Showing papers in "Stochastics An International Journal of Probability and Stochastic Processes in 2008"

Risk minimizing portfolios and HJBI equations for stochastic differential games

Abstract: In this paper, we consider the problem to find a market portfolio that minimizes the convex risk measure of the terminal wealth in a jump diffusion market. We formulate the problem as a two player (zero-sum) stochastic differential game. To help us find a solution, we prove a theorem giving the Hamilton–Jacobi–Bellman–Isaacs (HJBI) conditions for a general zero-sum stochastic differential game in a jump diffusion setting. We then use the theorem to study particular risk minimization problems. Finally, we extend our approach to cover general stochastic differential games (not necessarily zero-sum), and we obtain similar HJBI equations for the Nash equilibria of such games.

The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion

Abstract: The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index H>1/2 can be estimated by , where δ is the diameter of partition used for discretization. For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is .

A global construction of homogeneous random planar tessellations that are stable under iteration

Abstract: Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are constructed which have the characteristic property to be stable under the operation of iteration (or nesting), STIT for short. It is based on a Poisson point process on the space of lines that are endowed with a time of birth. A new approach is presented that describes the tessellation in the whole plane. So far, an explicit geometrical construction for those tessellations was only known within bounded windows.

Predicting the Last Zero of Brownian Motion with Drift

Abstract: Given a standard Brownian motion with drift μ ∈ IR and letting g denote the last zero of before T, we consider the optimal prediction problem where the infimum is taken over all stopping times τ of B μ. Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we show that the following stopping time is optimal: where the function t↦b − (t) is continuous and increasing on [0, T] with b − (T) = 0, the function t↦b +(t) is continuous and decreasing on [0, T] with b +(T) = 0, and the pair b − and b + can be characterised as the unique solution to a coupled system of nonlinear Volterra integral equations. This also yields an explicit formula for V * in terms of b − and b +. If μ = 0 then and there is a closed form expression for b ± as shown in Shiryaev (Theory Probab. Appl. in press) using the method of time change from Graversen et al. (2001, Theory Probab. Appl. 45, 125–136). The latter method cannot be extended to the case when ...

Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws

Abstract: We attempt to bring some modest unity to three subareas of heavy tail analysis and extreme value theory: limit laws for componentwise maxima of iid random variables; hidden regular variation and asymptotic independence; conditioned limit laws when one component of a random vector is extreme. The common theme is multivariate regular variation on a cone and the three cases cited come from specifying the cones and [0, ∞] × (0, ∞].

Sequential multi-hypothesis testing for compound Poisson processes

Abstract: Suppose that there are finitely many simple hypotheses about the unknown arrival rate and mark distribution of a compound Poisson process, and that exactly one of them is correct The objective is to determine the correct hypothesis with minimal error probability and as soon as possible after the observation of the process starts This problem is formulated in a Bayesian framework, and its solution is presented Provably convergent numerical methods and practical near-optimal strategies are described and illustrated on various examples

Optimal control of stochastic functional differential equations with a bounded memory

Abstract: This paper treats a finite time horizon optimal control problem in which the controlled state dynamics are governed by a general system of stochastic functional differential equations with a bounded memory. An infinite dimensional Hamilton–Jacobi–Bellman (HJB) equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.

Robust Optimization of Consumption with Random Endowment

Abstract: We consider the problem of optimal consumption for an investor who is risk and uncertainty averse. We model these preferences of the investor with the help of a convex risk-measure. Apart from consumption the agent has the possibility to invest initial capital and random endowment in a market where stock-prices are semimartingales. We formulate this as a maximin problem that will be solved by duality methods.

Martingales and first passage times of AR(1) sequences

Abstract: Using the martingale approach we find sufficient conditions for exponential boundedness of first passage times over a level for ergodic first order autoregressive sequences. Further, we prove a martingale identity to be used in obtaining explicit bounds for the expectation of first passage times.

