Let $X$ be a Brownian semimartingale $$ X_t = X_0+\int_0^t\mu_s\,ds+\int_0^t\sigma_s\,dW_s $$ with $\mu$ and $\sigma$ bounded and cadlag, $W$ Brownian motion and $$ \sigma_s = \sigma_0+\int_0^t\mu^{\prime}_s\,ds+\int_0^t\sigma^{\prime}_s\,dW_s+\int_0^{t}\sigma^{\prime\prime}_s\,dV_s $$ where $V$ is a Brownian motion independent of $W$ and $\mu^{\prime}$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are bounded and cadlag too. Assume that the filtration to which all elements are adapted is generated by a finite number of Brownian motions $$ \mathcal{F}_t = \sigma\left(W,V,Z^{(1)},\cdots,Z^{(p)}\right), $$ where $Z^{(1)},\dots,Z^{(p)}$ are independent Brownian motions. Consider the stopping time $$ \tau = \inf\{t>0| \left|X_t-X_0\right|\notin(-b,a)\} $$ with $a$ and $b$ positive real constants. It is possible to prove that the following representation holds $$ \tau = \frac{a\,b}{\sigma_0^2}+\int_0^{\tau}A_s\,dW_s+\int_0^{\tau}B_s\,dV_s+\int_0^{\tau}C_s\,ds\quad(1) $$ for suitable processes $A\neq 0$, $B\neq 0$ and $C\neq 1$$C\neq const$ (the case $A=0$, $B=0$ and $C=const$ is trivially true). I am wondering whether a representation like the (1) is valid for any $\mathcal{F}_t$-stopping time. Of course, replacing the term $\frac{a\,b}{\sigma_0^2}$ with a suitable $\mathcal{F}_0$ random variable.
Let $X$ be a Brownian semimartingale $$ X_t = X_0+\int_0^t\mu_s\,ds+\int_0^t\sigma_s\,dW_s $$ with $\mu$ and $\sigma$ bounded and cadlag, $W$ Brownian motion and $$ \sigma_s = \sigma_0+\int_0^t\mu^{\prime}_s\,ds+\int_0^t\sigma^{\prime}_s\,dW_s+\int_0^{t}\sigma^{\prime\prime}_s\,dV_s $$ where $V$ is a Brownian motion independent of $W$ and $\mu^{\prime}$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are bounded and cadlag too. Assume that the filtration to which all elements are adapted is generated by a finite number of Brownian motions $$ \mathcal{F}_t = \sigma\left(W,V,Z^{(1)},\cdots,Z^{(p)}\right), $$ where $Z^{(1)},\dots,Z^{(p)}$ are independent Brownian motions. Consider the stopping time $$ \tau = \inf\{t>0| \left|X_t-X_0\right|\notin(-b,a)\} $$ with $a$ and $b$ positive real constants. It is possible to prove that the following representation holds $$ \tau = \frac{a\,b}{\sigma_0^2}+\int_0^{\tau}A_s\,dW_s+\int_0^{\tau}B_s\,dV_s+\int_0^{\tau}C_s\,ds\quad(1) $$ for suitable processes $A\neq 0$, $B\neq 0$ and $C\neq 1$ (the case $A=0$, $B=0$ and $C=const$ is trivially true). I am wondering whether a representation like the (1) is valid for any $\mathcal{F}_t$-stopping time. Of course, replacing the term $\frac{a\,b}{\sigma_0^2}$ with a suitable $\mathcal{F}_0$ random variable.
Let $X$ be a Brownian semimartingale $$ X_t = X_0+\int_0^t\mu_s\,ds+\int_0^t\sigma_s\,dW_s $$ with $\mu$ and $\sigma$ bounded and cadlag, $W$ Brownian motion and $$ \sigma_s = \sigma_0+\int_0^t\mu^{\prime}_s\,ds+\int_0^t\sigma^{\prime}_s\,dW_s+\int_0^{t}\sigma^{\prime\prime}_s\,dV_s $$ where $V$ is a Brownian motion independent of $W$ and $\mu^{\prime}$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are bounded and cadlag too. Assume that the filtration to which all elements are adapted is generated by a finite number of Brownian motions $$ \mathcal{F}_t = \sigma\left(W,V,Z^{(1)},\cdots,Z^{(p)}\right), $$ where $Z^{(1)},\dots,Z^{(p)}$ are independent Brownian motions. Consider the stopping time $$ \tau = \inf\{t>0| \left|X_t-X_0\right|\notin(-b,a)\} $$ with $a$ and $b$ positive real constants. It is possible to prove that the following representation holds $$ \tau = \frac{a\,b}{\sigma_0^2}+\int_0^{\tau}A_s\,dW_s+\int_0^{\tau}B_s\,dV_s+\int_0^{\tau}C_s\,ds\quad(1) $$ for suitable processes $A\neq 0$, $B\neq 0$ and $C\neq const$ (the case $A=0$, $B=0$ and $C=const$ is trivially true). I am wondering whether a representation like the (1) is valid for any $\mathcal{F}_t$-stopping time. Of course, replacing the term $\frac{a\,b}{\sigma_0^2}$ with a suitable $\mathcal{F}_0$ random variable.
