In the following sections, we consider retarded SDE driven by an fBm in the form:

\mathrm{d}X(t)=({\int}_{-\tau}^{0}X(t+\theta )\rho (\mathrm{d}\theta ))\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+\sigma (t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}{W}^{H}(t),\phantom{\rule{1em}{0ex}}t>0

(2.1)

with the initial value X(t)=\xi (t), t\in [-\tau ,0], where {W}^{H}(t) is a RLfBm with Hurst parameter H>\frac{1}{2}, \rho (\cdot ) is a finite signed measure defined on [-\tau ,0].

In order to give the explicit form of (2.1), we first consider the following deterministic retarded differential equation:

\mathrm{d}Y(t)=({\int}_{-\tau}^{0}Y(t+\theta )\rho (\mathrm{d}\theta ))\phantom{\rule{0.2em}{0ex}}\mathrm{d}t

(2.2)

with the initial value Y(t)=\xi (t), t\in [-\tau ,0]. It is easy to see that the characteristic equation of (2.2) is

h(\lambda ):=\lambda -{\int}_{-\tau}^{0}{e}^{\lambda \theta}\rho (\mathrm{d}\theta )=0.

Denote Z(t) as the fundamental solution of (2.2) with initial value Z(0)=1 and Z(\theta )=0, \theta \in [-\tau ,0). By the variation-of-constants formula (see, *e.g.*, [11]), the solution of (2.2) has a unique explicit form:

Y(t)=Z(t)\xi (0)+{\int}_{-\tau}^{0}{\int}_{\theta}^{0}Z(t+\theta -s)\xi (s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\rho (\mathrm{d}\theta ),\phantom{\rule{1em}{0ex}}t\ge 0.

According to Hale [12], for any \alpha >{\alpha}_{0}:=max\{Re(\lambda ):h(\lambda )=0\}, there exists {k}_{\alpha}>0 such that the fundamental solution Z(t) satisfies the inequality

|Z(t)|\le {k}_{\alpha}{e}^{\alpha t},\phantom{\rule{1em}{0ex}}t\ge -\tau .

Before we introduce the explicit representation of (2.1), we first present a lemma that is useful in later parts.

**Lemma 2.1** [13]

*For every* \u03f5>0, *denote by* \beta =H-\frac{1}{2}, *define*

{W}^{H,\u03f5}(t)={\int}_{0}^{t}{(t-s+\u03f5)}^{\beta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}W(s).

*Then* {\{{W}^{H,\u03f5}(t)\}}_{t\ge 0} *is a semimartingale with the following decomposition*:

{W}^{H,\u03f5}(t)={\u03f5}^{\beta}W(t)+{\int}_{0}^{t}{\varphi}^{\u03f5}(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,

(2.3)

*where* {\varphi}^{\u03f5}(s)={\int}_{0}^{s}\beta {(s+\u03f5-u)}^{\beta -1}\phantom{\rule{0.2em}{0ex}}\mathrm{d}W(u). *Furthermore*, {W}^{H,\u03f5}(t) *converges to* {W}^{H}(t) *in* {L}^{2}(\mathrm{\Omega}) *uniformly with respect to* t\in [0,T] *when* \u03f5\to 0.

**Theorem 2.2** *There is a unique strong solution* {\{X(t)\}}_{t\ge 0} *to* (2.1) *which reads*

\begin{array}{rcl}X(t)& =& Z(t)\xi (0)+{\int}_{-\tau}^{0}{\int}_{\theta}^{0}Z(t+\theta -s)\xi (s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\rho (\mathrm{d}\theta )\\ +{\int}_{0}^{t}Z(t-s)\sigma (s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}{W}^{H}(s),\phantom{\rule{1em}{0ex}}t\ge 0,\end{array}

(2.4)

*where* Z(t) *is the fundamental solution of* (2.2) *with initial value* Z(0)=1 *and* Z(\theta )=0, \theta \in [-\tau ,0).

