Positive Harris recurrence of the CIR process and its applications

In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference.

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On the saturation rule for the stability of queues

Journal of Applied Probability ◽

10.1017/s0021900200102931 ◽

1995 ◽

Vol 32 (02) ◽

pp. 494-507 ◽

Cited By ~ 9

Author(s):

François Baccelli ◽

Serguei Foss

Keyword(s):

Queueing Networks ◽

Stability Region ◽

Queueing Systems ◽

Main Tool ◽

Stationary Regime ◽

Arrival Process ◽

State Variables ◽

Arrival Times ◽

Harris Recurrence ◽

The Stability

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: ‘saturate' the queues which are fed by the external arrival stream; look at the ‘intensity' μ of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say λ satisfies the condition λ < μ, whereas the network is unstable if λ > μ. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes-type backward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is subadditive ergodic theory. It is shown that, for various problems, the proposed method provides an alternative to the methods based on Harris recurrence and regeneration; this is particularly true in the Markov case, where we show that the distributional assumptions commonly made on service or arrival times so as to ensure Harris recurrence can in fact be relaxed.

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On non-singular Markov renewal processes with an application to a growth–catastrophe model

Journal of Applied Probability ◽

10.1017/s0021900200037736 ◽

1985 ◽

Vol 22 (02) ◽

pp. 253-266

Author(s):

Seppo Niemi

Keyword(s):

Markov Process ◽

Total Variation ◽

Population Model ◽

Limit Theorems ◽

Renewal Processes ◽

Regenerative Processes ◽

Catastrophe Model ◽

Markov Renewal Processes ◽

Harris Recurrence ◽

Forward Process

The paper is concerned with Markov renewal processes satisfying a certain non-singularity condition. The relation of this condition to irreducibility, Harris recurrence and regularity of the associated forward Markov process is studied. This enables one to prove limit theorems of a total variation type for Markov renewal processes and semi-regenerative processes by applying Orey's theorem to the forward process. The results are applied to a GI/G/1 queue and a growth-catastrophe population model.

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A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance

Risks ◽

10.3390/risks7040103 ◽

2019 ◽

Vol 7 (4) ◽

pp. 103 ◽

Cited By ~ 3

Author(s):

Angelos Dassios ◽

Jiwook Jang ◽

Hongbiao Zhao

Keyword(s):

Diffusion Process ◽

Jump Diffusion ◽

Insurance Premium ◽

General Process ◽

Numerical Examples ◽

Second Moments ◽

Distributional Properties ◽

The Laplace Transform ◽

Cir Process ◽

Mean Variance

In this paper, we study a generalised CIR process with externally-exciting and self-exciting jumps, and focus on the distributional properties and applications of this process and its aggregated process. The aim of the paper is to introduce a more general process that includes many models in the literature with self-exciting and external-exciting jumps. The first and second moments of this jump-diffusion process are used to calculate the insurance premium based on mean-variance principle. The Laplace transform of aggregated process is derived, and this leads to an application for pricing default-free bonds which could capture the impacts of both exogenous and endogenous shocks. Illustrative numerical examples and comparisons with other models are also provided.

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A new numerical scheme for the CIR process

Monte Carlo Methods and Applications ◽

10.1515/mcma-2015-0101 ◽

2015 ◽

Vol 21 (3) ◽

Cited By ~ 2

Author(s):

Nikolaos Halidias

Keyword(s):

Numerical Scheme ◽

Order Of Convergence ◽

Positivity Preserving ◽

Cir Process ◽

Well Posed

AbstractIn this paper we generalize an explicit numerical scheme for the CIR process that we have proposed before. The advantage of the new proposed scheme is that preserves positivity and is well posed for a (little bit) broader set of parameters among the positivity preserving schemes. The order of convergence is at least logarithmic in general and for a smaller set of parameters is at least 1/4.

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