On the differentiability of solutions of stochastic evolution equations with respect to their initial values (original) (raw)
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arXiv (Cornell University), 2017
In the recent years there has been an increased interest in studying regularity properties of the derivatives of stochastic evolution equations (SEEs) with respect to their initial values. In particular, in the scientific literature it has been shown for every natural number n ∈ AE that if the nonlinear drift coefficient and the nonlinear diffusion coefficient of the considered SEE are n-times continuously Fréchet differentiable, then the solution of the considered SEE is also n-times continuously Fréchet differentiable with respect to its initial value and the corresponding derivative processes satisfy a suitable regularity property in the sense that the n-th derivative process can be extended continuously to n-linear operators on negative Sobolev-type spaces with regularity parameters δ 1 , δ 2 ,. .. , δ n ∈ [0, 1 /2) provided that the condition n i=1 δ i < 1 /2 is satisfied. The main contribution of this paper is to reveal that this condition can essentially not be relaxed. Contents 1 Introduction 1 2 Counterexamples to regularities for the derivative processes associated to stochastic evolution equations 4 2.
Stochastic Partial Differential Equations: Analysis and Computations, 2013
We prove pathwise uniqueness for an abstract stochastic reaction-diffusion equation in Banach spaces. The drift contains a bounded Hölder term; in spite of this, due to the space-time white noise it is possible to prove pathwise uniqueness. The proof is based on a detailed analysis of the associated Kolmogorov equation. The model includes examples not covered by the previous works based on Hilbert spaces or concrete SPDEs. * Key words and phrases: Stochastic reaction-diffusion equations, Kolmogorov equations in infinite dimension, pathwise uniqueness.
2006
The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts I, II. In Part I (this paper), we establish a general existence and compactness theorem for C k -cocycles of semilinear see's and spde's. Our results cover a large class of semilinear see's as well as certain semilinear spde's with non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite-dimensional noise. We adopt a notion of stationarity employed in previous work of one of the authors with M. Scheutzow ([M-S.2], cf. [E-K-M-S]). In Part II of this work ([M-Z-Z]), we establish a local stable manifold theorem for non-linear see's and spde's.
Evolution equation of a stochastic semigroup with white-noise drift
The Annals of Probability, 2000
We study the existence and uniqueness of the solution of a functionvalued stochastic evolution equation based on a stochastic semigroup whose kernel p(s; t; y; x) is Brownian in s and t. The kernel p is supposed to be measurable with respect to the increments of an underlying Wiener process in the interval [s; t]. The evolution equation is then anticipative and choosing the Skorohod formulation we establish existence and uniqueness of a continuous solution with values in L 2 (R d). As an application we prove the existence of a mild solution of the stochastic parabolic equation du t = x udt + v (dt; x) ru + F (t; x; u) W (dt; x) where v and W are Brownian in time with respect to a common …ltration. In this case, p is the formal backward heat kernel of x + v (dt; x) r x .
2014
In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results. t L p (P;H) Θ λ. (5) We note that the right hand side of (5) might be infinite. Moreover, we would like to emphasize that Y 1 and Y 2 in (5) are arbitrary (F t) t∈[0,T ]-predictable stochastic processes which satisfy (4) and, in particular, we emphasize that Y 1 and Y 2 do not need to be solution processes of some SEEs. Estimate (5) follows from an appropriate application of a generalized Gronwall-type inequality (see the proof of Proposition 2.7 below for details). We use inequality (5) to establish an existence, uniqueness, and regularity result for SEEs with singularities at the initial time. More precisely, in Theorem 2.9 below we prove that for all δ ∈ −
Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations
Potential Analysis, 2009
Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rates of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.
Fr\'{e}chet derivatives of expected functionals of solutions to stochastic differential equations
2021
In the analysis of stochastic dynamical systems described by stochastic differential equations (SDEs), it is often of interest to analyse the sensitivity of the expected value of a functional of the solution of the SDE with respect to perturbations in the SDE parameters. In this paper, we consider path functionals that depend on the solution of the SDE up to a stopping time. We derive formulas for Fréchet derivatives of the expected values of these functionals with respect to bounded perturbations of the drift, using the CameronMartin-Girsanov theorem for the change of measure. Using these derivatives, we construct an example to show that the map that sends the change of drift to the corresponding relative entropy is not in general convex. We then analyse the existence and uniqueness of solutions to stochastic optimal control problems defined on possibly random time intervals, as well as gradient-based numerical methods for solving such problems.
2002
The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts I, II. In Part I (this paper), we establish a general existence and compactness theorem for C k -cocycles of semilinear see's and spde's. Our results cover a large class of semilinear see's as well as certain semilinear spde's with non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite-dimensional noise. We adopt a notion of stationarity employed in previous work of one of the authors with M. Scheutzow ([M-S.2], cf. [E-K-M-S]). In Part II of this work ([M-Z-Z]), we establish a local stable manifold theorem for non-linear see's and spde's.
Applied Mathematics and Optimization, 2005
We study a forward-backward system of stochastic differential equations in an infinite dimensional framekork and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction-diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black-Scholes or Hamilton-Jacobi-Bellman type.