Time Finite State Markov Chain Research Articles

Optimal Investment and Proportional Reinsurance in a Regime-Switching Market Model under Forward Preferences

In this paper, we study the optimal investment and reinsurance problem of an insurance company whose investment preferences are described via a forward dynamic exponential utility in a regime-switching market model. Financial and actuarial frameworks are dependent since stock prices and insurance claims vary according to a common factor given by a continuous time finite state Markov chain. We construct the value function and we prove that it is a forward dynamic utility. Then, we characterize the optimal investment strategy and the optimal proportional level of reinsurance. We also perform numerical experiments and provide sensitivity analyses with respect to some model parameters.

Mean-Variance Asset-Liability Management in a Non-Markovian Regime-Switching Jump-Diffusion Market with Random Horizon

This paper investigates the problem of mean-variance asset-liability management (ALM) in a market where the dynamics of assets are non-Markovian regime-switching models driven by a Brownian motion, a Poisson random measure and a continuous time finite-state Markov chain. It is assumed that the time horizon is uncertain relying not only on asset prices and liability values, but also on other factors. The insurer aims to minimize the variance of the terminal surplus given an expected terminal surplus subject to the risk of paying out random liabilities of an extensive Cramer-Lundberg model. By further developing the solvability of a linear backward stochastic differential equation (BSDE) with two types of jumps, namely jumps modelled by a Poisson random measure and by basic martingales related to a Markov chain, closed-form expressions for both the efficient strategy and the efficient frontier are obtained and represented by unique solutions to several relevant BSDEs.

Equilibrium investment and risk control for an insurer with non-Markovian regime-switching and no-shorting constraints

In this paper, we study the time-consistent equilibrium investment and risk control policies for an insurer under the mean-variance criterion. The dynamics of liability and assets are given by non-Markovian regime-switching models driven by a Brownian motion and a continuous time finite-state Markov chain. It is assumed that the insurer has a time-varying preference of risk depending dynamically on current wealth, and is not allowed to short sell the risky assets. With the aid of backward stochastic differential equations and bounded mean oscillation martingales, we obtain feedback representations of the open-loop equilibrium strategies. Finally, the result is also applied to solve an example of the Markovian regime-switching model.

Analysis of a hybrid switching SVIR epidemic model with vaccination and Lévy noise

This paper is concerned with the dynamical behavior of a hybrid switching SVIR epidemic model with vaccination and Lévy noise. Besides a standard geometric Brownian motion, another two driving processes are considered: a stationary Poisson point process and a continuous time finite-state Markov chain. To begin with, we obtain sufficient conditions for persistence in the mean of the disease. Then we establish sufficient conditions for extinction of the disease. Furthermore, in the case of persistence, we also obtain sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic Lyapunov function with regime switching.

N-intertwined SIS epidemic model with Markovian switching

Epidemic dynamics is often subject to environmental noise and uncertainty. In this paper, we investigate the effect of color noise on the spread of epidemic in complex networks, which is modeled by stochastic switched differential equations based on the [Formula: see text]-intertwined SIS model using a continuous time finite-state Markov chain. Applying Lyapunov functions, we prove that the model has a unique global positive solution and establish sufficient conditions for stochastic extinction and permanence of the epidemic. We also show that the solution is stochastically ultimately bounded and the variance of the solution is bounded too. Furthermore, we discuss the limit of the time average of the solution. Finally, numerical simulations are carried out to illustrate our theoretical results.

Dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps

In this paper, the dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps is considered. Besides a standard geometric Brownian motion, another two driving processes are taken into account: a stationary Poisson point process and a continuous time finite-state Markov chain. Firstly, we establish sufficient conditions for persistence in the mean of the disease. Then we obtain sufficient conditions for extinction of the disease. In addition, we also establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.

