AN INTEGRAL EQUATION FOR THE DISTRIBUTION OF THE FIRST EXIT TIME OF A REFLECTED BROWNIAN MOTION

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1 ANZIAM J. (29, 1 1 AN INTEGRAL EQUATION FOR THE DISTRIBUTION OF THE FIRST EXIT TIME OF A REFLECTED BROWNIAN MOTION VICTOR DE-LA-PEÑA 1, GERARDO HERNÁNDEZ-DEL-VALLE 2 and CARLOS G. PACHECO-GONZÁLEZ 3 (Received 15 March, 29 Abstract The reflected Brownian motion is being used in areas such as physiology, electrochemistry and nuclear magnetic resonance. We study the first passage time problem of this process which is relevant in applications; specifically, we find a Volterra integral equation for the distribution of the first time that a reflected Brownian motion reaches a non-decreasing barrier. Additionally, we make mention of how a numerical procedure can be used to handle the integral equation. 2 Mathematics subject classification: 6G7, 6G17, 45D5, 81Q, 6G17. Keywords and phrases: reflected Brownian motion, first passage time, Volterra integral equation. 1. Introduction The reflected Brownian motion has received a considerable amount of attention due to its capacity to model several real phenomena in physics, biology and chemistry; the reader can refer to Grebenkov [12] where it is mentioned several applications. It is known that this model is very suitable for modeling the interaction between a particle diffusing in a medium and an interface, where the particle may suffer a reflection (see for example [12, 18]. First passage time problems are important in these kind of applications (see for instance Levitz et al. [18], which motivates us to study the distribution of the time when the process surpasses a increasing varying level/barrier, i.e. the time when the particle interacting with the interface reaches a specific level 1 Statistics Department, Columbia University, 1255 Amsterdam Ave. Mail Code 443, New York, N.Y.; vp@stat.columbia.edu. 2 Statistics Department, Columbia University; gerardo@stat.columbia.edu. 3 Departamento de Matematicas, CINVESTAV-IPN, A. Postal 14-74, Mexico D.F. 7, MEXICO; cpacheco@math.cinvestav.mx. c Australian Mathematical Society 29, Serial-fee code /9 1

2 2 An integral equation for the distribution of the first exit time of a reflected Brownian motion [2] in the medium (separated from the interface. Hence, we concentrate on finding the distribution of T= inf{t : B t = f (t}, (1.1 where B={B t ; t } is a standard Brownian motion (BM and f (t is a non-decreasing function with f (>. However, this issue can be seen as a particular case of the first time that the BM exits a moving-range. That is, let f (t and g(t be two positive functions such that f (, g(>(that is, the BM starts in between the barriers. The general probabilistic question is: what is the distribution of T := inf{t : B t ( g(t, f (t}? (1.2 There are indeed works available in the literature concerning first exit times problems from varying regions, with single and double barriers. Considering a single barrier, see for example Ricciardi et al. [24], Peskir [22], De-la-Peña and Hernándezdel-Valle [1], Darling and Siegert [7]. Regarding two-sided barriers, as in (1.2, there has been also a lot of interest, mainly from a theoretical point of view. In Lifshits and Shi [19], it is concerned on the tail behavior of the exit-time-distribution from parabolic domains of a planar Brownian motion, see also [1, 2, 21]. Deblassie [8] studies the probability that a Brownian motion (with dimension higher that 1 remains in what he calls horn-shapes domains. Other related works worth to look at are [5, 9, 2]. Notice that the reflected Brownian motion is a one-dimensional Bessel process, in Betensky [3] it is treated passage issues for Bessel processes. Here we focus on characterizing the distribution of T, variable (1.1. The paper is organized as follows. In Section 2 we give preliminary results. In Section 3 we find an integral equation for the first exit time distribution of the reflected Brownian motion, and in Section 4 we state the integral equation for the density. At the end, we use a numerical method to solve integral equations. We also give conclusions and mention some open problems. 2. Hitting times of a Brownian motion To start studying exit times distribution, we define common variables that we use. Process B will be the Brownian motion (BM throughout the paper. The following hitting times are of interest T f := inf{t : B t = f (t}, T g := inf{t : B t = g(t}, where f and g are two functions, such that f ( > and g( >. The first fact we have to notice is that T f d = T f, by symmetry of the BM ( d = stands for equality in distribution. We readily see that the exit time T, defined in (1.2, satisfies T= T f T g. So, the exit time is the first time B hits any of the barriers.

