Singular equations driven by an additive noise and applications

Abstract

In the pathwise stochastic calculus framework, the paper deals with the general study of equations driven by an additive Gaussian noise, with a drift function having an infinite limit at point zero. An ergodic theorem and the convergence of the implicit Euler scheme are proved. The Malliavin calculus is used to study the absolute continuity of the distribution of the solution. An estimation procedure of the parameters of the random component of the model is provided. The properties are transferred on a class of singular stochastic differential equations driven by a multiplicative noise. A fractional Heston model is introduced. ContentsKey words and phrases. Ergodic theorem, Fractional Brownian motion, Gaussian processes, Heston model, Sensitivities, Stochastic differential equations.The second section is devoted to deterministic properties of Equation (1) : the global existence and the uniqueness of the solution, the regularity of the Itô map, the convergence of the implicit Euler scheme and some estimates.The third section is devoted to probabilistic and statistical properties of the solution X(x 0 ) of Equation (1), obtained via its deterministic properties proved at Section 2 and various additional conditions on the signal B. In order to ensure the integrability of estimates, B is a Gaussian process in the major part of Section 3. Subsection 3.1 deals with the ergodicity of X(x 0 ), studied in the random dynamical systems framework (see Arnold [1]). By assuming that B is a fractional Brownian motion, the existence of an attracting stationary solution of Equation (1) and an ergodic theorem are proved. Subsection 3.2 deals with applications of the Malliavin calculus (see Nualart [28]) to the absolute continuity of the distribution of X t (x 0 ) for every t ∈]0, T ]. Via Nourdin and Viens [27], a density with a suitable expression is provided. Subsection 3.3 deals with the integrability and the convergence of the implicit Euler scheme. A rate of convergence is provided. Subsection 3.4 deals with a relationship between X(x 0 ) and an Ornstein-Uhlenbeck process. By assuming that B is a fractional Brownian motion of Hurst parameter H ∈]1/2, 1[, an estimation procedure of (H, σ) is provided by using Melichov [24], Brouste and Iacus [3], and Berzin and León [2]. On the fractional Ornstein-Uhlenbeck process, see Cheridito et al. [5] and Garrido-Atienza et al. [10].The fourth section is devoted to the transfer of the properties established at sections 2 and 3 on a class of singular stochastic differential equations driven by a multiplicative noise. In particular, it covers and completes Marie [21] on a generalized Cox-Ingersoll-Ross model. Subsection 4.2 deals with a Heston model (see Heston [14]) in which the volatility is modeled by a fractional Cox-Ingersoll-Ross equation in order to take benefits of the long memory and of the regularity of the paths of the fractional Brownian motion as in Comte, Coutin and Renault [6].Notations. Let J ⊂ R be a compact interval.• The space C 0 (J, R) of the continuous functions from ...

On a Constrained Fractional Stochastic Volatility Model

“…One of them consists to choose µ and σ having a singular behavior around zero, and to prove that the components of the solution stay positive because they cannot hit zero. About these models, in Itô's calculus framework, see Karlin and Taylor [15], and in the pathwise stochastic calculus framework, see for instance Marie [17]. A famous example is the Cox-Ingersoll-Ross equation, which models the volatility in the Heston model for instance.…”

“…A famous example is the Cox-Ingersoll-Ross equation, which models the volatility in the Heston model for instance. About the Heston model, in Itô's calculus framework see Heston [14], and in the pathwise stochastic calculus framework, see for instance Comte et al [9] or Marie [17]. Another way to constrain the solution of a differential equation, which is used in this paper, is to assume that µ and σ satisfy a viability condition.…”