Conditions for optimality in the infinite-horizon portfolio-cum-saving problem with semimartingale investments

A model of optimal accumulation of capital and portfolio choice over an infinite horizon in continuous time is formulated in which the vector process representing returns to investments isa general semimartingale. Methods of stochastic calculus and calculus of variations are used to obtain necessary and sufficient conditions for optimality involving martingale properties ofthe shadow price processes associated with alternative portfolio cum saving plans.The relationship between such conditions and portfolio equations is investigated.The results are appliedtospecial cases where the returns process has stationary independent increments and the utility function has the discounted relative risk aversion form     

Pricing Equity Swaps in an Economy with Jumps

Abstract

Empirical evidence confirms that asset price processes exhibit jumps and that asset returns are not Gaussian. We provide a pricing model for equity swaps including quanto equity swaps for a non-Gaussian market. The market is driven by a general marked point process as well as by a standard multidimensional Wiener process. In order to obtain closed-form solutions of the swap values, we assume that all parameters in the asset price processes are deterministic, but possibly functions of time. We derive swap prices using martingale methods rather than replicating portfolios, and we show how to calculate the convexity correction term analytically. Our results are an extension of the results of Liao and Wang (2003 Liao, M. and Wang, M. 2003. Pricing models of equity swaps. The Journal of Futures Markets, 23(8): 751–772. [Crossref], [Web of Science ®] , [Google Scholar]; Pricing models of equity swaps, The Journal of Futures Markets, 23(8), pp. 751–772). The martingale method is the key that enables the extension.     

Stability of exponential utility maximization with respect to market perturbations

We investigate the continuity of expected exponential utility maximization with respect to perturbation of the Sharpe ratio of markets. By focusing only on continuity, we impose weaker regularity conditions than those found in the literature. Specifically, we require, in addition to the V$V$-compactness hypothesis of Larsen and ?itkovi? (2007) [13], a local bmo$bmo$ hypothesis, a condition which is essentially implicit in the setting of [13]. For markets of the form S=M+∫λd〈M〉$S=M+\int \lambda d〈M〉$, these conditions are simultaneously implied by the existence of a uniform bound on the norm of λ⋅M$\lambda \cdot M$ in a suitable bmo$bmo$ space.     

Filtering,Smoothing and M-ary Detection with Discrete Time Poisson Observations

Abstract

In this article, we solve a class of estimation problems, namely, filtering smoothing and detection for a discrete time dynamical system with integer-valued observations. The observation processes we consider are Poisson random variables observed at discrete times. Here, the distribution parameter for each Poisson observation is determined by the state of a Markov chain. By appealing to a duality between forward (in time) filter and its corresponding backward processes, we compute dynamics satisfied by the unnormalized form of the smoother probability. These dynamics can be applied to construct algorithms typically referred to as fixed point smoothers, fixed lag smoothers, and fixed interval smoothers. M-ary detection filters are computed for two scenarios: one for the standard model parameter detection problem and the other for a jump Markov system.     

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.     

Duality theorem for B‐valued martingale Orlicz–Hardy spaces associated with concave functions

The dual space of B ‐valued martingale Orlicz–Hardy space with a concave function Φ, which is associated with the conditional p‐variation of B ‐valued martingale, is characterized. To obtain the results, a new type of Campanato spaces for B ‐valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p‐uniform smoothness and q‐uniform convexity of the underlying Banach space.     

On the effect of selection in genetic algorithms

To study the effect of selection with respect to mutation and mating in genetic algorithms, we consider two simplified examples in the infinite population limit. Both algorithms are modeled as measure valued dynamical systems and are designed to maximize a linear fitness on the half line. Thus, they both trivially converge to infinity. We compute the rate of their growth and we show that, in both cases, selection is able to overcome a tendency to converge to zero. The first model is a mutation‐selection algorithm on the integer half line, which generates mutations along a simple random walk. We prove that the system goes to infinity at a positive speed, even in cases where the random walk itself is ergodic. This holds in several strong senses, since we show a.s. convergence, L^p convergence, convergence in distribution, and a large deviations principle for the sequence of measures. For the second model, we introduce a new class of matings, based upon Mandelbrot martingales. The mean fitness of the associated mating‐selection algorithms on the real half line grows exponentially fast, even in cases where the Mandelbrot martingale itself converges to zero. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 185–200, 2001     

Reflected BSDE Driven by a Lévy Process

In this paper, we study the reflected solution of one-dimensional backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove the existence and uniqueness of the solution using a penalization method combined with Snell envelope theory.      

A large closed queueing network with autonomous service and bottleneck

This paper studies the queue-length process in a closed Jackson-type queueing network with the large number N of homogeneous customers by methods of the theory of martingales and by the up- and down-crossing method. The network considered here consists of a central node (hub), being an infinite-server queueing system with exponentially distributed service times, and k single-server satellite stations (nodes) with generally distributed service times with rates depending on the value N. The service mechanism of these k satellite stations is autonomous, i.e., every satellite server j serves the customers only at random instants that form a strictly stationary and ergodic sequence of random variables. Assuming that the first k-1 satellite stations operate in light usage regime the paper considers the cases where the kth satellite station is a bottleneck node. The approach of the paper is based both on development of the method from the paper by Kogan and Liptser [16], where a Markovian version of this model has been studied, and on development of the up- and down-crossing method. This revised version was published online in June 2006 with corrections to the Cover Date.     

ON THE UNIFORM CONSISTENCY OF THEKAPLAN-MEIER ESTIMATOR UNDER HEAVY CENSORING

Let X,i.i.d. and Y1i. i.d. be two sequences of random variables with unknown distribution functions F(x) and G(y) respectively. X, are censored by Y1. In this paper we study the uniform consistency of the Kaplan-Meier estimator under the case ey=sup(t:F(t)＜1)＞to=sup(t2G(t)＜1) The sufficient condition is discussed.