Elements of stochastic analysis for the case of Grassmann variables. III. Correlation functions

The present paper continues earlier studies [1,2], in which analogs were proposed in the case of Grassmann variables for concepts such as classical stochastic analysis, stochastic integrals, random processes, and stochastic partial differential equations and their solutions. This was done for the special case when the classical objects are functionals of a so-called smoothed Wiener process onR ₊×R ^v . In the present paper, the correlation functions of the solution of a stochastic partial differential equation are studied together with some applications.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 323–335, December, 1993.     

ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS

This article concerns the construction of approximate solutions for a general stochastic integrodifferential equation which is not explicitly solvable and whose coeffcients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L^p-th norm, uniformly on the time interval. The convergence with probability one is also given.     

Stochastic partial differential equations for some measure-valued diffusions

Summary We consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R ¹, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.     

Large Deviations for Stochastic Volterra Equations in the Plane

We prove a Large Deviation Principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof uses the method developped by Azencott for diffusion processes. The main ingredients are exponential inequalities for different classes of two-parameter stochastic integrals; these integrals are related to the representation of the stochastic term in the differential equation as a representable semimatringale.     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

Backward SDEs and Cauchy problem for semilinear equations in divergence form

 We extend the definition of solutions of backward stochastic differential equations to the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces. Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002 Research supported by KBN Grant 0253 P03 2000 19. Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55 Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Anticipated backward stochastic differential equations with non-Lipschitz coefficients

This paper deals with a class of anticipated backward stochastic differential equations. We extend results of Peng and Yang (2009) to the case in which the generator satisfies non-Lipschitz condition. The existence and uniqueness of solutions for anticipated backward stochastic differential equations as well as a comparison theorem are obtained. The existence and uniqueness of L^p(p>2) solutions for anticipated backward stochastic differential equations are also studied.     

Mean-field backward stochastic differential equations driven by G-Brownian motion and related partial differential equations

In this paper, we study mean-field backward stochastic differential equations driven by G-Brownian motion (G-BSDEs). We first obtain the existence and uniqueness theorem of these equations. In fact, we can obtain local solutions by constructing Picard contraction mapping for Y term on small interval, and the global solution can be obtained through backward iteration of local solutions. Then, a comparison theorem for this type of mean-field G-BSDE is derived. Furthermore, we establish the connection of this mean-field G-BSDE and a nonlocal partial differential equation. Finally, we give an application of mean-field G-BSDE in stochastic differential utility model.     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675     

Développments terminaux des graphes infinis III. Arbres maximaux sans rayon,cardinalité maximum des ensembles disjoints de rayons

A class of FELLER 's one-dimemsional continuous strong MARKOV processes generated by the generalized second order differential operator D_mD₈⁺ is considered. In the case of natural boundaries of the state space ?? and an identical road map s(x) = x, these diffusion processes are martingales. In a first part of this note some earlier results concerning the representation of such processes as weak solutions of stochastic differential equations are improved. The second part concerns with diffusions absolutely continuous with respect to a given one, determined by the generator D_mD. Such absolutely continuous diffusions on the line were first described analytically by S. OREY in terms of the corresponding speed measures and road maps. By the aid of the derived stochastic equations an explicit expression for the corresponding RADON -NIKODYM derivatives is possible. This allows a characterization of diffusions with non-identical scale functions by stochastic differential equations.     

Sur l'existence de solutions d'équations différentielles stochastiques progréssives rétrogrades couplées

Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamadème, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amongst others.

Unfortunately, most existence or uniqueness results on solutions of forward–backward stochastic differential equations need regularity assumptions. The coefficients are required to be at least continuous which is somehow too strong in some applications. To the best of our knowledge, our work is the first to prove existence of a solution of a forward–backward stochastic differential equation with discontinuous coefficients and degenerate diffusion coefficient where, moreover, the terminal condition is not necessary bounded.

The aim of this work is to find a solution of a certain class of forward–backward stochastic differential equations on an arbitrary finite time interval. To do so, we assume some appropriate monotonicity condition on the generator and drift coefficients of the equation.

The present paper is motivated by the attempt to remove the classical condition on continuity of coefficients, without any assumption as to the non-degeneracy of the diffusion coefficient in the forward equation.

The main idea behind this work is the approximating lemma for increasing coefficients and the comparison theorem. Our approach is inspired by recent work of Boufoussi and Ouknine, 2003, On a SDE driven by a fractional brownian motion and with monotone drift, Electronic Communications in Probability, 8, 122–134; combined with that of Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569. Pursuing this idea, we adopt a one-dimensional framework for the forward and backward equations and we assume a monotonicity property both for the drift and for the generator coefficient.

At the end of the paper we give some extensions of our result.     

A relative compactness criterion in Wiener–Sobolev spaces and application to semi-linear Stochastic PDEs

We prove a relative compactness criterion in Wiener–Sobolev space which represents a natural extension of the compact embedding of Sobolev space H¹ into , at the level of random fields. Then we give a specific statement of this criterion for random fields solutions of semi-linear stochastic partial differential equations with coefficients bounded in an appropriate way. Finally, we employ this result to construct solutions for semi-linear stochastic partial differential equations with distribution as final condition. We also give a probabilistic interpretation of this solution in terms of backward doubly stochastic differential equations formulated in a weak sense.     

Qp-valued jump processes associated with linear and nonlinear pseudo-differential equations

We consider stochastic differential equations in the space Q_p of p-adic numbers associated with nonlinear pseudo-differential equations and systems. We construct both scalar and matrix-valued multiplicative functionals of Q_p-valued Markov processes and derive probabilistic representations of solutions to the Cauchy problem for nonlinear pseudo-differential equations and systems which extend the Feynman-Kac formula. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 5–37.     

Exceptional symmetry reductions of Burgers' equation in two and three spatial dimensions

Lie point symmetry analysis of the general class of nonlinear diffusion-convection equations in two and three dimensions has shown that only for Burgers' equation (that isD(u)=const,K(u)=quadratic) is a full symmetry reduction to an ordinary differential equation possible. The optimal system of symmetry operators is determined to ensure that a minimal complete set of reductions is obtained. For each reduced partial differential equation, classical Lie group analysis has been performed and further reductions obtained. In this manner, all possible reductions to an ordinary differential equation are found, leading to exact solutions to both the two and three dimensional Burgers' equation.     

Stochastic partial differential equations in M-type 2 Banach spaces

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     

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Quantum stochastic differential equations of the form

L p -Theory of Parabolic SPDE s Degenerating on the Boundary of C 1 Domains

Degenerate stochastic partial differential equations are considered in C ¹ domains. Existence and uniqueness results are given in weighted Sobolev spaces, and H?lder estimates of the solutions are presented. This work was supported by a Korea University grant.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Representation theorems for generators of BSDEs in L p spaces

In this paper,we prove that the generator g of a class of backward stochastic differential equations (BSDEs) can be represented by the solutions of the corresponding BSDEs at point (t,y,z),when the terminal data is in L p spaces,for 1 < p ≤ 2.