An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.     

The Ostrovsky–Vakhnenko equation: A Riemann–Hilbert approach

We present an inverse scattering transform approach for the (differentiated) Ostrovsky–Vakhnenko equation:

u_txx−3u_x+3u_xu_xx+uu_xxx=0.$u_{txx}-3u_x+3u_x{u}_{xx}+u{u}_{xxx}=0.$

This equation can also be viewed as the short-wave model for the Degasperis–Procesi equation. The approach is based on an associated Riemann–Hilbert problem, which allows us to give a representation for the classical (smooth) solution of the Cauchy problem, to get the principal term of its long-time asymptotics, and also to find, in a natural way, loop soliton solutions.     

Singular ergodic control for multidimensional Gaussian–Poisson processes

Singular control for multidimensional Gaussian–Poisson processes with a long-run (or ergodic) and a discounted criteria is discussed. The corresponding Hamilton–Jacobi–Bellman equations are discussed. Complete details on the proofs and further extensions are left for future works.     

Differentiating σ-fields for Gaussian and shifted Gaussian processes

We study the notions of differentiating and non-differentiating σ-fields in the general framework of (possibly drifted) Gaussian processes and characterize their invariance properties, when changing to an equivalent probability measure. As an application, we investigate the class of stochastic derivatives associated with shifted fractional Brownian motions. We finally establish conditions for the existence of a jointly measurable version of the differentiated process and we outline a general framework for stochastic embedded equations.     

An iterative formula for the Kostka–Foulkes polynomials

Journal of Algebraic Combinatorics - An iterative formula for the Kostka–Foulkes polynomials is given using the vertex operator realization of the Hall–Littlewood polynomials. The...     

Sobolev derivatives and Itô decomposition formula for Gaussian processes using properties of their RKHSs

Malliavin calculus has been extensively developed for abstract Wiener spaces. It is of interest to develop the basic concepts of an infinite-dimensional calculus for an arbitrary Gaussian processX=(X_t), wheret T (T being a multiparameter set or, more generally, a complete separable metric space), bringing into evidence the properties of the covariance kernel (or, equivalently, the reproducing kernel Hilbert space) ofX. In this paper a definition of thekth Sobolev derivative is given and thekth chaos expansion of a functional is shown to be thekth-order divergence operator. An extension of Itô's decomposition formula is derived.     

A Bass–Heller–Swan formula for pseudoisotopies

The Bass–Heller–Swan formula is a basic calculational tool in pseudoisotopy K-theory. We describe the Nil-groups and the Bass–Heller–Swan splitting for the group of the pseudoisotopies of a closed manifold. We use the methods of controlled topology used in the Bass–Heller–Swan splitting in K-theory.     

A Nekrasov–Okounkov formula for Macdonald polynomials

We construct the irreducible unipotent modules of the finite general linear groups from actions on tableaux. Our approach is analogous to that of James (Bull Lond Math Soc 8:229–232, 1976) for the symmetric groups, answering an open question as to whether such a construction exists. We show that our modules are isomorphic to those previously constructed by James (Representations of general linear groups, London Mathematical Society Lecture Note Series, vol. 94. Cambridge University Press, Cambridge, 1984. doi: 10.1017/CBO9780511661921) , although the two presentations are quite different. Key to our construction are the generalized Gelfand–Graev representations of Kawanaka (Generalized Gel’fand-Graev representations and Ennola duality. In: Algebraic groups and related topics (Kyoto/Nagoya, 1983), advanced studies in pure math., vol. 6, pp. 175–206. North-Holland, Amsterdam 1985).     

Wick–Itô formula for regular processes and applications to the Black and Scholes formula

We derive a Wick–Itô formula, that is, an Itô-type formula based on Wick integration. We derive it in the context of regular Gaussian processes which include Brownian motion and fractional Brownian motion with Hurst parameter greater than 1/2. We then consider applications to the Black and Scholes formula for the pricing of a European call option. It has been shown that using Wick integration in this context is problematic for economic reasons. We show that it is also problematic for mathematical reasons because the resulting Black and Scholes formula depends only on the variance of the process and not on its dependence structure.     

A Riemann–Hilbert problem for skew-orthogonal polynomials

We find a local (d+1)×(d+1)$(d+1)\times (d+1)$ Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d  . Our Riemann–Hilbert problem is similar to a local d×d$d\times d$ Riemann–Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann–Hilbert problems, and brings us closer to finding full asymptotic expansions of the skew-orthogonal polynomials.     

Riemann–Hilbert approach for the Camassa–Holm equation on the line

We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation $u_t?u_{txx}+2\omega u_x+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}$ on the line (CH). We show that: (i) for all $\omega >0$, the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann–Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for $\omega =0$. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A Feynman–Kac formula for stochastic Dirichlet problems

A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Itô sense.     

An MPCC approach for stochastic Stackelberg–Nash–Cournot equilibrium

In this article, we investigate a Stochastic Stackelberg–Nash–Cournot Equilibrium problem by reformulating it as a Mathematical Program with Complementarity Constraints (MPCC). The complementarity constraints are further reformulated as a system of nonsmooth equations. We characterize the followers’ Nash–Cournot equilibria by studying the implicit solution of a system of equations. We outline numerical methods for the solution of a stochastic Stackelberg–Nash–Cournot Equilibrium problem with finite distribution of market demand scenarios and propose a discretization approach based on implicit numerical integration to deal with stochastic Stackelberg–Nash–Cournot Equilibrium problem with continuous distribution of demand scenarios. Finally, we discuss the two-leader Stochastic Stackelberg–Nash–Cournot Equilibrium problem.     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.