Convergence of delay differential equations driven by fractional Brownian motion

By Marco Ferrante, C. Rovira. 2009

60 citations recorded by the source corpus.

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 Lp, to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/112285e8edd9acf46a05a59693053ec9815c08b0

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In Applied Mathematics

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Citations in the corpus

Malliavin Calculus for Fractional Delay Equations (2009, background): “The kind of delay stochastic differential system described by (1) is widely studied when driven by a standard Brownian motion (see [26] for a nice survey), but the results in the fractional Brownian case are scarce: we are only aware of [12] for the case H > 1/2 and f (Z ) = σ(Z y(−r)), 0 ≤ r ≤ h, t”
The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion (2011, result): “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 Răşcanu [18], and the same sort of results have been shown recently for non-constant delay in Bo”