I am faced with the task of calculating the distribution of the following integral: $$I(\omega) := \int_0^1 B_{G_-(t)}(\omega)\,\mathrm dF(t),$$ where $(B_t)_{t\in[0,1]}$ is a Brownian bridge, i.e. a Gaussian process with mean function $t\mapsto 0$ and variance-covariance function $(s,t)\mapsto t\wedge s - st$, and $F,G:[0,1]\rightarrow[0,1]$ are a monotone increasing càdlàg functions, and $G_{-}$ denotes the left-continuous version of $G$, i.e. $G_-(x) = \lim_{z\uparrow x}G(z)$ for all $x\in[0,1]$.
Let's define $X_t := B_{G_-(t)}$ and $Y_t = F(t)$. So that $Y_t$ is deterministic. Then $I(\omega)$ could be seen as a Itô integral: $$I(\omega) = \int_0^1 X_t(\omega)\,\mathrm dY_t(\omega),$$ which is defined by $$\int_0^1 X_t(\omega)\,\mathrm dY_t(\omega) := \lim_{n\rightarrow\infty}\sum_{i=1}^n X_{\frac{i-1}n}(\omega)\left(Y_{\frac{i}n}(\omega) - Y_{\frac{i-1}n}(\omega)\right).$$ To show that the expression is well-defined and the Itô integral exists, I thought about showing that the sequence of partial sums, $(S_n)_{n\in\mathbb N}$, where $$S_n :=\sum_{i=1}^n X_{\frac{i-1}n}(\omega)\left(Y_{\frac{i}n}(\omega) - Y_{\frac{i-1}n}(\omega)\right),$$ forms a Cauchy sequence in $L_2$. So let $n,m\in\mathbb N$ and consider $\Delta_{n,m} = \vert S_m - S_n \vert$. Then it's straightforward to show that \begin{align*} \mathbb E[\Delta_{n,m}^2] = \mathbb E[\vert S_m - S_n\vert^2] = \mathbb E[S_n^2 - 2S_nS_m + S_m^2] = \mathbb E[S_n^2] - 2\mathbb E[S_nS_m] + \mathbb E[S_m^2]. \end{align*}
I was able to show that $\mathbb E[S_n^2], \mathbb E[S_m^2], \mathbb E[S_nS_m] \leq 1$, so $\mathbb E[\Delta_{n,m}^2] = 0$ by using $\mathbb E[X_tX_s] = G_-(t\wedge s) - G_-(s)G_-(t)) \leq 1$ for all $s,t\in[0,1]$ as well \begin{align*} \sum_{i=1}^n\sum_{j=1}^m\left(Y_{\frac{i}n}(\omega) - Y_{\frac{i-1}n}(\omega)\right)\left(Y_{\frac{j}m}(\omega) - Y_{\frac{j-1}m}(\omega)\right) &= \left(\sum_{i=1}^nY_{\frac{i}n}(\omega) - Y_{\frac{i-1}n}(\omega)\right)\left(\sum_{j=1}^mY_{\frac{j}m}(\omega) - Y_{\frac{j-1}m}(\omega)\right) \\ &= (F(1) - F(0))(F(1) - F(0)) \\ &= (1 - 0)(1- 0) \\ &= 1.\end{align*} So the limit should exist in the $L_2$ sense and hence $I$ should be well-defined. Is my reasoning correct? I am unsure, mainly because of three things:
- I obtain $\mathbb E[\Delta_{n,m}] \leq 0$ for all $n,m\in\mathbb N$ ─ I don't even have to consider the limit case. This raises some serious doubts ...
- So far I have seen only the Itô integral defined for constant integrads with respect to a Brownian motion. Is it possible to obtain an integral of the form $\int_0^1 F(t)\,\mathrm dB_{G_-(t)}(\omega)$ from the given $I(\omega)$?
- I haven't used the left-continuity of $G_-$ in any way. This feels strange. However, I feel like the left-continuity may be just useful for simplifing the notation when summing over the partition and hence is not necessary to assume.