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

Abstract

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.

Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises

“…Having this in mind, we shall say that the drift-implicit scheme is sandwich preserving. We note that a similar approximation scheme was studied in [21] and [18,22] for processes of the type (0.4) driven by a fractional Brownian motion with H > 1/2. Our work can be seen as an extension of those.…”

“…In this work, we develop a numerical approximation (both pathwise and in L r (Ω; L ∞ ([0, T ]))) for sandwiched processes (0.1) which is similar to the drift-implicit (also known as backward ) Euler scheme constructed for the classical Cox-Ingersoll-Ross process in [2,3,13] and extended to the case of the fractional Brownian motion with H > 1 2 in [18,21,22]. In this drift-implicit scheme, in order to generate Y (t k+1 ), one has to solve the equation of the type…”

“…However, we emphasize that our results have several elements of novelty. In particular, the paper [21] discusses only pathwise convergence and not convergence in L r (Ω; L ∞ ([0, T ])). The approach of [18] and [22] is very noise specific as both use Malliavin calculus techniques in the spirit of [19,Proposition 3.4] to estimate inverse moments of the considered process (which turns out to be crucial to control explosive growth of the drift).…”

Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS Model

“…As for the fractional CKLS model, Marie [19] discussed the local existence of solution in rough paths sense. Kubilius and Medži ūnas [20] found the sufficiently simple conditions when the solution of fractional CKLS for γ > 1 and H ∈ (1/2, 1) is positive, and obtained the convergence rate of the backward Euler approximation method for solutions of fractional CKLS model.…”

Parameter estimation in CKLS model by continuous observations

“…The model (1) with β = 2/3 is used as a model for stochastic volatility, see Carr and Sun (2007); Lewis (2000). The fractional generalization of this model was investigated in Kubilius and Medžiūnas (2021).…”