On Random Discretization for Ito Diffusion Processes

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We tested the newly generated random numbers for some moments of τ whichis obtained from the exponential martingale exp (λW t − λ 2 t/2), λ ∈ R and relatedpolynomial type martingales - see [1] for the first and second moment andthe third moment is also obtained similarly using the polynomial martingalewhich are for h = 1,W 6t − 15tW 4t + 45t 2 W 2t − 15t 3 , t ≥ 0,Eτ = 1, Eτ 2 = 5/3, and Eτ 3 = 61/15.We included ±30 terms from the infinite series (7). We took n = 1000 andM = 5000 and 7000. See Table 1 for the test results.5 A Comparison for ErrorsFor fixed equidistant discretization, usually, the uniform error for Brownianmotion W t is[]E max |W t − Y t |0≤t≤1for the approximate process Y t i.e., the average of the maximum error. But forour adaptive random discretization (4), the uniform error has the deterministicupper bound and not uniform just in the average sense. In this section, wecompare the errors for these two discretizations i.e. the average sense uniformerror for the equidistant discretization (AE) and the deterministic uniformerror for the random discretization (4)(DR).Since Eτ h = h 2 , Eτ h · EN ≈ 1, and the deterministic error bound is 2h forstandard Brownian motion, we have2h ≈ 2 √EN. (8)We let m = EN and compare the deterministic error 2h with the averageerror for m-equidistant discretization.Let Y (m) (t) be the linearly interpolated Brownian motion for the equidistantnodes {i/m} 1≤i≤m and B i , i = 1, 2, . . . be independent Brownian bridges on8
the unit interval, then[]E max |W t − Y (m) (t)| = 1 []√ · E max ||B l|| ∞ ,0≤t≤1 m 0≤l≤mwhere ||B l || ∞ = max 0≤s≤1 |B l (s)|. According to Corollary 1 of [4], we haveand hencelimm→∞[1√ · E ln m]max ||B l|| ∞0≤l≤m= 1 √2[]E max |W t − Y (m) (t)| ≈ 1 √ln m√ · √ (9)0≤t≤1 2 mwhen m is large. In Table 2, we compare the two kinds of errors (DR) and(AE) by the values (8) and (9). When m increases, we can observe that DRbecomes even smaller than AE.>References[1] R. Durrett, Probability: Theory and Examples, (2nd edn. Duxbury, 1996).[2] J. G. Gaines and T. J. Lyons, Variable step size control in the numerical solutionof stochastic differential equations, SIAM J. Appl. Math. 57, no. 5, (1997) 1455–1484.[3] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic DifferentialEquations, (Springer, Berlin, 1992).[4] N. Hofmann, T. Müller-Gronbach and K. Ritter, Step size control for the uniformapproximation of systems of stochastic differential equations with additive noise,Ann. Appl. Probab. 10, no. 2, (2000) 616–633.[5] N. Hofmann, T. Müller-Gronbach and K. Ritter, The optimal discretization ofstochastic differential equations, J. Complexity. 17, no. 1, (2001) 117–153.[6] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, (2nd.edn. Springer 1991).[7] S. Mauthner, Step size control in the numerical solution of stochastic differentialequations, J. Comput. Appl. Math. 100 no. 1, (1998) 93–109.[8] T. Müller-Gronbach, The optimal uniform approximation of systems ofStochastic differential equations, preprint (2000).9
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