Nearest-neighbour modelling of reciprocal chains

Abstract: This paper focuses on the class of finite-state, discrete-index, reciprocal processes (reciprocal chains). Such a class of processes seems to be a suitable setup in many applications and, in particular, it appears well-suited for image-processing. While addressing this issue, the aim is 2-fold: theoretic and practical. As to the theoretic purpose, some new results are provided: first, a general stochastic realization result is provided for reciprocal chains endowed with a known, arbitrary, distribution. Such a model has the form of a fixed-degree, nearest-neighbour polynomial model. Next, the polynomial model is shown to be exactly linearizable, which means it is equivalent to a nearest-neighbour linear model in a different set of variables. The latter model turns out to be formally identical to the Levi–Frezza–Krener linear model of a Gaussian reciprocal process, although actually non-linear with respect to the chain's values. As far as the practical purpose is concerned, in order to yield an example of ...

Stability of stochastic jump-parameter semi-Markov linear systems of differential equations

Abstract: The asymptotic stability of stochastic Ito-type jump-parameter semi-Markov systems of linear differential equations is examined. A system of integral matrix equations is derived which has the property that the existence of a positive definite solution of the system implies the asymptotic stability of the stochastic semi-Markov system. Finally, an illustrative example is presented.

A note on the supremum of a stable process

Abstract: If X is a spectrally positive stable process of index α ∈ (1, 2) whose Levy measure has density on (0,∞), and it is known that as x → ∞. It is also known that S 1 has a continuous density, s say. The point of this note is to show that as x → ∞.

Expectations and martingales in metric spaces

Abstract: We develop a theory of martingales with values in metric spaces of nonpositive curvature. Our main results state existence of (continuous-time) martingales with given terminal data X T . These processes will be constructed via time discretization. The notion of discrete-time martingale is based on the concept of iterated conditional barycenters. Moreover, in more specific cases we present martingale characterizations.

On Financial Markets Based on Telegraph Processes

Abstract: The paper develops a new class of financial market models. These models are based on generalised telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black–Scholes fundamental differential equation is derived, but, in contrast with the Black–Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging.

Discrete approximation of stochastic integrals with respect to fractional Brownian motion of Hurst index H>1/2

Abstract: In this paper we provide a discrete approximation for the stochastic integral with respect to the fractional Brownian motion of Hurst index H>1/2 defined in terms of the divergence operator. To determine the suitable class of integrands for which the approximation holds, we also investigate the relations among the spaces of Malliavin differentiable processes in the fractional and standard case.

Perpetual convertible bonds with credit risk

Abstract: A convertible bond is a security that the holder can convert into a specified number of underlying shares. We enrich the standard model by introducing some default risk of the issuer. Once default has occured payments stop immediately. In the context of a reduced form model with infinite time horizon driven by a Brownian motion, analytical formulae for the no-arbitrage price of this American contingent claim are obtained and characterised in terms of solutions of free boundary problems. It turns out that the default risk changes the structure of the optimal stopping strategy essentially. Especially, the continuation region may become a disconnected subset of the state space.

Statistical analysis of redundant systems with ‘warm’ stand-by units

Abstract: Mathematical formulation of fluent switching from ‘warm’ to ‘hot’ conditions of standby units is given using the well known Sedyakin's and accelerated failure time models. Non-parametric estimators of cumulative distribution function and mean failure time of a redundant system with several stand-by units are proposed. Goodness-of-fit tests for two given models are given.

Itô's formula for linear fractional PDEs

Abstract: In this paper we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Ito-type formula for the process X.

Tauberian theorems and large deviations

Abstract: The link between Tauberian theorems and large deviations is surveyed, with particular reference to regular variation.

Estimates for first exit times of non-Markovian Itô processes

Abstract: First exit times and their path-wise dependence on trajectories are studied for non-Markovian Ito processes. Estimates of distances between two exit times are obtained. In particular, it follows that first exit times of two Ito processes are close if their trajectories are close.

