NUMERICAL ANALYSIS OF REFLECTING BROWNIAN MOTION AND A NEW MODEL OF SEMI-REFLECTING BROWNIAN MOTION WITH SOME DOMAINS

We investigate the error of the Euler-Maruyama approximate solution of the multidimensional reflecting Brownian motion and semi-reflecting Brownian motion using the penalty method and show their numerical simulations. Furthermore we consider a new model of semireflecting Brownian motion with some domains. AMS (MOS) Subject Classification. 60G50, 60H10, 65C30 1. PRELIMINARIES Recently, from the viewpoint of applications for financial mathematics, statistical physics and so on, one of important problems in stochastic analysis is to consider stochastic differential equations with reflecting boundaries on multi-dimensional domains (so-called Skorohod SDE). For example, in the case of mathematical finance, when a stock continues to climb or drop investors often suppose that its ceiling height or bottom, respectively. Therefore such fluctuations are often considered to have a kind of reflecting barrier. In this paper we investigate the behavior of reflecting Brownian motion with some conditions for reflecting boundaries on multidimensional domains using numerical simulation. We also investigate sample paths of multidimensional reflecting Brownian motion with a semi-reflecting boundary, i.e. any path of the Brownian motion does not reflect at the boundary immediately but is absorbed for a short period according to the speed of the path getting out of the boundary. Such situation is observed when electromagnetic wave reaches the ionosphere of the earth. Furthermore, in the case of mathematical finance, the fluctuations of some financial assets, e.g. stocks, bonds etc., are often considered to have a kind of reflecting barrier which seems to have a kind of flexibility, not perfect elasticity. In the last section we propose a new model of semi-reflecting Brownian motion with semi-reflecting domains in which the semi-reflecting Brownian motion moves Received March 28, 2008 1083-2564 $15.00 c ©Dynamic Publishers, Inc. REFLECTING BROWNIAN MOTION 231 from a domain to another domain, one after another. We realize a semi-reflecting Brownian motion by a modification of the penalty function which defines an usual reflecting Brownian motion. For observing numerical data by computer simulation of Skorohod SDE (stochastic differential equation) we define a discretized approximate solution of the SDE and obtain its mean square error estimation. There are two approaches to define an approximate solution of such SDE. [19], and [11] and [15] constructed Skorohod equations using the projection π on the boundary. Their method is called projection scheme. Roughly speaking, the reflecting path x(t) is defined for given function w(t) by the following manner: Define a step function wn(t) by discretization of w(t) and construct the reflecting step function xn(t) for wn(t) as follows. For a domain D, if xn(t0) ∈ D and yn(s) ≡ xn(t0)+wn(s)−wn(t0) ∈ D, t0 ≤ s ≤ t for some t0 and t, put xn(s) = yn(s), t0 ≤ s ≤ t. On the other hand, if yn(s) ∈ D, t0 ≤ s < t and yn(t) / ∈ D, put xn(t) = π(yn(t)). We can show xn tends to x uniformly in t. For non-convex domains, this projection can be defined in r0 neighborhood of the domain for some finite constant r0. A second one is the penalty method type, that is, we approximate Skorohod equations by equations with coefficients of gradient type. For the purpose of obtaining numerical data by computer simulation of Skorohod SDE’s we need discretized approximate solutions of them. In this paper, we focus our attention on reflecting Brownian motion on multi-dimensional domains and give a new penalty method type approximation. Our scheme is separated into two steps: we first approximate reflecting Brownian motions by standard Brownian motions with a drift term whose coefficients are gradient type (penalty method) in the pathwise sense and secondly we approximate the Brownian motion with the drift term by the Euler-Maruyama scheme. Here we do not need to assume boundedness and convexity for domains. We make use of a slight modification of the result in [2] to obtain the rate of convergence. As for approximation of solutions of general SDE, see e.g. [13], [18], and [8]. 2. STRONG APPROXIMATION OF MULTIDIMENSIONAL REFLECTING BROWNIAN MOTION 2.1. Skorohod equation. In the section we consider a strong approximation of multidimensional reflecting Brownian motion according to [9] and [5] in which stochastic differential equations with some boundary conditions on multidimensional domains (so-called Skorohod SDE) are considered using a penalty method. It had been known that the Skorohod SDE with smooth, convex and bounded domain has a unique solution until Saisho [15] extended the result to prove the existence of it for nonconvex and non-smooth domain (see Figure 1). Now we show some results which are obtained in [17] and [15]. We first consider two essential conditions for a domain 232 S. KANAGAWA D ⊂ R. Define some sets of inward normal vectors at x ∈ ∂D by Nx,r = {n : |n| = 1, B (x − rn, r) ∩ D = ∅} , r > 0, Nx = ⋃ r>0 Nx,r, x ∈ ∂D, where B(y, r) = {z ∈ R : |z − y| < r}, y ∈ R, r > 0. Condition (A) There exists a constant r0 > 0 such that for all x ∈ ∂D Nx = Nx,r0 6= ∅. Condition (B) There exist constants δ > 0 and β ∈ [1,∞) such that the following holds: for any x ∈ ∂D there is a unit vector lx with 〈lx, n〉 ≥ 1/β forany n ∈ ⋃ y∈B(x,δ)∩∂D Ny, where 〈·, ·〉 is the usual inner product in R. Figure 1. Non-smooth domain, Star shaped domain, Normal vector Remark 2.1. A star shape domain in Fig. 1 does not satisfy Condition (A). Remark 2.2. For any convex domain D, Condition (A) is satisfied for all r0 > 0. Condition(A) means that for any x ∈ ∂D there exists a ball with the radius larger than r0 which touches D at x but is disjoint with D as in the cases of Figures 1-2 and 7-12. On the other hand, from Fig. 1, at the roots of hands of the star there does not exist such ball. Therefore the star shaped domain does not satisfy Condition (A). If D satisfies Condition (A) and dist(x, D) < r0 there exists a unique π(x) ≡ x ∈ D with dist(x, D) = |x− x|. Especially, we have x = x for x ∈ D and if x / ∈ D we have x ∈ ∂D. Skorohod problem/equation Skorohod problem is means to find a pair (x, φ) satisfying the equation (called Skorohod equation) (2.1) { x (t) = w (t) + φ (t) , t ≥ 0 w (0) ∈ D̄ under conditions · x (t) ∈ D, iscontinuous, REFLECTING BROWNIAN MOTION 233 · φ (t) : iscontinuousandboundedvariation, φ (0) = 0, · φ (t) = ∫ t 0 n (s)d |φ|s , |φ|t = ∫ t 0 I{x(s)∈∂D}(s)d |φ|s, · n (s) ∈ Nx(s) if x (s) ∈ ∂D for given continuous function w. Here, |φ|t is the total variation of φ on [0, t] and IA (s) = { 1, s ∈ A 0, s / ∈ A . Existence of the unique solution of the Skorohod equation (2.1) is guaranteed by the following theorem. Theorem 1. [15] Suppose that D satisfies Conditions (A) and (B). Then for any continuous w, there exists a unique solution (x, φ) of the Skorohod problem. 2.2. Euler-Maruyama scheme. Let σ : R → R ⊗ R, b : R → R be bounded Lipschitz continuous functions and △n be a partition of the interval [0, T ]: △n : 0 = t0 < t1 < t2 < · · · < tn = T and put |△n| = max 1≤k≤n |tk − tk−1|, △B(tk) = B(tk) − B(tk−1), △tk = tk − tk−1. We note that {B(t)} is a standard Brownian motion. According to Maruyama [13] the Euler-Maruyama scheme for the SDE