Itô integrand of a finite sum of Gaussian exponentials ($f_t={\sum }_j{c}_j{h}_j(t)\exp ({\int }_0^t{h}_{j}dB-\frac{1}{2}{\int }_0^t{h}_j^{2}ds)$)

For deterministic continuous functions $h_j$ on a finite interval, their stochastic exponentials $M(h_j)$ are square-integrable martingales and satisfy $dM(h_j)=h_{j}M(h_j)dB$. Thus $F={\sum }_j{c}_{j}M(h_j{)}_T$ has mean ${\sum }_j{c}_j$ and stochastic-integral representation with the displayed integrand. Itô isometry proves uniqueness up to time-probability null sets among square-integrable predictable integrands. It does not give pointwise uniqueness, since changing a value at one deterministic time changes no integral.