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

Abstract: We derive a Wick–Ito formula, that is, an Ito-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.

Discrete-time Markov chains with two-time scales and a countable state space: limit results and queueing applications

Abstract: This work is concerned with discrete-time Markov chains having countable state spaces and two-time-scale structures. We examine two classes of Markov chains. In the first class, the state space of ...

Random walks systems with killing on ℤ

Abstract: We study random walks systems on ℤ whose general description follows. At time zero, there is a number of particles at each vertex of ℕ, all being inactive, except for those placed at the vertex one. Each active particle performs a simple random walk on ℤ and, up to the time it dies, it activates all inactive particles that it meets along its way. An active particle dies at the instant it reaches a certain fixed total of jumps (L ≥ 1) without activating any particle, so that its lifetime depends strongly on the past of the process. We investigate how the probability of survival of the process depends on L and on the jumping probabilities of the active particles.

On the pathwise uniqueness of solutions of stochastic differential equations driven by symmetric stable Lévy processes1

Abstract: We investigate a sufficient condition for pathwise uniqueness property for 1D stochastic differential equation driven by symmetric α-stable Levy process, where α ∈ (1, 2).

Rapid paths in von Neumann-Gale dynamical systems

Abstract: The paper examines random dynamical systems related to the classical von Neumann and Gale models of economic growth. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of von Neumann–Gale dynamics is played by a special class of paths called rapid (they maximise properly defined growth rates). Up to now the theory lacked quite satisfactory results on the existence of such paths. This work provides a general existence theorem holding under assumptions analogous to the standard deterministic ones. The result solves a problem that remained open for more than three decades.

Differential equations driven by Lévy white noise in spaces of Hilbert space-valued stochastic distributions

Abstract: We develop a white noise framework and the theory of stochastic distribution spaces for Hilbert space-valued Levy processes. We then apply these concepts to the study of generalized solutions of st...

Sur l'existence de solutions d'équations différentielles stochastiques progréssives rétrogrades couplées

Abstract: Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadene, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamademe, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amon...

Domains of attraction of the random vector (X, X 2) and applications

Abstract: Many statistics are based on functions of sample moments. Important examples are the sample variance , the sample coefficient of variation SV(n), the sample dispersion SD(n) and the non-central t-statistic t(n). The definition of these quantities makes clear that the vector defined by plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector (X, X 2) belongs to a bivariate domain of attraction of a stable law. This approach is new, uniform and simple. Our main results include a full discussion of the asymptotic behaviour of SV(n), SD(n) and t 2(n). For simplicity, in restrict ourselves to positive random variables X.

On the implicit Black–Scholes formula

Abstract: We prove that, if the Black–Scholes formula holds implicitly then the model is the Black–Scholes model. In a recent paper, [Hamza K. and Klebaner F. C. On nonexistence of non-constant volatility in the Black–Scholes formula. Discrete and Continuous Dynamical Systems, 6 (2006), 829–834] it is shown that, models with non-constant implied volatility θ t , assumed to be a function (possibly random) of time t, are not compatible with the Black–Scholes formula, unless θ t is a constant. Here, we give generalizations by allowing θ t to also depend on the maturity .

Optimality of estimators for misspecified semi-Markov models

Abstract: Suppose we observe a geometrically ergodic semi-Markov process and have a parametric model for the transition distribution of the embedded Markov chain, for the conditional distribution of the inter-arrival times, or for both. The first two models for the process are semiparametric, and the parameters can be estimated by conditional maximum likelihood estimators. The third model for the process is parametric, and the parameter can be estimated by an unconditional maximum likelihood estimator. We determine heuristically the asymptotic distributions of these estimators and show that they are asymptotically efficient. If the parametric models are not correct, the (conditional) maximum likelihood estimators estimate the parameter that maximizes the Kullback–Leibler information. We show that they remain asymptotically efficient in a nonparametric sense.