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mnmn1993
mnmn1993
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The derivativedifferent quotient of the first exit time with respect to the initial state

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mnmn1993
mnmn1993
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Let $B_t$ be a 1-dimensional Brownian motion and $t \in [0,T]$. Suppose we have a diffusion process $X_t$ such that $$ dX_t = u(X_t,t) dt + v(X_t,t) dt \hspace{10pt} \text{ and } \hspace{10pt} X_0= x \in (a,b) \subset \mathbb{R}$$ for some smooth $u$ and $v$. Let $\tau(x,t) = \inf \big\{ s \in [t,T]: X_s\notin (a,b) \big\}$ be the first exit time. Is there any method to computeobtain some estimate about the derivative $\dfrac{\partial \tau}{\partial x}$?expectation of different quotient of (suppose it exists$\dfrac{\tau(x+h) - \tau (x)}{h}$?) I can not find it some standard textbooks.

Let $B_t$ be a 1-dimensional Brownian motion and $t \in [0,T]$. Suppose we have a diffusion process $X_t$ such that $$ dX_t = u(X_t,t) dt + v(X_t,t) dt \hspace{10pt} \text{ and } \hspace{10pt} X_0= x \in (a,b) \subset \mathbb{R}$$ for some smooth $u$ and $v$. Let $\tau(x,t) = \inf \big\{ s \in [t,T]: X_s\notin (a,b) \big\}$ be the first exit time. Is there any method to compute the derivative $\dfrac{\partial \tau}{\partial x}$? (suppose it exists?) I can not find it some standard textbooks.

Let $B_t$ be a 1-dimensional Brownian motion and $t \in [0,T]$. Suppose we have a diffusion process $X_t$ such that $$ dX_t = u(X_t,t) dt + v(X_t,t) dt \hspace{10pt} \text{ and } \hspace{10pt} X_0= x \in (a,b) \subset \mathbb{R}$$ for some smooth $u$ and $v$. Let $\tau(x,t) = \inf \big\{ s \in [t,T]: X_s\notin (a,b) \big\}$ be the first exit time. Is there any method to obtain some estimate about the expectation of different quotient of $\dfrac{\tau(x+h) - \tau (x)}{h}$? I can not find it some standard textbooks.

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mnmn1993
mnmn1993
  • 54
  • 1
  • 13

The derivative of first exit time with respect to the initial state

Let $B_t$ be a 1-dimensional Brownian motion and $t \in [0,T]$. Suppose we have a diffusion process $X_t$ such that $$ dX_t = u(X_t,t) dt + v(X_t,t) dt \hspace{10pt} \text{ and } \hspace{10pt} X_0= x \in (a,b) \subset \mathbb{R}$$ for some smooth $u$ and $v$. Let $\tau(x,t) = \inf \big\{ s \in [t,T]: X_s\notin (a,b) \big\}$ be the first exit time. Is there any method to compute the derivative $\dfrac{\partial \tau}{\partial x}$? (suppose it exists?) I can not find it some standard textbooks.