# Derivation of Wiener process first passage times using probability generating function?

I would like to find the distribution of first passage times in a simple Wiener process using the idea of probability generating functions. Thus there will be, at certain point, a limiting step to go from the discrete method of pgf's to a distribution function. I have seen the derivation of this distribution using the reflection principle, but I am keen to do it by the pgf method, although I am not sure if it can work. I am doing the following:

Let $\tau(m)$ be the first time a level $m>0$ is reached after n steps and $S_0=0$: $$\tau(m)=min\{n\geq0:S_n=m\}\$$ The pgf of $\tau(m =1)$ is $$f_{\tau}(s)=\sum_{n=1}^{\infty}s^nP\{\tau=n\}$$ Similarly for m>1. Then noticing that the pgf of a sum of random variables is the product of the pgf's I get the following $$f_{\tau}(s)=\frac{(1-\sqrt{1-s^2})}{s}$$ Now I suspect I need to do some sort of limiting step to get a distribution for $\tau$, but I do not know how proceed from here. My aim is to recover the well known distribution for first passage times $f(\tau)=\frac{a}{\sqrt{2\pi}}t^{-3/2}exp(-a^2/2t)$
Any advice will be highly appreciated

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This doesn't make any sense to me. You define $\tau$ dependent on a parameter $m$; then you define $f_\tau(s)$ without mentioning $m$, which I would have interpreted as suppressing $m$ in the notation but still having it as a parameter; but then you give a probability generating function for $\tau$ that doesn't depend on $m$ (and also I don't understand your argument for it). And then you say you want to obtain a distribution for first passage times that's looks like a function from a continuous context -- how did that get in here? – joriki Aug 5 '12 at 19:30
I am sorry if I was not so explicit. m is the value that the random walk reaches at certain time $\tau$.For example, to reach m=1 we write $\tau(1)$, to reach $m\leq>1$, we write $\tau(m)$. So then to reach m the random walk must reach +1 first, etc. S – miles Aug 5 '12 at 19:38
Please clarify the question itself; people shouldn't have to delve into the comments to understand the question. Also you haven't responded to my second question, how a function from a continuous context entered into it. Do you want to obtain this from the discrete result through a limiting process? – joriki Aug 5 '12 at 19:46
Again, please update the question to clarify all these points. The question starts out with "I would like to find the distribution of first passage times in a simple random walk". I gather from your comments that what you want to find is in fact not the distrubution of first passage times in a simple random walk, but the distrubution of first passage times in a Wiener process. – joriki Aug 5 '12 at 20:00
No need to be sorry about not being an expert. The only thing to be sorry about is that you keep writing clarifications only in the comments instead of clarifying the question. This is in your own interest; very few people will bother to read all these lengthy comments when they don't understand the question. – joriki Aug 5 '12 at 20:06

As Robert showed, the probability generating function for the first passage time $\tau(n)$ to $n$ is

$$g_n(s) = \left(\frac{1-\sqrt{1-s^2}}{s}\right)^n\;.$$

I'm using $n$ instead of $m$ to match the variables in Theorem VIII.8 (p. 587). First we need to bring this into a form in which we can apply the theorem, that is, we need to satisfy conditions $\mathbf L_1$ and $\mathbf L_2$ in Section VIII.8.1 (p. 586):

$$g_n(s)=s^n\left(\frac{1-\sqrt{1-s^2}}{s^2}\right)^n=:s^nB(s^2)^n$$

with

$$B(z)=\frac{1-\sqrt{1-z}}z\;.$$

To find the distribution of the first passage time $t$ at $a$, we need to take the step sizes $\Delta x$ and $\Delta t$ to zero while keeping $\Delta x^2/\Delta t=:\beta$ constant. Then $n=a/\Delta x$ and $N=t/(2\Delta t)=\beta t/(2\Delta x^2)$, where the factor $2$ arises from $z=s^2$. We need to find the asymptotic behaviour of the coefficients of $B(z)$ as $\Delta x$ goes to zero and thus both $n$ and $N$ go to infinity.

First note that the spread $T$ defined in Section VIII.8.1 is infinite in this case, so we don't have to worry about restrictions on the ratio $\lambda=N/n=\beta t/(2a\Delta x)$, which goes to infinity as $\Delta x\to0$. We have to find the root $\zeta$ of

$$\zeta\frac{B'(\zeta)}{B(\zeta)}=\lambda\;,$$

which turns out to have the nice form

$$\zeta=1-\frac1{(2\lambda+1)^2}$$

with

$$B(\zeta)=1-\frac1{2(\lambda+1)}\;.$$

We also need

$$\xi=\frac{\mathrm d^2}{\mathrm d\zeta^2}\left(\log B(\zeta)-\lambda\log\zeta\right)=\frac{B''(\zeta)}{B(\zeta)}+O\left(\lambda^2\right)=(2\lambda)^3+O\left(\lambda^2\right)\;.$$

