Dual representations for general multiple stopping problems

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by integer valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers (2010), Bender (2011a), Bender (2011b), Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cashflow structures than the additive structure in the above references. For example some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for the price of multiple exercise options and exemplify it by a numerical study on the pricing of a swing option in an electricity market.Comment: This is an updated version of WIAS preprint 1665, 23 November 201

Wigner chaos and the fourth moment

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer-Major theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AOP657 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Measure and integral : new foundations after one hundred years

The present article aims to describe the main ideas and developments in the theory of measure and integral in the course and at the end of the first century of its existence

Frobenius actions on the de Rham cohomology of Drinfeld modules

We study the action of endomorphisms of a Drinfeld A-module \phi on its de Rham cohomology H_{DR}(\phi,L) and related modules, in the case where \phi is defined over a field L of finite A-characteristic \mathfrak{p}. Among others, we find that the nilspace H_{0} of the total Frobenius Fr_{DR} on H_{DR}(\phi,L) \mathfrak{p} has dimension h = height of \phi. We define and study a pairing between the \mathfrak{p}-torsion _{\mathfrak{p}}\phi of \phi and H_{DR}(\phi,L), which becomes perfect after dividing out H_{0}

On the torsion of optimal elliptic curves over function fields

For an optimal elliptic curve E over \mathbb{F}_{q}(t) of conductor \mathfrak{p}\cdot\infty, where \mathfrak{p} is prime, we show that E(F)_{tor} is generated by the image of the cuspidal divisor group

A formula for the probability of the exponents of finite p-groups

In this paper, I will introduce a link between the volume of a finite p-group in the Cohen-Lenstra measure and partitions of a certain type. These partitions will be classified by the output of an algorithm. As a corollary, I will give a formula for the probability of a p-group to have a specific exponent

Estimates of the second-order derivatives for solutions to the two-phase parabolic problem

The L_{\infty}-estimates of the second derivatives for solutions of the parabolic free boundary problem with two phases \Delta u-\partial_{t}u=\lambda^{+}\chi_{\left\{ u>0\right\} }-\lambda^{-}\chi_{\left\{ u<0\right\} }\textrm{in }B_{1}^{+}\times]-1,0],\textrm{ }\lambda^{\pm}\geq0,\lambda^{+}+\lambda^{-}>0, satisfying the non-zero Dirichlet condition on \Pi_{1}:=\left\{ (x,t):\left|x\right|\leq1,x_{1}=0,-1<t\leq0\right\}, are proved

Partial regularity for local minimizers of splitting-type variational integrals

We consider local minimizers u:\mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{N} of anisotropic variational integrals of (p,q)-growth with exponents 2\leq p\leq q\leq\mbox{min}\left\{ 2+p,p\frac{n}{n-2}\right\}. If the integrand is of splitting-type, then partial C^{1}-regularity of u is established