Martingale Representation Theorem

Theorem5.2Martingale representation

Every square-integrable martingale $(M_t)$ adapted to the filtration of Brownian motion can be represented as:

$M_t = M_0 + \int_0^t H_s \, dB_s$

for some adapted process $H$ with $\mathbb{E}[\int_0^T H_s^2 \, ds] < \infty$ .

This theorem says that every martingale driven by Brownian motion is a stochastic integral. It is fundamental to derivative pricing: any contingent claim can be replicated by trading in the underlying asset.

ExampleHedging in Black-Scholes

In the Black-Scholes model, the discounted price of a European option is a martingale, hence can be written as $V_t = V_0 + \int_0^t \Delta_s \, dS_s$ , where $\Delta_s$ is the delta hedge (number of shares to hold). This is the foundation of dynamic hedging.