Mathematics and Computers in Simulation 57 001 9 34 Second order weak Runge Kutta type methods for Itô equations Vigirdas Mackevičius, Jurgis Navikas Department of Mathematics and Informatics, Vilnius University, Naugarduko 4, 600 Vilnius, Lithuania Received 1 October 000; accepted 13 November 000 Abstract A standard second order weak Runge Kutta method for a stochastic differential equation can be applied only in the case where the equation is understood in the Stratonovich sense. To adapt Runge Kutta type methods for Itô equations, we propose to use a rather simple additional derivative-free term. 001 IMACS. Published by Elsevier Science B.V. All rights reserved. MSC: 60H35; 65C30; 68U0 Keywords: Stochastic differential equation; Runge Kutta method; Weak approximation 1. Introduction: Runge Kutta type methods for SDEs Consider a one-dimensional stochastic differential equation of the form dx t = bx t dt + σx t db t, X 0 = x, 1 with a driving Brownian motion B ={B t,t 0}. By standard four-stage Runge Kutta type approximations of its solution we mean approximations X h ={Xkh h,k = 0, 1,,...}, h>0, of the form where X h 0 = x, Xh k+1h = axh kh,h, B k, B k = B k+1h B kh, ax,s,y = x + 3 q i F i s + i=0 3 r i G i y, i=0 Corresponding author. E-mail address: vigirdas.mackevicius@maf.vu.lt V. Mackevičius. 0378-4754/01/$0.00 001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S0378-4754000084-6
30 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation 57 001 9 34 F 0 = bx, G 0 = σx + α 00 F 0 s, H 1 = x + α 10 F 0 s + β 10 G 0 y, F 1 = bh 1, G 1 = σh 1, H = x + α 0 F 0 + α 1 F 1 s + β 0 G 0 + β 1 G 1 y, F = bh, G = σh, H 3 = x + α 30 F 0 + α 31 F 1 + α 3 F s + β 30 G 0 + β 31 G 1 + β 3 G y, F 3 = bh 3, G 3 = σh 3, In [3], it is shown that if Eq. 1 is understood in the Stratonovich sense, then, under some boundedness and smoothness conditions, 1 unknown parameters α ij, β ij, q i, and r i can be chosen so that the approximation X h has the second order accuracy in the weak sense or in the sense of distributions. This means that see, e.g. [,4,5], for each fixed T>0 EfX h T EfX T = Oh, h 0, for a rather wide class of functions f : R R including polynomials. Unfortunately, this cannot be done when Eq. 1 is understood in the Itô sense. The reason is that, in the Itô case, the system of 15 equations to be satisfied by the parameters has no solutions. For example, one of the equations is q 0 + q 1 + q + q 3 bx = bx 1/σ σ x which, in Stratonovich case, becomes q 0 + q 1 + q + q 3 bx = bx, i.e. q 0 + q 1 + q + q 3 = 1. Our main idea is adding to the method-defining function a = ax,s,y a derivative-free two-stage correction term, which neutralizes bad terms in the equations for parameters such as 1/σ σ x in the example above. More precisely, we consider the approximations of the form with X h 0 = x, Xh k+1h = AXh kh,h, B k, 3 Ax,s,y= ax,s,y + Kx,s, where a is of the form and K is of the form Kx,s = γ 1 σ x + γ σxs σx + γ 3 σxs. In this paper, we show that, with this refinement, there exist second order weak Runge Kutta approximations for Itô equations and give a simulation example.. Second order conditions for weak Runge Kutta approximations In [3], we derived the conditions sufficient for the second order accuracy of a weak approximation of the general form 3 not necessarily Runge Kutta type of the solution of the Itô Eq. 1 A y x = σx, A s x = bx 1 σσ x, A yy x = σσ x, 4 A ys + A yyy x = bσ x + 1 σ σ x, A ss + A yys + 1 4 A yyyy x = bb x + 1 σ b x,
V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation 57 001 9 34 31 where x = x, 0, 0. Using this equations system, let us try to derive the relations for the parameters of the Runge Kutta type approximation. We first need to calculate the derivatives A y, A s, A yy, A ys, A yyy, A ss, A yys, and A yyyy of the function A that appear in Eq. 4. 1 The expressions of derivatives are rather long, but we are actually interested on their values at the points x only. Denoting α i = j α ij and β i = j β ij, we have A y x = r 0 + r 1 + r + r 3 σ x, A yy x = r 1β 1 + r β + r 3 β 3 σ σ x, A yyy x = 3r 1β1 + r β + r 3β3 σ σ x + 6r β 1 β 1 + r 3 β 31 β 1 + r 3 β 3 β σ σ x, A yyyy x = 4r 1β1 3 + r β 3 + r 3β3 3 σ 3 σ x + [1r β 1 β1 + r 3β 31 β1 + r 3β 3 β +4r β 1 β 1 β + r 3 β 31 β 1 β 3 + r 3 β 3 β β 3 ]σ σ σ x + 4r 3 β 3 β 1 β 1 σ σ 3 x, A s x = q 0 + q 1 + q + q 3 bx, A ss x = q 1α 1 + q α + r 3 α 3 bb x, A ys x = r 0α 0 + r 1 α 1 + r α + r 3 α 3 bσ x + q 1 β 1 + q β + q 3 β 3 b σx, A yys x = [r 1β 1 α 0 + r β 0 α 0 + β 1 α 1 + r 3 β 30 α 0 + β 31 α 1 + β 3 α ]bσ x +r 1 α 1 β 1 + r α β + r 3 α 3 β 3 bσ σ x + q β 1 β 1 + q 3 β 31 β 1 + q 3 β 3 β +r α 1 β 1 + r 3 α 31 β 1 + r 3 α 3 β b σσ x + q 1 β 1 + q β + q 3β 3 b σ x. Note that the partial derivatives of A with respect to y contain only the diffusion coefficient σ, the derivatives with respect to s contain the shift coefficient b, and, finally, the mixed partial derivatives with respect to both y and s contain both coefficients. Substituting these values into Eq. 4, we get the following 17 equations for comparison, in the parentheses, we indicate the corresponding values of the right sides in the Stratonovich case r 0 + r 1 + r + r 3 = 1, 1 5.1 r 1 β 1 + r β + r 3 β 3 = 1, 1 5. r 1 β1 + r β + r 3β3 = 1 6, 1 3 5.3 r β 1 β 1 + r 3 β 31 β 1 + r 3 β 3 β = 0, 1 6 5.4 r 1 β1 3 + r β 3 + r 3β3 3 = 0, 1 4 5.5 r β 1 β1 + r 3β 31 β1 + r 3β 3 β + r β 1 β 1 β + r 3 β 31 β 1 β 3 + r 3 β 3 β β 3 = 0, 1 3 5.6 r 3 β 3 β 1 β 1 = 0, 5.7 1 4 q 0 + q 1 + q + q 3 = 1, 1 5.8 q 1 α 1 + q α + r 3 α 3 = 1, 1 5.9 r 0 α 0 + r 1 α 1 + r α + r 3 α 3 = 1, 1 5.10 q 1 β 1 + q β + q 3 β 3 = 1, 1 5.11 1 For calculations, we used MAPLE.
