On the reflection of a stochastic integral

Question

Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by $$ I_t=\int_{0}^{t}\theta_sdW_t, $$ where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq 0},\mathbb{P})$ and $\theta$ a stochastic process adapted to $\mathcal{F}_t$ satisfying the follows condition of integrability $$ E\left(\int_{0}^{t}\theta_s^2 ds\right)<\infty\;\;\ \forall t> 0. $$

We define the first passage time at $a$ for Brownian motion $W$ by the following random variable $$ \tau_a = \inf\{t\geq 0,W_t\geq a\}, $$ where $a>0$.

It is possible to show that $\tau_a$ is a stopping time. Moreover, By virtue of the reflection principle, we know that the following process

\begin{equation*} Z_t = \begin{cases} W_t \qquad & if \qquad 0 \leq t \leq \tau_a \\ 2a-W_t \qquad & if \qquad t > \tau_a \end{cases} \end{equation*}

also follows a standard Brownian motion under $\mathbb{P}$.

My question is as follows : Is it possible to rewrite the process $I$ in relation to the process $Z$?

I would like your opinion on this issue, thank you in advance.

Answer 1

Set $$X_t=\exp\left(-\int_{0}^{t}\theta_sdW_s^{\mathbb{P}}-\frac{1}{2}\int_{0}^{t}\theta_s^{\,2}ds\right)$$ By application of Gisanov theorem , we have

• $X_t$ is a $\mathbb{P}-$ martingale.

• By changing the measure $\mathbb{P}$ to $\mathbb{Q}$ such that $$\mathbb{E^P}\left[\frac{d\mathbb{Q}}{d\mathbb{P}}\Big{|}\mathcal{F_t}\right]=\frac{d\mathbb{Q}}{d\mathbb{P}}\Big{|}_\mathcal{F_t}=X_t$$ then

  $$W_t^{\mathbb{Q}}=W_t^{\mathbb{P}}+I_t$$ is a $\mathbb{Q}$ standard wiener process. We have \begin{equation*} Z_t = \begin{cases} W_t^{\mathbb{Q}}-I_t \qquad & , \qquad 0 \leq t \leq \tau_a \\ 2a-W_t^{\mathbb{Q}}+I_t \qquad & , \qquad t > \tau_a \end{cases} \end{equation*} then (If I am right) \begin{equation*} dZ_t = \begin{cases} dW_t^{\mathbb{Q}}-dI_t \qquad & , \qquad 0 \leq t \leq \tau_a \\ -dW_t^{\mathbb{Q}}+dI_t \qquad & , \qquad t > \tau_a \end{cases} \end{equation*}