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

Abstract

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.

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…”

Sandwiched SDEs with unbounded drift driven by Hölder noises

“…In this section, we present simulated paths of the sandwiched process based on a semi-heuristic approximation approach. One must note that it does not have the virtue of giving sandwiched discretized process and has worse convergence type in comparison to some alternative schemes (see, for example, [21,33] for the case of fractional Brownian motion, but, on the other hand, allows much weaker assumptions on both the drift and the noise and is much simpler from the implementation point of view. Let ∆ = {0 = t 0 < t 1 < ... < t N = T } be a uniform partition of [0, T ], t k = T k N , k = 0, 1, ..., N , |∆| := T N .…”

“…Furthermore, some examples of possible noises (including Gaussian Volterra processes, multifractional Brownian motion and continuous martingales) are provided. In Section 2, we prove existence and uniqueness of the solution to (0.2) in the case of ψ ≡ ∞, derive upper and lower bounds for the solution in terms of the noise and study finiteness of E sup t∈[0,T ] |Y t | r and E sup t∈[0,T ] (Y t − ϕ(t)) −r , r ≥ 1, which is crucial for the numeric schemes to control the increments of the drift (see, for example, [21,33] for the case of fractional Brownian motion). Full details of the proof of the existence are provided in the Appendix A.…”

Malliavin differentiability and regularity of densities in semi-linear stochastic delay equations driven by weighted fractional Brownian motion

“…The problem of existence and uniqueness of solution is considered in [22] for a Hurst parameter H > 1/3, and extended to H > 1/4 in [37]. Recently, when the drift is locally Lipschitz and unbounded in the neighborhood of the origin, particularly to the mean-reverting stochastic volatility models in finance, the existence and positivity of a unique solution have studied in [41].…”