Now we can apply Theorem VIII.8 to obtain the distribution function as the limit

\begin{align} \lim_{\Delta x\to0}\frac{[s^{2N}]g_n(s)}{\Delta t} &= \lim_{\Delta x\to0}\frac{[z^{N-n/2}]B(z)^n}{\Delta t} \\ &= \lim_{\Delta x\to0}\frac{[z^N]B(z)^n}{\Delta t} \\ &= \lim_{\Delta x\to0}\frac{B(\zeta)^n}{\zeta^{N+1}\sqrt{2\pi n\xi}\Delta t} \\ &= \lim_{\Delta x\to0}\frac{\left(1-\dfrac1{2(\lambda+1)}\right)^n}{\left(1-\dfrac1{(2\lambda+1)^2}\right)^{N+1}\sqrt{2\pi (2\lambda)^3\dfrac a{\Delta x}}\,\Delta t} \\ &= \lim_{\Delta x\to0}\frac{\left(1-\dfrac{a\Delta x}{\beta t}\right)^{a/\Delta x}}{\left(1-\left(\dfrac{a\Delta x}{\beta t}\right)^2\right)^{\beta t/2(\Delta x^2)+1}\sqrt{2\pi\left(\dfrac{\beta t}{a\Delta x}\right)^3\dfrac a {\Delta x}}\,\Delta x^2/\beta} \\ &= \frac{\exp\left(-\dfrac{a^2}{\beta t}\right)}{\exp\left(-\dfrac{a^2}{2\beta t}\right)\sqrt{\dfrac{2\pi\beta t^3}{a^2}}} \\ &= \frac{a\exp\left(-\dfrac{a^2}{2\beta t}\right)}{\sqrt{2\pi\beta t^3}}\;. \end{align}

Then setting $\beta=1$ gives the desired result; it appears you forgot a square root around $2\pi$.

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Thank you for your explicit answer, it seems I was on the right track but lost in details. The referenced book will be very handy. Also, thank you for the initial comments, this was my first attmpt on Stack Exchange, I should be more careful in future when asking questions. Regards. – miles Aug 6 '12 at 14:35
@miles: You're welcome! – joriki Aug 6 '12 at 14:41
+1. (And you know I am a big fan of AC.) In the case at hand, a shortcut might be to show that $g_{a\sqrt{n}}(\mathrm e^{-\lambda/n})\to\mathrm e^{-a\sqrt{2\lambda}}=\mathrm E(\mathrm e^{-\lambda\tau_a})$, where $\tau_a$ denotes the first hitting time of the level $a$ by a standard Brownian motion, and to identify the RHS as the Laplace transform of the correct density. – Did Aug 6 '12 at 14:58
@did: Interesting, could you spell that out a bit? By the way, it's also interesting that you seem to be getting comment notifications; I'm not; I was going to file a bug but perhaps it's just my browser then. – joriki Aug 6 '12 at 15:17
Using $s=\mathrm e^{-\lambda/n}$ and $k=a\sqrt{n}$ in the expression of $g_k(s)$ stated in your post yields $1-s^2\sim2\lambda/n$, $(1-\sqrt{1-s^2})^{a\sqrt{n}}\to\mathrm e^{-a\sqrt{2\lambda}}$ and $s^{a\sqrt{n}}\to1$, hence $g_k(s)\to\mathrm e^{-a\sqrt{2\lambda}}$. Now, using $x=a\sqrt{2\lambda}$ and introducing $H(x)=\int\limits_0^{+\infty}\mathrm e^{-x^2s/2-1/(2s)}\mathrm ds/\sqrt{2\pi s^3}$, one sees that $H(0)=1$ (use $s=1/t^2$) and $H'(x)=-H(x)$ (use $x^2s=1/u$) hence $H(x)=\mathrm e^{-x}$, QED. – Did Aug 6 '12 at 17:35

Let $g_m(s) = E[s^{\tau(m)}]$ be the probability generating function for the first passage time $\tau(m)$ to $m$. Since $\tau(0) = 0$ with probability $1$, $g_0(s) = 1$. Otherwise, there must be a first step which with equal probabilities takes you to $\pm 1$, so $g_m(s) = s (g_{m+1}(s) + g_{m-1}(s))/2$. The general solution of this homogeneous linear difference equation is $g_m(s) = A {r_+}^m + B {r_-}^m$ where $r_\pm = \frac{1 \pm \sqrt{1-s^2}}{s}$ and $A$, $B$ are constants. However, since we need $\lim_{s \to 0+} g_m(s)$ to exist, $r_+$ is excluded. Since $g_0(s) = 1$, we conclude that $$g_m(s) = {r_-}^m = \left(\frac{1-\sqrt{1-s^2}}{s}\right)^m$$

