In the following sections, we consider retarded SDE driven by an fBm in the form:
with the initial value , , where is a RLfBm with Hurst parameter , is a finite signed measure defined on .
In order to give the explicit form of (2.1), we first consider the following deterministic retarded differential equation:
with the initial value , . It is easy to see that the characteristic equation of (2.2) is
Denote as the fundamental solution of (2.2) with initial value and , . By the variation-of-constants formula (see, e.g., ), the solution of (2.2) has a unique explicit form:
According to Hale , for any , there exists such that the fundamental solution satisfies the inequality
Before we introduce the explicit representation of (2.1), we first present a lemma that is useful in later parts.
Lemma 2.1 
For every , denote by , define
Then is a semimartingale with the following decomposition:
where . Furthermore, converges to in uniformly with respect to when .
Theorem 2.2 There is a unique strong solution to (2.1) which reads
where is the fundamental solution of (2.2) with initial value and , .
Proof According to Dung , the solution of (2.1) can be approximated by solving the following equation:
with the initial datum , . By the decomposition (2.3), we can rewrite the above equation as
Multiplying this equation by , sufficiently large, integrating from 0 to ∞, denoting by the Laplace transform of , we obtain
An application of the Laplace inversion theorem (see, for example, ) yields
where is the fundamental solution of
with the initial value and , .
With the definition of a convolution of the Laplace transform, we get
Therefore, by (2.3), it follows that
Let , the representation of is obvious. □