Strong Approximation of Reflecting Brownian Motion Using Penalty Method and its Application to Cumputer Simulation

— We deal with Skorohod problem (Skorohod equation), and state the EulerMaruyama scheme for SDE. We introduce a penalty method, and propose a new approximation of reflecting Brownian motion. Examples of numerical computations are given. Introduction One of important problems in stochastic analysis is to consider stochastic differential equations with boudary conditions on multi-dimensional domains (so-called Skorohod SDE). There are two approaches to define approximate Solutions of such stochastic differential equations. Tanaka [18], Lions and Sznitman [4], and Saisho [13] constructed Skorohod equations using the projection π on the boundary. Roughly speaking, the reflecting path χ (t) is defined for a 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] s follows. For a domain D, if zn(i0) G ~D and yn(s) = xn(tQ) + wn(s) wn(to) G ~D, i0 < s < t for some ίο and i, put xn(s) = yn(s)j to < s < i. On the other hand, if 106 S.Kanagawa and Y.Saisho yn(s) e D, i0 < s < t and yn(t) £ ~D, put xn(t) = π(τ/η(<)). We can show that xn tends to χ uniformly in t ([13]). For non-convex domains, this projection can be defined in ΓΟ neighbourhood of the domain for some finite constant ΓΟ (see Remark 1.2). This type of approximations is often called a projection scheme ([2],[9],[11],[16],[17]). Second one is the penalty method type, that is, we approximate Skorohod equations by equations with coefficients of gradient type ([5],[7],[8],[10],[15]) (for more details of penalty method, see Sect.3). In [5] and [10], the convergence rate of the approximation in LP is studied under assumptions of boundedness and convexity of domains. 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 usual Brownian motions with a drift term whose coefficients are of gradient type (penalty method) in pathwise sence and secondly we approximate the Brownian motion with such drift term by Euler-Maruyama scheme, Here we don't need t o assume boundedness and convexity of domains. We make use of a slite modification of the result in Kanagawa [1] to obtain the rate of convergence. In Sect.l, we refer the Skorohod problem (Skorohod equation). In Sect.2, we state Euler-Maruyama scheme for SDE and in Sect.3, we introduce the penalty method. A new approximation of reflecting Brownian motion is proposed in Sect.4. In Sect.5, examples of numerical computation with Computer System are given. 1. Skorohod problem Following [12] we pose two conditions for a domain D C R. Define sets of inward normal vectors at χ Ε dD by MXyr = {n: \n\ = l, B (x rn,r) Π D = 0} , r > 0, -Λ& = U Λ4. Γ, x € dD, r>0 where B(y,r) = {z G R : \z y\ < r}, y G R, r > 0, and pose two conditions for domains: Strang Approximation of Reflecting Brownian Motion Using Penalty Method 107 Condition (A): There exists a constant ΓΟ > 0 such that for all χ e dD Condition (B): There exist constants δ > 0 and β 6 [Ι,οο) such that the following holds: for any χ 6 dD there is a unit vector ix with for any n £ |J y where (·, ·} is the usual inner product in R. Rernark 1.1 In case when D is convex, the Condition (A) is satisfied for any r0 > 0. Remark 1.2 ([13], Remark 1.3) // D satisfies the Condition (A) and dist(x,-D) < r*o, then there exists a unique π(χ) = x£D with dist(x, D) = \x — x\. Especially we have χ = χ for χ € D; if χ ^ D we have χ Ε dD. Skorohod problem/equation. Skorohod problem is to find a pair (z, φ) satisfmg the equation (called Skorohod equation) , ( > under the conditions • χ (i) e D is continuous, • φ (t) is continuous and of bounded Variation, φ (0) = 0, • φ (t) = ji n (s)d \φ\3 , \(p\t = ji I{x(8)€dD}(s)d \· R ® , b : R -» d be bounded Lipschitz continuous functions and Δη be a partition of the interval [0,T]: Δη : 0 = ί0 < ti < t2 < · · · «n = T,