Scalar Stochastic Differential Equations Research Articles

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all p∈[1,∞) a transformed Milstein-type scheme reaches an Lp-error rate of at least 3/4 when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate 3/4 is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound from Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

The modified truncated Euler–Maruyama method for stochastic differential equations with concave diffusion coefficients

Influenced by Gyöngy and Rásonyi (2011), many scholars established the strong convergence of several numerical methods for scalar stochastic differential equations (SDEs) with superlinearly growing drift and Hölder continuous diffusion coefficients. However, their methods depend on the Yamada-Watanabe method and therefore fail to work for multi-dimensional SDEs. In this paper, we study the strong Lp−convergence, for all p⩾2, of the modified truncated Euler–Maruyama method for multi-dimensional SDEs with superlinearly growing drift and concave diffusion coefficients satisfying the Osgood condition. We also discuss an example with computer simulations to illustrate our theoretical results.

Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and nondegenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best Lp-error rate obtained so far for approximation of scalar SDEs at the final time is 3/4 in terms of the number of evaluations of the driving Brownian motion. In the present article, we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the Lp-error rate 3/4 can not be improved. For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the Lp-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the Lp-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables (Y,Z) by providing explicit lower bounds for the covariance Cov(f(Y),g(Z)) in case of piecewise Lipschitz continuous functions f and g. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals ∫01f(Wt)dt of a Brownian motion W, which substantially extends known results for the case of f being an indicator function.

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Kr\"uger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.

An adaptive strong order 1 method for SDEs with discontinuous drift coefficient

In recent years, a number of results has been proven in the literature for strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. For scalar SDEs of that type the best Lp-error rate known so far for approximation of the solution at the final time point is 3/4 in terms of the number of evaluations of the driving Brownian motion and it is achieved by the transformed equidistant quasi-Milstein scheme, see [24]. Recently in [27] it has been shown that for such SDEs the Lp-error rate 3/4 can not be improved in general by no numerical method based on evaluations of the driving Brownian motion at fixed time points. In the present article we construct for the first time in the literature a method based on sequential evaluations of the driving Brownian motion, which achieves an Lp-error rate of at least 1 in terms of the average number of evaluations of the driving Brownian motion for such SDEs.

A strong order 3/4 method for SDEs with discontinuous drift coefficient

Abstract In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently, it has been shown in Müller-Gronbach & Yaroslavtseva (2020, On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. Henri Poincaré Probab. Stat., 56, 1162–1178) that for scalar SDEs with a piecewise Lipschitz drift coefficient, and a Lipschitz diffusion coefficient that is nonzero at the discontinuity points of the drift coefficient the classical Euler–Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty )$. Up to now this was the best $L_p$-error rate available in the literature for equations of that type. In the present paper we construct a method based on finitely many evaluations of the driving Brownian motion that even achieves an $L_p$-error rate of at least $3/4$ for all $p\in [1,\infty )$ under additional piecewise smoothness assumptions on the coefficients. This is the first higher order method known in the literature for SDEs of that type. We add that the $L_p$-error rate $3/4$ cannot be improved in general. To obtain the upper error bound for our new method, we prove in particular that a quasi-Milstein scheme achieves an $L_p$-error rate of at least $3/4$ in the case of coefficients that are both Lipschitz continuous and piecewise differentiable with Lipschitz continuous derivatives, which is of interest in itself. The latter error rates are obtained via a detailed analysis of the average size of increments of the time-continuous quasi-Milstein scheme over time intervals in which the scheme crosses a point of nondifferentiability of the coefficients.

On the MS-stability of predictor–corrector schemes for stochastic differential equations

Predictor–corrector schemes are designed to be a compromise to retain the stability properties of the implicit schemes and the computational efficiency of the explicit ones. In this paper a complete analytical study for the linear mean-square stability of the two-parameter family of Euler predictor–corrector schemes for scalar stochastic differential equations is given. The analyzed family is given in terms of two parameters that control the degree of implicitness of the method. For each selection of the parameters the stability region is obtained, letting its comparison. Particular cases of the counter-intuitive fact of losing numerical stability by reducing the step size, is confirmed and proved. Figures of the MS-stability regions and numerical examples that confirm the theoretical results are shown.

