On the Wiener-Hopf factorization for Levy processes with bounded positive jumps

We study the Wiener-Hopf factorization for Levy processes with bounded positive jumps and arbitrary negative jumps. Using the results from the theory of entire functions of Cartwright class we prove that the positive Wiener-Hopf factor can be expressed as an infinite product in terms of the solutions to the equation $\psi(z)=q$, where $\psi$ is the Laplace exponent of the process. Under some additional regularity assumptions on the Levy measure we obtain an asymptotic expression for these solutions, which is important for numerical computations. In the case when the process is spectrally negative with bounded jumps, we derive a series representation for the scale function in terms of the solutions to the equation $\psi(z)=q$. To illustrate possible applications we discuss the implementation of numerical algorithms and present the results of several numerical experiments.Comment: 29 pages, 4 figure

Generating functions for Wilf equivalence under generalized factor order

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that the $i$-th character of $v$ is greater than or equal to the $i$-th character of $u$ for all $i$. This subword $v$ is called an embedding of $u$ into $w$. For the case where $P$ is the positive integers with the usual ordering, they defined the weight of a word $w = w_1\ldots w_n$ to be $\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}$, and the corresponding weight generating function $F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)$. They then defined two words $u$ and $v$ to be Wilf equivalent, denoted $u \backsim v$, if and only if $F(u;t,x) = F(v;t,x)$. They also defined the related generating function $S(u;t,x) = \sum_{w \in \mathcal{S}(u)} \text{wt}(w)$ where $\mathcal{S}(u)$ is the set of all words $w$ such that the only embedding of $u$ into $w$ is a suffix of $w$, and showed that $u \backsim v$ if and only if $S(u;t,x) = S(v;t,x)$. We continue this study by giving an explicit formula for $S(u;t,x)$ if $u$ factors into a weakly increasing word followed by a weakly decreasing word. We use this formula as an aid to classify Wilf equivalence for all words of length 3. We also show that coefficients of related generating functions are well-known sequences in several special cases. Finally, we discuss a conjecture that if $u \backsim v$ then $u$ and $v$ must be rearrangements, and the stronger conjecture that there also must be a weight-preserving bijection $f: \mathcal{S}(u) \rightarrow \mathcal{S}(v)$ such that $f(u)$ is a rearrangement of $u$ for all $u$.Comment: 23 page

Wiener-Hopf factorization and distribution of extrema for a family of L\'{e}vy processes

In this paper we introduce a ten-parameter family of L\'{e}vy processes for which we obtain Wiener-Hopf factors and distribution of the supremum process in semi-explicit form. This family allows an arbitrary behavior of small jumps and includes processes similar to the generalized tempered stable, KoBoL and CGMY processes. Analytically it is characterized by the property that the characteristic exponent is a meromorphic function, expressed in terms of beta and digamma functions. We prove that the Wiener-Hopf factors can be expressed as infinite products over roots of a certain transcendental equation, and the density of the supremum process can be computed as an exponentially converging infinite series. In several special cases when the roots can be found analytically, we are able to identify the Wiener-Hopf factors and distribution of the supremum in closed form. In the general case we prove that all the roots are real and simple, and we provide localization results and asymptotic formulas which allow an efficient numerical evaluation. We also derive a convergence acceleration algorithm for infinite products and a simple and efficient procedure to compute the Wiener-Hopf factors for complex values of parameters. As a numerical example we discuss computation of the density of the supremum process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP673 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Soliton Content of Self Dual Yang-Mills Equations

Exploiting the formulation of the Self Dual Yang-Mills equations as a Riemann-Hilbert factorization problem, we present a theory of pulling back soliton hierarchies to the Self Dual Yang-Mills equations. We show that for each map \C^4 \to \C^{\infty } satisfying a simple system of linear equations formulated below one can pull back the (generalized) Drinfeld-Sokolov hierarchies to the Self Dual Yang-Mills equations. This indicates that there is a class of solutions to the Self Dual Yang-Mills equations which can be constructed using the soliton techniques like the $\tau$ function method. In particular this class contains the solutions obtained via the symmetry reductions of the Self Dual Yang-Mills equations. It also contains genuine 4 dimensional solutions . The method can be used to study the symmetry reductions and as an example of that we get an equation exibiting breaking solitons, formulated by O. Bogoyavlenskii, as one of the $2 + 1$ dimensional reductions of the Self Dual Yang-Mills equations.Comment: 11 pages, plain Te

Cluster algebras and Poisson geometry

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over real numbers.Comment: minor change