Steven R. Dunbar
Department of Mathematics
203 Avery Hall
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

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Hitting Times and Ruin Probabilities

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Note: These pages are prepared with MathJax. MathJax is an open source JavaScript display engine for mathematics that works in all browsers. See http://mathjax.org for details on supported browsers, accessibility, copy-and-paste, and other features.

_______________________________________________________________________________________________ ### Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

_______________________________________________________________________________________________ ### Section Starter Question

What is the probability that a simple random walk with $p=1∕2=q$ starting at the origin will hit value $a>0$ before it hits value $-b<0$, where $b>0$? What do you expect in analogy for the standard Wiener process and why?

_______________________________________________________________________________________________ ### Key Concepts

1. With the Reﬂection Principle, we can derive the p.d.f of the hitting time ${T}_{a}$.
2. With the hitting time, we can derive the c.d.f. of the maximum of the Wiener Process on the interval $0\le u\le t$.

__________________________________________________________________________ ### Vocabulary

1. The Reﬂection Principle says the Wiener process reﬂected about a ﬁrst passage has the same distribution as the original motion.
2. The hitting time ${T}_{a}$ is the ﬁrst time the Wiener process assumes the value $a$. In notation from analysis
${T}_{a}=inf\left\{t>0:W\left(t\right)=a\right\}.$

__________________________________________________________________________ ### Mathematical Ideas

#### Hitting Times

Consider the standard Wiener process $W\left(t\right)$, which starts at $W\left(0\right)=0$. Let $a>0$. The hitting time ${T}_{a}$ is the ﬁrst time the Wiener process hits $a$. In notation from analysis

${T}_{a}=inf\left\{t>0:W\left(t\right)=a\right\}.$

Note the very strong analogy with the duration of the game in the gambler’s ruin.

Some Wiener process sample paths will hit $a>0$ fairly directly. Others will make an excursion to negative values and take a long time to ﬁnally reach $a$. Thus ${T}_{a}$ will have a probability distribution. We determine that probability distribution by a heuristic procedure similar to the ﬁrst step analysis we made for coin-ﬂipping fortunes.

Speciﬁcally, we consider a probability by conditioning, that is, conditioning on whether or not ${T}_{a}\le t$, for some given value of $t$.

$ℙ\left[W\left(t\right)\ge a\right]=ℙ\left[W\left(t\right)\ge a|{T}_{a}\le t\right]ℙ\left[{T}_{a}\le t\right]+ℙ\left[W\left(t\right)\ge a|{T}_{a}>t\right]ℙ\left[{T}_{a}>t\right]$

Now note that the second conditional probability is $0$ because it is an empty event. Therefore:

$ℙ\left[W\left(t\right)\ge a\right]=ℙ\left[W\left(t\right)\ge a|{T}_{a}\le t\right]ℙ\left[{T}_{a}\le t\right].$

Now, consider Wiener process “started over” again at the time ${T}_{a}$ when it hits $a$. By the shifting transformation from the previous section, the “started-over” process has the distribution of Wiener process again, so

$\begin{array}{llll}\hfill ℙ\left[W\left(t\right)\ge a|{T}_{a}\le t\right]& =ℙ\left[W\left(t\right)\ge a|W\left({T}_{a}\right)=a,{T}_{a}\le t\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =ℙ\left[W\left(t\right)-W\left({T}_{a}\right)\ge 0|{T}_{a}\le t\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =1∕2.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

This argument is a speciﬁc example of the Reﬂection Principle for the Wiener process. It says that the Wiener process reﬂected about a ﬁrst passage has the same distribution as the original motion.

Thus

$ℙ\left[W\left(t\right)\ge a\right]=\left(1∕2\right)ℙ\left[{T}_{a}\le t\right].$

or

$\begin{array}{llll}\hfill ℙ\left[{T}_{a}\le t\right]& =2ℙ\left[W\left(t\right)\ge a\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2}{\sqrt{2\pi t}}{\int }_{a}^{\infty }exp\left(-{u}^{2}∕\left(2t\right)\right)\phantom{\rule{3.26288pt}{0ex}}\phantom{\rule{0.3em}{0ex}}du\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2}{\sqrt{2\pi }}{\int }_{a∕\sqrt{t}}^{\infty }exp\left(-{v}^{2}∕2\right)\phantom{\rule{3.26288pt}{0ex}}\phantom{\rule{0.3em}{0ex}}dv\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

(note the change of variables $v=u∕\sqrt{t}$ in the second integral) and so we have derived the c.d.f. of the hitting time random variable. One can easily diﬀerentiate to obtain the p.d.f

${f}_{{T}_{a}}\left(t\right)=\frac{a}{\sqrt{2\pi }}{t}^{-3∕2}exp\left(-{a}^{2}∕\left(2t\right)\right).$

Actually, this argument contains a serious logical gap, since ${T}_{a}$ is a random time, not a ﬁxed time. That is, the value of ${T}_{a}$ is diﬀerent for each sample path, it varies with $\omega$. On the other hand, the shifting transformation deﬁned in the prior section depends on having a ﬁxed time, called $h$ in that section. To ﬁx this logical gap, we must make sure that “random times” act like ﬁxed times. Under special conditions, random times can act like ﬁxed times. Speciﬁcally, this proof can be ﬁxed and made completely rigorous by showing that the standard Wiener process has the strong Markov property and that ${T}_{a}$ is a Markov time corresponding to the event of ﬁrst passage from $0$ to $a$.

