Large deviations for stochastic evolution equations in the critical variational setting

Previous article Next article A Simple Proof of the Existence of a Solution of Itô’s Equation with Monotone CoefficientsN. V. KrylovN. V. Krylovhttps://doi.org/10.1137/1135082PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. V. Krylov and , B. L. Rozovskii, Stochastic evolution equationsCurrent problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, 71–147, 256 81m:60116 Google Scholar[2] I. Gyongy and , N. V. Krylov, On stochastic equations with respect to semimartingales. I, Stochastics, 4 (1980/81), 1–21 82j:60104 CrossrefGoogle Scholar[3] N. V. Krylov, Extremal properties of solutions of stochastic equations, Theory Probab. Appl., 29 (1984), 205–217 10.1137/1129033 0557.60042 LinkGoogle Scholar[4] L. A. Alyushina, Euler polygonal lines for Itô equations with monotone coefficients, Theory Probab. Appl., 32 (1987), 340–346 10.1137/1132046 0663.60049 LinkGoogle Scholar[5] Gisirō Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo (2), 4 (1955), 48–90 17,166f 0053.40901 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo modelApplied Numerical Mathematics | 1 Apr 2022 Cross Ref Approximation of BSDEs with super-linearly growing generators by Euler’s polygonal line method: A simple proof of the existenceSystems & Control Letters, Vol. 153 | 1 Jul 2021 Cross Ref Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic & 2D-Navier–Stokes Equations: The General CaseJournal of Mathematical Fluid Mechanics, Vol. 22, No. 4 | 4 September 2020 Cross Ref A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditionsInternational Journal of Computer Mathematics, Vol. 5 | 9 March 2020 Cross Ref Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility ModelsAndrei Cozma and Christoph ReisingerSIAM Journal on Numerical Analysis, Vol. 56, No. 6 | 11 December 2018AbstractPDF (1095 KB)Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded DomainsJournal of Mathematical Fluid Mechanics, Vol. 18, No. 1 | 22 October 2015 Cross Ref An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz CoefficientsJean-François Chassagneux, Antoine Jacquier, and Ivo MihaylovSIAM Journal on Financial Mathematics, Vol. 7, No. 1 | 14 December 2016AbstractPDF (602 KB)A New Type of Stability Theorem for Stochastic Systems With Application to Stochastic StabilizationIEEE Transactions on Automatic Control, Vol. 61, No. 1 | 1 Jan 2016 Cross Ref Loss of regularity for Kolmogorov equationsThe Annals of Probability, Vol. 43, No. 2 | 1 Mar 2015 Cross Ref Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay Under Local Lipschitz ConditionStochastic Analysis and Applications, Vol. 32, No. 2 | 28 February 2014 Cross Ref A Note on Euler Approximations for Stochastic Differential Equations with DelayApplied Mathematics & Optimization, Vol. 68, No. 3 | 24 August 2013 Cross Ref Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equationsThe Annals of Applied Probability, Vol. 23, No. 5 | 1 Oct 2013 Cross Ref Rate of convergence of Euler’s approximations for SDEs with non-Lipschitz coefficientsActa Mathematica Sinica, English Series, Vol. 29, No. 8 | 15 July 2013 Cross Ref Stochastic Itô inclusion with upper separated multifunctionsJournal of Mathematical Analysis and Applications, Vol. 400, No. 2 | 1 Apr 2013 Cross Ref Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficientsThe Annals of Applied Probability, Vol. 22, No. 4 | 1 Aug 2012 Cross Ref Stochastic 3D tamed Navier–Stokes equations: Existence, uniqueness and small time large deviation principlesJournal of Differential Equations, Vol. 252, No. 1 | 1 Jan 2012 Cross Ref On the Strong Solution for the 3D Stochastic Leray-Alpha ModelBoundary Value Problems, Vol. 2010, No. 1 | 1 Jan 2010 Cross Ref ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTSStochastics and Dynamics, Vol. 09, No. 04 | 21 November 2011 Cross Ref Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicityProbability Theory and Related Fields, Vol. 145, No. 1-2 | 19 July 2008 Cross Ref Large deviations in the Langevin dynamics of a random field Ising modelStochastic Processes and their Applications, Vol. 105, No. 2 | 1 Jun 2003 Cross Ref Viable solutions of set-valued stochastic equationOptimization, Vol. 48, No. 2 | 1 Jan 2000 Cross Ref Volume 35, Issue 3| 1991Theory of Probability & Its Applications411-623 History Submitted:05 September 1988Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135082Article page range:pp. 583-587ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

This chapter is devoted to presenting some results on controllability for forward and backward stochastic linear evolution equations with possibly unbounded control operators. By the duality argument, these controllability problems can be reduced to suitable observability for the dual equations. Explicit forms of controls for the controllability problems are also provided. Finally, the controllability of some forward stochastic evolution equations are shown to be equivalent to that of suitably chosen backward stochastic evolution equations.

