Let
- $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces
- $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace
- $f:[0,\infty)\times H\to\mathbb R$ be continuously Fréchet differentiable in time (first argument) and twice Fréchet differentiable in space (second argument)
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $X_0$ and $Y$ be $H$-valued random variables on $(\Omega,\mathcal A,\operatorname P)$
- $Z$ be a $\mathfrak L(U,H)$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$
- $(W_t)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$
Assume that $$X_t:=X_0+tY+ZW_t\;\;\;\text{for }t>0.\tag 1$$ Let $t\ge 0$ and $t_0,\ldots,t_n\ge 0$ with $$0=t_0<\cdots<t_n=t$$ for some $n\in\mathbb N_0$. Moreover, let $$S_n:=\sum_{i=1}^n\frac{\partial f}{\partial t}\left(t_i,X_{t_i}\right)\left(t_i-t_{i-1}\right)$$ and $$T_n:=\sum_{i=1}^n\langle X_{t_i}-X_{t_{i-1}},\frac{\partial f}{\partial x}\left(t_{i-1},X_{t_{i-1}}\right)\rangle\;.$$ I want to show that $$\lim_{n\to\infty}S_n=\int_0^t\frac{\partial f}{\partial t}\left(s,X_s\right){\rm d}s\;\;\;\operatorname P\text{-almost surely}\tag 2$$ and $$\lim_{n\to\infty}T_n=\int_0^t\langle Y,\frac{\partial f}{\partial x}\left(s,X_s\right)\rangle\;{\rm d}s+\int_0^t\langle Z{\rm d}W_s,\frac{\partial f}{\partial x}\left(s,X_s\right)\rangle\;\;\;\operatorname P\text{-almost surely}\;.\tag 3$$
If I'm not mistaken, $(2)$ follows by definition of the Riemann integral, since $\partial_tf$ is assumed to be continuous and $W$ is $\operatorname P$-almost surely continuous (please correct me, if I'm wrong).
How can we show $(3)$?
$^1$ Let $\mathfrak L(A,B)$ be the space of bounded, linear operators from $A$ to $B$.