3 [3] How to use the class filecupaus.cls 3 The following cases are known (refer to [11, 14 16, 23]. If f (t=a for all t, the densityϕ a (s of T f is given by { } a ϕ a (s= exp a2, s. 2πs 3 2s If f (t=ct+ a, with a>, the density becomes { } a ϕ f (s= exp (cs+a2, t. (2.1 2πs 3 2s For the case of constant barriers f (t=g(t=a, it is known that the Laplace transform of T ([23, Proposition 2.3.7] is: E ( e θt = 1 cosh(a 2θ. Thus, the density is given by the inverse function. In this paper, we shall work with a more general form: Proposition 2.1 ([4, Page 172]. Consider a Brownian motion B starting at x. The density of T, the first time B exits [ a, b] with x [ a, b], is given by where P(T dt= ss (b x,b+a (t+ ss (x+a,b+a (t, (2.2 ss (u,v (t= k= v u+2kv e (v u+2kv2 /(2t. 2πt 3 Furthermore, the expected value of T is ([23, Exercise ] E (T=(x+a(b x. (2.3 We denote byϕ ( a,b x (t the density function P(T dt in ( Integral equation When the functions f and g in (1.2 are non-decreasing, T is said to be the first time that B exits the horn-shape {( g(t, f (t, t }. In this section, we shall derive an integral equation to compute the distribution of the first time a BM leaves a region determined by reflective barriers, that is, when g= f. First we need some notation Notation Let f and g be two positive functions We denote by T f the first time a BM hits f, and with T g when it hits g. The notation is the same when f and g are constants, namely T b or T a. We denote by T g(x f the first time a BM starting at x hits f or g, where x [ g(, f (]. So T a(x b denotes the hitting time when the barriers are constants. When there is not ambiguity we write T g( f only as T.

4 4 An integral equation for the distribution of the first exit time of a reflected Brownian motion [4] The distribution of T g(x f is denoted byφ ( g, x f (t := P(T g(x<t, f or onlyφ(t when x=. Notice that if the barriers are constants, the distribution is known, and the density ( g(t, f (s x is given byϕ ( a,b x (u in (2.2. For example, if we fix t and s, thenϕ (u denotes the density of the first time a BM hits either of the fixed points g(t or f (s (the functions evaluated at specific nodes t and s Methodology As part of the technique, we shall take approximations of the barrier. So, given a function f, we consider a partition of the time domain [, t], namelyπ n :={=t,n < t 1,n < <t n,n = t} then the approximating barrier is f n (t := n f (t i 1,n I [ti 1,n,t i,n (t, (3.1 i=1 whereiis the indicator function. We take the partitions such thatπ m Π n for m<n, and max t i,n t j,n as n. We have that i, j n Proposition 3.1. For any pair of non-decreasing functions f and g onr + with f (> and g( >, we have P ( T f n g n < t P ( T f g< t, for any t>, as n, where, f n and g n are the approximations as in (3.1 on the same partitions {Π n, n=1, 2,...}. Proof. By construction f n (s f (s and g n (s g(s for all s, thus, the sequence P ( T f n g n < t, n=1, 2,... is decreasing, and is bounded from below by P(T< t. Since we have { T f n g n < t } { T g< f t } as n, the convergence holds. Exploiting well known properties of the BM, in the following result we obtain an integral equation of Volterra type for the exit time distribution. We use similar argumentation as in [1] or [13]. Theorem 3.2. Let f be a non-decreasing function such that f ( >. Then, the distributionφof the first exit time T f f ( obeys the following integral equation t t ( s ( f (u, f (u ( f (s f (u, f (s f (u Φ(t= ϕ (udu ϕ (s uφ(du ds. (3.2 where functionϕ (a,b x (u is given by (2.2.

