Mixed Stochastic Delay Differential Equations
نویسنده
چکیده
where W is a Wiener process, Z is a Hölder continuous process with Hölder exponent greater than 1/2, the coefficients a, b, c depend on the past of the process X . The integral with respect to W is understood in the usual Itô sense, while the one with respect to Z is understood in the pathwise sense. (A precise definition of all objects is given in Section 2.) We will call this equation a mixed stochastic delay differential equation; the word mixed refers to the mixed nature of noise, while the word delay is due to dependence of the coefficients on the past. In the pure Wiener case, where c = 0, this equation was considered by many authors, often by the name “stochastic functional differential equation”. For overview of their results we refer a reader to [9, 12], where also the importance of such equations is explained, and several particular results arising in applications are given. In the pure “fractional” case, where b = 0, there are only few results devoted to such equations, considering usually the case where Z = B is a fractional Brownian motion (for us, it is also the most important example of the driver Z). In [4, 5], the existence of a solution is shown for the coefficients of the form a(t,X) = a(X(t)), b(t,X) = b(X(t − r)), and H > 1/2. It is also proved that the solution has a smooth density, and the convergence of solutions is established for a vanishing delay. A similar equation constrained to stay non-negative is considered in [1]. Existence and uniqueness of solution for an equation with general coefficients, also in the case H > 1/2, are established in [2, 8]. For such equation, it is proved in [8] that the solution possesses infinitely differentiable density, and in [3], that the solution generates a continuous random dynamical system. In [13], the unique solvability is established for an equation with H > 1/3 and coefficients of the form f(X(t), X(t− r1), X(t− r2), . . . ). Concerning mixed stochastic delay differential equations, there are no results known to author. There are some literature devoted to mixed equations without
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