Regime-switching Jump Diffusion Process Research Articles

Feller property of regime-switching jump diffusion processes with hybrid jumps

The transition kernel of an ℝ n -valued diffusion or jump diffusion process {X t } is known to satisfy the Feller property if {X t } is the solution of an SDE whose coefficients are Lipschitz continuous. This Lipschitz route to Feller falls short if {X t } is the solution of an SDE whose coefficients depend on a state-dependent regime-switching process {θ t }. In this paper it is shown that pathwise uniqueness and the Feller property are satisfied under mild conditions for a regime-switching jump diffusion process {X t , θ t } with hybrid jumps, i.e. jumps in {X t } that occur simultaneously with {θ t } switching.

A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes

We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different types of constraints, which lead us to find real-valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with the stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type called Bancassurance.

American options pricing under regime-switching jump-diffusion models with meshfree finite point method

In an incomplete market construction and by no-arbitrage assumption, the American options pricing problem under the jump-diffusion regime-switching process is formulated by a variational form of coupled partial integro-differential equations. In this paper, a valuation algorithm is developed for American options when the dynamics of underlying assets follow the regime-switching jump-diffusion processes. Using the fact that the price of an American option under jump-diffusion regime-switching processes is formulated by a collection of coupled variational partial integro-differential equations with the free boundary characteristic, we combine the moving least-squares approximation with an operator splitting method to treat American constraints. Numerical experiments with American options under three, five, and seven regimes demonstrate the efficiency and effectiveness of our computational scheme for pricing American options under the regime-switching models.

On Foster–Lyapunov criteria for exponential ergodicity of regime-switching jump diffusion processes with countable regimes

AbstractThis paper is devoted to the study of regime-switching jump diffusion processes with countable regimes. It aims to establish Foster–Lyapunov-type criteria for exponential ergodicity of such processes. After recalling results concerning the petiteness of compact sets, this paper presents sufficient conditions for the existence of a Foster–Lyapunov function; this, in turn, helps to establish sufficient conditions for the desired exponential ergodicity for regime-switching jump diffusion processes. Finally, an application to feedback control problems is presented.

Optimal portfolio and consumption for a Markovian regime-switching jump-diffusion process

We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases. doi:10.1017/S1446181121000122

Optimal oil production and taxation under mean reverting jump diffusion models

This paper studies the optimal extraction policy of an oil field as well as the efficient taxation of the revenues generated in light of various economic restrictions and constraints. Taking into account the fact that the oil price in worldwide commodity markets fluctuates randomly following global and seasonal macroeconomic parameters, we model the evolution of the oil price as a mean reverting regime-switching jump diffusion process. Moreover, taking into account the fact that oil producing countries rely on oil sale revenues as well as taxes levied on oil companies for a good portion of the revenue side of their budgets, we formulate this problem as a differential game where the two players are the mining company whose aim is to maximize the revenues generated from its extracting activities and the government agency in charge of regulating and taxing natural resources. We prove the existence of a Nash equilibrium and characterize the value functions of this stochastic differential game as the unique viscosity solutions of the corresponding Hamilton Jacobi Isaacs equations. We propose a numerical approximation of the value functions and prove its convergence. Furthermore, optimal extraction and fiscal policies that should be applied when the equilibrium is reached are derived. A numerical example is presented to illustrate these results.

On Feller and strong Feller properties and irreducibility of regime-switching jump diffusion processes with countable regimes

This work focuses on a class of regime-switching jump diffusion processes with a countably infinite state space for the discrete component. Such processes can be used to model complex hybrid systems in which both structural changes, small fluctuations as well as big spikes coexist and are intertwined. The paper provides weak sufficient conditions for Feller and strong Feller properties and irreducibility for such processes. The conditions are presented in terms of the coefficients of the associated stochastic differential equations.

A new lattice-based scheme for swing option pricing under mean-reverting regime-switching jump–diffusion processes

Swing options are complex path-dependent contracts, granting their holders a prefixed number of transaction rights to buy/sell a variable amount of the underlying asset (e.g. energy commodities) subject to daily or periodic constraints. Stating the swing option price as the solution of a stochastic optimal control problem, we employ a dynamic programming formulation in which the underlying asset price is modeled by a mean-reverting regime-switching jump–diffusion process. We explore a newly devised lattice-based pricing framework to find the premium of swing options in a cost-effective and easily implementable manner. We compare the performance of the proposed tree building procedure with a simulation-based Least-Squares Monte-Carlo (LSM) approach.

Stability of regime-switching jump diffusion processes

This work studies the stability of regime-switching jump diffusion processes in a finite or a countably infinite state space. Some criteria with sufficient conditions for stability and instability are provided based on characterizing the stability property of the processes in any fixed state through constants under common measurements. Also, some variational formula of these constants are given. Moreover, some examples of nonlinear regime-switching jump diffusion processes are provided to show the usefulness and sharpness of these criteria.

The maximum principles for partially observed risk-sensitive optimal controls of Markov regime-switching jump-diffusion system

ABSTRACTThis paper studies partially observed risk-sensitive optimal control problems with correlated noises between the system and the observation. It is assumed that the state process is governed by a continuous-time Markov regime-switching jump-diffusion process and the cost functional is of an exponential-of-integral type. By virtue of a classical spike variational approach, we obtain two general maximum principles for the aforementioned problems. Moreover, under certain convexity assumptions on both the control domain and the Hamiltonian, we give a sufficient condition for the optimality. For illustration, a linear-quadratic risk-sensitive control problem is proposed and solved using the main results. As a natural deduction, a fully observed risk-sensitive maximum principle is also obtained and applied to study a risk-sensitive portfolio optimization problem. Closed-form expressions for both the optimal portfolio and the corresponding optimal cost functional are obtained.

