Numerical Solution Of Stochastic Differential Equations

[Rosalyn Pomposo]

MonteCarlo methods are numerical methods, where random numbers are used to conduct a computational experiment. Numerical solution of stochastic differential equations can be viewed as a type of Monte Carlo calculation. Monte Carlo simulation is perchance the most common technique for propagating the incertitude in the various aspects of a system to the predicted performance.

In Monte Carlo simulation, the entire system is simulated a large number of times. So, a set of suitable sample paths is produced on \([t_0,T]\). Each simulation is equally likely, referred to as a realization of the system. For each realization, all of the uncertain parameters are sampled. For each sample, we produce a sample path solution to the SDE on \([t_0,T]\). This is generally obtained from the stochastic Taylor formula, which was derived in [13], for the solution X of the SDE, on a small subinterval of \([t_0,T]\) [5, 14]. From the Ito-Taylor expansion, we can construct numerical schemes for (1) over the interval \([t_i,t_i+1]\).

The Taylor formula plays a very significant role in numerical analysis. We can obtain the approximation of a sufficiently smooth function in a neighborhood of a given point to any desired order of accuracy with the Taylor formula.

for \(i=0,1,2,\ldots,N-1\) with the initial value \(X(t_0)=X_0\). Euler-Maruyama approximation converges with strong order 0.5 under Lipschitz and bounded growth conditions on the coefficients f and g, which were shown in [15]. [16] and [17] showed that an Euler-Maruyama approximation of an Ito process converges with weak order 1.0 under conditions of sufficient smoothness. It is clear that weak order of convergence is greater than strong order of convergence in the Euler-Maruyama method.

In Table 1, our EM and Milstein approximations of this example were evaluated for 10,000 sample paths for \(N=2^9,2^10,2^11,2^12\) and 213 over \([0,1]\) to estimate \(E[X(1)]\approx\frac110\text,000\sum_i=1^10\text,000X_N^i\), where \(X_N^i\) is the estimate of X at the end time \(T=1\) for the ith sample path using N subinterval.

are calculated with the EM and Milstein methods for each value of N, where \(X_N^i\) is the estimate of X at the end time \(T=1\) for the ith sample path using N subinterval.

In Figure 2, the exact solution and the Milstein approximation are plotted for 10,000 sample paths and along 50 individual paths on the interval \([0,1]\). \(X\mathitmilstein_\mathitmean\) holds average of the Milstein solution, which is plotted as blue asterisks connected with dashed lines. Xmilstein keeps Milstein approximations, which is plotted as red straight lines.

In Figure 3, exact solution, EM and Milstein approximations are plotted on the same graph. The first graph is plotted for \(N=2^9\) subintervals, and the second one is plotted for \(N=2^13\) subintervals. \(X_EM\) denotes EM approximation, \(X_\mathitMilstein\) denotes Milstein approximation and \(X_\mathitexact\) denotes exact solution, which are plotted as blue, yellow and red straight lines, respectively.

In this paper we have studied the Euler and Milstein schemes which are obtained from the truncated Ito-Taylor expansion already proposed in [7]. Then we implemented these schemes to a nonlinear stochastic differential equation for comparing the EM and Milstein methods to each other while illustrating efficiency. Moreover, we calculated estimation values for Euler-Maruyama and Milstein methods so as to analyze similarities between the exact solution and numerical approximations. Then we investigated approximations for 29, 210, 211, 212 and 213 discretization in the interval \([0,1]\) with 10,000 different sample paths. According to our results, we can say that when the discretization value N is increasing, numerical solutions achieved from Euler-Maruyama and Milstein schemes are close to exact solution, and our results in the tables show that the Milstein method is more effective than the Euler-Maruyama method.

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Inspired by path integral solutions to the quantum relaxation problem, we develop a numerical method to solve classical stochastic differential equations with multiplicative noise that avoids averaging over trajectories. To test the method, we simulate the dynamics of a classical oscillator multiplicatively coupled to non-Markovian noise. When accelerated using tensor factorization techniques, it accurately estimates the transition into the bifurcation regime of the oscillator and outperforms trajectory-averaging simulations with a computational cost that is orders of magnitude lower.

Schematic derivation of the method. (a) Break the propagation into small steps and apply symmetric Trotter splitting to each. (b) Expand each resulting stochastic matrix using a spectral decomposition to separate the propagator into deterministic matrices G0 and stochastic, scalar exponentials S. (c) Compute the average of the scalar part analytically, which generates a high-rank influence tensor I that couples different state-time points.

Comparing different solution methods for the noise-averaged position of the stochastic parametric oscillator [Eq. (3), γ=ω0=1/τc]. Exhaustive trajectory averaging (dashed dotted orange lines) is arbitrarily accurate and is the standard to which all other methods should be compared. ASPEN agrees quantitatively (solid blue line) with trajectory averaging for all noise strengths and all times but with orders of magnitude lower computational cost. Time local (red alternating dashed lines) and time nonlocal (green dashed line) are the results of the perturbation theories described in the text that correspond to different types of cumulant expansions. (a) At low noise strength κ=0.5, all methods agree. (b) At a moderate noise strength κ=1, the perturbative methods show moderate disagreement with trajectory averaging and with one another for the transients (inset). (c) At high noise strength, near the bifurcation transition (κ=1.581), the oscillator coordinate reaches a nonzero steady state because it cannot dissipate the power of the noise. Perturbation theories fail qualitatively in this regime. The time nonlocal perturbation theory predicts a bifurcation, but at an incorrect Kubo number κc=3 (shown in gray).

The current state of the art is to tame the forward Euler-Maruyama scheme in an explicit way, and completely avoid tricky convergence questions that may come up when using drift-implicit Euler-Maruyama. The tamed Euler-Maruyama scheme is easy to implement and provably works if the drift coefficient is globally one-sided Lipschitz continuous and the noise coefficient is globally Lipschitz continuous. There are several variants of this scheme. For the drift-tamed and increment-tamed Euler-Maruyama schemes check out (1.5) and (3.122):

To follow-up on your answer, if you look at the implicit schemes in Kloden's Numerical Solution of Stochastic Differential Equations you will notice that all of the implicit schemes are "semi-implicit" where the only implicit part is on the drift (or deterministic) term. This is because of the following:

We also saw in Section 8 of Chapter 9 that difficulties can arise in applying fully implicit schemes to obtain strong approximations of solutions of stochastic differential equations, because they usually involve reciprocals of Gaussian random variables which do not have finite absolute moments. Consequently, finite absolute moments generally will not exist for fully implicit strong approximations and a strong convergence analysis would not make sense. For mainly this reason we shall restrict our attention her to "semi-implicit" strong approximations, which we shall call implicit

So this is actually a more general idea that the proper implicit form of a method for SDEs is for it to be semi-implicit by only making the drift term implicit. The diffusion term should be kept the same to make sure there are finite moments.

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs.

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