Brownian First-Passage Probabilities and the Heat Equation

Summary

The document poses a problem about a function defined through a Gaussian tail integral and asks for boundary values and a differential equation associated with Brownian motion. The requested properties include a zero value at the starting boundary, a unit value at a terminal boundary, and satisfaction of the backward heat equation. These are characteristic ingredients of a Brownian hitting-probability problem, where the Gaussian distribution describes the process displacement over a time interval.

However, the document provides no derivation, solution, or discussion of Dynkin's formula. Its notation also appears inconsistent: the displayed definition labels the function with different arguments from those in the requested boundary conditions, and the integral's time parameter is not clearly aligned with the variables in the differential equation. Readers can recognize the connection between Gaussian transition probabilities and Brownian boundary-value problems, but would need to clarify the function definition and work out the proof independently. No application to trading or empirical evidence is supplied.

Key ideas

• The prompt links a Gaussian tail integral to a Brownian boundary-value problem.
• It asks for boundary conditions and the backward heat equation.
• Dynkin's formula is named as the intended proof technique, but no proof is included.
• The function's arguments are inconsistent, so the setup requires clarification before solving.

Tags

# Prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$


# Prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$












Using the Dynkin's formula, prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$

where $F(s,t)=2\int_{x-x_0}^{\infty}\frac{1}{\sqrt{2\pi t}}e^{-\frac{u^2}{2t}}du-1$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.