Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

Abstract

In this paper we consider stochastic differential equations with nonnegativity constraints, driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first study an ordinary integral equation, where the integral is defined in the Young sense, and we prove an existence result and the boundedness of the solutions. Then we apply this result pathwise to solve the stochastic problem.

Generalized fractional BSDE with non Lipschitz coefficients

“…Remark 3.1 This problem is solved in Ferrante and Rovira [9] for a fractional Brownian motion and a set G = (0, ∞). Pardoux and Zhang [8] proved that for H = 1/2 and for each…”

SDEs with constraints driven by semimartingales and processes with bounded p-variation

“…This kind of equations have many applications, for instance in queueing systems, seismic reliability analysis and finance (see, e.g., [1,10,16,25]). In recent papers by Besalu and Rovira [2] and Ferrante and Rovira [13] SDE with non-negativity constraints driven by fractional Brownian motion B H with Hurst index H > 1/2 and A t = t, t ∈ R + , is studied. This equation is a particular case of (1.1) because B H has locally bounded p-variation for p > 1/H.…”

“…This equation is a particular case of (1.1) because B H has locally bounded p-variation for p > 1/H. In the main theorem of [13] the existence of a solution is proved under the assumption that the coefficients f, g are Lipschitz continuous. The proof is based on a quite natural in the context of SDEs driven by B H technics based on λ-Hölder norms.…”

“…The proof is based on a quite natural in the context of SDEs driven by B H technics based on λ-Hölder norms. Unfortunately, in [13] it is only shown that the solution is unique for some small time interval. To our knowledge, global uniqueness for fractional SDEs with constraints is still an open problem.…”

“…To our knowledge, global uniqueness for fractional SDEs with constraints is still an open problem. In contrast to [13], in our paper we use p-variation norm (for the theory of functions of p-variation and its various applications see, e.g., [7,8]). In our paper we consider two conditions: continuity and linear growth of f and Hölder continuity of g (condition (H1)) and local Lipschitz continuity of f and local Hölder continuity of the derivative of each component g i,j i, j = 1, .…”