on Stochastic Differential Equations: Theory, Numerics and Applications

(01/01/2022-31/12/2023)

Zoom meeting ID: 865 5854 2412

Passcode: RSESDE

Abstract: We examine how adaptive time-stepping can be used to solve stochastic (partial) differential equations where the drift/diffusion terms do not satisfy a global Lipschitz condition. It was shown in \cite{Hutzenthaler1} that the standard Euler method does not converge strongly because of lack of control on the moments of the numerical method and in \cite{Hutzenthaler2} fixed step tamed Euler methods were proposed. As an alternative to taming we show we can use timestep adaptivity to control moments of the numerical method. Although not designed to control local error directly we do observe in numerical simulations an improved error constant and that the adaptivity is good at capturing the local behaviour. The talk will introduce the issue with standard Euler methods for non-globally Lipschitz coefficients and introduce two different approaches for the adaptivity. We will briefly examine the proof of strong convergence and will compare the adaptive methods to other approaches in the literature such as the tamed Euler and other variants. We conclude with a comment on the deterministic simulations.

Abstract: In principle, once the existence of the stationary distribution of an SDE is assured, we may compute its probability density function by solving the corresponding Kolmogorov-Fokker-Planck equation. However, this is nontrivial in practice. Our primary objective is to construct easily implementable numerical solutions and prove that the probability measures induced by the numerical solutions will converge weakly (in the Kantorovich metric) to the stationary distribution of the true solution of the SDE. We will only concentrate on three numerical methods: the Euler-Maruyama, backward Euler-Maruyama and truncated Euler-Maruyama.

Abstract: We consider the numerical approximation of a stochastic differential equation (SDE) whose drift is the gradient of a convex and nonnegative potential. An important class of such SDEs consists of nonlinear Langevin-type equations. Other examples naturally arise from the discretization of stochastic PDEs as, for instance, the stochastic p-Laplace equation. In the first part of the talk, we review an interesting approach for the error analysis of the backward Euler method, which has been developed by [Nochetto, SavarÃ©, Verdi, 2000] for deterministic evolution equations with multivalued and maximal monotone operators. Thereafter, we discuss how these techniques can be adapted to stochastic gradient flows with possibly discontinuous drift coefficients. As our main result, we show that the order of convergence of the backward Euler-Maruyama method is at least 1/4 under these low regularity assumptions. The main advantage of this approach is that it avoids the application of Gronwall-type arguments and does not require any pre-knowledge about the temporal regularity of the exact solution. This is joint work with M. Eisenmann, M. Kovacs, S. Larsson and J. Weinberger.

Abstract: When designing new discretisation methods for stochastic differential equations (SDEs), we are often concerned with convergence, either in the strong or weak sense. In their 2011 article, Hutzenthaler, Jentzen, & Kloeden showed that the Euler-Maruyama method fails to converge in either the strong or weak sense for SDEs where either the drift or diffusion coefficients are non-Lipschitz continuous and satisfy a polynomial growth condition. This demonstration sparked considerable activity, and in the last decade there have been several new explicit numerical methods proposed as a result, including the tamed and truncated classes of discretisation method. Our interest in new methods is motivated not only by the need for convergence, but also by the requirement for dynamical consistency: a good method should reproduce qualitative structures such as pathwise stability and instability, domain invariance including preservation of positivity, and the presence of any finite-time explosions. In this talk we will explore the use of path-dependent adaptive timestepping with standard explicit and semi-implicit Euler methods to overcome the instability and convergence issues that arise for these methods when applied to some nonlinear systems of SDEs over uniform grids. We will also look at adaptive strategies for ensuring dynamical consistency with such systems, and identify some open problems.

Abstract: This talk is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is established by using the Faedo-Galerkin approximation and an idea analogous to Da Prato and Zabczyk. Second, we show the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the steady-state solutions of the deterministic problem is also solution of the stochastic one, or when this does not happen. Next, to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation of the noise given by Wong-Zakai's technique using the so called colored noise. For this model, we can use the power of the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the cases of additive and multiplicative noise, it is proved that the Wong-Zakai approximation models possess random attractors which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem.

Abstract: Approximate Bayesian Computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed as an established tool for modelling time dependent, real world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise. First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. In this talk we consider SDEs having an invariant density and apply measure-preserving splitting schemes for the synthetic data generation. We illustrate the results of the parameter estimation with the corresponding ABC algorithm with simulated data. This talk is based on joined work with Massimiliano Tamborrino, University of Warwick, and Irene Tubikanec, Johannes Kepler University Linz.

Abstract: In this talk, I would like to discuss around the approximation of the solution of the following class of 1D SDEs : dX(t) = b(X(t)) dt + X^a(t) dW(t), when a > 1. I will first present some motivating examples where such SDEs arise in modelling approaches. Of course, a priori knowledge about the conditions on the coefficients ensuring well-posedness and some control on the moments is required, not only for use in modelling but also to study the convergence of approximation schemes. Here, we focus on the weak convergence rate. First, a set of conditions for the C^{1,4}-regularity of the Feynman-Kac formula is proposed using stochastic tools. In particular, we show how to avoid the costly control of the moments of the successive derivatives of the flow process by using a change of measure technique. We then introduce an exponential scheme for the time integration of the SDE, which reproduces well the control of the moments of the exact process and for which we prove a convergence rate of order one. This talk is based on two recent papers with Kerlyns Martinez Rodrigez (University of Valparaiso) and Jean Francois Jabir (HSE University Moscow).

Abstract: The long--run behaviour of the mean square of linear stochastic differential and difference equations without memory has been well--understood for many decades. The key contribution was made in the 1950's by Richard Bellman. The situation with continuous--time stochastic equations with memory has proved more challenging. This is in contrast to the situation with linear deterministic equations in continuous time, whose stability can be characterised by results such as the Paley--Wiener theorem. Indeed, even exponential rates of growth and decay can be identified by the solutions of transcendental characteristic equations. Existing results in continuous time which attempt to give a stability characterisation, find a characteristic equation, or determine characteristic exponents, all have certain limitations. The results we present in this talk attempt to improve the situation in the scalar case at least. The key observation which makes this possible is that the mean square of the solution of the SFDE can be written in terms of the solution of deterministic Volterra equations. The main difficulty is that the kernels and forcing functions in these auxiliary equations no longer depend directly on the problem data of the original SFDE. Nevertheless, these difficulties can be surmounted, so that necessary and sufficient conditions for stability in terms of the problem data, as well as a characterisation of convergence for forced equations and exponential decay, can be obtained. For certain equations, we are even able to identify a transcendental equation whose solution is the top Lyapunov exponent of the system. Time permitting, results for Volterra and difference equations will also be discussed. In the latter case, the results are of independent interest, but also open up the prospect of seeing how stability or instability is preserved by numerical methods. The work is joint with Emmet Lawless (DCU).

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