Using Feynman–Kac to Find Drifted Brownian Motion Hitting Probabilities

Summary

The document explores how to calculate the probability that Brownian motion with constant drift reaches an upper barrier before a lower one. It focuses on the drifted case, where the process is no longer a martingale, and explains that the hitting probability can still be represented as a function of the current state because the process is time-homogeneous and Markovian. Applying the generator gives a second-order ordinary differential equation, with boundary values set by which barrier has been reached.

The central question is how this boundary-value formulation relates to the conditional-expectation version of Feynman–Kac, which usually specifies a terminal payoff. For a hitting event, the relevant payoff is determined by the boundary outcome, and the stopping time replaces a fixed terminal date. The document asks for clarification rather than supplying a full derivation or numerical solution. It is a conceptual prompt, so readers may need a stochastic-process reference for rigorous conditions and treatment of the unbounded stopping time.

Key ideas

• A drifted Brownian motion remains Markovian even though it is no longer a martingale.
• The probability of reaching one barrier before another satisfies an equation formed from the process generator.
• Boundary values encode the outcome associated with hitting each barrier.
• For a hitting event, the relevant payoff is tied to the boundary reached at the stopping time.
• The document poses the connection to Feynman–Kac but does not develop a complete derivation.

Tags

# Hitting time of Brownian motion with drift using Feynman-Kac


# Hitting time of Brownian motion with drift using Feynman-Kac












I was studying this question from "A Practical Guide to Quantitative Finance Interviews" and was having some trouble understanding one solution. Please advise if misunderstood anything or if there is any reference paper I could turn to.

Here is the question:

Suppose that X is a Brownian motion with no drift, i.e., dX(t) = dW(t). If X starts at 0, what is the probability that X hits 3 before hitting -5? What if X has drift m, i.e., dX(t)=mdt+dW(t)?

My question is only for the second half and here goes the solution (I'm simply copying from the book):

When X has drift m, the process is no longer a martingale. Let P(t,x) be the probatility that the process hits 3 before hitting -5 when X=x at time t. Although X is no longer a martingale process, it is still a Markov process. So P(t,x)=P(x) is actually independent of t. Applying the Feynman-Kac equation, we have: mPx(x)+1/2Pxx(x)=0, boundary conditions:P(3)=1, P(-5)=0

What I'm trying to figure out is how to construct and define the functions that serve as the bridge to connects the PDE and conditional expectation? Namely, Feynman-Kac goes as:

If X is an Ito process such that dX(t)=$\beta$(t,X)dt+$\gamma$(t,X)dW and f(x) is a function of X. Define function V(t,x)=E[f(XT)|Xt=x], then V(t,x) is a martingale process that satisfies the partial differential equation $\frac{\partial V}{\partial t}+\beta(t,x)\frac{\partial V}{\partial S}+\frac{1}{2}\gamma(t,x)\frac{\partial^2 V}{\partial S^2}=0$ and terminal condition V(T,x)=f(x).

If P(x) is the V here, then how do we find and define f(x)?

Many thanks!

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