Let $X$ be a Brownian semimartingale $$ X_t = X_0+\int_0^t\mu_s\,ds+\int_0^t\sigma_s\,dW_s $$ with $\mu$ and $\sigma$ bounded and cadlag, $W$ Brownian motion and $$ \sigma_s = \sigma_0+\int_0^t\mu^{\prime}_s\,ds+\int_0^t\sigma^{\prime}_s\,dW_s+\int_0^{t}\sigma^{\prime\prime}_s\,dV_s $$ where $V$ is a Brownian motion independent of $W$ and $\mu^{\prime}$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are bounded and cadlag too. ConsiderAssume that the filtration to which all elements are adapted is generated by a finite number of Brownian motions $$ \mathcal{F}_t = \sigma\left(W,V,Z^{(1)},\cdots,Z^{(p)}\right), $$ where $Z^{(1)},\dots,Z^{(p)}$ are independent Brownian motions. Consider the stopping time $$ \tau = \inf\{t>0| \left|X_t-X_0\right|\notin(-b,a)\} $$ with $a$ and $b$ positive real constants. It is possible to prove that the following representation holds $$ \tau = \frac{a\,b}{\sigma_0^2}+\int_0^{\tau}A_s\,dW_s+\int_0^{\tau}B_s\,dV_s+\int_0^{\tau}C_s\,ds\quad(1) $$ for suitable processes $A$$A\neq 0$, $B$$B\neq 0$ and $C$$C\neq 1$ (the case $A=0$, $B=0$ and $C=const$ is trivially true). I am wondering whether a representation like the (1) is valid for any $\mathcal{F}^X$$\mathcal{F}_t$-stopping time, where $\{\mathcal{F}^X_t\}$ indicates the filtration generated by $X$: $$ \mathcal{F}_t^X=\sigma\left(X_s|0\leq s\leq t\right). $$ Of. Of course, replacing the term $\frac{a\,b}{\sigma_0^2}$ with a suitable $\mathcal{F}_0$ random variable.
Let $X$ be a Brownian semimartingale $$ X_t = X_0+\int_0^t\mu_s\,ds+\int_0^t\sigma_s\,dW_s $$ with $\mu$ and $\sigma$ bounded and cadlag, $W$ Brownian motion and $$ \sigma_s = \sigma_0+\int_0^t\mu^{\prime}_s\,ds+\int_0^t\sigma^{\prime}_s\,dW_s+\int_0^{t}\sigma^{\prime\prime}_s\,dV_s $$ where $V$ is a Brownian motion independent of $W$ and $\mu^{\prime}$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are bounded and cadlag too. Consider the stopping time $$ \tau = \inf\{t>0| \left|X_t-X_0\right|\notin(-b,a)\} $$ with $a$ and $b$ positive real constants. It is possible to prove that the following representation holds $$ \tau = \frac{a\,b}{\sigma_0^2}+\int_0^{\tau}A_s\,dW_s+\int_0^{\tau}B_s\,dV_s+\int_0^{\tau}C_s\,ds\quad(1) $$ for suitable processes $A$, $B$ and $C$. I am wondering whether a representation like the (1) is valid for any $\mathcal{F}^X$-stopping time, where $\{\mathcal{F}^X_t\}$ indicates the filtration generated by $X$: $$ \mathcal{F}_t^X=\sigma\left(X_s|0\leq s\leq t\right). $$ Of course, replacing the term $\frac{a\,b}{\sigma_0^2}$ with a suitable $\mathcal{F}_0$ random variable.
Let $X$ be a Brownian semimartingale $$ X_t = X_0+\int_0^t\mu_s\,ds+\int_0^t\sigma_s\,dW_s $$ with $\mu$ and $\sigma$ bounded and cadlag, $W$ Brownian motion and $$ \sigma_s = \sigma_0+\int_0^t\mu^{\prime}_s\,ds+\int_0^t\sigma^{\prime}_s\,dW_s+\int_0^{t}\sigma^{\prime\prime}_s\,dV_s $$ where $V$ is a Brownian motion independent of $W$ and $\mu^{\prime}$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ are bounded and cadlag too. Assume that the filtration to which all elements are adapted is generated by a finite number of Brownian motions $$ \mathcal{F}_t = \sigma\left(W,V,Z^{(1)},\cdots,Z^{(p)}\right), $$ where $Z^{(1)},\dots,Z^{(p)}$ are independent Brownian motions. Consider the stopping time $$ \tau = \inf\{t>0| \left|X_t-X_0\right|\notin(-b,a)\} $$ with $a$ and $b$ positive real constants. It is possible to prove that the following representation holds $$ \tau = \frac{a\,b}{\sigma_0^2}+\int_0^{\tau}A_s\,dW_s+\int_0^{\tau}B_s\,dV_s+\int_0^{\tau}C_s\,ds\quad(1) $$ for suitable processes $A\neq 0$, $B\neq 0$ and $C\neq 1$ (the case $A=0$, $B=0$ and $C=const$ is trivially true). I am wondering whether a representation like the (1) is valid for any $\mathcal{F}_t$-stopping time. Of course, replacing the term $\frac{a\,b}{\sigma_0^2}$ with a suitable $\mathcal{F}_0$ random variable.