Note that deriving the p.d.f. of the hitting time is much stronger than the analogous result for the duration of the game until ruin in the coin-ﬂipping game. There we were only able to derive an expression for the expected value of the hitting time, not the probability distribution of the hitting time. Now we are able to derive the probability distribution of the hitting time fairly intuitively (although strictly speaking there is a gap). Here is a place where it is simpler to derive a quantity for Wiener process than it is to derive the corresponding quantity for random walk.

Let us now consider the probability that the Wiener process hits $a>0$, before hitting $-b<0$ where $b>0$. To compute this we will make use of the interpretation of the standard Wiener process as being the limit of the symmetric random walk. Recall from the exercises following the section on the gambler’s ruin in the fair ($p=1∕2=q$) coin-ﬂipping game that the probability that the random walk goes up to value $a$ before going down to value $b$ when the step size is $\Delta x$ is

Thus, the probability of hitting $a>0$ before hitting $-b<0$ does not depend on the step size, and also does not depend on the time interval. Therefore, passing to the limit in the scaling process for random walks, the probabilities should remain the same. Here is a place where it is easier to derive the result from the coin-ﬂipping game and pass to the limit than to derive the result directly from Wiener process principles.

#### The Distribution of the Maximum

Let $t$ be a given time, let $a>0$ be a given value, then

$\begin{array}{llll}\hfill ℙ\left[\underset{0\le u\le t}{max}W\left(u\right)\ge a\right]& =ℙ\left[{T}_{a}\le t\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2}{\sqrt{2\pi }}{\int }_{a∕\sqrt{t}}^{\infty }exp\left(-{y}^{2}∕2\right)\phantom{\rule{3.26288pt}{0ex}}\phantom{\rule{0.3em}{0ex}}dy\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

#### Sources

This section is adapted from: Probability Models, by S. Ross, and A First Course in Stochastic Processes Second Edition by S. Karlin, and H. Taylor, Academic Press, 1975. The technical note about the algorithm combines discussion about Donkser’s principle from Freedman’s Brownian Motion and Diﬀusions, Breiman’s, Probability, Khoshnevisan’s notes, and Peled’s notes

_______________________________________________________________________________________________ ### Algorithms, Scripts, Simulations

#### Algorithm

Set a time interval length $T$ suﬃciently long to have a good chance of hitting the ﬁxed value $a$ before ﬁxed time $t. Set a large value $n$ for the length of the random walk used to create the Wiener process, then ﬁll an $n×k$ matrix with Bernoulli random variables. Cumulatively sum the Bernoulli random variables to create a scaled random walk approximating the Wiener process. For each random walk, ﬁnd when the hitting time is encountered. Also ﬁnd the maximum value of the scaled random walk on the interval $\left[0,t\right]$. Since the approximation is piecewise linear, only the nodes need to be examined. Compute the fraction of the $k$ walks which have a hitting time less than $t$ or a maximum greater than $a$ on the interval $\left[0,t\right]$. Compare the fraction to the theoretical value.

Technical Note: The careful reader will note that the hitting time ${T}_{a}=inf\left\{t>0:W\left(t\right)=a\right\}$ and the events $\left[{T}_{a}\le t\right]$ and $\left[\underset{0\le u\le t}{max}W\left(u\right)\ge a\right]$ are “global” events on the set of all Wiener process paths. However, the deﬁnition of Wiener process only has prescribed probability distributions on the values at speciﬁed times. Implicit in the deﬁnition of the Wiener process is a probability distribution on the “global” set of Wiener processes, but the proof of the existence of the probability distribution is beyond the scope of this text. Moreover, there is no easy distribution function to compute the probability of such events as there is with the normal probability distribution.

The algorithm approximates the probability by counting how many of $k$ scaled binomial random variables hit a value greater than or equal to $a$ at a node which corresponds to a time less than or equal to $t$. Convergence of the counting probability to the global Wiener process probability requires justiﬁcation with Donsker’s Invariance Principle. The principle says that the piecewise linear processes ${Ŵ}_{N}\left(t\right)$ converge in distribution to the Wiener process.

