This paper provides an introduction to stochastic calculus and stochastic differential equations, in both theory and applications, emphasizing the numerical methods needed to solve such equations

R. Horváth Bokor

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Mathematical Communications 4(1999), 251-256 251Numerical methods for solving stochasticdifferential equations∗Ro´zsa Horva´th Bokor†Abstract. This paper provides an introduction to stochastic calculusand stochastic differential equations, in both theory and applications,emphasizing the numerical methods needed to solve such equations.Key words: stochastic differential equations, strong solutions, nu-merical schemes1. IntroductionThe aim of this paper is to present efficient numerical methods to compute cer-tain quantities depending on the unknown process (X(t)), with algoritms based onsimulations on a computer of the other processes.Before going on, it must be said that the numerical analysis of stochastic dif-ferential equations is at its very beginning. Nevertheless, it already appears thatthis field is not at all direct continuation of what has been done for the numericalsolving of ordinary differential equations. It is often unuseful to try to approximatethe stochastic differential equation on the space of trajectories, when one wants tocompute a quantity which depends on law.We refer here to the books of Kloeden and Platen ([4]), Kloeden, Platen andSchurz ([5]) and Schurz ([7],[8]).Consider the simple population growth modeldXdt= a(t)X(t), X(0) = A, (1)where X(t) is the size of the population at time t, and a(t) is the relative rategrowth at time t. It might happen that a(t) is not completely known, but subjectto some random environmental effects, so that we havea(t) = r(t) + ” noise ”, (2)where r(t) is non-random function. So the equation (1) becomesdXdt= r(t)X(t) +X(t) · ” noise ” (3)∗The lecture presented at the Mathematical Colloquium in Osijek organized by CroatianMathematical Society - Division Osijek, November 27, 1998.†Department of Mathematics, University of Zagreb, Bijenicˇka 30, HR-10 000 Zagreb, Croatia,e-mail: bokor@math.hr252 R. Horva´th Bokoror more generally equation of the formdXtdt(ω) = a(t,Xt(ω))dt+ b(t,Xt(ω))ξt(ω) (4)where a and b are some given functions, and (ξt) were standard Gaussian randomvariables for each t and b(t, x) a (generally) time-space dependent intensity factor.This symbolic differential was interpreted as an integral equationXt(ω) = Xt0(ω) +∫ tt0a(s,Xs(ω))ds +∫ tt0b(s,Xs(ω))ξs(ω)ds (5)for each sample path. For the special case of (5) with a ≡ 0, b ≡ 1 we see that ξtshould be derivative of pure Brownian motion, that is the derivative of a Wienerprocess Wt, thus we suggest that we could write (5) alternatively asXt(ω) = Xt0(ω) +∫ tt0a(s,Xs(ω))ds +∫ tt0b(s,Xs(ω))dWs(ω). (6)The problem with this is that a Wiener process Wt is nowhere differentiable, sostrictly speaking the white noise process ξt does not exist as a conventional functionof t. Thus the second integral in (6) cannot be an ordinary Riemann or Lebesqueintegral. Worse still, the continuous sample paths of a Wiener process are not ofbounded variation on any bounded time interval, so the second integral cannot evenbe interpreted as a Riemann-Stieltjes integral for each sample path.2. The main resultsDefinition 1. A processW=(Wt) defined on a complete probability space (Ω,F ,P)is called a standard Brownian motion if it has the following properties:1.) W0 = 0 (a.s),2.) For a.a.