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1.
István Gyöngy 《Potential Analysis》1995,4(2):157-171
We prove a limit theorem for non-degenerate quasi-linear parabolic SPDEs driven by space-time white noise in one space-dimension, when the diffusion coefficient is Lipschitz continuous and the nonlinear drift term is only measurable. Hence we obtain an existence and uniqueness and a comparison theorem, which generalize those in [2], [4], [5] to the case of non-degenerate SPDEs with measurable drift and Lipschitz continuous diffusion coefficients.Research supported by the Hungarian National Foundation of Scientific Research No. 2290. 相似文献
2.
Summary We prove existence and uniqueness of the solution of a parabolic SPDE in one space dimension driven by space-time white noise, in the case of a measurable drift and a constant diffusion coefficient, as well as a comparison theorem.and INRIAPartially supported by DRET under contract 901636/A000/DRET/DS/SR 相似文献
3.
István Gyöngy 《Potential Analysis》1993,2(2):101-113
We present a general framework of treating SPDEs on manifolds by adapting the notion of well-weighted Sobolev spaces from [1]. Using this we extend the theory of SPDEs to the case of manifolds.Research supported by the Hungarian National Foundation of Scientific Research No. 2290. 相似文献
4.
Summary We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.The work of the first author was supported in part by the NSF grant DMS-91-01360 相似文献
5.
6.
Jong Uhn Kim 《Journal of Differential Equations》2011,250(3):1650-1684
We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for a quasi-linear symmetric hyperbolic system with random noise in Rd. When the noise is multiplicative satisfying some nondegenerate conditions and the initial data are sufficiently small, we show that the solution exists globally in time in probability, i.e., the probability of global existence can be made arbitrarily close to one if the initial date are small accordingly. 相似文献
7.
We present a direct approach to existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone. The proof of uniqueness is very elementary, based on a new method of applying Itô’s formula for the -norm. The proof of existence relies on a recent regularity result and is direct in the sense that it does not rely on the stochastic compactness method. 相似文献
8.
In [R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations. A limit approach. Ann. Probab. (2007) (in press). Available online: http://www.imstat.org/aop/future_papers.htm] the authors obtained mean-field Backward Stochastic Differential Equations (BSDE) associated with a mean-field Stochastic Differential Equation (SDE) in a natural way as a limit of a high dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying them in a more general framework, with general coefficient, and to discuss comparison results for them. In a second step we are interested in Partial Differential Equations (PDE) whose solutions can be stochastically interpreted in terms of mean-field BSDEs. For this we study a mean-field BSDE in a Markovian framework, associated with a McKean–Vlasov forward equation. By combining classical BSDE methods, in particular that of “backward semigroups” introduced by Peng [S. Peng, J. Yan, S. Peng, S. Fang, L. Wu (Eds.), in: BSDE and Stochastic Optimizations; Topics in Stochastic Analysis, Science Press, Beijing (1997) (Chapter 2) (in Chinese)], with specific arguments for mean-field BSDEs, we prove that this mean-field BSDE gives the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated with mean-field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions. 相似文献
9.
Peter Hepperger 《Stochastic Processes and their Applications》2012,122(2):600-622
The basic contracts traded on energy exchanges are swaps involving the delivery of electricity for fixed-rate payments over a certain period of time. The main objective of this article is to solve the quadratic hedging problem for European options on these swaps, known as electricity swaptions. We consider a general class of Hilbert space valued exponential jump-diffusion models. Since the forward curve is an infinite-dimensional object, but only a finite set of traded contracts are available for hedging, the market is inherently incomplete. We derive the optimization problem for the quadratic hedging problem under the risk neutral measure and state a representation of its solution, which is the starting point for numerical algorithms. 相似文献
10.
In this paper, we prove the existence and uniqueness of the solution for a class of backward stochastic partial differential equations (BSPDEs, for short) driven by the Teugels martingales associated with a Lévy process satisfying some moment conditions and by an independent Brownian motion. An example is given to illustrate the theory. 相似文献
11.
Tusheng Zhang 《Stochastic Processes and their Applications》2011,121(6):1356-1372
In this paper, we establish the existence and uniqueness of solutions of systems of stochastic partial differential equations (SPDEs) with reflection in a convex domain. The lack of comparison theorems for systems of SPDEs makes things delicate. 相似文献
12.
Variable coefficient and Wick-type stochastic nonlinear Schrödinger (NLS) equations are investigated. By using white noise analysis, Hermite transform and extended F-expansion method, we obtain a number of Wick versions of periodic-like wave solutions and periodic wave solutions expressed by various Jacobi elliptic functions for Wick-type stochastic and variable coefficient NLS equations, respectively. In the limit cases, the soliton-like wave solutions are showed as well. Since Wick versions of functions are usually difficult to evaluate, we get some nonWick versions of the solutions for Wick-type stochastic NLS equations in special cases. 相似文献
13.
Second-order stochastic partial differential equations of parabolic type are considered. Generalizations of known theorems on existence, uniqueness and on approximations are presented. Thus, in particular the case of unbounded coefficients in investigated. Some examples illustrating the usefulness of the results are also given. 相似文献
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15.
Zdzisław Brzeźniak 《Potential Analysis》1995,4(1):1-45
We study abstract stochastic evolution equations in M-type 2 Banach spaces. Applications to stochastic partial differential equations inL
p
spaces withp2 are given. For example, solutions of such equations are Hölder continuous in the space variables.The author is an Alexander von Humboldt Stiftung fellow 相似文献
16.
In this paper we study the well-posedness and regularity of the adapted solutions to a class of linear, degenerate backward stochastic partial differential equations (BSPDE, for short). We establish new
a priori estimates for the adapted solutions to BSPDEs in a general setting, based on which the existence, uniqueness, and regularity
of adapted solutions are obtained. Also, we prove some comparison theorems and discuss their possible applications in mathematical
finance.
Received: 24 September 1997 / Revised version: 3 June 1998 相似文献
17.
Krystyna Twardowska 《Acta Appl Math》1996,43(3):317-359
The aim of this paper is to give a wide introduction to approximation concepts in the theory of stochastic differential equations. The paper is principally concerned with Zong-Zakai approximations. Our aim is to fill a gap in the literature caused by the complete lack of monographs on such approximation methods for stochastic differential equations; this will be the objective of the author's forthcoming book. First, we briefly review the currently-known approximation results for finite- and infinite-dimensional equations. Then the author's results are preceded by the introduction of two new forms of correction terms in infinite dimensions appearing in the Wong-Zakai approximations. Finally, these results are divided into four parts: for stochastic delay equations, for semilinear and nonlinear stochastic equations in abstract spaces, and for the Navier-Stokes equations. We emphasize in this paper results rather than proofs. Some applications are indicated.The author's research was partially supported by KBN grant No. 2 P301 052 03. 相似文献
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19.
In this article we prove new results concerning the long-time behavior of random fields that are solutions in some sense
to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state
that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non-random
global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that
the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that
they set out to oscillate between them. In the first case we can also determine exactly the corresponding Lyapunov exponents.
In the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two
asymptotic states in a finite time with probability one. In both cases we also interpret our results in terms of stability
properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods
of the asymptotic states. Our methods of proof rest upon the use of a suitable regularization of the Brownian motion along
with a related Wong-Zaka? approximation procedure.
Received: 8 April 1997/Revised version: 30 January 1998 相似文献
20.
We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established.
Numerical examples are also presented to examine our theoretical results.
This research is supported by Air Force Office of Scientific Research under the grant number FA9550-05-1-0133 and 985 Project
of Jilin University. 相似文献