Analysis of some ruin-related quantities in a Markov-modulated risk model

In this paper, we study the joint Laplace transform and probability generating function of some random quantities that occur in each environment state by the time of ruin in a Markov-modulated risk process. These quantities include the duration spent in each state, the number of claims and the aggregate amount of claims that occurred in each state by the time of ruin. Explicit formulae for the joint transforms, given the initial surplus, and the initial and terminal environment states, are expressed in terms of a matrix version of the scale function. Moments and covariances of these ruin-related quantities are obtained and numerical illustrations are presented. The joint transform of the duration spent in each state, the number of claims, and the aggregate amount of claims that occurred in each state by the time the surplus attains a certain level are also investigated.     

Pricing power options with a generalized jump diffusion

In this article, the valuation of power option is investigated when the dynamic of the stock price is governed by a generalized jump-diffusion Markov-modulated model. The systematic risk is characterized by the diffusion part, and the non systematic risk is characterized by the pure jump process. The jumps are described by a generalized renewal process with generalized jump amplitude. By introducing NASDAQ Index Model, their risk premium is identified respectively. A risk-neutral measure is identified by employing Esscher transform with two families of parameters, which represent the two parts risk premium. In this article, the non systematic risk premium is considered, based on which the price of power option is studied under the generalized jump-diffusion Markov-modulated model. In the case of a special renewal process with log double exponential jump amplitude, the accurate expressions for the Esscher parameters and the pricing formula are provided. By numerical simulation, the influence of the non systematic risk’s price and the index of the power options on the price of the option is depicted.     

Matrix-analytic Models and their Analysis

We survey phase-type distributions and Markovian point processes, aspects of how to use such models in applied probability calculations and how to fit them to observed data. A phase-type distribution is defined as the time to absorption in a finite continuous time Markov process with one absorbing state. This class of distributions is dense and contains many standard examples like all combinations of exponential in series/parallel. A Markovian point process is governed by a finite continuous time Markov process (typically ergodic), such that points are generated at a Poisson intensity depending on the underlying state and at transitions; a main special case is a Markov-modulated Poisson process. In both cases, the analytic formulas typically contain matrix-exponentials, and the matrix formalism carried over when the models are used in applied probability calculations as in problems in renewal theory, random walks and queueing. The statistical analysis is typically based upon the EM algorithm, viewing the whole sample path of the background Markov process as the latent variable.     

Ruin Probabilities of a Dual Markov-Modulated Risk Model

This article is devoted to studying a dual Markov-modulated risk model, which can properly represent, to some extent, surplus processes of companies that pay costs continuously and have occasional gains. We consider both the finite and infnite horizon ruin probabilities under this dual model. Upper and lower bounds of Lundberg type are derived for these ruin probabilities. We also obtain a time-dependent version of Lundberg type inequalities.     

Slowing time: Markov-modulated Brownian motions with a sticky boundary

We define a new family of stochastic processes called Markov modulated Brownian motions with a sticky boundary at zero. Intuitively, each process is a regulated Markov-modulated Brownian motion whose boundary behavior is modified to slow down at level zero.

To determine the stationary distribution of a sticky MMBM, we follow a Markov-regenerative approach similar to the one developed with great success in the context of quasi-birth-and-death processes and fluid queues. Our analysis also relies on recent work showing that Markov-modulated Brownian motions arise as limits of a parametrized family of fluid queues.     

An (s,k, S) fluid inventory model with exponential leadtimes and order cancellations

We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.     

Total shift during the first passages of Markov-modulated Brownian motion with bilateral ph-type jumps: Formulas driven by the minimal solution matrix of a Riccati equation

This article describes our study of the total shift during the first passages (one-sided and two-sided exit times) of Markov-modulated Brownian motion with bilateral ph-type jumps, which is referred to as MMBM. The total shift is defined as the value of a so-called shift process at the first passage epochs of the MMBM. The shift process, introduced by Bean and O’Reilly, behaves like a continuous Markovian fluid process; that is, it increases or decreases linearly with slopes regulated by the underlying Markov process that determines the path of the MMBM. Hence, the notion of total shift, which includes the first passage times of the MMBM as special cases, is useful for describing various performance measures of systems modeled by the MMBM. In this article, we present formulas for the Laplace–Stieltjes transform matrices of the total shift during various first passages of the MMBM. In particular, a Riccati equation is derived so that a matrix associated with the Laplace–Stieltjes transform of the total shift during the first return time of the MMBM is its minimal non-negative solution matrix. With this solution matrix, the Laplace–Stieltjes transform matrices can be obtained without much additional work. Furthermore, it is shown that the Riccati equation satisfies the conditions for the Newton scheme to have quadratic convergence, which enables us to use algorithms with quadratic convergence, such as Newton’s method and the Stochastic Doubling Algorithm, to compute the presented matrix-driven formulas. For the analyses, we take an approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows with bilateral ph-type jumps, referred to as MMFF, that weakly converge to the MMBM. Another contribution of this article is that duality results are derived in relation to the MMBM, which is an extension of the duality theorems developed by Ahn and Ramaswami for an MMFF without a jump.     

Hedging of contingent claims written on non traded assets under Markov-modulated models

ABSTRACT

This paper studies the hedging problem of European contingent claims when the underlying asset is non traded. We assume that the share prices of the assets are governed by Markov-modulated processes; that is, the market parameters switch over the time according to a finite-state continuous time Markov chain. Due to the presence of Markov chain the non traded asset, the market which we consider is incomplete, we shall use the local risk minimization method to obtain an optimal hedging strategy in a closed-form for an investor. Finally, numerical illustrations of an optimal hedging strategy are given by the Monte Carlo simulation.     

Scheduling and performance analysis under a stochastic model for electric vehicle charging stations

Wide-spread infrastructures for electric vehicle battery charging stations are essential in order to significantly increase the implementation of electric vehicles (EVs) in the foreseeable future. Therefore, we propose a stochastic model and charge scheduling methods for an EV battery charging system. We utilize a flexible Poisson process with a hidden Markov chain for modeling the complexity of the time-varying behavior of the EV stream into the system. Relevant random factors and constraints, which include parking times, requested amounts of electricity, the number of parking lots (charging facilities), and maximal demand level, are considered within the proposed stochastic model. Performance measures for the proposed charge scheduling are analytically derived by obtaining stationary distributions of states concerning the number of inbound EVs, waiting time distributions, and the joint distributions of parking time and electricity charged during random parking times.     

A quadratically convergent algorithm for first passage time distributions in the Markov-modulated Brownian motion

A Markov-modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian Motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As with Brownian Motion, the time-dependent analysis of the MMBM becomes easy once the first passage times between levels are determined. However, in the MMBM those distributions cannot be obtained explicitly, and we need efficient algorithms to compute them. In this article, we provide a powerful approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows without Brownian components that weakly converge to the MMBM. Our main result is a Riccati equation for an associated matrix of transforms that satisfies conditions for the Newton scheme to have quadratic convergence and thus yields a very practical tool. The solution of that Riccati equation determines needed first passage times in the MMBM without much additional work. The success of our approach, which is based essentially on first-order fluid flows and a stochastic limit process, is argued to be due to the way we have isolated certain terms involving the quadratic variation effects of the Brownian. As an illustration of our algorithm, we present a numerical example of time-dependent results for a MMBM considered by Asmussen for which he determined (only) the eventual first return probabilities which we use here as an accuracy check.