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On jump-diffusion processes with regime switching: martingale approach  [PDF]
Antonio Di Crescenzo,Nikita Ratanov
Mathematics , 2015,
Abstract: We study jump-diffusion processes with parameters switching at random times. Being motivated by possible applications, we characterise equivalent martingale measures for these processes by means of the relative entropy. The minimal entropy approach is also developed. It is shown that in contrast to the case of L\'evy processes, for this model an Esscher transformation does not produce the minimal relative entropy.
Solutions to a partial integro-differential parabolic system arising in the pricing of financial options in regime-switching jump diffusion models  [cached]
Ionut Florescu,Ruihua Liu,Maria Cristina Mariani
Electronic Journal of Differential Equations , 2012,
Abstract: We study a complex system of partial integro-differential equations (PIDE) of parabolic type modeling the option pricing problem in a regime-switching jump diffusion model. Under suitable conditions, we prove the existence of solutions of the PIDE system in a general domain by using the method of upper and lower solutions.
Computing Quantiles in Regime-Switching Jump-Diffusions with Application to Optimal Risk Management: a Fourier Transform Approach  [PDF]
Alessandro Ramponi
Quantitative Finance , 2012,
Abstract: In this paper we consider the problem of calculating the quantiles of a risky position, the dynamic of which is described as a continuous time regime-switching jump-diffusion, by using Fourier Transform methods. Furthermore, we study a classical option-based portfolio strategy which minimizes the Value-at-Risk of the hedged position and show the impact of jumps and switching regimes on the optimal strategy in a numerical example. However, the analysis of this hedging strategy, as well as the computational technique for its implementation, is fairly general, i.e. it can be applied to any dynamical model for which Fourier transform methods are viable.
Fourier Transform Methods for Regime-Switching Jump-Diffusions and the Pricing of Forward Starting Options  [PDF]
Alessandro Ramponi
Quantitative Finance , 2011,
Abstract: In this paper we consider a jump-diffusion dynamic whose parameters are driven by a continuous time and stationary Markov Chain on a finite state space as a model for the underlying of European contingent claims. For this class of processes we firstly outline the Fourier transform method both in log-price and log-strike to efficiently calculate the value of various types of options and as a concrete example of application, we present some numerical results within a two-state regime switching version of the Merton jump-diffusion model. Then we develop a closed-form solution to the problem of pricing a Forward Starting Option and use this result to approximate the value of such a derivative in a general stochastic volatility framework.
Stability and recurrence of regime-switching diffusion processes  [PDF]
Jinghai Shao,Fubao Xi
Mathematics , 2015,
Abstract: We provide some criteria on the stability of regime-switching diffusion processes. Both the state-independent and state-dependent regime-switching diffusion processes with switching in a finite state space and an infinite countable state space are studied in this work. We provide two methods to deal with switching processes in an infinite countable state space. One is a finite partition method based on the nonsingular M-matrix theory. Another is an application of principal eigenvalue of a bilinear form. Our methods can deal with both linear and nonlinear regime-switching diffusion processes. Moreover, the method of principal eigenvalue is also used to study the recurrence of regime-switching diffusion processes.
Stability of Nonlinear Regime-switching Jump Diffusions  [PDF]
Zhixin Yang,G. Yin
Mathematics , 2014, DOI: 10.1016/j.na.2012.02.007
Abstract: Motivated by networked systems, stochastic control, optimization, and a wide variety of applications, this work is devoted to systems of switching jump diffusions. Treating such nonlinear systems, we focus on stability issues. First asymptotic stability in the large is obtained. Then the study on exponential p-stability is carried out. Connection between almost surely exponential stability and exponential p-stability is exploited. Also presented are smooth-dependence on the initial data. Using the smooth-dependence, necessary conditions for exponential p-stability are derived. Then criteria for asymptotic stability in distribution are provided. A couple of examples are given to illustrate our results.
Criteria for transience and recurrence of regime-switching diffusion processes  [PDF]
Jinghai Shao
Mathematics , 2014,
Abstract: We provide some on-off type criteria for recurrence and transience of regime-switching diffusion processes using the theory of M-matrix and the Perron-Frobenius theorem. State-independent and state-dependent regime-switching diffusion processes in a finite space and a countable space are both studied. We put forward a finite partition method to deal with switching process in a countable space. As an application, we improve the known criteria for recurrence of linear regime-switching diffusion processes, and provide an on-off type criterion for a kind of nonlinear regime-switching diffusion processes.
Stability of Numerical Methods for Jump Diffusions and Markovian Switching Jump Diffusions  [PDF]
Zhixin Yang,G. Yin,Haibo Li
Mathematics , 2014,
Abstract: This work focuses on stability analysis of numerical solutions to jump diffusions and jump diffusions with Markovian switching. Due to the use of Poisson processes, using asymptotic expansions as in the usual approach of treating diffusion processes does not work. Different from the existing treatments of Euler-Maurayama methods for solutions of stochastic differential equations, we use techniques from stochastic approximation. We analyze the almost sure exponential stability and exponential $p$-stability. The benchmark test model in numerical solutions, namely, one-dimensional linear scalar jump diffusion is examined first and easily verifiable conditions are presented. Then Markovian regime-switching jump diffusions are dealt with. Moreover, analysis on stability of numerical methods for linearizable and multi-dimensional jump diffusions is carried out.
Approximate analysis of biological systems by hybrid switching jump diffusion  [PDF]
Alessio Angius,Gianfranco Balbo,Marco Beccuti,Enrico Bibbona,Andras Horvath,Roberta Sirovich
Computer Science , 2014,
Abstract: In this paper we consider large state space continuous time Markov chains (MCs) arising in the field of systems biology. For density dependent families of MCs that represent the interaction of large groups of identical objects, Kurtz has proposed two kinds of approximations. One is based on ordinary differential equations, while the other uses a diffusion process. The computational cost of the deterministic approximation is significantly lower, but the diffusion approximation retains stochasticity and is able to reproduce relevant random features like variance, bimodality, and tail behavior. In a recent paper, for particular stochastic Petri net models, we proposed a jump diffusion approximation that aims at being applicable beyond the limits of Kurtz's diffusion approximation, namely when the process reaches the boundary with non-negligible probability. Other limitations of the diffusion approximation in its original form are that it can provide inaccurate results when the number of objects in some groups is often or constantly low and that it can be applied only to pure density dependent Markov chains. In order to overcome these drawbacks, in this paper we propose to apply the jump-diffusion approximation only to those components of the model that are in density dependent form and are associated with high population levels. The remaining components are treated as discrete quantities. The resulting process is a hybrid switching jump diffusion. We show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original Markov chains or starting from a higher level description language, like stochastic Petri nets. The proposed approach is illustrated on three models: one modeling the so called crazy clock reaction, one describing viral infection kinetics and the last considering transcription regulation.
Pricing Participating Products under a Generalized Jump-Diffusion Model  [PDF]
Tak Kuen Siu,John W. Lau,Hailiang Yang
International Journal of Stochastic Analysis , 2008, DOI: 10.1155/2008/474623
Abstract: We propose a model for valuing participating life insurance products under a generalized jump-diffusion model with a Markov-switching compensator. It also nests a number of important and popular models in finance, including the classes of jump-diffusion models and Markovian regime-switching models. The Esscher transform is employed to determine an equivalent martingale measure. Simulation experiments are conducted to illustrate the practical implementation of the model and to highlight some features that can be obtained from our model.
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