Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
Hitting Times and Ruin Probabilities
Mathematically Mature: may contain mathematics beyond calculus with proofs.
What is the probability that a simple random walk with starting at the origin will hit value before it hits value , where ? What do you expect in analogy for the standard Wiener process and why?
Consider the standard Wiener process , which starts at . Let . The hitting time is the ﬁrst time the Wiener process hits . In notation from analysis
Note the very strong analogy with the duration of the game in the gambler’s ruin.
Some Wiener process sample paths will hit fairly directly. Others will make an excursion to negative values and take a long time to ﬁnally reach . Thus 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 , for some given value of .
Now note that the second conditional probability is because it is an empty event. Therefore:
Now, consider Wiener process “started over” again at the time when it hits . By the shifting transformation from the previous section, the “started-over” process has the distribution of Wiener process again, so
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.
(note the change of variables 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
Actually, this argument contains a serious logical gap, since is a random time, not a ﬁxed time. That is, the value of is diﬀerent for each sample path, it varies with . On the other hand, the shifting transformation deﬁned in the prior section depends on having a ﬁxed time, called 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 is a Markov time corresponding to the event of ﬁrst passage from to .
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 , before hitting where . 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 () coin-ﬂipping game that the probability that the random walk goes up to value before going down to value when the step size is is
Thus, the probability of hitting before hitting 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.
Let be a given time, let be a given value, then
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
Set a time interval length suﬃciently long to have a good chance of hitting the ﬁxed value before ﬁxed time . Set a large value for the length of the random walk used to create the Wiener process, then ﬁll an 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 . Since the approximation is piecewise linear, only the nodes need to be examined. Compute the fraction of the walks which have a hitting time less than or a maximum greater than on the interval . Compare the fraction to the theoretical value.
Technical Note: The careful reader will note that the hitting time and the events and 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 scaled binomial random variables hit a value greater than or equal to at a node which corresponds to a time less than or equal to . 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 converge in distribution to the Wiener process.
To apply Donsker’s Principle, consider the functional
where is a bounded continuous function. The convergence in distribution from the Invariance Principle implies that
Now taking to be an approximate indicator function on the interval shows that the counting probability converges to the global Wiener process probability. The details require careful analysis.
R script for hitting time..
 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.
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