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Mini-Workshop: Stochastic Differential Equations: Regularity and Numerical Analysis in Finite and Infinite Dimensions
This Mini-Workshop is devoted to regularity and numerical analysis of stochastic ordinary and partial differential equations (SDEs for both). The standard assumption in the literature on SDEs is global Lipschitz continuity of the coefficient functions. However, many SDEs arising from applications fail to have globally Lipschitz continuous coefficients. Recent years have seen a prosper growth of the literature on regularity and numerical approximations for SDEs with non-globally Lipschitz coefficients. Some surprising results have been obtained – e.g., the Euler–Maruyama method diverges for a large class of SDEs with super-linearly growing coefficients, and the limiting equation of a spatial discretization of the stochastic Burgers equation depends on whether the discretization is symmetric or not. Several positive results have been obtained. However the regularity of numerous important SDEs and the closely related question of convergence and convergence rates of numerical approximations remain open. The aim of this workshop is to bring together the main contributers in this direction and to foster significant progress
Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity
We consider a time inhomogeneous jump Markov process with state
dependent jump intensity, taking values in Its infinitesimal generator
is given by \begin{multline*} L_t f (x) = \sum_{i=1}^d \frac{\partial
f}{\partial x_i } (x) b^i ( t,x) - \sum_{ i =1}^d \frac{\partial f}{\partial
x_i } (x) \int_{E_1} c_1^i ( t, z, x) \gamma_1 ( t, z, x ) \mu_1 (dz ) \\ +
\sum_{l=1}^3 \int_{E_l} [ f ( x + c_l ( t, z, x)) - f(x)] \gamma_l ( t, z, x)
\mu_l (dz ) , \end{multline*} where are sigma-finite measurable spaces describing three different jump
regimes of the process (fast, intermediate, slow).
We give conditions proving that the long time behavior of can be related
to the one of a time homogeneous limit process Moreover, we
introduce a coupling method for the limit process which is entirely based on
certain of its big jumps and which relies on the regeneration method. We state
explicit conditions in terms of the coefficients of the process allowing to
control the speed of convergence to equilibrium both for and for $\bar X.
Stochastic ordinary differential equations in applied and computational mathematics
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation
Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems
In this paper, we establish that for a wide class of controlled stochastic
differential equations (SDEs) with stiff coefficients, the value functions of
corresponding zero-sum games can be represented by a deep artificial neural
network (DNN), whose complexity grows at most polynomially in both the
dimension of the state equation and the reciprocal of the required accuracy.
Such nonlinear stiff systems may arise, for example, from Galerkin
approximations of controlled stochastic partial differential equations (SPDEs),
or controlled PDEs with uncertain initial conditions and source terms. This
implies that DNNs can break the curse of dimensionality in numerical
approximations and optimal control of PDEs and SPDEs. The main ingredient of
our proof is to construct a suitable discrete-time system to effectively
approximate the evolution of the underlying stochastic dynamics. Similar ideas
can also be applied to obtain expression rates of DNNs for value functions
induced by stiff systems with regime switching coefficients and driven by
general L\'{e}vy noise.Comment: This revised version has been accepted for publication in Analysis
and Application
Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift
In this paper, we first establish well-posedness results for one-dimensional
McKean-Vlasov stochastic differential equations (SDEs) and related particle
systems with a measure-dependent drift coefficient that is discontinuous in the
spatial component, and a diffusion coefficient which is a Lipschitz function of
the state only. We only require a fairly mild condition on the diffusion
coefficient, namely to be non-zero in a point of discontinuity of the drift,
while we need to impose certain structural assumptions on the
measure-dependence of the drift. Second, we study fully implementable
Euler-Maruyama type schemes for the particle system to approximate the solution
of the one-dimensional McKean-Vlasov SDE. Here, we will prove strong
convergence results in terms of the number of time-steps and number of
particles. Due to the discontinuity of the drift, the convergence analysis is
non-standard and the usual strong convergence order known for the
Lipschitz case cannot be recovered for all schemes.Comment: 33 pages, 4 figures, revised introduction and Section
Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
In this paper, we derive error estimates of the backward Euler-Maruyama
method applied to multi-valued stochastic differential equations. An important
example of such an equation is a stochastic gradient flow whose associated
potential is not continuously differentiable, but assumed to be convex. We show
that the backward Euler-Maruyama method is well-defined and convergent of order
at least with respect to the root-mean-square norm. Our error analysis
relies on techniques for deterministic problems developed in [Nochetto,
Savar\'e, and Verdi, Comm.\ Pure Appl.\ Math., 2000]. We verify that our
setting applies to an overdamped Langevin equation with a discontinuous
gradient and to a spatially semi-discrete approximation of the stochastic
-Laplace equation
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