Martingale representation theorem

Let ${\displaystyle B_{t}}$ be a Brownian motion on a standard filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},P)}$ and let ${\displaystyle {\mathcal {G}}_{t}}$ be the augmented filtration generated by ${\displaystyle B}$. If X is a square integrable random variable measurable with respect to ${\displaystyle {\mathcal {G}}_{\infty }}$, then there exists a predictable process C which is adapted with respect to ${\displaystyle {\mathcal {G}}_{t},}$ such that

${\displaystyle X=E(X)+\int _{0}^{\infty }C_{s}\,dB_{s}.}$

Consequently,

${\displaystyle E(X|{\mathcal {G}}_{t})=E(X)+\int _{0}^{t}C_{s}\,dB_{s}.}$

The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that ${\displaystyle \left(M_{t}\right)_{0\leq t<\infty }}$ is a Q-martingale process, whose volatility ${\displaystyle \sigma _{t}}$ is always non-zero. Then, if ${\displaystyle \left(N_{t}\right)_{0\leq t<\infty }}$ is any other Q-martingale, there exists an ${\displaystyle {\mathcal {F}}}$-previsible process ${\displaystyle \varphi }$, unique up to sets of measure 0, such that ${\displaystyle \int _{0}^{T}\varphi _{t}^{2}\sigma _{t}^{2}\,dt<\infty }$ with probability one, and N can be written as:

${\displaystyle N_{t}=N_{0}+\int _{0}^{t}\varphi _{s}\,dM_{s}.}$

The replicating strategy is defined to be:

• hold ${\displaystyle \varphi _{t}}$ units of the stock at the time t, and
• hold ${\displaystyle \psi _{t}B_{t}=C_{t}-\varphi _{t}Z_{t}}$ units of the bond.

where ${\displaystyle Z_{t}}$ is the stock price discounted by the bond price to time ${\displaystyle t}$ and ${\displaystyle C_{t}}$ is the expected payoff of the option at time ${\displaystyle t}$.

At the expiration day T, the value of the portfolio is:

${\displaystyle V_{T}=\varphi _{T}S_{T}+\psi _{T}B_{T}=C_{T}=X}$

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 ${\displaystyle \left(dV_{t}=\varphi _{t}\,dS_{t}+\psi _{t}\,dB_{t}\right)}$.

• Backward stochastic differential equation

• Montin, Benoît. (2002) "Stochastic Processes Applied in Finance" [1]
• 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