Some Methods In The Numerical Analysis Of Stochastic Ordinary Differential Equations

Physical systems are often modelled by ordinary differential equations(ODEs). However, such models may represent idealized situations, as theyignore stochastic effects. By incorporating random elements in the differentialequation system (either in the initial or boundary conditions for theproblem, or in the function describing the physical system), a system ofstochastic differential equations (SDEs) arises. Some areas where SDEs areused in modelling include investment finance, turbulent diffusion, populationdynamics, polymer dynamics, genetic regulation, biological waste treatment,hydrology and very large scale integration (VLSI) circuit design. These modelscan offer a far more realistic representation of the physical system but,whereas there is a rich theory for designing effective numerical methods forsolving ODEs, the stochastic counterparts are less well developed. The aimof this thesis is to construct some more effict numerical methods for solvingSDEs based on existing methods, which are strong numerical approximationsbe close to the strong solution of stochastic initial value problems.In this thesis, we discuss the numerical methods for solving It(?) andStratonovich stochastic differential equations (stiff and nonstiff cases), respectively.In Chapter 2, by introducing split-step techniques besed on Euler-Maruyama method and Milstein method, we study split-step Euler methods,two-stage Milstein methods and three-stage Milstein methods for nonstiffIt(?) stochastic systems. In Chapter 3, we construct split-step backward Milstienmethods and split-step backward balanced Milstein methods for stiff It(?) stochastic systems. In Chapter 4, we construct three-stage stochastic Runge-Kutta methods for stiff and nonstiff Stratonovich stochastic systems. Strongconvergence order of these methods is proved. The analyses of the stabilityproperties and numerical results show the effectiveness of these methods inthe pathwise approximation of stochastic differential systems. Now let usintroduce the main results of the thesis.Split-step Euler methods for It(?) SDEsFirstly we consider the It(?) stochastic initial value problemwhere W(t) is a Wiener process, whose incrementâ–³W(t) = W(t+â–³t)- W(t)is a Gaussian random variable N(0,â–³t). For above problem, we present twofully explicit methods, the drifting split-step Euler (DRSSE) methodand the diffused split-step Euler (DISSE) methodAssumption 1 The functions f and g in the problem satisfy the Lipschitzcondition for constant K > 0, i. e.and linear growth bound, i. e.We obtain the following result.Theorem 1 Let y_k be the numerical approximation to y(t_k) at time T after ksteps with step size h = T/N, N = 1,2,…. Apply one of DRSSE and DISSE methods to the It(?) SDEs under Assumption 1, then for all k = 0,1,…,N,we haveSplit-step Milstein methods for It(?) SDEsUsing more splitting techniques to Milstein method, we present six fullyexplicit methods, the first two-stage Milstein (TSM1) methodthe second two-stage Milstein (TSM2) methodthe third two-stage Milstein (TSM3) methodthe fourth two-stage Milstein (TSM4) methodthe fifth two-stage Milstein (TSM5) methodand the sixth two-stage Milstein (TSM6) methodAssumption 2 The functions f, g and g(?)g/(?)y in the problem satisfy theLipschitz condition for constant K > 0, i. e. and linear growth bound, i. e.We obtain the following result.Theorem 2 Let y_k be the numerical approximation to y(t_k) at time T after ksteps with step size h = T/N, N = 1,2,…. Apply one of two-stage Milsteinmethods to the It(?) SDEs under Assumption 2, then for all k = 0,1,…, N,we haveUsing similar splitting techniques to Milstein method, we can obtain sixthree-stage Milstein methods, namely the TSM1a methodthe TSM1b methodthe TSM1c methodthe TSM1d method the TSM1e methodthe TSM1f methodStrong convergence order of these methods is proved under Assumption 2.Theorem 3 Let y_k be the numerical approximation to y(t_k) at time T after ksteps with step size h = T/N, N = 1,2,…. Apply one of three-stage Milsteinmethods to the It(?) SDEs under Assumption 2, then for all k = 0,1,…,N?we haveSplit-step backward Milstein methods for stiff It(?) systemsWe also consider the implicit methods for stiff It(?) stochastic systems.Similar to the above implicit splitting technique, we present the drifting splitstepbackward Milstein (DSSBM) method, given byIn addition, using the fully splitting technique for deterministic terms, weobtain the following modified split-step backward Milstein (MSSBM) method,namelyIn order to avoid possible non-existence of solution in realization of abovemethods, the following assumption must is satisfied. Assumption 3 For the functions f, g in the problem, matrixeshave inverses and satisfy the conditionHere I is the unit matrix and K is positive constant.We obtain the following result.Theorem 4 Let y_k be the numerical approximation to y(t_k) at time T afterk steps with step size h = T/N, N = 1,2,…. If applying one of split-stepbackward Milstein methods to the It(?) SDEs under Assumption 3, then forall k=0,1,…,N, we haveSplit-step backward balanced Milstein methods for stiff It(?) systemsUsing some splitting techniques to balanced Milstein method, we presenta family of drifting split-step backward balanced Milstein (DSSBBM) methodswherea family of modified split-step backward balanced Milstein (MSSBBM) methods whereIn addition, adding balanced terms for split-step stage, we obtain again fol-lowing two families of methods, a family of drifting split-step backward doublebalanced Milstein (DSSBDBM) methodswherea family of modified split-step backward double balanced Milstein (MSSBDBM)methodswhereFor these methods, the control functions c₀ and c₂ must satisfy the followingassumption.Assumption 4 The c₀ and c₂ represent bounded m×m-matrix-valuedfunctions. For any real numbers a₀∈[0,(?)₁], a₂∈[-(?)₂,(?)₂], where (?)₁≥h,|(?)₂|≥|(â–³W_n)²-h| for all step sizes h considered and (t, x)∈[0,∞]×R^m,the matrixhas an inverse and satisfies the condition Here I is the unit matrix, K is a positive constant.We obtain the following result.Theorem 5 Let y_k be the numerical approximation to y(t_k) at time T afterk steps with step size h = T/N, N = 1, 2,…. If applying one of split-stepbackward balanced Milstein methods to the It(?) SDEs under Assumption 2and Assumption 4, then for all k = 0,1,…, N, we haveThree-stage stochastic Runge-Kutta methods for Stratonovich SDEsNext we consider the Stratonovich stochastic initial value problemwhere W(t) is a Wiener process, whose incrementâ–³W (t) = W(t+â–³t)-W(t)is a Gaussian random variable N(0,â–³t). For above problem, we present aclass of three-stage stochastic Runge-Kutta methods, namelywhere J₁～N(0, h) is a Gaussian random variable, h is constant stepsize,I is the identity matrix and (?) is Kronecker product such that A(?) I is theblock diagonal matrix with the matrix A on the diagonal, We present a three-stage explicit (E3) method with the following Butchertableauand a three-stage semi-implicit (SI3) method with the following coefficientsThree-stage stochastic Runge-Kutta methods for stiff StratonovichSDEsWe also consider the implicit methods for stiff Stratonovich stochasticsystems. In order to construct efficient stiffly accurate methods, the matrixA will be chosen so that the deterministic component of three-stage stochasticRunge-Kutta methods is the classical Runge-Kutta method of Alexander givenbywhereθis the root of x³ - 3x² + 3/2 -1/6 = 0 lying in (1/6, 1/2), l = (1 +θ)/2,m₁=-(6θ²-16θ+1)/4, m₂=(6θ²-20θ+5)/4. Thus we present a stiffly accurate semi-implicit (SASI3) method with the following coefficientsand a stiffly accurate diagonally implicit (SADI3) method with the followingcoefficients...