Simulation Of Stochastic Differential Equations Research Articles

Balanced implicit Patankar-Euler methods for positive solutions of stochastic differential equations of biological regulatory systems.

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar-Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar-Euler method has much better stability properties than our recently designed composite Patankar-Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar-Euler methods. Numerical results suggest that the proposed balanced implicit Patankar-Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems.

Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry.

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

Some open problems in exact simulation of stochastic differential equations

Some open problems in exact simulation of stochastic differential equations

The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations

We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.

Numerical Simulations of Stochastic Differential Equations with Multiple Conserved Quantities by Conservative Methods

Numerical Simulations of Stochastic Differential Equations with Multiple Conserved Quantities by Conservative Methods

CoinDy – A Library for Computing Simulations of Stochastic Differential Equations in Symbolic Form

CoinDy – A Library for Computing Simulations of Stochastic Differential Equations in Symbolic Form

Book Review: “An Introduction to the Numerical Simulation of Stochastic Differential Equations” by Desmond J. Higham and Peter E. Kloeden

Book Review: “An Introduction to the Numerical Simulation of Stochastic Differential Equations” by Desmond J. Higham and Peter E. Kloeden

Book Review: “An Introduction to the Numerical Simulation of Stochastic Differential Equations” by Desmond J. Higham and Peter E. Kloeden

Book Review: “An Introduction to the Numerical Simulation of Stochastic Differential Equations” by Desmond J. Higham and Peter E. Kloeden

Variational quantum simulations of stochastic differential equations

Stochastic differential equations (SDEs), which models uncertain phenomena as the time evolution of random variables, are exploited in various fields of natural and social sciences such as finance. Since SDEs rarely admit analytical solutions and must usually be solved numerically with huge classical-computational resources in practical applications, there is strong motivation to use quantum computation to accelerate the calculation. Here, we propose a quantum-classical hybrid algorithm that solves SDEs based on variational quantum simulation (VQS). We first approximate the target SDE by a trinomial tree structure with discretization and then formulate it as the time-evolution of a quantum state embedding the probability distributions of the SDE variables. We embed the probability distribution directly in the amplitudes of the quantum state while the previous studies did the square-root of the probability distribution in the amplitudes. Our embedding enables us to construct simple quantum circuits that simulate the time-evolution of the state for general SDEs. We also develop a scheme to compute the expectation values of the SDE variables and discuss whether our scheme can achieve quantum speed-up for the expectation-value evaluations of the SDE variables. Finally, we numerically validate our algorithm by simulating several types of stochastic processes. Our proposal provides a new direction for simulating SDEs on quantum computers.

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Efficient Simulation of Stochastic Differential Equations Based on Markov Chain Approximations With Applications

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms \textit{both biased and unbiased} simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory. Note: this is an earlier version of the work Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations.

Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations

In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.

On the long-term simulation of stochastic differential equations for predicting effective dispersion coefficients

This article discusses some numerical pitfalls associated with the classical solution of stochastic Langevin equations via the Euler–Langevin algorithm for long-term dispersion features. A simple operator-splitting algorithm is proposed to overcome these short-comings and, starting from it, a simple and computationally consistent algorithm for the simulation of stochastic differential equations is presented. Dispersion properties of tracer particles in cylindrical wavy channels and cellular flows are addressed in detail. Numerical results of the two algorithms proposed are compared with numerical results obtained from Brenner’s macrotransport theory.

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.

Quasi-Monte Carlo Methods Applied to Tau-Leaping in Stochastic Biological Systems

Quasi-Monte Carlo methods have proven to be effective extensions of traditional Monte Carlo methods in, amongst others, problems of quadrature and the sample path simulation of stochastic differential equations. By replacing the random number input stream in a simulation procedure by a low-discrepancy number input stream, variance reductions of several orders have been observed in financial applications. Analysis of stochastic effects in well-mixed chemical reaction networks often relies on sample path simulation using Monte Carlo methods, even though these methods suffer from typical slow {mathcal {O}}(N^{-1/2}) convergence rates as a function of the number of sample paths N. This paper investigates the combination of (randomised) quasi-Monte Carlo methods with an efficient sample path simulation procedure, namely tau -leaping. We show that this combination is often more effective than traditional Monte Carlo simulation in terms of the decay of statistical errors. The observed convergence rate behaviour is, however, non-trivial due to the discrete nature of the models of chemical reactions. We explain how this affects the performance of quasi-Monte Carlo methods by looking at a test problem in standard quadrature.

Method of Iterated Integrals for Solution of Stochastic Integrals with Applications to Monte Carlo Simulation of Stochastic Differential Equations.

In this article we suggest a new method for solutions of stochastic integrals where the dynamics of the variables in integrand are given by some stochastic differential equation. We also propose numerical simulation of stochastic differential equations which is based on iterated integrals method employing Ito change of variable expansion of the drift and volatility function of the SDE. If a function has to be evaluated along time, it would be possible to calculate its value in time using the initial value of the function and an integral of the time derivative of the function till terminal time. Method of iterated integrals is a method of solution of integrals in which we repeatedly expand different time dependent terms in the integral as an initial value and integral of its derivative along time so that their sum would equal the original term in the integrand that was expanded. We continue to repeatedly expand the integrand in time dependent terms into initial value terms and integral of further time dependent higher derivative and continue until we have enough accuracy in the terms evaluated at initial values so as to be able to neglect the time dependent highest derivative terms. We evaluate all these iterated integrals terms at the initial time of the simulation along with the solution of the stochastic integrals which is found in terms of Hermite polynomials and variance of the integrals. We apply the method of iterated integrals to simulation of various stochastic differential equations. We find that the accuracy of simulation sharply increases using the method of iterated integral as compared to naive and simple monte carlo simulation methods.

Numerical methods for simulation of stochastic differential equations

In this paper we are concerned with numerical methods to solve stochastic differential equations (SDEs), namely the Euler-Maruyama (EM) and Milstein methods. These methods are based on the truncated Ito-Taylor expansion. In our study we deal with a nonlinear SDE. We approximate to numerical solution using Monte Carlo simulation for each method. Also exact solution is obtained from Ito’s formula. To show the effectiveness of the numerical methods, approximation solutions are compared with exact solution for different sample paths. And finally the results of numerical experiments are supported with graphs and error tables.

Unbiased simulation of stochastic differential equations

We propose an unbiased Monte-Carlo estimator for E[g(X t 1 , · · · , X tn)], where X is a diffusion process defined by a multi-dimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are updated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by Bismu-Elworthy-Li formula from Malliavin calculus, as exploited by Fournie et al. [14] for the simulation of the Greeks in financial applications. In particular, this algorithm can be considered as a variation of the (infinite variance) estimator obtained in Bally and Kohatsu-Higa [3, Section 6.1] as an application of the parametrix method.

A Micro-Macro Acceleration Method for the Monte Carlo Simulation of Stochastic Differential Equations

We present and analyze a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with separation between the (fast) time scale of individual trajectories and the (slow) time scale of the macroscopic function of interest. The algorithm combines short bursts of path simulations with extrapolation of a number of macroscopic state variables forward in time. The new microscopic state, consistent with the extrapolated variables, is obtained by a matching operator that minimizes the perturbation caused by the extrapolation. We provide a proof of the convergence of this method, in the absence of statistical error, and we analyze various strategies for matching, as an operator on probability measures. Numerical experiments we show illustrate the effects of the different approximations on the resulting error in macroscopic predictions.