The pricing of defaultable bonds under a regime-switching jump-diffusion model with stochastic default barrier

ABSTRACTIn this paper, we investigate the price for the zero-coupon defaultable bond under a structural form credit risk with regime switching. We model the value of a firm and the default threshold by two dependent regime-switching jump-diffusion processes, in which the Markov chain represents the states of an economy. The price is associated with the Laplace transform of the first passage time and the expected discounted ratio of the firm value to the default threshold at default. Closed-form results used for calculating the price are derived when the jump sizes follow a regime-switching double exponential distribution. We present some numerical results for the price of the zero-coupon defaultable bond via Gaver-Stehfest algorithm.

Optimal dividend payment strategies with debt constraint in a hybrid regime-switching jump–diffusion model

This paper develops an optimal dividend policy for an insurance company, whose asset and liability have different dynamics, and whose surplus follows a regime-switching jump–diffusion process. The insurer aims to maximize the expected total discounted value of dividends paid out under the debt management constraint. By using the dynamic programming principle, a generalized system of integro-differential Hamilton–Jacobi–Bellman equations is derived. Moreover, this paper studies two cases where the liability of the insurer follows a uniform claim density and a generalized Pareto density in a two-regime economy, respectively. Closed-form solutions of the value functions and the optimal dividend payment strategies are obtained in both cases. A numerical example is provided to illustrate the relationship between the key parameters in the model and the value functions, as well as some interesting economic insights.

On Feller and Strong Feller Properties and Exponential Ergodicity of Regime-Switching Jump Diffusion Processes with Countable Regimes

This work focuses on a class of regime-switching jump diffusion processes, in which the switching component has countably infinite many states or regimes. The existence and uniqueness of the underlying process are obtained by an interlacing procedure. Then the Feller and strong Feller properties of such processes are derived by the coupling method and an appropriate Radon-Nikodym derivative. Finally the paper studies exponential ergodicity of regime-switching jump-diffusion processes.

Optimal oil production under mean-reverting Lévy models with regime switching

This paper is concerned with the problem of finding optimal extraction policies for an oil field in light of various financial and economical restrictions and constraints. Taking into account the fact that the oil price in worldwide commodity markets fluctuates randomly following global and seasonal macroeconomic parameters, we model the evolution of the oil price as a mean-reverting regime-switching jump-diffusion process. We formulate this problem as a finite-time horizon optimal control problem, which we solve using the method of viscosity solutions. Moreover, we construct and prove the convergence of a numerical scheme for approximating the optimal reward function and the optimal extraction policy. A numerical example illustrating these results is presented.

Kolmogorov-type systems with regime-switching jump diffusion perturbations

Population systems are often subject to various different types of environmental noises. This paper considers a class of Kolmogorov-type systems perturbed by three different types of noise including Brownian motions, Markovian switching processes, and Poisson jumps, which is described by a regime-switching jump diffusion process. This paper examines these three different types of noises and determines their effects on the properties of the systems. The properties to be studied include existence and uniqueness of global positive solutions, boundedness of this positive solution, and asymptotic growth property, and extinction in the senses of the almost sure and the $p$th moment. Finally, this paper also considers a stochastic Lotka-Volterra system with regime-switching jump diffusion processes as a special case.

A high order finite element scheme for pricing options under regime switching jump diffusion processes

This paper considers the numerical pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. In an incomplete market structure and using the no-arbitrage pricing principle, the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro-differential equations describing different states of a Markov chain. We develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration. Various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model.

Numerical schemes for pricing Asian options under state-dependent regime-switching jump–diffusion models

We study the pricing problem of Asian options when the underlying asset price follows a very general state-dependent regime-switching jump–diffusion process via a partial differential equation approach. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs). We prove existence of the solution to this system of PIDEs by the method of upper and lower solutions via constructing a monotonic sequence of approximating solutions whose limit is a strong solution of the PIDE system. We then propose several numerical schemes for solving the system of PIDEs. One of the proposed schemes is built upon the constructive proof, hence its results are provably convergent to the solution of the system of PIDEs. We illustrate the accuracy of the proposed methods by several numerical examples.

Numerical Schemes for Pricing Asian Options Under State-Dependent Regime-Switching Jump-Diffusion Models

We propose numerical schemes for pricing Asian options when the underlying asset price follows a very general state-dependent regime-switching jump-diffusion process. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs) via iterative techniques. One of the proposed schemes is provably convergent to the solution of the system of PIDEs. In addition, by treating the coupling and integral terms explicitly, over each iteration of the scheme, the pricing problem under this scheme can be partitioned into independent pricing sub-problem, with communication at the end of the iteration. Hence, this method allows for a very natural and easy-to-implement, yet efficient, parallelization of the solution process on multi-core architectures. We illustrate the accuracy and efficiency of the proposed methods by several numerical examples.

Feynman–Kac formulas for regime-switching jump diffusions and their applications

This work develops Feynman–Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated with a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.

Lookback option pricing for regime-switching jump diffusion models

In this paper, we will introduce a numerical method to price theEuropean lookback floating strike put options where the underlyingasset price is modeled by a generalized regime-switching jumpdiffusion process. In the Markov regime-switching model, the optionvalue is a solution of a coupled system of nonlinearintegro-differential partial differential equations. Due to thecomplexity of regime-switching model, the jump process involved, andthe nonlinearity, closed-form solutions are virtually impossible toobtain. We use Markov chain approximation techniques to construct adiscrete-time Markov chain to approximate the option value.Convergence of the approximation algorithms is proved. Examples arepresented to demonstrate the applicability of the numerical methods.