ω, the trajectories t→Wt(ω) are continuous t ∈ [0, T ],3.) For all 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4, ti ∈ [0, T ] the random variables Wt4 −Wt3 andWt2 −Wt1 are independent,4.) For every s and t (s ≤ t) Wt −Ws has Gaussian distribution with zero meanand variance equal to t-s, t, s,∈ [0, T ].We suppose that we have a complete probability space (Ω,F ,P), a Brownianmotion W = (Wt, t ≥ 0) and a filtration {Ft, t ≥ 0}.Definition 2. A process W =(Wt) is called a (Ft)-Brownian motion if it has thefollowing properties:1.) (Wt) is (Ft)-adapted,2.) (Wt) is a standard Brownian motion,3.) For every t ≥ 0 and s > 0 Wt+s −Wt is independent with Ft.Numerical solving of stochastic differential equations 253Definition 3. Let M[0,T ] be the class of all processes f =(ft) satisfying the condi-tions:1.) (ft) is B(R)/B([0, T ])× F-measurable,2.) (ft) is (Ft)-adapted,3.) E∫ T0||ft ||2 dt < +∞.Definition 4. The Ito integral∫ T0f(t, ω)dWt(ω), sign by I(f), for an integrandf ∈M[0,T ] is equal to the mean-square limit of the sumsSn(ω) =n∑j=1f(ξ(n)j , ω)(Wt(n)j+1(ω) −Wt(n)j(ω))with evalution points ξ(n)j = t(n)j for partitions 0 = t(n)1 < t(n)2 < . . . < t(n)n+1 = T forwhichδn = max1≤j≤n(t(n)j+1 − t(n)j )→ 0 as n→ +∞.Other choices of evalution points t(n)j ≤ ξ(n)j ≤ t(n)j+1 are possible, but generallylead to different random variables in the limit. While arbitrarily chosen evalutionpointsξ(n)j = (1− λ)t(n)j + λt(n)j+1for the same fixed 0 ≤ λ ≤ 1 lead to limits, which we shall denote here by(λ)∫ T0f(t, ω)dWt(ω).As an indication of how the value of these integrals vary with λ we observe that(λ)∫ T0Wt(ω)dWt(ω) =12W 2T (ω) + (λ −12)T.Lemma 1. For any f, g ∈M[0,T ] and α, β ∈ R the Ito stochastic integral satisfies1.) I(f) is FT -measurable,2.) E[I(f)] = 0,3.) E[I(f)]2 =∫ T0E(f(t, ·))2dt.The problem considered here is that of approximating strong solutions of thefollowing type of the Ito stochastic differential equation:dXt = a(t,Xt)dt+ b(t,Xt)dWt, for 0 ≤ t ≤ T. (7)We suppose that E ||X0 ||2 < +∞ and X0 is independent of Ft = σ{Ws, 0 ≤s ≤ t}, the σ-algebra generated by the underlying process.Also, suppose that the coefficients a(t, x) and b(t, x) satisfy conditions whichguarantee the existence of the unique, strong solution of the stochastic differentialequation.254 R. Horva´th BokorTheorem 1. Let U : [0, T ]×R→ R have continuous partial derivatives ∂U∂t , ∂U∂x , ∂2U∂x2and define the process (Yt, 0 ≤ t ≤ T ) by Yt = U (t,Xt) w.p.1, where Xt satisfiesthe differential (7). Then the stochastic differential equation for Yt is given bydYt = [∂U∂t(t,Xt) + a(t,Xt)∂U∂x(t,Xt) +12b2(t,Xt)∂2U∂x2(t,Xt)]dt+ b(t,Xt)∂U∂x(t,Xt)dWt.Details about this stochastic object and the corresponding calculus can be found inKaratzas and Shreve([3])Let us consider the 1-dimensional stochastic differential equation in an integralformXt = X0 +∫ t0a(Xs)ds+∫ t0b(Xs)dWs,for t ∈ [0, T ], where the second integral is an Ito stochastic integral and the co-efficients a and b are sufficiently smooth real-valued functions satisfying a lineargrowth bound. Then, for any twice continuously differentiable function f : R→ Rthe Ito formula givesf(Xt) = f(X0) +∫ t0(a(Xs)∂∂xf(Xs) +12b2(Xs)∂2∂x2f(Xs))ds+∫ t0b(Xs)∂∂xf(Xs)dWs= f(X0) +∫ t0L0f(Xs)ds +∫ t0L1f(Xs)dWs.So if we apply the Ito formula to the functions f = a and f = b we obtainXt = X0 +∫ t0a(Xz)dz +∫ t0b(Xz)dWz= X0 +∫ t0[a(X0) +∫ z0L0a(Xs)ds +∫ z0L1a(Xs)dWs]dz+∫ t0[b(X0) +∫ z0L0b(Xs)ds +∫ z0L1b(Xs)dWs]dWz= X0 + a(X0)∫ t0ds+ b(X0)∫ t0dWs +R.So far a given discretization 0 = t1 < t2 < . . . < tN = T at the time interval [0, T ],an Euler scheme is given byYn+1 = Yn + a(Yn)(tn+1 − tn) + b(Yn)(Wtn+1 −Wtn),for n = 0, 1, 2, . . ., N − 1 with the initial value Y0 = X0.We can continue the Ito-Taylor expansion by applying the Ito formula to f =L1b, in which case we getXt = X0 + a(X0)∫ t0ds+ b(X0)∫ t0dWs + L1b(X0)∫ t0∫ s0dWzdWs +R.Numerical solving of stochastic differential equations 255For a given discretization mentioned earlier a Milstein scheme is given byYn+1 = Yn + a(Yn)(tn+1 − tn) + b(Yn)(Wtn+1 −Wtn)+12b (Yn)∂∂xb(Yn)((Wtn+1 −Wtn)2 − (tn+1 − tn)),for n = 0, 1, 2, . . ., N − 1 with the initial value Y0 = X0.Usually, we consider equidistant discretization times tn = n∆, where ∆ = TN .Theorem 2. Let us suppose that the functions a and b are of class C2, with boundedderivatives of first and second orders. Then the Euler scheme satisfies: for anyintegration time T, there exists a positive constant C(T) such that, for any step size∆ of type TN, N = 1, 2, 3, . . .[E‖X(T )− YN‖2] 12 ≤ C(T )√∆.For the Milstein scheme, we can substitute the following bound for the error:[E‖X(T ) − YN‖2] 12 ≤ C(T )∆.To get a better rate of convergence in the mean-square sense that these schemes,one must involve multiple stochastic integrals of order strictly larger than 1, forexample:∫ t0∫ s0dWzdWs,∫ t0∫ s0∫ z0dWudWzdWs and∫ t0∫ s0∫ z0dudWzdWs.Of course, most of those integrals have probability laws which seem difficult tosimulate. There are some other two-step methods for numerical solutions of stochas-tic differential equations. Details about this stochastic object and the correspondingcalculus can be found in ([1]) and ([2]).References[1] R.Horva´t Bokor, On two-step methods for SDE, Acta Cybernetica13(1997), 197–207.[2] R.Horva´t Bokor, On stability of two-step methods for stochastic differentialequations, in: Proceedings of the 7th International Conference on OperationalReserch KOI’98, (I. Aganovic´, T.Hunjak and R.Scitovski, eds.), Osijek, 1999,267–279.[3] I.Karatzas, S. E. Shreve,. Brownian Motion and Stochastic Calculus,Springer-Verlag, New York, 1988.[4] P.E.Kloeden, E.Platen, Numerical Solution of Stochastic DifferentialEquations, Springer-Verlag, New York, 1992.[5] P.E.Kloeden, E. Platen, H. Schurz, Numerical Solution of StochasticDifferential Equations through Numerical Experiments, Universitext, Springer,New York, 1994.[6] W.Ru¨melin, Numerical treatment of stochastic differential eqautions, SIAMJ. Numer. Anal. 19(1982), 604–613.256 R. Horva´th Bokor[7] H.Schurz, Mean square stability analysis of some numerical method solvingbilinear SDEs, Proceedings of Workshop on ’Stochastic and Finance’, Sept.1994, Humboldt University, Berlin, Special Press, 1994, 77-78.[8] H.Schurz, Construction of nonegative numerical solutions of SDEs: Explo-sions, life time, nonegativity and uniform boundedness, Discussion Paper 40,SFB 373, Humboldt University, Berlin, 1994

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