Randomized derivative-free Milstein algorithm for efficient approximation of solutions of SDEs under noisy information

We deal with pointwise approximation of solutions of scalar stochastic differential equations in the presence of informational noise about underlying drift and diffusion coefficients. We define a randomized derivative-free version of Milstein algorithm Āndf−RM and investigate its error. We also study the lower bounds on the error of arbitrary algorithm. It turns out that in some case the scheme Āndf−RM is the optimal one. Finally, in order to test the algorithm Āndf−RM in practice, we report performed numerical experiments.

The Euler–Maruyama scheme for SDEs with irregular drift: convergence rates via reduction to a quadrature problem

Abstract We study the strong convergence order of the Euler–Maruyama (EM) scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev–Slobodeckij-type regularity of order $\kappa \in (0,1)$ for the nonsmooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min \{3/4,(1+\kappa )/2\}-\epsilon$ for the equidistant EM scheme (for arbitrarily small $\epsilon>0$). The cut-off of the convergence order at $3/4$ can be overcome by using a suitable nonequidistant discretization, which yields the strong convergence order of $(1+\kappa )/2-\epsilon$ for the corresponding EM scheme.

General multilevel Monte Carlo methods for pricing discretely monitored Asian options

We describe general multilevel Monte Carlo methods that estimate the price of an Asian option monitored at m fixed dates. For a variety of processes that can be simulated exactly, we prove that, for the same computational cost, our method yields an unbiased estimator with variance lower than the variance of the standard Monte Carlo estimator by a factor of order m. We show how to combine our approach with the Milstein scheme for processes driven by scalar stochastic differential equations, and with the Euler scheme for processes driven by multidimensional stochastic differential equations. Numerical experiments confirm that our method outperforms the conventional Monte Carlo algorithm by a factor proportional to m.

Performing Parallel Monte Carlo and Moment Equations Methods for Itô and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc

We introduce Sim.DiffProc, an R package for symbolic and numerical computations on scalar and multivariate systems of stochastic differential equations (SDEs). It provides users with a wide range of tools to simulate, estimate, analyze, and visualize the dynamics of these systems in both forms, Ito and Stratonovich. One of Sim.DiffProc key features is to implement the Monte Carlo method for the iterative evaluation and approximation of an interesting quantity at a fixed time on SDEs with parallel computing, on multiple processors on a single machine or a cluster of computers, which is an important tool to improve capacity and speed-up calculations. We also provide an easy-to-use interface for symbolic calculation and numerical approximation of the first and central second-order moments of SDEs (i.e., mean, variance and covariance), by solving a system of ordinary differential equations, which yields insights into the dynamics of stochastic systems. The final result object of Monte Carlo and moment equations can be derived and presented in terms of LATEX math expressions and visualized in terms of LATEX tables. Furthermore, we illustrate various features of the package by proposing a general bivariate nonlinear dynamic system of Haken-Zwanzig, driven by additive, linear and nonlinear multiplicative noises. In addition, we consider the particular case of a scalar SDE driven by three independent Wiener processes. The Monte Carlo simulation thereof is obtained through a transformation to a system of three equations. We also study some important applications of SDEs in different fields.

Path integral solutions of the governing equation of SDEs excited by Lévy white noise

In this paper, the probability density functions (PDFs) of scalar stochastic differential equations (SDEs) subject to α-stable Lévy white noise are investigated. The path integral (PI) method is extended to solve one-dimensional space fractional Fokker-Planck-Kolmogorov (FPK) equations, which are the governing equations corresponded to scalar SDEs excited by α-stable Lévy white noise. First, we derive a short time solution of the one-dimensional space fractional FPK equation, which is used in the Chapman-Kolmogorov-Smoluchowski (CKS) equation to obtain the PI solution. Then, the accuracy of the PI solution is analyzed theoretically in terms of its characteristic function. Our results demonstrate that the PI method has a higher accuracy than the first order finite difference method for one step iteration in time. Finally, several illustrative examples are carried out in detail to verify the feasibility and effectiveness of the PI method for solving one-dimensional space fractional FPK equations. We find that the PI solution agrees well with the exact solution or the Monte Carlo one.

