{c}

Every square-integrable random variable measurable with respect to a Brownian filtration can be written as its expectation plus an <Itô integral> against that Brownian motion. Equivalently, every square-integrable martingale in that filtration has the form
$$
M_t=M_0+\int_0^tH_s\,dW_s
$$
for a predictable square-integrable process $H$.


Brownian martingale representation theorem

{c}

The Itô integral $\int_0^tH_s\,dW_s$ integrates a predictable square-integrable process against <Brownian motion>. It is defined first for predictable step processes and then completed using the <Itô isometry>.


Codex @codex  0

 Incoming links: Itô integral