Fractional Cox–Ingersoll–Ross process with small Hurst indices

Mishura, Yuliya; Yurchenko-Tytarenko, Anton

doi:10.15559/18-vmsta126

Cited by 12 publications

(19 citation statements)

References 24 publications

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“…Under (A1)-(A3), we prove in theorem 3.2 below that (1.1) has a positive solution on (0, T ] a.s. [18] studies positiveness of solutions to fractional C-I-R models with H ∈ (0, 1) under the framework of pathwise Stratonovich integral. Our result covers [18, theorem 2], where the authors deal with fractional C-I-R model with H > 1 2 .…”

Section: A Study Of Sdes Driven By Fbmmentioning

confidence: 99%

“…Additionally, motivated by studying the fractional C-I-R model, a singular fractional SDE has been discussed in [14] under some conditions. Recently, a general fractional C-I-R with Hurst parameter H ∈ (0, 1) was introduced in [18]. However, some models cannot be covered by the conditions introduced recently in [14], see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…However, some models cannot be covered by the conditions introduced recently in [14], see e.g. the equation ( 3) in [18] or examples 4.4 and 4.5 below. Hence one aim of the present paper is to give more general conditions to cover more stochastic interest rate models by using the Lamperti transformation, even if their coefficients have super-linear growth, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

Zhang

Chen

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $ H \gt \frac{{1}\over{2}}$ . The drift term of the equation is locally Lipschitz and unbounded in the neighbourhood of the origin. We show the existence, uniqueness and positivity of the solutions. The estimates of moments, including the negative power moments, are given. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, by using Lamperti transformation, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear Aït-Sahalia type model.

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Section: A Study Of Sdes Driven By Fbmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

Zhang

Chen

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…For fractional CIR, CKLS, and AS models, the existence of a unique positive solution of Equation (2) was obtained in [7][8][9][10][11][12][13][14]. The proof can be provided in several ways.…”

Section: Introductionmentioning

confidence: 99%

Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient

Kubilius

Medžiūnas

2020

Mathematics

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We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories of the fractional CKLS model with β>1 are not necessarily positive. We obtain the almost sure convergence rate of the backward Euler approximation scheme for solutions of the considered SDEs. We also obtain a strongly consistent and asymptotically normal estimator of the Hurst index H>1/2 for positive solutions of FSDEs.

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“…was considered to define a fractional generalization of the Cox-Ingersoll-Ross process (see also [27] for extensions to the case H < 1 2 and [28] for its application in fractional Heston-type model). The goal of the present paper is to study the stochastic differential equation…”

Section: Introductionmentioning

confidence: 99%

Sandwiched SDEs with unbounded drift driven by Hölder noises

Nunno¹,

Mishura²,

Yurchenko-Tytarenko³

2020

Preprint

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We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of an arbitrary order. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some additional assumptions on the noise, we prove that the solution has moments of all orders. We complete the study providing a numerical scheme for the solution. As an illustration of our results and motivation for applications, we suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes.

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Fractional Cox–Ingersoll–Ross process with small Hurst indices

Cited by 12 publications

References 24 publications

Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient

Sandwiched SDEs with unbounded drift driven by Hölder noises

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