Stochastic evolution equations with general white noise disturbance

A topological theorem and correlations, within the context of stochastic evolution

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

The classical Pontryagin maximum principle is a first-order necessary condition (FONC for short) for optimal controls of ordinary differential equations (ODEs for short). When the FONC degenerates, people introduce second-order necessary conditions (SONCs for short) for optimal controls. SONCs are well studied for optimal controls of systems described by ODEs and partial differential equations (PDEs for short). Some results for SONC of optimal controls for systems governed by stochastic differential equations (SDEs for short) are also obtained. However, nothing is known about the SONC for optimal controls of systems described by stochastic (infinite dimensional) evolution equations (SEEs for short). This paper aims to give a solution to this difficult problem, under some mild conditions.

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations

A method for numerical approximation of a class of fractional parabolic stochastic evolution equations is introduced and analysed. This class of equations has recently been proposed as a space-time extension of the SPDE-method in spatial statistics. A truncation of the spectral basis function expansion is used to discretise in space, and then a quadrature is used to approximate the temporal evolution of each basis coefficient. Strong error bounds are proved both for the spectral and temporal approximations. The method is tested, and the results are verified by several numerical experiments.

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

This paper addresses general discounted stochastic linear-quadratic (LQ) optimal control problems, where the control cost weighting operator may be singular, and the system is governed by controlled stochastic evolution equations in a Hilbert space. In the time-consistent setting, we establish conditions on the problem coefficients that guarantee the well-posedness of the generalized Riccati equation, despite the singularity of the control weighting operator. For the time-inconsistent case, a necessary and sufficient condition for the existence of a closed-loop equilibrium operator is established in terms of the solvability of the equilibrium Riccati evolution equation with certain regularity. In the last part of the paper, the solvability of the equilibrium Riccati evolution equation is established through the application of the multi-person differential game method. These results extend the recent work of Lü and Ma [Science China Mathematics, 67 (2024), 211–236], which addressed this problem for controlled stochastic differential equations.

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

Fractional Brownian Motions, Fractional Noises and Applications

Stochastic evolution equations are investigated using a new approach to the group analysis of stochastic differential equations. It is shown that the proposed approach reduces the problem of group analysis for this type of equations to the same problem of group analysis for evolution equations of special form without stochastic integrals.

We consider a problem of optimal control of the stochastic evolution equation $d\xi = (A(t)\xi + B(t)u)dt + \sigma (t)dw$, on a separable Hilbert space, where $\{ A(t),B(t),\sigma (t),t \geqq 0\} $ are progressively measurable operator-valued random processes with A generally unbounded. We prove the existence and uniqueness of (weak) solutions of the evolution equation. Then we present the existence of optimal controls and necessary conditions of optimality for a quadratic (random) cost function. For optimal feedback controls we solve a random operator Riccati equation and a backward stochastic evolution equation. The backward equation is solved by transposing a random isomorphism generated from a forward evolution equation. The optimal feedback control is given by a random affine transformation of the state. Some examples are presented to indicate usefulness of the results. This work is a partial extension of the results of Bismut [SIAM J. Control Optim., 14 (1976), pp. 419–444; 15 (1977), pp. 1–4] and Bensoussan and Voit [SIAM J. Control Optim., 13 (1975), pp. 904–926].

Preface Picture of Raphael Hoegh-Krohn Bibliography of Raphael Hoegh-Krohn 1. On the scientific work of Raphael Hoegh-Krohn Part I. Stochastic Analysis 2. Some Euclidean integer-valued random fields with Markov properties 3. A Beurling-Deny type structure theorem for Dirichlet forms on general state spaces 4. Log-concavity of radial Schroedinger wave functions and convergence of planetesimal diffusions 5. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals 6. Large deviations for weak solutions of stochastic differential equations 7. On a class of probabilistic integrodifferential equations 8. Operator integrals and martingale integrals with general parameter 9. Wick multiplication and Ito-Skorohod stochastic differential equations 10. Multiscale analysis in random dynamics 11. A limiting distribution connected with fractional parts of linear forms 12. Rapport sur les representations chaotiques 13. Markov properties of solutions of stochastic partial differential equations in a finite volume 14. On stochastic evolution equations with non-homogeneous boundary conditions 15. Grey noise 16. Existence of invariant measures for diffusion process with infinite dimensional state space Part II. Infinite Dimensional Groups 17. On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds 18. Fields of left-invariant standard Brownian motion processes on a smooth bundle of compact-simple Lie groups 19. Unitary highest weight representations of gauge groups Part III. Operator Algebras 20. Some non-commutative orbifolds 21. Ergodic actions of non-abelian compact groups 22. Positive projections onto Jordan algebras and their enveloping von Neumann algebras Part IV. Nonlinear Analysis and Applications 23. How many singularities can there be in an energy minimizing map from the ball to the sphere? 24. Front tracking for petroleum reservoirs 25. Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equations 26. A new representation of soliton solutions of the Kadomtsev-Petviashvili equation 27. On scalar conservation laws in one dimension.