Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be real, and let $f,g:[0,T]\to\mathbb{R}$ be continuous functions such that, with the Riemann integral (existing by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval),

All identities between random variables below are almost sure identities, and degenerate intervals follow the conventions of Mean-Square Riemann Integral of a Family of Random Variables.

1. (Wiener integrals) Let $B=(B_t)_{t\ge0}$ be a standard Brownian motion on $(\Omega,\mathcal{F},P)$, let $k:[0,T]\to\mathbb{R}$ be continuous, and for $t\in[0,T]$ let

be a fixed choice of versions of the Wiener integrals of $k$, with $V_0=0$. Then the family $(V_t)_{t\in[0,T]}$ is mean-square continuous on $[0,T]$; consequently the family $(g(u)V_u)_{u\in[0,T]}$ is mean-square continuous by claim 2 of Basic Properties of the Mean-Square Riemann Integral, and its mean-square Riemann integrals exist by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families. For every $t\in[0,T]$,

where the left-hand side is the Wiener integral of the continuous function $fk$. In particular, taking $k$ identically $1$, so that $V_t=B_t$ almost surely by claim 1 of Wiener Integrals of Continuous Functions are Jointly Gaussian:

2. (Mean-square Riemann integrals) Let $(H_t)_{t\in[0,T]}$ be a mean-square continuous family of square-integrable random variables, and for $t\in[0,T]$ let $Y_t=\int_0^t H_u\,du$ be a fixed choice of versions of the mean-square Riemann integrals, with $Y_0=0$. Then $(Y_t)_{t\in[0,T]}$ is mean-square continuous on $[0,T]$ (claim 6 of Basic Properties of the Mean-Square Riemann Integral), the families $(f(u)H_u)_{u\in[0,T]}$ and $(g(u)Y_u)_{u\in[0,T]}$ are mean-square continuous by claim 2 of Basic Properties of the Mean-Square Riemann Integral, their mean-square Riemann integrals exist by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families, and for every $t\in[0,T]$,