We recall that Brownian motion, denoted X ( t ) , is a time-homogeneous Markovian process of independent increments which has two parameters, d (drift) and c (diffusion coefficient); the main result states that the distribution of X ( t ) , given that X ( 0 ) = a , is Normal with the mean of a + d ⋅ t and variance of c ⋅ t [1]. Graphically, t is typically displayed using the horizontal axis while the values of X ( t ) are shown on the vertical scale (of the so-called state space).

It is intuitively obvious (and formally proven below) that, when a is positive and d is negative, the process must reach, sooner or later, State 0; it is a major and complex task (see [2][3]) to prove that the exact time (denoted T ) of this event happening for the first time (i.e. of the first passage) is a random variable having inverse Gaussian distribution [4]. Simplifying the usual arguments and finding the corresponding probability density function (PDF) of T by elementary means is the goal of this article.

This is achieved by first discretizing both time (making each of its units into c k 2 discrete time steps) and state space (making each spatial unit into k discrete vertical steps), resulting in a much simpler process called random walk [5]. We can then easily solve the same problem using the discrete version to find the probability mass function (PMF) of a similarly defined first-passage time T ˜ (now, an integer-valued random variable), and then convert this discrete PMF to the continuous PDF of T by taking the k → ∞ limit.

To underline the importance of first-passage distributions, we mention that they arise naturally whenever one is interested in the time required for a fluctuating quantity to reach a prescribed threshold, including problems in physics, finance, reliability, and biology.

Note that paragraphs and sections starting with the ■ prefix are optional and can be omitted without affecting our derivation of the main result (only in those sections, we occasionally use concepts and formulas which may be deemed less elementary).

The discrete version of the process has the following interesting interpretation: consider a gambler who starts with i monetary units (we call them dollars) and keeps on betting $1 on a flip of a biased coin (the probability of winning an individual round is p < 1 2 , of losing is q : = 1 − p ) until losing all his money (i.e. reaching State 0), assuming that his opponent has infinite resources. Each round of this game takes one discrete time step, while one dollar is the state-space unit.

To derive a formula for the probability that the game (which ends by the gambler getting broke) will take exactly n rounds, we consider all possible paths which take the gambler from State i to State 0 without visiting 0 on any previous occasion, and compute their total probability. The individual probabilities are easy: each such path must consist of exactly n − i 2 wins and n + i 2 losses (note that n must be even/odd when i is even/odd respectively, while n ≥ i ); therefore, it has the probability of p ( n − i ) / 2 q ( n + i ) / 2 .

To count how many such paths are there is more difficult: to avoid visiting State 0 before completing n rounds (or time steps), the last step must be a loss; we are then left with an easier (yet equivalent) task of establishing the number of paths that connect State i to State 1 in n − 1 steps, while avoiding State 0. This is done in two steps: ignoring the last condition (i.e.allowing visits to 0), the number of such paths is ( n − 1 n − i 2 ) , since any combination of n − i 2 wins and n + i − 2 2 losses will do. We then need to subtract the number of those paths (from i to 1, in n − 1 steps) which have visited State 0, at least once, prior to reaching State 1.

To count these, we use the following classical reflection principle (or André’s reflection method, see [5][6]), which provides a convenient way of counting paths constrained by a boundary: visualize flipping the first part (up to and including the first visit to State 0) of any such path with respect to the time axis, while keeping its second part (from that point on) intact (see Figure 1).

Figure 1.Original (blue) and flipped (red) path.

This results in a path which starts at − i and ends up, after n − 1 steps, in State 1 (necessarily visiting State 0); the important thing is that there is a one-to-one correspondence between paths of the newly created (red) set and those of the original (blue) set. Counting the number of paths of the former is easy: this time we need exactly n − i − 2 2 wins and n + i 2 losses; the answer is thus ( n − 1 n + i 2 ) . The number of paths contributing to the probability of the game taking exactly n rounds is thus

The resulting PMF of T ˜ is then

when n = i , i + 2 , i + 4 , i + 6 , i + 8 , ⋯ (zero otherwise). This classical first-passage probability for a one-dimensional random walk can be found, in various forms, in standard treatments of random walks and probability theory, such as [5][7][8].

■ Note that an alternate way of expressing the same result is

where j = 0 , 1 , 2 , 3 , ⋯ and w ( j ) : = w ( w + 1 ) ( w + 2 ) ⋯ ( w + j − 1 ) . This can be shown by replacing n by i + 2 j in (2) and expanding:

To convert (2) into a PDF of the first-passage time T of a Brownian motion with drift d (assumed to be negative, i.e. d < 0 ), diffusion coefficient c > 0 , and starting in State a > 0 , we proceed (as indicated in the Introduction) as follows:

change the amount of a single bet to 1 k , where k is a (large) integer; the initial State i is then replaced by a ⋅ k ,play c ⋅ k 2 rounds of the new game each unit ofreal time t ,while setting the player’s probability of winning each round at

This means that, in a single round, the distribution of change in the value of X ( t ) , i.e. of its increment Δ , is

having the expected value of d c ⋅ k 2 and variance equal to 1 k 2 − d 2 c 2 k 4 . This further implies that, at time t (after c k 2 t independent rounds have been played) the expected value of the total increment is d ⋅ t , with the corresponding variance of c ⋅ t − d 2 t c ⋅ k 2 ; Brownian motion is then the limit of this process as k → ∞ . Central Limit Theorem tells us that the distribution of X ( t ) is Normal, with the mean of a + d ⋅ t and variance equal to c ⋅ t .

We have shown how the first-passage-time distribution of one-dimensional Brownian motion can be obtained from the corresponding problem for a simple discrete random walk. The essential idea is to discretize both space and time, solve the resulting gambler’s-ruin problem exactly, and then take a carefully chosen limit in which the spatial step tends to zero while the number of discrete time steps tends to infinity. In this way, a result that is not particularly transparent when derived directly from Brownian motion emerges naturally from a much simpler discrete problem.

The derivation also illustrates the close connection between random walks and Brownian motion. The discrete first-passage probability is obtained by elementary path counting, while the Brownian density follows from the appropriate diffusive scaling. The resulting density is the familiar inverse Gaussian first-passage distribution. The calculation therefore provides not a new first-passage formula, but a relatively direct derivation of a classical result and, in particular, an alternative route for readers who may find derivations based on differential equations or more advanced stochastic-process methods less accessible.

Several extensions of the present approach would be interesting. One natural direction is to consider a second absorbing boundary, leading to first-passage problems in a finite interval. Another is to replace the fixed boundary by a time-dependent one. It would also be interesting to investigate whether analogous discrete constructions can provide useful derivations for first-passage problems associated with other stochastic processes, or for related quantities such as the distribution of the maximum of a random walk or Brownian motion.

The restriction to one spatial dimension is another important limitation of the present treatment. Higher-dimensional first-passage problems are considerably richer, and it is not obvious how far the simple path-counting argument used here can be extended. Exploring discrete approximations to such problems may nevertheless provide an interesting bridge between random-walk methods and the corresponding Brownian-motion results.

Finally, the present derivation establishes the limiting distribution but does not address the rate at which the discrete first-passage distribution approaches its Brownian limit. Quantifying this convergence, and determining how large the number of discrete steps must be for the continuous approximation to be accurate, would be a useful subject for further work.

Thus, although the present calculation is deliberately restricted in scope, it suggests that discrete random walks can provide a particularly transparent starting point for understanding a wider class of first-passage problems.