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Thank you very much for your answer. Can I ask, from g_m(s) is it possible to get to the standard continuous first passage distribution by some limiting step? – miles Aug 5 '12 at 20:44

This is to expand on some comments on @joriki's answer, which might complete it. The story began when I (somewhat foolishly) suggested:

In the case at hand, a shortcut might be to show that $g_{a\sqrt{n}}(\mathrm e^{-\lambda/n})\to\mathrm e^{-a\sqrt{2\lambda}}=\mathrm E(\mathrm e^{-\lambda\tau_a})$, where $\tau_a$ denotes the first hitting time of the level $a$ by a standard Brownian motion, and to identify the RHS as the Laplace transform of the correct density.

Prompted (quite rightfully) for some less cursory explanations, I then added:

Using $s=\mathrm e^{-\lambda/n}$ and $k=a\sqrt{n}$ in the expression of $g_k(s)$ stated in your post yields $1-s^2\sim2\lambda/n$, $(1-\sqrt{1-s^2})^{a\sqrt{n}}\to\mathrm e^{-a\sqrt{2\lambda}}$ and $s^{a\sqrt{n}}\to1$, hence $g_k(s)\to\mathrm e^{-a\sqrt{2\lambda}}$. Now, using $x=a\sqrt{2\lambda}$ and introducing $H(x)=\int\limits_0^{+\infty}\mathrm e^{-x^2s/2-1/(2s)}\mathrm ds/\sqrt{2\pi s^3}$, one sees that $H(0)=1$ (use $s=1/t^2$) and $H'(x)=-H(x)$ (use $x^2s=1/u$) hence $H(x)=\mathrm e^{-x}$, QED.

To which @joriki (pitilessly but legitimately) replied:

Where does the exponent in the integral come from?

The following is my best try to answer this question.

To sum up, at this point one knows that $\mathrm e^{-a\sqrt{2\lambda}}=\mathrm E(\mathrm e^{-\lambda\tau_a})=\int\limits_0^{+\infty}\mathrm e^{-\lambda t}f_a(t)\mathrm dt$ for some density $f_a$ and the goal is to determine $f_a$. The changes of variables $x=a\sqrt{2\lambda}$ and $t=sa^2$ yield that the function $f$ defined by $f(s)=a^2f_a(a^2s)$ is independent of $a$ and solves $$\mathrm e^{-x}=\int\limits_0^{+\infty}\mathrm e^{-x^2s/2}f(s)\mathrm ds.$$ My problem here is that I know what is the solution $f$ of this identity, as a distribution canonically associated to the linear Brownian motion, but that what @joriki asked for is a route to find the density $f$. Such a route might be as follows.

First, multiply both sides by $\mathrm e^x$, then complete the square in the exponential. This yields $2x-x^2s=-xz^2+1/s$ with $z=\sqrt{xs}-1/\sqrt{xs}$. The transformation $s\mapsto\zeta(s)=\sqrt{xs}-1/\sqrt{xs}$ is a diffeomorphism from $(0,+\infty)$ to $\mathbb R$ with inverse $z\mapsto\sigma(z)$ and one asks that $$1=\int\mathrm e^{-xz^2/2}\mathrm e^{1/(2s)}f(s)\mathrm ds=\int\limits_{-\infty}^{+\infty}\mathrm e^{-xz^2/2}\mathrm e^{1/(2\sigma(z))}f(\sigma(z))\sigma'(z)\mathrm dz.$$ This is a Gaussian integral with respect to $z$ hence, for every nonnegative function $\eta$ such that $\eta(z)+\eta(-z)=1$, the identity $$\sqrt{x}\eta(z)=2\sqrt{2\pi}\mathrm e^{1/(2\sigma(z))}f(\sigma(z))\sigma'(z),$$ yields a solution $f$. Thus, $$f(s)=\frac{\sqrt{x}}{\sqrt{2\pi}}\mathrm e^{-1/(2s)}\zeta'(s)2\theta(xs)=\frac1{\sqrt{2\pi}}\mathrm e^{-1/(2s)}\frac1{s\sqrt{s}}(1+xs)\theta(xs),$$ for some function $\theta$ such that $\theta(1/u)+\theta(u)=1$. Choosing $\theta(u)=1/(1+u)$ yields, as required, a solution $f$ independent of $x$, namely, $$f(s)=\frac1{\sqrt{2\pi}}\mathrm e^{-1/(2s)}\frac1{s\sqrt{s}}.$$

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Thanks for spelling this out! It's a cool approach, and good to know about; but I'm afraid it has now lost at least some of its appeal as a "shortcut" :-) – joriki Aug 12 '12 at 13:54
@joriki: You might have a point here... – Did Aug 12 '12 at 13:55