When are Local Martingales/Martingales(stochastic integrals wrt BM) with the same quadratic variation same?

Question

I was reading a paper on SIS epidemiological models where the authors do following Given a 2-d BM $W=(W_1,W_2)$ and nice functions $f$ and $g$ such that $\int f(s)dW_1(s)$ and $\int g(s)dW_2(s)$ are martingales. The authors claim that it follows from the martingale representation theorem that $$\int f(s)dW_1(s)+\int g(s)dW_2(s)= \int \sqrt{f(s)^2+g(s)^2}dW_3(s)$$ for some new B.M $W_3$.

I thought that the Martingale representation theorem doesnt give us the integrand but only that an integrand exists and moreover the integrator is supposed to be the same brownian motion which generated the filtration(before augmentation with P-Null sets).

I see that the quadratic variation of the LHS and RHS is the same but does it also imply in this case that the LHS=RHS almost surely?

Answer 1

There are several martingale representation theorems; the author of the paper is refering to one of the following statements:

Let $(M_t,\mathcal{F}_t)_{t \leq T}$ be a continuous $L^2$-martingale such that $\langle M \rangle_t = \int_0^t m(s)^2 \, ds$ for some (almost everywhere) strictly positive process $m$. Then there exists a Brownian motion $(W_t)_{t \leq T}$ such that $(\mathcal{F}_t)_{t \leq T}$ is admissible and $$M_t-M_0 = \int_0^t m(s) \, dW_s.$$

See for instance Theorem 18.13 in Brownian Motion - An Introduction to Stochastic Processes by Schilling & Partzsch for a proof.

The assumption that $m$ is stricly positive can be weakened:

Let $(M_t,\mathcal{F}_t)_{t \leq T}$ be a continuous $L^2$-martingale such that $\langle M \rangle_t = \int_0^t m(s)^2 \, ds$. Then there exists an enlargement of the underlying probability space and a Brownian motion $(W_t)_{t \leq T}$ (on the enlarged probability space) such that

$$M_t-M_0 = \int_0^t m(s) \, dW_s.$$