To apply Donsker’s Principle, consider the functional

$\varphi \left(f\right)=h\left(\underset{0\le u\le t}{max}f\left(u\right)\right)$

where $h\left(\cdot \right)$ is a bounded continuous function. The convergence in distribution from the Invariance Principle implies that

$𝔼\left[\varphi \left({Ŵ}_{N}\left(t\right)\right)\right]\to 𝔼\left[\varphi W\right].$

Now taking $h\left(\cdot \right)$ to be an approximate indicator function on the interval $\left[a,\infty \right)$ shows that the counting probability converges to the global Wiener process probability. The details require careful analysis.

Geogebra
R
1T <- 10
2a <- 1
3time <- 2
4
5p <- 0.5
6n <- 10000
7k <- 1000
8
9Delta = T/n
10
11winLose <- 2 * (array( 0+(runif(n*k) <= p), dim=c(n,k))) - 1
12# 0+ coerces Boolean to numeric
13totals <- apply( winLose, 2, cumsum)
14
15paths <- array( 0 , dim=c(n+1, k) )
16paths[2:(n+1), 1:k] <- sqrt(Delta)*totals
17
18hitIndex <- apply( 0+(paths <= a), 2, (function(x) match(0, x, nomatch=n+2)))
19# If no hiting on a walk, nomatch=n+2 sets the hitting
20# time to be two more than the number of steps, one more than
21# the column length.  Without the nomatch option, get NA which
22# works poorly with the comparison
23
24hittingTime = Delta*(hitIndex-1)
25## subtract 1 since vectors are 1-based
26
27probHitlessTa <- sum( 0+(hittingTime <= time))/k
28probMax = sum( 0+( apply(paths[1:((time/Delta)+1),], 2, max) >= a ) )/k
29theoreticalProb = 2*pnorm(a/sqrt(time), lower=FALSE)
30
31cat(sprintf("Empirical probability Wiener process paths hit %f before %f: %f \n", a, time, probHitlessTa ))
32cat(sprintf("Empirical probability Wiener process paths greater than %f before %f: %f \n", a, time, probMax ))
33cat(sprintf("Theoretical probability: %f \n", theoreticalProb ))
Octave
1T = 10;
2a = 1;
3time = 2;
4
5p = 0.5;
6n = 10000;
7k = 1000;
8
9Delta = T/n;
10
11winLose = 2 * (rand(n,k) <= p) - 1;
12# -1 for Tails, 1 for Heads
13totals = cumsum(winLose);
14# -n..n (every other integer) binomial rv sample
15
16paths = sqrt(Delta)*[zeros(1,k); totals];
17hittingTime = zeros(1,k);
18for j = 1:k
19  hitIndex = find(paths(:,j) >= a);
20  if ( !rows(hitIndex) )
21    hittingTime(j) = Delta * (n+2);
22    ## If no hitting on a walk, set hitting time to be two
23    ##  more than the number of steps, one more than the column length.
24  else
25    hittingTime(j) = Delta * (hitIndex(1)-1);
26    ## some hitting time
27    ## subtract 1 since vectors are 1-based
28  endif
29endfor
30
31probHitlessTa = sum( (hittingTime <= time) ) /k;
32probMax = sum( max(paths(1:((time/Delta)+1),:)) >= a)/k;
33theoreticalProb = 2*(1 - normcdf(a/sqrt(time)));
34
35printf("Empirical probability Wiener process paths hit %f before %f: %f \n", a, time, probHitlessTa )
36printf("Empirical probability Wiener process paths greater than %f before %f: %f \n", a, time, probMax )
37printf("Theoretical probability: %f \n", theoreticalProb )
Perl
1use PDL::NiceSlice;
2
3sub pnorm {
4    my ( $x,$sigma, $mu ) = @_; 5$sigma = 1 unless defined($sigma); 6$mu    = 0 unless defined($mu); 7 8 return 0.5 * ( 1 + erf( ($x - $mu ) / ( sqrt(2) *$sigma ) ) );
9}
10
11$T = 10; 12$a    = 1;
13$time = 2; 14 15$p = 0.5;
16$n = 10000; 17$k = 1000;
18
19$Delta =$T / $n; 20 21$winLose = 2 * ( random( $k,$n ) <= $p ) - 1; 22$totals = ( cumusumover $winLose->xchg( 0, 1 ) )->transpose; 23 24$paths = zeroes( $k,$n + 1 );
25
26# use PDL:NiceSlice on next line
27$paths ( 0 : ($k - 1 ), 1 : $n ) .= sqrt($Delta) * $totals; 28 29$hita = $paths->setbadif($paths <= $a ); 30 31$hitIndex =
32    ( $hita (,)->xchg( 0, 1 )->minimum_ind )->inplace->setbadtoval($n + 1 );
33
34$hittingTime =$Delta * $hitIndex; 35 36$probHitlessTa = sumover( $hittingTime <$time ) / $k; 37$probMax       = sumover(
38    maximum(
39        $paths ( 0 : ($k - 1 ), 0 : ( $time /$Delta ) - 1 )->xchg( 0, 1 )
40        ) >= $a 41) /$k;
42$theoreticalProb = 2 * ( 1 - pnorm($a / sqrt($time) ) ); 43 44print "Empirical probability Wiener process paths hit ",$a, "before ", $time, 45 "is ",$probHitlessTa, "\n";
46print "Empirical probability Wiener process paths greater than ", $a, 47 "before ",$time, "is ", $probMax, "\n"; 48print "Theoretical probability:",$theoreticalProb, "\n";
SciPy
1import scipy
2
3T = 10.0
4# note type float
5a = 1
6time = 2
7
8p = 0.5
9n = 1000
10k = 50
11
12Delta = T/n
13
14winLose = 2*( scipy.random.random((n,k)) <= p ) - 1
15totals = scipy.cumsum(winLose, axis = 0)
16
17paths = scipy.zeros((n+1,k), dtype=float)
18paths[ 1:n+1, :] = scipy.sqrt(Delta) * totals
19
20def match(x,arry,nomatch=None):
21    if arry[scipy.where( (arry >= x))].any():
22        return  scipy.where( (arry >= x) ) - 1
23    else:
24        return nomatch
25# arguments: x is a scalar, arry is a python list, value of nomatch is scalar
26# returns the first index of first  of its first argument in its second argument
27# but if a is not there, returns the value nomatch
28# modeled on the R function "match", but with less generality
29
30hitIndex = scipy.apply_along_axis(lambda x:( match(a,x,nomatch=n+2)), 0, paths)
31# If no ruin or victory on a walk, nomatch=n+2 sets the hitting
32# time to be two more than the number of steps, one more than
33# the column length.
34
35hittingTime = Delta * hitIndex
36
37probHitlessTa = (scipy.sum( hittingTime < time).astype(float))/k
38probMax = (scipy.sum( scipy.amax( paths[ 0:scipy.floor(time/Delta)+1, : ], axis=0) >= a).astype(float))/k
39from scipy.stats import norm
40theoreticalProb = 2 * (1 - norm.cdf(a/scipy.sqrt(time)))
41
42print "Empirical probability Wiener process paths hit ", a, "before ", time, "is ", probHitlessTa
43print "Empirical probability Wiener process paths greater than ", a, "before ", time, "is ", probMax
44print "Theoretical probability:", theoreticalProb