5 [5] How to use the class filecupaus.cls 5 Proof. Consider an approximation{ f n, n=1, 2,...} for f as in (3.1. We note Φ n (t=p ( T f n < t = n i=1 P ( t i 1,n T f n < t i,n. (3.3 Here,Φ n ( =Φ (, f n (, and for simplicity we only write T f g instead of T f g(. By Proposition 3.1 we have thatφ n Φas n. The proof is divided into three parts. Part 1. We analyze the right hand side of identity (3.3. For notational convenience, set A i,n = ( t i 1,n T f (t i 1,n f (t i 1,n < t i,n. Then for each term we observe that P ( t i 1,n T f n < t i,n = P(Ai,n P ( A i,n, T f n < t i 1,n. (3.4 Part 2. The first part of the right hand side of (3.4 can be obtained from (2.2 by integrating on the interval [t i 1,n, t i,n. So, it reads as follows P(A i,n = ti,n ϕ ( f (t i 1,n, f (t i 1,n (udu. t i 1,n By the Mean Value Theorem for integrals (consult for instance [2] the previous equation becomes P(A i,n =ϕ ( f (t i 1,n, f (t i 1,n (t i,n (t i,n t i 1,n, for some value ti,n [t i 1,n, t i,n. Thus, in Equation (3.3 we actually have a Riemann sum which converges to the desired quantity: lim n n i=1 φ ( f (t i 1,n, f (t i 1,n (t i,n (t i,n t i 1,n = t ( f (u, f (u ϕ (udu. Hence, the second element of Equation (3.2 is obtained. Part 3. For the last term in (3.4, we have the following analysis, P ( A i,n, T f n < t i 1,n = ti 1,n i 1 = i 1 = tk,n P ( A i,n T f n = u Φ n (du t k 1,n P ( A i,n T f n = u Φ n (du tk,n i 1 + t k 1,n P ( A i,n T f n < T fn, T f n = u P ( T f n < T fn T f n = u Φ n (du tk,n t k 1,n P ( A i,n T f n T fn, T f n = u P ( T f n T fn T f n = u Φ n (du. (3.5

6 6 An integral equation for the distribution of the first exit time of a reflected Brownian motion [6] Now, we analyze the quantities involved in the sums. We know that P ( T f n < T fn T f n = u + P ( T f n T fn T f n = u = 1. By the symmetry of the BM, the probabilities are equal: P ( T f n < T fn T f n = u = P ( T f n T fn T f n = u = 1 2. (3.6 On the other hand, for each integral in the sums, let u [t k 1,n, t k,n with k i 1. From the regenerative properties of the BM we have that P ( A i,n T f n T fn, T f n = u = P ( t i 1,n u T f (t i 1,n f (t k 1,n f (t i 1,n f (t k 1,n < t i,n u = ti,n u t i 1,n u ϕ ( f (t i 1,n f (t k 1,n, f (t i 1,n f (t k 1,n (wdw, (3.7 (for this last step, the assumption of non-decreasing barriers is important then, after a change of variable and applying the Mean Value Theorem for integrals for each, 2,...,i 1the probability (3.7 becomes ϕ ( f (t i 1,n f (t k 1,n, f (t i 1,n f (t k 1,n (t k,i,n u(t k,n t k 1,n, (3.8 for some tk,i,n [t i 1,n, t i,n, and k<i 1and i=1, 2,...,n. Furthermore, for the last sums at (3.5, due to the symmetry of the BM we have that P ( t i 1,n T f (t i 1,n f (t i 1,n < t T f n < T fn, T f n = u = P ( t i 1,n T f (t i 1,n f (t i 1,n < t T f n T fn, T f n = u. (3.9 So, using also (3.6 and (3.8, probability (3.5 ends up being i 1 2 tk,n ϕ ( f (t i 1,n f (t k 1,n, f (t i 1,n f (t k 1,n (tk,i,n u(t k,n t k 1,n 1 t k 1,n 2 Φ n(du. Substitution in equations (3.4 and (3.3 yields n i 1 i=1 tk,n ϕ ( f (t i 1,n f (t k 1,n, f (t i 1,n f (t k 1,n (tk,i,n u(t k,n t k 1,n Φ n (du. (3.1 t k 1,n Upon applying again the Mean Value Theorem, there are t i,k,n [t k 1,n, t k,n for each i<n, such that sum (3.1 equals n i 1 i=1 ϕ ( f (t i 1,n f (t k 1,n, f (t i 1,n f (t k 1,n (t k,i,n t i,k,n (t k,n t k 1,n ( Φ n (t i,n Φ n (t i 1,n. (3.11