Lower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficients

We study pathwise approximation of scalar stochastic differential equations at a single time point or globally in time by means of methods that are based on finitely many observations of the driving Brownian motion. We prove lower error bounds in terms of the average number of evaluations of the driving Brownian motion that hold for every such method under rather mild assumptions on the coefficients of the equation. The underlying simple idea of our analysis is as follows: the lower error bounds known for equations with coefficients that have sufficient regularity globally in space should still apply in the case of coefficients that have this regularity in space only locally, in a small neighborhood of the initial value. Our results apply to a huge variety of equations with coefficients that are not globally Lipschitz continuous in space including Cox–Ingersoll–Ross processes, equations with superlinearly growing coefficients, and equations with discontinuous coefficients. In many of these cases, the resulting lower error bounds even turn out to be sharp.

SDE models with exponential drift and diffusion for approximating fatigue crack growth dynamics

A scalar stochastic differential equation (SDE) is studied for modeling fatigue crack growth dynamics. The drift and diffusion coefficients in the SDE model are exponential functions of the number of fatigue cycles. Exponential coefficients are shown to be reasonable based on experimental, physical, and theoretical arguments. The SDE model is examined for two fatigue crack growth datasets. For each dataset, the four parameters in the SDE model are fit using observed crack lengths for two different low values for the number of cycles. Then, using the SDE model with the fitted parameters, probability distributions for several higher values of the number of cycles and several first-passage probability distributions are calculated and compared to the observed distributions. The comparisons indicate that the proposed SDE model provides a good approximation for fatigue crack growth.

Stochastic Dynamics of Eukaryotic Flagellar Growth.

We study the dynamics of flagellar growth in eukaryotes where intraflagellar transporters (IFT) play a crucial role. First we investigate a stochastic version of the original balance point model where a constant number of IFT particles move up and down the flagellum. The detailed model is a discrete event vector-valued Markov process occurring in continuous time. First the detailed stochastic model is compared and contrasted with a simple scalar ordinary differential equation (ODE) model of flagellar growth. Numerical simulations reveal that the steady-state mean value of the stochastic model is well approximated by the ODE model. Then we derive a scalar stochastic differential equation (SDE) as a first approximation and obtain a "small noise" approximation showing flagellar length to be Gaussian with mean and variance governed by simple ODEs. The accuracy of the small noise model is compared favorably with the numerical simulation results of the detailed model. Secondly, we derive a revised SDE for flagellar length following the revised balance point model proposed in 2009 in which IFT particles move in trains instead of in isolation. Small noise approximation of the revised SDE yields the same approximate Gaussian distribution for the flagellar length as the SDE corresponding to the original balance point model.

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain D of the phase space R and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.

Existence of almost periodic solutions of stochastic differential equations with periodic coefficients

Abstract In this work, under some conditions, we will prove that a scalar stochastic differential equation with periodic coefficients admits almost periodic solutions.

Theorems of comparison and stability with probability 1 for one-dimensional stochastic differential equations

We prove the comparison theorems for scalar stochastic differential equations in the case of different diffusion coefficients. Conditions are given of stability with probability 1 with respect to the trivial solution to stochastic differential equations with random coefficients. The results remain valid for deterministic analogs of stochastic differential equations with symmetric integrals.

LIQUIDATION OF AN INDIVISIBLE ASSET WITH INDEPENDENT INVESTMENT

AbstractWe provide an extension of the explicit solution of a mixed optimal stopping–optimal stochastic control problem introduced by Henderson and Hobson. The problem examines whether the optimal investment problem on a local martingale financial market is affected by the optimal liquidation of an independent indivisible asset. The indivisible asset process is defined by a homogeneous scalar stochastic differential equation, and the investor's preferences are defined by a general expected utility function. The value function is obtained in explicit form, and we prove the existence of an optimal stopping–investment strategy characterized as the limit of an explicit maximizing strategy. Our approach is based on the standard dynamic programming approach.

Construction of some accelerated methods for solving scalar stochastic differential equations

The purpose of this short communication is to contribute in constructing some new solvers for the strong solution of stochastic ordinary differential equations. It is derived that new methods could be obtained as new variants of the high order scheme of Platen 1987. A simple technique to accelerate their speed will also be pointed out. Finally, numerical simulations are brought forward to show the efficiency of the derived methods.