Intuition for Martingale Representation Theorem

Let me give my intuition as a former Electrical Engineer. This is going to be very sloppy.

Suppose you have a Brownian Motion with increments (or "noise term" in EE language) $dB_t$. Obviously you can generate a martingale by integrating these noise terms $B_t=int_0^tdB_t$. But you can also generate other martingales by varying the "amplitude" with which the same increments are applied $M_t=int_0^t A(t)dB_t$. This is similar to changing the volume A(t) on a radio while the music is playing, you get a "different music" from the same sounds. You can even generate stochastic processes which are not martingales by adding a "level" term which controls the expected value $X_t= L(t)+int_0^t A(t)dB_t$. For example if $L(t)=sin(omega t+phi)$ you can get a process which goes up and down (seasonality), or if $L(t)=k t$ (linear trend) you can get a submartingale which rises over time.

The Martingale Representation Theorem says that indeed you can get a very large class of random processes in this way (starting with $dB(t)$, integrating it in a time-varying manner and adding an external predictable input). The only ones you cannot get are pathological cases such as processes which are not adapted to the same filtration, i.e. are dependent on a different set of random events altogether. The precise technical conditions are of course very important, and I am leaving them out. But the point is many interesting processes can be decomposed ("represented") in this way (integral of $dB_t$ plus something else).

First, let us be clear with the fact that if a process is a martingale for some Probability measure, it may not be a martingale under a different probability measure. (refer Girsanov's theorem).

Now intuitively, the Martingale Representation theorem (MRT) says that if a process $M(t)$ is a martingale with respect to filtration generated by Brownian Motion ($W$) (filtration can be intuitively understood as the path generated by Brownian motion till some time 's' ). Then $M(t)$ can be written as:

$dM(t) = D(u)dW(u)$
There is no $dt$ term in the above equation, which means that the process $M(t)$ is driftless, it has no tendency to rise or fall (precisely what a martingale is).
Integrating the above,

$M(T) = M(0) + $$int_0^T D(u) ,dW(u)$
here, W represents the brownian motion.
$D(u)$ is an adapted process to the same filtration generated by the brownian motion.

($D(u)$ is adapted basically means knowing the filtration till time $t$ implies that we know $D(u)$ at some time $t$)