
The martingale representation theorem can be used to establish the existenceof a hedging strategy.Suppose that
\left(Mt\right)0
\sigmat
\left(Nt\right)0
l{F}
\varphi
T | |
\int | |
0 |
2 | |
\varphi | |
t |
2 | |
\sigma | |
t |
dt<infty
Nt=N0+
t | |
\int | |
0 |
\varphisdMs.
The replicating strategy is defined to be:
\varphit
\psitBt=Ct-\varphitZt
where
Zt
t
Ct
t
At the expiration day T, the value of the portfolio is:
VT=\varphiTST+\psiTBT=CT=X
and it is 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
\left(dVt=\varphitdSt+\psitdBt\right)