_______________________________________________________________________________________________ ### Problems to Work for Understanding

1. Diﬀerentiate the c.d.f. of ${T}_{a}$ to obtain the expression for the p.d.f of ${T}_{a}$.
2. Show that $𝔼\left[{T}_{a}\right]=\infty$ for $a>0$.
3. Suppose that the ﬂuctuations of a share of stock of a certain company are well described by a Wiener process. Suppose that the company is bankrupt if ever the share price drops to zero. If the starting share price is $A\left(0\right)=5$, what is the probability that the company is bankrupt by $t=25$? What is the probability that the share price is above $10$ at $t=25$?
4. Suppose you own one share of stock whose price changes according to a Wiener process. Suppose you purchased the stock at a price $b+c$, $c>0$ and the present price is $b$. You have decided to sell the stock either when it reaches the price $b+c$ or when an additional time $t$ goes by, whichever comes ﬁrst. What is the probability that you do not recover your purchase price?
5. Modify the scripts by setting $p>0.5$ or $p<0.5$. What happens to the hitting time?
1. Modify the scripts to plot the probability that the hitting time is less than or equal to $a$ as a function of $a$.
2. Modify the scripts to plot the probability that the hitting time is less than or equal to $a$ as a function of $t$. On the same set of axes plot the theoretical probability as a function of $t$.

__________________________________________________________________________ ### References

   Leo Breiman. Probability. Addison Wesley, 1968.

   David Freedman. Brownian Motion and Diﬀusions. Holden-Day, 1971. QA274.75F74.

   S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.

   Davar Khoshnevisan. Leture notes on donsker’s theorem. http://www.math.utah.edu/~davar/ps-_pdf-_files/donsker.pdf, 2005. accessed June 30, 2013.

   Ron Peled. Random walks and brownian motion. http://www.math.tau.ac.il/~peledron/Teaching/RW_and_BM_2011/index.htm, 2011.

   Sheldon M. Ross. Introduction to Probability Models. Elsevier, 8th edition, 2003.

__________________________________________________________________________ 1. Russell Gerrard, City University, London, Stochastic Modeling . Notes for the MSc in Actuarial Science, 2003-2004. Contributed by S. Dunbar October 30, 2005.

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