Convergence of delay differential equations driven by fractional Brownian motion

Abstract

In this note we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.

The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion

“…In Neuenkirch et al [16], using rough path theory, the authors prove existence and uniqueness of solutions to fractional equations with delays when H > 1/3. More recently, Ferrante and Rovira [5] established the existence and uniqueness of solutions to delayed SDEs with fBm for H > 1/2 and constant delay, by extending the results established in Nualart and Rȃşcanu [18], and the same sort of results have been shown recently for non-constant delay in Boufoussi and Hajji [2].…”

Convergence of solutions of mixed stochastic delay differential equations with applications

“…. These equations are similar to (5), but they do not contain a part with the process Z. Another difference is that the coefficients of (A.1) are random.…”

“…Fractional stochastic delay differential equations, in which b = 0 and Z = B H , were considered in only few papers. For H > 1/2, the existence and uniqueness of solution under different sets of assumptions was established in [1,2,3,4,5,8]. In the case H > 1/3, the existence and uniqueness of solution was shown in [14] for coefficients of the form f (X(t), X(t − r 1 ), X(t − r 2 ), .…”

Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space

“…Neuenkirch, Nourdin, and Tindel (2008) studied the problem by using rough path analysis. Ferrante and Rovira (2010) studied the existence and convergence when the delay goes to zero by using the Riemann-Stieltjes integral. Using also the Riemann-Stieltjes integral, Boufoussi and Hajji (2011) and Boufoussi, Hajji, and Lakhel (2012) proved the existence and uniqueness of a mild solution and studied the dependence of the solution on the initial condition in finite and infinite dimensional space.…”