Estimating First-Passage Probabilities for Moving Barriers

Summary

The document asks how to calculate the probability that a stochastic process reaches one of two time-varying boundaries first. It starts from the familiar fixed-boundary result, which expresses the upper-boundary hitting probability through a process’s scale function and applies, for example, to geometric Brownian motion under stated parameter conditions.

The question proposes substituting the moving boundary values at the random exit time into that fixed-boundary expression, then averaging over exit times. It does not include an answer or establish that this proposed procedure is valid. In particular, the exit time and boundary locations evolve together, so the fixed-boundary formula cannot simply be assumed to apply at each random time. The text is useful as a statement of the problem and a reminder of the fixed-barrier framework, but offers no solution, derivation, or evidence for the proposed expectation. Analysis of moving boundaries would require additional stochastic-process methods and assumptions.

Key ideas

• For fixed barriers, the scale function gives the probability of reaching the upper boundary before the lower one.
• The document frames the challenge of extending that hitting-probability result to time-varying barriers.
• It proposes averaging a fixed-boundary expression over the random exit time, but does not justify the step.
• No solution or validation is provided for the suggested moving-barrier method.

Tags

# Simple way to get the crossing probabilities of a moving barrier


# Simple way to get the crossing probabilities of a moving barrier












Hello Quant Finance StackExchange,

Is there a simple way to find the crossing probabilities of a moving barrier, namely a barrier written in the form $U(t)=\alpha_1t^2 + \beta_1t + \gamma_1$ and $L(t)=\alpha_2t^2 + \beta_2t + \gamma_2$. (or if no solution, reduce the complexity to linear polynomial $\alpha t + \beta$.

I understand that there's a simple solution if the barrier is fixed, $U(t) = U$ and $L(t) = L$ and that $P(U$before$L)$, written here as $H(x)$, for a process $X_t$is given by

$H(x)=\frac{s(x)-s(L)}{s(U)-s(L)}$

where $x$, $L\leq x \leq U$ is the starting position and $s(x)$ is the scaling function such that $Y_t = s(X_t)$ is a martingale. Example, for GBM with $2\mu \neq\sigma^2$,

$s(x)=\frac{x^{1-2\mu/\sigma^2}}{1-2\mu/\sigma^2}$.

Now, if I were to apply optional stopping for a moving barrier as I did a fixed barrier, it would seem that I'll get something like,

$H(x,\tau)=\frac{s(x)-s(L(\tau))}{s(U(\tau))-s(L(\tau))}$

which would make it a random variable as the first exit time $\tau$ is random. Would getting the crossing probability as simple as taking the expectation, namely

$H(x)=\text{E}_\tau(H(x,\tau))=\sum_{t=0}^T\left[H(x,t)P(\tau=t)\right]$

Is this method reasonably logical?

(My rusty notation comes from me being two years removed from doing stochastic calculus at college. A rough explanation would do. Yes, I know there's tons of mathematical intricacies I conveniently overlooked.)

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