3 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation 57 001 9 34 r 1 β 1 α 0 + r β 0 α 0 + β 1 α 1 + r 3 β 30 α 0 + β 31 α 1 + β 3 α = 0, 1 4 q β 1 β 1 + q 3 β 31 β 1 + q 3 β 3 β + r α 1 β 1 + r 3 α 31 β 1 + r 3 α 3 β = 0, 1 4 r 1 α 1 β 1 + r α β + r 3 α 3 β 3 = 0, 1 4 q 1 β1 + q β + q 3β3 = 1, 1 γ 1 γ γ 3 = 1, γ 1 γ γ 3 = 0. 5.1 5.13 5.14 5.15 5.16 5.17 Here we have again grouped the equations into groups according to the participation of the coefficients. The first stochastic or diffusion group 5.1 5.7 contains the parameters r i and β ij related to the diffusion coefficient σ ; the second deterministic or drift one 5.8 and 5.9 contains the parameters q i and α ij related to the drift coefficient b; the third mixed group 5.10 5.15 contains all the parameters just mentioned; finally, the last two equations connects the additional parameters γ i. 3. Example We first easily choose the values of the parameters γ i, i = 1,, 3, satisfying Eqs. 5.16 and 5.17 γ 1 = 1 4, γ = 1, γ 3 = 1. It is convenient to put the remaining parameters into the Butcher-type array of the form cf. [1] From several solutions of Eqs. 5.1 5.15 that we succeeded to find using MAPLE, we have chosen the following rather nice one having many zeros
V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation 57 001 9 34 33 Thus, our approximation is given by the following method-defining function Ax,s,y= x + 1 bxs + 1 5 bx + bxs + σxys 1 σ x 4 5 bxs y + 1 4 σxy + 4 3 σ x + 1 4 bxs + 1 σxy y 1 σx + bxs + σxyy 6 1 σσx + σxs σx σxs. 4 It is interesting to compare it with the classical Milstein [4] approximation defined by the function ax,s,y= x + σxy + bx 1 σxσ x s + 1 σxσ xy + 1 bσ x + 1 4 σ xσ x ys + 1 bxb x + 1 4 σ xb x s Fig. 1. A comparison of Runge Kutta type and Euler approximations. Solid line: exact values of EfX t. Solid polygonal line: the Runge Kutta approximation. Dashed polygonal line: the Euler approximation. Number of simulated trajectories of approximations n = 10 000.
34 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation 57 001 9 34 containing the derivatives of b and σ up to the second order. For a simulation example, we have chosen the equation dx t = 1 X t + Xt + 1 dt + Xt + 1dB t, X t = 0, having the solution X t = sin ht + B t, t 0. To compare approximations with the exact solution, we have to chose the test function f such that EfX t could be explicitly found. To produce, for example, a third order polynomial of t, we take fx = Parcsinh x, where Py = y 3 6y + 8y. Then ft:= EfX t = E[t + B t 3 6t + B t + 8t + B t ] = t 3 3t + t = tt 1t. In Fig. 1, we see the simulation results for the constructed Runge Kutta type approximation. The approximation steps were taken h = 0.3, 0., 0.1. As usual, approximations of the function f were obtained by averaging the values fx h,i kh, where Xh,i, i = 1,,...,n, are n independent simulated trajectories of the approximation. The number of trajectories was taken n = 10 000. For comparison, we have simultaneously generated trajectories of the Euler approximation defined by ax,s,y = x + bxs + σxy. We see that the Runge Kutta type approximation behaves significantly better, especially, for large values of h. We have also compared the Runge Kutta type approximation with the Milstein one. It is, however, difficult to show this graphically, since the corresponding graphs appeared to be visually indistinguishable! References [1] J. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge Kutta and General Linear Methods, Wiley, New York, 1987. [] P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 199. [3] V. Mackevičius, Second order weak approximations for Stratonovich stochastic differential equations, Lith. Math. J. 34 1994 183 00. [4] G.N. Milshtein, A method of second-order accuracy for the integration of stochastic differential equations, Theor. Probab. Appl. 3 1978 396 401. [5] D. Talay, Discrétization d une équation différentielle stochastique et calcul approché d espérences de fonctionnelles de la solution, Modélisation Mathématique et Analyse Numérique 0 1986 141 179.