Fractional Brownian Motion and Sheet as White Noise Functionals

In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and Фksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed.     

Existence and Uniqueness of Solution for a Fourth Order Differential－Difference Equation

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On a Cameron–Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

We present a Cameron–Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion.     

Existence of Positive Solutions to Fourth Order Quasilinear Boundary Value Problems

In this paper, we generalize the fixed point theorem of cone expansion and compression of norm type to the theorem of functional type. As an application, the existence of positive solutions for some fourth-order beam equation boundary value problems is obtained. The emphasis is put on that the nonlinear term is dependent on all lower order derivatives.     

Generalized Γ Calculus and Application to Interacting Particles on a Graph

Potential Analysis - The classical Bakry-Émery calculus is extended to study, for degenerated (non-elliptic, non-reversible, or non-diffusive) Markov processes, questions such as...     

An abstract Nash–Moser theorem with parameters and applications to PDEs

We prove an abstract Nash–Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the “tame” estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large “clusters of small divisors”, due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.     

Spectral-Parameter Dependent Yang–Baxter Operators and Yang–Baxter Systems From Algebra Structures

For any algebra, two families of colored Yang–Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang–Baxter equation. An open problem about a system of functional equations is stated. The matrix forms of these operators for two and three dimensional algebras are computed. A FRT bialgebra for one of these families is presented. Solutions for the one-parameter quantum Yang–Baxter equation are derived and a Yang–Baxter system constructed.     

Forward–backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs

We obtain an existence and uniqueness theorem for fully coupled forward–backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al. (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial–boundary value problem for non-local quasilinear parabolic second-order PDEs.     

An Itô Formula of Generalized Functionals and Local Time for Fractional Brownian Sheet

Abstract

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H ₁, H ₂ ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.     

From Sturm–Liouville problems to fractional and anomalous diffusions

Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those equations, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one’s efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm–Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of Bochner’s subordination rule and we establish some connections between subordination and space-fractional operators.     

Rank–Crank-type PDEs and generalized Lambert series identities

We show how Rank–Crank-type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank–Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson, respectively. The first author’s Lambert series identity is a common generalization. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic functions can be extended to prove these generalized Lambert series identities.     

Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics

We consider a mixed Brownian–fractional-Brownian model of a financial market. The class of self-financing strategies is restricted to Markov-type smooth functions. It is proved that such strategies satisfy a parabolic equation that can be reduced to heat equation. Then it is proved that the mixed model is arbitrage-free. Finally, the capital of the model is presented as the limit of a sequence of semimartingales.     

Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation

This paper studies forward and backward versions of the random Burgers equation (RBE) with stochastic coefficients. First, the celebrated Cole–Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of FBSDEs. Exploiting this connection, we derive a generalization of the Cole–Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman–Kac representations for the solutions are provided. Explicit solutions are constructed and applications in stochastic control and mathematical finance are discussed.     

Recurrency and Transiency of Markov Processes Governed by Coupled Systems of Second–Order Elliptic Operators

Markov processes corresponding to coupled systems of the form k=1,2,...,m, are considered. Here L_k are second–order linear elliptic operators and d_k,j are non–negative functions. We prove sufficient conditions for the recurrency and transiency of the Markov processes corresponding to such systems and alos provide some examples which show that it is possible to have a recurrent system even if some or all its components are transient; where by components, we mean the Markov processes engendered by the uncoupled equations. We also provide all example which shows that a system composed of recurrent processes may itself be transient. Finally, we consider necessary and sufficient conditions for the exterior Dirichlet problem of the coupled system to have a bounded solution as.     

Local Functional Law of the Iterated Logarithm for Increments of a Brownian Sheet北大核心CSCD

本文利用Brown单与Brown单增量的大偏差,得到了Brown单与Brown单增量的局部泛函重对数律.