Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost everywhere on the boundary with respect to the surface measure. Furthermore define $X_t$ to be a process which behaves like the standard Brownian Motion in $(0,1)^2$ and has normal reflection on the boundary (the classical reflected brownian motion)

It is known, that $X_t$ is a Semimartingale and can be expressed as: $X_t=X_0+W_t+\frac{1}{2}\int_0^t v(X_s)d L_s$ where $W^t$ is the Standard Brownian Motion, $L$ the boundary local time and $v$ is the inword normal vektor field on the boundary. Obviosuly $V_t=\frac{1}{2}\int_0^t v(X_s)d L_s$ is thus a process of bounded Variation.

Question: would $f(X_t)=f(X_0)+\int_0^t \nabla f(X_s)dW_s+\int_0^t \nabla f(X_s)dV_s+\int_0^t \Delta f(X_s)dt$ be still vaild if $X_0=x\in \partial [0,1]^2$ ?

My main concern is that neither $\nabla f$ nor $\Delta f$ are defined on $\partial [0,1]^2$ - is there a way to modify the Defintion of $f$ so that Itô's formula coul still be used ? - so basically how does one define Derivatives on the boundary - is there a standard way? (have been studying math for a while and am wondering why this question has never occured to me before)

Thanks in Advance :)

EDIT: My actual goal is to show the following result (perhaps one might get there without using Itô's Formula)

Let $X_t$ be a process with statepace $[0,1]^2$ having normal reflection on the $x$- and $y$-Axis and which is being absorbed when hitting the rest of $[0,1]^2$ (the upper and right side of the square are thus absorbing) Let $f\in\mathcal{C}^0([0,1]^2)\cap \mathcal{C}^2((0,1)^2)$ and having normal derivative $0$ on the reflecting part of the boundary. Then $f$ can be represented as: $f(x)=\mathbb{E}^xf(X_{\tau})$ with $\tau$ being the first hitting tim of the absorbing boundary.

Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost everywhere on the boundary with respect to the surface measure. Furthermore define $X_t$ to be a process which behaves like the standard Brownian Motion in $(0,1)^2$ and has normal reflection on the boundary (the classical reflected brownian motion)

Hi together - this question has arisen while I was contemplating a possible solution to another question I have posted (see Harmonic functions and Itô‘s Formula with respect to the generator of a Reflecting Brownian Motion)

Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost everywhere on the boundary with respect to the surface measure. Furthermore define $X_t$ to be a process which behaves like the standard Brownian Motion in $(0,1)^2$ and has normal reflection on the boundary (the classical reflected brownian motion)

It is known, that $X_t$ is a Semimartingale and can be expressed as: $X_t=X_0+W_t+\frac{1}{2}\int_0^t v(X_s)d L_s$ where $W^t$ is the Standard Brownian Motion, $L$ the boundary local time and $v$ is the inword normal vektor field on the boundary. Obviosuly $V_t=\frac{1}{2}\int_0^t v(X_s)d L_s$ is thus a process of bounded Variation.

Question: would $f(X_t)=f(X_0)+\int_0^t \nabla f(X_s)dW_s+\int_0^t \nabla f(X_s)dV_s+\int_0^t \Delta f(X_s)dt$ be still vaild if $X_0=x\in \partial [0,1]^2$ ?

My main concern is that neither $\nabla f$ nor $\Delta f$ are defined on $\partial [0,1]^2$ - is there a way to modify the Defintion of $f$ so that Itô's formula coul still be used ? - so basically how does one define Derivatives on the boundary - is there a standard way? (have been studying math for a while and am wondering why this question has never occured to me before)

Thanks in Advance :)

EDIT: My actual goal is to show the following result (perhaps one might get there without using Itô's Formula)

Let $X_t$ be a process with statepace $[0,1]^2$ having normal reflection on the $x$- and $y$-Axis and which is being absorbed when hitting the rest of $[0,1]^2$ (the upper and right side of the square are thus absorbing) Let $f\in\mathcal{C}^0([0,1]^2)\cap \mathcal{C}^2((0,1)^2)$ and having normal derivative $0$ on the reflecting part of the boundary. Then $f$ can be represented as: $f(x)=\mathbb{E}^xf(X_{\tau})$ with $\tau$ being the first hitting tim of the absorbing boundary.

Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost everywhere on the boundary with respect to the surface measure. Furthermore define $X_t$ to be a process which behaves like the standard Brownian Motion in $(0,1)^2$ and has normal reflection on the boundary (the classical reflected brownian motion)

It is known, that $X_t$ is a Semimartingale and can be expressed as: $X_t=X_0+W_t+\frac{1}{2}\int_0^t v(X_s)d L_s$ where $W^t$ is the Standard Brownian Motion, $L$ the boundary local time and $v$ is the inword normal vektor field on the boundary. Obviosuly $V_t=\frac{1}{2}\int_0^t v(X_s)d L_s$ is thus a process of bounded Variation.

