scholarly journals Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching

2018 ◽  
Vol 23 (6) ◽  
pp. 2433-2455 ◽  
Author(s):  
Zhenzhong Zhang ◽  
◽  
Enhua Zhang ◽  
Jinying Tong
Keyword(s):  
Markov Switching ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stable Processes ◽  
Cir Model ◽  
Model Driven ◽  
Necessary And Sufficient

Population dynamics driven by truncated stable processes with Markovian switching

2021 ◽  
Vol 58 (2) ◽  
pp. 505-522
Author(s):  
Zhenzhong Zhang ◽  
Jinying Tong ◽  
Qingting Meng ◽  
You Liang
Keyword(s):  
Population Dynamics ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Markovian Switching ◽  
Stable Processes ◽  
Necessary And Sufficient ◽  
The Impact

AbstractWe focus on the population dynamics driven by two classes of truncated $\alpha$-stable processes with Markovian switching. Almost necessary and sufficient conditions for the ergodicity of the proposed models are provided. Also, these results illustrate the impact on ergodicity and extinct conditions as the parameter $\alpha$ tends to 2.

Necessary and sufficient conditions for ergodicity of CIR type SDEs with Markov switching

2019 ◽  
Vol 19 (03) ◽  
pp. 1950023 ◽  
Author(s):  
Zhenzhong Zhang ◽  
Hongqian Yang ◽  
Jinying Tong ◽  
Liangjian Hu
Keyword(s):  
Interest Rate ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Wasserstein Distance ◽  
Necessary And Sufficient Conditions ◽  
Exponential Rate ◽  
Transition Semigroup ◽  
Rate Model ◽  
Interest Rate Model ◽  
Necessary And Sufficient

In this paper, we consider the ergodicity and transience of the Cox–Ingersoll–Ross (CIR) interest rate model with Markov switching. Using the theory of [Formula: see text]-matrices, we give some necessary and sufficient conditions for ergodicity of the CIR interest rate model with Markov switching. Besides, we show that the transition semigroup converges to the stationary distribution at an exponential rate in the Wasserstein distance. Finally, two examples are presented to illustrate our theory.

WEAK CONVERGENCE TO LÉVY STABLE PROCESSES IN DYNAMICAL SYSTEMS

2010 ◽  
Vol 10 (02) ◽  
pp. 263-289 ◽  
Author(s):  
MARTA TYRAN-KAMIŃSKA
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Stable Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stable Processes ◽  
Skorohod Space ◽  
Necessary And Sufficient

We study convergence of normalized ergodic sum processes to Lévy stable process in the Skorohod space with J1-topology. Our necessary and sufficient conditions allow us to prove or disprove such convergence for specific examples.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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