On quasi-Monte Carlo simulation of stochastic differential equations

In a number of problems of mathematical physics and other fields stochastic differential equations are used to model certain phenomena. Often the solution of those problems can be obtained as a functional of the solution of some specific stochastic differential equation. Then we may use the idea of weak approximation to carry out numerical simulation. We analyze some complexity issues for a class of linear stochastic differential equations (Langevin type), which can be given by dXt = -α:Xtdt + β(t)dWt, X0 ≔ 0, where α > 0 and β: [0, T] → ℝ. It turns out that for a class of input data which are not more than Lipschitz continuous the explicit Euler scheme gives rise to an optimal (by order) numerical method. Then we study numerical phenomena which occur when switching from (real) Monte Carlo simulation to quasi-Monte Carlo one, which is the case when we carry out the simulation on computers. It will easily be seen that completely uniformly distributed sequences yield good substitutes for random variates, while not all uniformly distributed (mod1) sequences are suited. In fact we provide necessary conditions on a sequence in order to serve for quasi-Monte Carlo purposes. This condition is expressed in terms of the measure of well distribution. Numerical examples complement the theoretical analysis

Quasi-random Simulation of Linear Kinetic Equations

AbstractWe study the improvement achieved by using quasi-random sequences in place of pseudo-random numbers for solving linear spatially homogeneous kinetic equations. Particles are sampled from the initial distribution. Time is discretized and quasi-random numbers are used to move the particles in the velocity space. Quasi-random points are not blindly used in place of pseudo-random numbers: at each time step, the number order of the particles is scrambled according to their velocities. Convergence of the method is proved. Numerical results are presented for a sample problem in dimensions 1, 2 and 3. We show that by using quasi-random sequences in place of pseudo-random points, we are able to obtain reduced errors for the same number of particles

On quasi-Monte Carlo simulation of stochastic differential equations

In a number of problems of mathematical physics and other fields stochastic differential equations are used to model certain phenomena. Often the solution of those problems can be obtained as a functional of the solution of some specific stochastic differential equation. Then we may use the idea of weak approximation to carry out numerical simulation. We analyze some complexity issues for a class of linear stochastic differential equations (Langevin type), which can be given by dX_t=-#alpha#X_tdt+#beta#(t)dW_t, X_0:=0, where #alpha# > 0 and #beta#:[0,T] #-># R. It turns out that for a class of input data which are not more than Lipschitz continuous the explicit Euler scheme gives rise to an optimal (by order) numerical method. Then we study numerical phenomena which occur when switching from (real) Monte Carlo simulation to quasi-Monte Carlo one, which is the case when we carry out the simulation on computers. It will easily be seen that completely uniformly distributed sequences yield good substitutes for random variates, while not all uniformly distributed (mod 1) sequences are suited. In fact we provide necessary conditions on a sequence in order to serve for quasi-Monte Carlo purposes. This condition is expressed in terms of the measure of well distribution. Numerical examples complement the theoretical analysis. (orig.)Available from TIB Hannover: RR 5549(166)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman