In this paper, ruin probabilities are examined in a discrete time
risk model in which the interest rates follow a Markov chain with a
denumerable state space and the net losses(the claim amount minus
the premium income) are assumed to have a dependent AR(1) structure.
An upper bound for ultimate ruin probability is obtained by
martingale approach. Recursive equations for both finite time ruin
probabilities and ultimate ruin probability are derived. By
integrating the inductive method and the recursive equation, an
upper bound is given for both finite time ruin probabilities and
ultimate ruin probability.
In this paper, we discuss a Markov chain approximation method to price European options, American options and barrier
options in a Markovian regime-switching environment. The model parameters are modulated by a continuous-time, finite-state,
observable Markov chain, whose states represent the states of an economy. After selecting an equivalent martingale measure
by the regime-switching Esscher transform, we construct a discrete-time, inhomogeneous Markov chain to approximate the
dynamics of the logarithmic stock price process. Numerical examples and empirical analysis are used to illustrate the practical implementation
of the method.
This paper investigates the optimal strategies for liability management and dividend payment in an insurance company. The surplus process is jointly determined by the reinsurance policies, liability levels, future claims and unanticipated shocks. The decision maker aims to maximize the total expected discounted utility of dividend payment in infinite time horizon. To describe the extreme scenarios when catastrophic events occur, a jump-diffusion Cox-Ingersoll-Ross process is adopted to capture the substantial claim rate hikes. Using dynamic programming principle, the value function is the solution of a second-order integro-differential Hamilton-Jacobi-Bellman equation. The subsolution--supersolution method is used to verify the existence of classical solutions of the Hamilton-Jacobi-Bellman equation. The optimal liability ratio and dividend payment strategies are obtained explicitly in the cases where the utility functions are logarithm and power functions. A numerical example is provided to illustrate the methodologies and some interesting economic insights.
We assume that the asset value process of some company is directly
related to its stock price dynamics, which can be modeled by
geometric Brownian motion. The company can control its asset by
paying dividends and injecting capitals, of course both procedures
imply proportional and fixed costs for the company. To maximize the
expected present value of the dividend payments minus the capital
injections until the time of bankruptcy, which is defined as the
first time when the asset value falls below the regulation
requirement $m $, we seek to find the joint optimal dividend payment
and capital injection strategy. By solving the Quasi-variational
inequalities, the optimal control problem is addressed, which
depends on the parameters of the model and the costs. The
sensitivities of transaction costs (such as tax, consulting fees) to
the optimal strategy, the expected growth rate and volatility of the
firm asset value are also examined, some interesting economic
insights are included.
This paper investigates an optimal dividend and capital injection problem for a spectrally positive Lévy process, where the dividend rate is restricted. Both the ruin penalty and the costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, the penalized discounted capital injections before ruin, and the expected discounted ruin penalty. By the fluctuation theory of Lévy processes, the optimal dividend and capital injection strategy is obtained. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Besides, a series of numerical examples are provided to illustrate our consults.
This paper extends the model in Riesner (2007) to a Markov modulated Lévy process. The parameters of the Lévy process switch over time according to the different states of an economy, which is described by a finite-state continuous time Markov chain. Employing the local risk minimization method, we find an optimal hedging strategy for a general payment process. Finally, we give an
example for single unit-linked insurance contracts with guarantee to
display the specific locally risk-minimizing hedging strategy.
In the framework of dual risk model, Yao et al. (Optimal dividend and capital injection problem in the dual
model with proportional and fixed transaction costs. European
Journal of Operational Research, 211, 568-576) show how to
determine optimal dividend and capital injection strategy when the
dividend rate is unrestricted and the bankruptcy is forbidden. In
this paper, we further include constrain on dividend rate and allow
for bankruptcy when it is in deficit. We seek the optimal strategy
for maximizing the expected discounted dividends minus the
discounted capital injections before bankruptcy. Explicit solutions
for strategy and value function are obtained when income jumps
follow a hyper-exponential distribution, the corresponding limit
results are presented, some known results are
We consider the optimal control problem with dividend
payments and issuance of equity in a dual risk model. Such a model
might be appropriate for a company that specializes in inventions
and discoveries, which pays costs continuously and has occasional
profits. Assuming proportional transaction costs, we aim at finding
optimal strategy which maximizes the expected present value of the
dividends payout minus the discounted costs of issuing new equity
before bankruptcy. By adopting some of the techniques and
methodologies in L$\phi$kka and Zervos (2008), we construct two
categories of suboptimal models, one is the ordinary dual model
without issuance of equity, the other one assumes that, by issuing
new equity, the company never goes bankrupt. We identify the value
functions and the optimal strategies corresponding to the suboptimal
models in two different cases. For exponentially distributed jump
sizes, closed-form solutions are obtained.
This paper focuses on stochastic investment games between two investors with incorporating the influence of the macro economical environment that modeled by a Markov chain with $d$ states. There are two correlated assets are available to two investors, each investor can only invest into one of assets and his opponent choose to invest the other one. The dynamic of the two assets are driven by two drifted Brownian motion with coefficients specified by the functions of the Markov chain. Thus the system considered in this paper is controlled SDEs with random coefficients. Only one payoff function is available to both investors, one investor wants to maximize the expected payoff function, while his opponent wants to minimize the quantity at the same time. As results, the existence of the saddle point of the game, a couple of equations satisfied by the value functions and optimal policies for both investors are derived. Based on finite-difference method and weak convergence theory, a vector-valued Markov chain is constructed for approximating the underlying risky process weakly, which enables us to obtain the value function and optimal policies numerically. To some extend, we can view this paper as a further research of the problems proposed in Wan .
In this paper, optimal problems for the insurer
who can invest on risky market and purchase reinsurance are considered.
The surplus process of the insurer is a kind of perturbed classical risk model
with stochastic premium income. The investment return generating process
of the risky market is a drifted Brownian motion plus a compound
Poisson process. The objective function in this paper is to maximize
the expected utility of wealth of the insurer at terminal time, say
$T$. By solving the Hamilton-Jacobi-Bellman equations related to our
optimal control problems, the closed form expression for optimal
strategy and the value function is derived, which indicates that
the value function for an insurer to purchase both investment
and reinsurance is always better than the one for the insurer to
purchase only either investment or reinsurance.