# Computing wedge probabilities: finite time horizon case

## TL;DR

This paper introduces an alternative infinite series formula for computing the probability that a Brownian motion stays within sloping boundaries over a finite time, offering faster convergence and high precision.

## Contribution

It provides a new series representation for wedge probabilities with improved convergence properties and practical rules for choosing the optimal formula based on parameters.

## Key findings

- Convergence rate depends on boundary slopes and time.
- Six terms of the series achieve precision of 10^{-16}.
- The new formula offers an efficient alternative to Anderson's formula.

## Abstract

We present an alternative to the well-known Anderson's formula for the probability that a first exit time from the planar region between two slopping lines -a_1 t -b_1 and a_2 t + b_2 by a standard Brownian motion is greater than T. As the Anderson's formula, our representation is an infinite series from special functions. We show that convergence rate of both formulas depends only on terms (a_1 + a_2)(b_1 + b_2) and (b_1 + b_2)^2 /T and deduce simple rules of appropriate representation's choose. We prove that for any given set of parameters a_1, b_1, a_2, b_2, T the sum of first 6 terms ensures precision 10^{-16}.