Constraining Brownian Paths to a Specified Maximum and Minimum

Summary

The document asks how to sample a Wiener process with specified starting and ending values as well as a prescribed maximum and minimum, extending the usual Brownian bridge setup. A response outlines an approximate construction for a fixed maximum: simulate a path with enough total variation, then use reflection after boundary crossings to keep the path within the desired maximum. If a path never reaches the maximum, the suggestion is to flip the signs of selected declining segments until it does, then apply the reflection procedure. The minimum constraint is described as a generalization, but its construction is not worked out.

The approach is based on the reflection principle and aims to match the constraint within a small tolerance. Its stated limitation is discrete sampling: sampled time points do not locate boundary crossings exactly. The document provides no derivation, proof that the procedure samples from the desired conditional distribution, or empirical validation. It therefore presents an idea for constrained path construction rather than a complete, validated simulation method.

Key ideas

• A Brownian bridge fixes the process endpoints, while the question adds constraints on its extrema.
• The response proposes reflection after crossings to constrain a path's maximum.
• Sign-flipping selected declining segments is suggested when a simulated path does not reach the target maximum.
• The minimum constraint is only described as a generalization, without construction details.
• Discrete time sampling limits the accuracy of the proposed method near boundary crossings.

Tags

# Sample Wiener process constrained to open (initial), high (max), low (min), close (final)


# Sample Wiener process constrained to open (initial), high (max), low (min), close (final)












With a Brownian bridge, one can sample a Wiener process constrained to a specified initial value and a final value.

Can the same be done when the process is constrained also to have a specified maximum and minimum value over the time interval?

One could select a very small subset of enough samples (simulations) that happen to have the desired minimum and maximum values (and final value). But is there a way to explicitly construct such samples?

## Answer by Brian B (score 1)

https://quant.stackexchange.com/a/79374

You can do this (within some small $\epsilon$) using the reflection principle.

Let's take the simpler case, where you only want to constrain the maximum. Adding the minimum is a fairly simple generalization.

Let's say you want your maximum to be 1.0 (and you have a standard BM starting at zero and running for 1 time unit).

First, construct sample paths until you have one whose total variation is greater than 1.0. Now we have two cases:

- The maximum on this sample path is greater than 1 at time $t$. In this case, at the point where the path exceeds 1, simply reflect the path after $t_1$ so that it turns back down. If the new path exceeds 1 at some later time $t_2>t_1$, reflect again.

- The maximum on the sample path does not exceed 1. Here, choose a subinterval on which the path has decreased, and flip all the signs in the subinterval. Repeat until the maximum exceeds 1 and revert to the case above.

This only comes within $\epsilon$ since your (finite) path sample times do not exactly identify the crossings of 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.