Measure distorted arrival rate risks and their rewards

Risks embedded in asset price dynamics are taken to be accumulations of surprise jumps. A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level. Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation. The motion is decomposed into a space dependent drift and a space dependent martingale component. Though there is some local mean reversion in the drift, space dependence of the martingale component renders the dynamics to be of the momentum type. Local risk is measured using market calibrated measure distortions that introduce risk charges into the lower and upper prices of two price economies. These risks are compensated by the exponential variation of space dependent arrival rates. Estimations are conducted for the S&P 500 index (SPX), the exchange traded fund for the financial sector (XLF), J. P. Morgan stock prices (JPM), the ratio of JPM to XLF, and the ratio of XLF to SPX.

Arrival Rate Functions

Asset price dynamics are taken to be accumulations of surprise jumps in the logarithm of prices. A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level. Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation. The motion is decomposed into a space dependent drift and a space dependent martingale component. Though there is some local mean reversion by and large the dynamics estimated is that of the momentum type. Risk compensation is seen by a linear relation between the exponential variation and measure distorted variations for the bid and ask prices of two price economies. Estimations are conducted for the S&P 500 index (SPX), the exchange traded fund for the financial sector (XLF), J. P. Morgan stock prices (JPM), the ratio of JPM to XLF and the ratio of XLF to SPX.

Monte Carlo Variance Reduction by Conditioning for Pricing with Underlying a Continuous-Time Finite State Markov Process

We consider the pricing of derivatives when the evolution of the underlying is given by a continuous time finite state Markov chain. We present a semianalytic approach that consists in (i) simulating the number of transitions of the underlying up to a given time horizon, (ii) computing via an explicit analytic formula the derivative price for each simulated number of transitions, and (iii) approximating the actual price by the empirical average over the values computed in (ii). This corresponds to a Monte Carlo approach with variance reduction by conditioning and, with respect to a plain Monte Carlo, it thus leads to a smaller variance in addition to more precise values. The method is applied, in particular, to path dependent derivatives, and numerical results are presented and discussed.

On the relation between reversibility and monotonicity of fluctuation spectra for discrete time finite state Markov chains

For the irreducible aperiodic discrete time finite state Markov chain, (i) if it is reversible, we provide a necessary and sufficient condition for all fluctuation spectra to be monotonic on [ 0 , π ] and (ii) if it is irreversible, then there exist some nonmonotonic fluctuation spectra.

Model robustness of finite state nonlinear filtering over the infinite time horizon

We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converges to zero uniformly over the infinite time interval as the misspecified model converges to the true model, provided the signal obeys a mixing condition. The filtering error is controlled through the exponential decay of the derivative of the nonlinear filter with respect to its initial condition. We allow simultaneously for misspecification of the initial condition, of the transition rates of the signal, and of the observation function. The first two cases are treated by relatively elementary means, while the latter case requires the use of Skorokhod integrals and tools of anticipative stochastic calculus.

An interest rate model with a Markovian mean reverting level

A two‐factor Vasicek model, where the mean reversion level changes according to a continuous time finite state Markov chain, is considered. This model could capture the behaviour of monetary authorities who normally set a reference rate which changes from time to time. We derive the term structure via the analytic expression of the bond price that involves a fundamental matrix. The validity of the bond price closed form solution is verified via the forward rate dynamics.

Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

Recently, Asmussen and Koole (Journal of Applied Probability 30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

Recently, Asmussen and Koole (Journal of Applied Probability30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

Approximation and decomposition of singularly perturbed stochastic hybrid systems

This paper considers a singularly perturbed stochastic hybrid system whose state equations are governed by a stochastic switching process, which is modelled as a nearly completely decomposable continuous time finite state Markov chain. Approximate models for the slow mode subsystem are derived over the intervals of rapid switching and their accuracy (the mean-squared errors between the approximate models and the original systems) is quantified. Aggregate models for the subsystem are obtained over a longer period. The almost surely exponential stability of the subsystem is studied. The behaviour of the fast subsystem which depends on the relative size of the perturbation parameters is analysed. The decomposition of the overall system and the switching process together into slow and fast subsystems is investigated. The results derived in the paper are shown to hold when the switching process is stationary and irreducible, and each group of strongly interacting states is irreducible and time reversible. Finally a simple example is detailed to illustrate the aforementioned techniques