7 [7] How to use the class filecupaus.cls 7 Equation (3.11 represents a Riemann Stieltjes sum of a continuous function. Thus, when n we obtain the limit t ( s ( f (s f (u, f (s f (u ϕ (s uφ(du ds, which finally yields the last part of equation (3.2. This concludes the proof. 4. Equations for the density In the previous section, Theorem 3.2 stated an integral equation for the distribution of T, the first exit time. There are results on first passage time requiring additional conditions in order to derive an integral equation for the density. In [1, 22, 24] the barrier needs to be differentiable. In our case, we can readily see in Equation (3.2 that Φ is differentiable, and thus we are able to obtain an integral equation for the density. Corollary 4.1. Under conditions of Theorem 3.2, we have that the density of T, which we denote byϕ, satisfies the integral equation ( f (t, f (t ϕ(t=ϕ (t t where functionϕ (a,b x (u is given by (2.2. ( f (t f (u, f (t f (u ϕ (t uϕ(udu, (4.1 Remark 1. As mentioned above, generally one asks to have differentiable barriers to ensure the existence of a density, however, since the barriers we use are nondecreasing, one also has that the distribution function is differentiable. Remark 2. Since functionϕ ( a,b x (u is continuous in all its arguments, it is known that Equation (4.1 has unique solution. This is a classical result in the theory of integral equations (for instance, see [6, Theorem 5, Page 183] Another expression for the exit time density From the proof of Theorem 3.2, we can see that it is possible to modify slightly the integral equation. Recall notation T g(x; f when looking at Equation (3.7, one also has ( P t i 1,n T f (t i 1,n f (t i 1,n (<t T f n T fn, T f n = u for u (t k 1,n, t k,n ]. Finally, it is written as = P ( t i 1,n u T f (t i 1,n f (t i 1,n ( f (t k 1,n<t i,n u, ti,n u t i 1,n u ϕ ( f (t i 1,n, f (t i 1,n f (t k 1,n (wdw. The idea is to consider a BM starting at f (t k 1,n, rather than at, as it was originally done in (3.7. The change gives a new expression for the integral equation, namely ( f (t, f (t ϕ(t=ϕ (t t ϕ ( f (t, f (t f (u (t uϕ(udu. (4.2

8 8 An integral equation for the distribution of the first exit time of a reflected Brownian motion [8] 4.2. Numerical solutions We now exploit a numerical procedure to solve (4.1. A numerical algorithm based on a recursive formulae is quite straight forward to implement; we summarize it in a brief manner (the interested reader can refer to [17]. We want to solve the Volterra integral equation F(t=G(t+ t K(t, sf(sds, where G and K are known functions, this last one is usually called the kernel. It is called of the second kind because function G is not naught; the fact that G is not zero is important for the method to work. Suppose that we want to obtain an approximation of F in the interval [, r], and we divide it into N equally spaced subinterval of size h. This means that we have N+ 1 nodes{t, t 1,...,t N } such that t n+1 t n = h, n=, 1,..., N 1, with t =. We denote by{f 1,..., F N } the approximation of f at the nodes{t 1,...,t N }. The following recursive formula can be used for n=,..., N 1. Notice that tk F n+1 = G(t n+1+ n F k K(t t n+1, sds k 1 1 t n+1, (4.3 K(t t n+1, sds n G(h F 1 = 1. h K(h, sds Example 1. Using Scilab 4.1.2, we compute the density function of T 1 1 ( and T f f (, where f (t=1+.1t. The two densities should be close from each other. The density of T 1 1 ( is known explicitly as (2.2, which is used to plot (Figure 1 a graph (line. The density of T f f ( is plotted (with dots using procedure ( Conclusion We studied the first exit time distribution of a reflected Brownian motion. We have found a Volterra integral equation for the density. The main result is derived from approximating the barriers by step functions and carrying a careful analysis of the paths. The solution of the integral equations does not seem a trivial task, however, it was shown how tractability is feasible using numerical methods. We briefly mention other possible direction to take: One may be willing to extend the result to the case of non-symmetric barriers, where the main technical difficulty is the non-symmetrical probabilities in (3.6. It remains open the problem of solving explicitly the integral equation.

9 [9] How to use the class filecupaus.cls Figure 1. Densities for the exit times. There is an interesting relation between the maximum and the reflected Brownian motion. Let M be the maximum of the Brownian motion, that is, M t = max(b s, s t. It is known the two processes {M t B t, t } and { B t,t } have the same law (see, for example, [14, Page 21]. One may want to use this to find the first passage distribution of M. Acknowledgement This research was partially supported by the CONACYT-NSF (Mexico-USA program. References [1] D. Bañuelos, R.D. DeBlassie, and R. Smits, The first exit time of a planar Brownian motion from the interior of a parabola. Ann. Probab. 29(2, (21. [2] R.G. Bartle, The Elements of Real Analysis, 2nd ed. John Wiley & Sons., New York (1976. [3] R.A. Betensky, A boundary crossing probability for the Bessel process. Adv. Appl. Prob. 3, (1998. [4] A.N. Borodin and P. Salminen, Handbook of Brownian Motion - Facts and Formulae, Birkhäuser, Basel (1996. [5] Burkholder, D. L., Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. Math. 26, (1977. [6] W. Cheney, Analysis for Applied Mathematics, Springer, New York (21. [7] D.A. Darling and A.J.F. Siegert, The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, (1953.

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