Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model

Consider an insurer who is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same asymptotic formula holds for the finite-time and infinite-time ruin probabilities. Restricting our attention to the so-called constant investment strategy, we show how the insurer adjusts his investment portfolio to maximize the expected terminal wealth subject to a constraint on the ruin probability.     

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R₊ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.     

Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem

In this paper we study a class of nonlinearities for which a nonlocal parabolic equation with Neumann-Robin boundary conditions, for p-Laplacian, has finite time blow-up solutions.     

Stochastic processes with proportional increments and the last-arrival problem

The notion of stochastic processes with proportional increments is introduced. This notion is of general interest as indicated by its relationship with several stochastic processes, as counting processes, Lévy processes, and others, as well as martingales related with these processes. The focus of this article is on the motivation to introduce processes with proportional increments, as instigated by certain characteristics of stopping problems under weak information. We also study some general properties of such processes. These lead to new insights into the mechanism and characterization of Pascal processes. This again will motivate the introduction of more general f-increment processes as well as the analysis of their link with martingales. As a major application we solve the no-information version of the last-arrival problem   which was an open problem. Further applications deal with the impact of proportional increments on modelling investment problems, with a new proof of the 1/e$1/e$-law of best choice, and with other optimal stopping problems.     

On the Dirichlet problem

A particular case of the Dirichlet problem is solved using the Convergence Theorem for discrete-time martingales and the mean value property of harmonic functions as the main tools.     

Convergence of the increments of a stochastic integral associated to the stochastic wave equation

We study a type of one-dimensional wave equation on the plane with non-linear random forcing. We are interested in the almost sure behaviour of the normalized increments of the solution process associated to this type of wave equation. Also we study the behaviour of the normalized increments of some other stochastic integral equation.     

Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups

This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size a$a$. As a consequence, we obtain the probabilities that a drawdown of size a$a$ precedes a drawup of size b$b$ and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein–Uhlenbeck process, and Cox–Ingersoll–Ross process.     

A stochastic heat equation with the distributions of Lévy processes as its invariant measures

We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation.     

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.     

Weakly dependent chains with infinite memory

We prove the existence of a weakly dependent strictly stationary solution of the equation _Xt=F(_Xt−1,_Xt−2,_Xt−3,…;_ξt)$X_t=F(X_{t-1},X_{t-2},X_{t-3},\dots ;{\xi }_t)$ called a chain with infinite memory. Here the innovations  _ξt${\xi }_t$ constitute an independent and identically distributed sequence of random variables. The function F$F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments, the rate of decay of the Lipschitz coefficients of the function F$F$ and the weak dependence properties. From these weak dependence properties, we derive strong laws of large number, a central limit theorem and a strong invariance principle.     

Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

Let F$F$ be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F$F$. The expansion is based on an expansion for the right Wiener–Hopf factor which we derive first. An application to ruin probabilities is developed.     

Atomic decompositions of Banach lattice-valued martingales

In this paper, atomic decompositions of Banach lattice-valued martingales are given. We discuss the relation between the LERMT property and atomic decompositions. With the help of atomic decompositions, the relation of the martingale spaces is investigated.     

Steiner polygons in the Steiner problem

A polygon, whose vertices are points in a given setA ofn points, is defined to be a Steiner polygon ofA if all Steiner minimal trees forA lie in it. Cockayne first found that a Steiner polygon can be obtained by repeatedly deleting triangles from the boundary of the convex hull ofA. We generalize this concept and give a method to construct Steiner polygons by repeatedly deletingk-gons,k n. We also prove the uniqueness of Steiner polygons obtained by our method.     

On the distribution tail of an integrated risk model: A numerical approach

We consider an insurance risk process with the possibility to invest the capital reserve into a portfolio consisting of a risky asset and a riskless asset. The stock price is modelled by an exponential Lévy process and the riskless interest rate is assumed to be constant. We aim at the risk assessment of the integrated risk process in terms of a high quantile or the far out distribution tail. We indicate an application to an optimal investment strategy of an insurer.     

The tug-of-war without noise and the infinity Laplacian in a wedge

Consider the ending time of the tug-of-war without noise in a wedge. There is a critical angle for finiteness of its expectation when player I maximizes the distance to the boundary and player II minimizes the distance. There is also a critical angle such that for smaller angles, player II can find a strategy where the expected ending time is finite, regardless of player I’s strategy. For larger angles, for each strategy of player II, player I can find a strategy making the expected ending time infinite. Using connections with the inhomogeneous infinity Laplacian, we bound this critical angle.     

Optimal proportional reinsurance and investment under partial information

In this paper, we study the optimal proportional reinsurance and investment strategy for an insurer that only has partial information at its disposal, under the criterion of maximizing the expected utility of the terminal wealth. We assume that the surplus of the insurer is governed by a jump diffusion process, and that reinsurance is used by the insurer to reduce risk. In addition, the insurer can invest in financial markets. We give a characterization for the optimal strategy within a non-Markovian setting. Malliavin calculus for Lévy processes is used for the analysis.     

Fluctuations of the front in a stochastic combustion model

We consider an interacting particle system on the one-dimensional lattice Z modeling combustion. The process depends on two integer parameters 2?a?M<∞. Particles move independently as continuous time simple symmetric random walks except that (i) when a particle jumps to a site which has not been previously visited by any particle, it branches into a particles, (ii) when a particle jumps to a site with M particles, it is annihilated. We start from a configuration where all sites to the left of the origin have been previously visited and study the law of large numbers and central limit theorem for rt, the rightmost visited site at time t. The proofs are based on the construction of a renewal structure leading to a definition of regeneration times for which good tail estimates can be performed.     

Explicit characterization of the super-replication strategy in financial markets with partial transaction costs

We consider a continuous time multivariate financial market with proportional transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi [B. Bouchard, N. Touzi, Explicit solution of the multivariate super-replication problem under transaction costs, The Annals of Applied Probability 10 (3) (2000) 685–708] except that some of the assets can be exchanged freely, i.e. without paying transaction costs. In this context, we generalize the result of the above paper and prove that the super-replication price is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time. Our proof relies on the introduction of a new auxiliary control problem whose value function can be interpreted as the super-hedging price in a model with unbounded stochastic volatility (in the directions where transaction costs are non-zero). In particular, it confirms the usual intuition that transaction costs play a similar role to stochastic volatility.     

Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for a semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle, the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We also prove regularity properties and approximation results for solutions of the problem.     

On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity

(P)