# Stochastic integral representation of $F(T-s,X_s)$-type equations

For $$T\in R$$ given and fixed consider: $${\rm d}F(T-t,X_t)=g(T-t,X_t)\,{\rm d}W_t.$$ where $$g(t,x)$$ is a given functions and $$X_t$$ is a given process driven by a brownian motion ($$dX_t=(...)dt+(...)dW_t$$). How does the stochastic integral representation of $$F$$ look like?

Are any of these two representations correct?

$$F(T-t,X_t)=F(T-0,X_0)+\int^t_0g(T-u)dW_u (1)$$ $$F(T-t,X_t)=F(T-0,X_0)+\int^{T-t}_0g(T-u,X)dW_u (2)$$

• I think the 2nd representation is correct. Feb 19, 2019 at 13:58