Question: would $f(X_t)=f(X_0)+\int_0^t \nabla f(X_s)dW_s+\int_0^t \nabla f(X_s)dV_s+\int_0^t \Delta f(X_s)dt$ be still vaild if $X_0=x\in \partial [0,1]^2$ ?

My main concern is that neither $\nabla f$ nor $\Delta f$ are defined on $\partial [0,1]^2$ - is there a way to modify the Defintion of $f$ so that Itô's formula coul still be used ? - so basically how does one define Derivatives on the boundary - is there a standard way? (have been studying math for a while and am wondering why this question has never occured to me before)

Thanks in Advance :)

EDIT: My actual goal is to show the following result (perhaps one might get there without using Itô's Formula)

Let $X_t$ be a process with statepace $[0,1]^2$ having normal reflection on the $x$- and $y$-Axis and which is being absorbed when hitting the rest of $[0,1]^2$ (the upper and right side of the square are thus absorbing) Let $f\in\mathcal{C}^0([0,1]^2)\cap \mathcal{C}^2((0,1)^2)$ and having normal derivative $0$ on the reflecting part of the boundary. Then $f$ can be represented as: $f(x)=\mathbb{E}^xf(X_{\tau})$ with $\tau$ being the first hitting tim of the absorbing boundary.

Hi together - this question has arisen while I was contemplating a possible solution to another question I have posted (see Harmonic functions and Itô‘s Formula with respect to the generator of a Reflecting Brownian Motion)

Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost everywhere on the boundary with respect to the surface measure. Furthermore define $X_t$ to be a process which behaves like the standard Brownian Motion in $(0,1)^2$ and has normal reflection on the boundary (the classical reflected brownian motion)

It is known, that $X_t$ is a Semimartingale and can be expressed as: $X_t=X_0+W_t+\frac{1}{2}\int_0^t v(X_s)d L_s$ where $W^t$ is the Standard Brownian Motion, $L$ the boundary local time and $v$ is the inword normal vektor field on the boundary. Obviosuly $V_t=\frac{1}{2}\int_0^t v(X_s)d L_s$ is thus a process of bounded Variation.

Question: would $f(X_t)=f(X_0)+\int_0^t \nabla f(X_s)dW_s+\int_0^t \nabla f(X_s)dV_s+\int_0^t \Delta f(X_s)dt$ be still vaild if $X_0=x\in \partial [0,1]^2$ ?

My main concern is that neither $\nabla f$ nor $\Delta f$ are defined on $\partial [0,1]^2$ - is there a way to modify the Defintion of $f$ so that Itô's formula coul still be used ? - so basically how does one define Derivatives on the boundary - is there a standard way? (have been studying math for a while and am wondering why this question has never occured to me before)

Thanks in Advance :)

EDIT: My actualy intention is to show that $f$ has the following representation:

$f(x)=\mathbb{E}^x(X_{\tau})$ ith $\tau$ being the first hitting time of the boundary.

Hi together - this question has arisen while I was contemplating a possible solution to another question I have posted (see Harmonic functions and Itô‘s Formula with respect to the generator of a Reflecting Brownian Motion)

Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost everywhere on the boundary with respect to the surface measure. Furthermore define $X_t$ to be a process which behaves like the standard Brownian Motion in $(0,1)^2$ and has normal reflection on the boundary (the classical reflected brownian motion)

It is known, that $X_t$ is a Semimartingale and can be expressed as: $X_t=X_0+W_t+\frac{1}{2}\int_0^t v(X_s)d L_s$ where $W^t$ is the Standard Brownian Motion, $L$ the boundary local time and $v$ is the inword normal vektor field on the boundary. Obviosuly $V_t=\frac{1}{2}\int_0^t v(X_s)d L_s$ is thus a process of bounded Variation.

Question: would $f(X_t)=f(X_0)+\int_0^t \nabla f(X_s)dW_s+\int_0^t \nabla f(X_s)dV_s+\int_0^t \Delta f(X_s)dt$ be still vaild if $X_0=x\in \partial [0,1]^2$ ?

My main concern is that neither $\nabla f$ nor $\Delta f$ are defined on $\partial [0,1]^2$ - is there a way to modify the Defintion of $f$ so that Itô's formula coul still be used ? - so basically how does one define Derivatives on the boundary - is there a standard way? (have been studying math for a while and am wondering why this question has never occured to me before)

Thanks in Advance :)

EDIT: My actual goal is to show the following result (perhaps one might get there without using Itô's Formula)

Let $X_t$ be a process with statepace $[0,1]^2$ having normal reflection on the $x$- and $y$-Axis and which is being absorbed when hitting the rest of $[0,1]^2$ (the upper and right side of the square are thus absorbing) Let $f\in\mathcal{C}^0([0,1]^2)\cap \mathcal{C}^2((0,1)^2)$ and having normal derivative $0$ on the reflecting part of the boundary. Then $f$ can be represented as: $f(x)=\mathbb{E}^xf(X_{\tau})$ with $\tau$ being the first hitting tim of the absorbing boundary.