*Proof* According to Dung [8], the solution X(t) of (2.1) can be approximated by solving the following equation:

\mathrm{d}{X}^{\u03f5}(t)=({\int}_{-\tau}^{0}{X}^{\u03f5}(t+\theta )\rho (\mathrm{d}\theta ))\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+\sigma (t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}{W}^{H,\u03f5}(t),\phantom{\rule{1em}{0ex}}t>0

with the initial datum {X}^{\u03f5}(t)=\xi (t), t\in [-\tau ,0]. By the decomposition (2.3), we can rewrite the above equation as

\mathrm{d}{X}^{\u03f5}(t)=({\int}_{-\tau}^{0}{X}^{\u03f5}(t+\theta )\rho (\mathrm{d}\theta )+\sigma (t){\varphi}^{\u03f5}(t))\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+\sigma (t){\u03f5}^{\beta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}W(t),\phantom{\rule{1em}{0ex}}t>0.

Multiplying this equation by {e}^{-\lambda t}, Re\lambda >c sufficiently large, integrating from 0 to ∞, denoting by \mathcal{L}({X}^{\u03f5})(\lambda ) the Laplace transform of {X}^{\u03f5}(t), we obtain

\begin{array}{rcl}h(\lambda )\mathcal{L}\left({X}^{\u03f5}\right)(\lambda )& =& \xi (0)+{\int}_{-\tau}^{0}{\int}_{\theta}^{0}{e}^{\lambda (\theta -t)}\xi (t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\rho (\mathrm{d}\theta )+{\int}_{0}^{\mathrm{\infty}}{e}^{-\lambda t}\sigma (t){\varphi}^{\u03f5}(t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ +{\int}_{0}^{\mathrm{\infty}}{e}^{-\lambda t}\sigma (t){\u03f5}^{\beta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}W(t).\end{array}

An application of the Laplace inversion theorem (see, for example, [12]) yields

\begin{array}{rcl}{X}^{\u03f5}(t)& =& {\int}_{(c)}{e}^{\lambda t}{h}^{-1}(\lambda )[\xi (0)+{\int}_{-\tau}^{0}{\int}_{\theta}^{0}{e}^{\lambda (\theta -t)}\xi (t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\rho (\mathrm{d}\theta )\\ +{\int}_{0}^{\mathrm{\infty}}{e}^{-\lambda t}\sigma (t){\varphi}^{\u03f5}(t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+{\int}_{0}^{\mathrm{\infty}}{e}^{-\lambda t}\sigma (t){\u03f5}^{\beta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}W(t)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}\lambda .\end{array}

Note that

\mathcal{L}\left({Z}^{\u03f5}\right)(\lambda )={h}^{-1}(\lambda ),

where {Z}^{\u03f5}(t) is the fundamental solution of

\mathrm{d}{Y}^{\u03f5}(t)=({\int}_{-\tau}^{0}{Y}^{\u03f5}(t+\theta )\rho (\mathrm{d}\theta ))\phantom{\rule{0.2em}{0ex}}\mathrm{d}t

with the initial value {Z}^{\u03f5}(0)=1 and {Z}^{\u03f5}(\theta )=0, \theta \in [-\tau ,0).

With the definition of a convolution of the Laplace transform, we get

\begin{array}{rcl}{X}^{\u03f5}(t)& =& {Z}^{\u03f5}(t)\xi (0)+{\int}_{-\tau}^{0}{\int}_{\theta}^{0}{Z}^{\u03f5}(t+\theta -s)\xi (s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\rho (\mathrm{d}\theta )\\ +{\int}_{0}^{t}{Z}^{\u03f5}(t-s)\sigma (s){\varphi}^{\u03f5}(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+{\int}_{0}^{t}{Z}^{\u03f5}(t-s)\sigma (s){\u03f5}^{\beta}\phantom{\rule{0.2em}{0ex}}\mathrm{d}W(s).\end{array}

Therefore, by (2.3), it follows that

\begin{array}{rcl}{X}^{\u03f5}(t)& =& {Z}^{\u03f5}(t)\xi (0)+{\int}_{-\tau}^{0}{\int}_{\theta}^{0}{Z}^{\u03f5}(t+\theta -s)\xi (s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\rho (\mathrm{d}\theta )\\ +{\int}_{0}^{t}{Z}^{\u03f5}(t-s)\sigma (s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}{W}^{H,\u03f5}(s),\phantom{\rule{1em}{0ex}}t\ge 0.\end{array}

Let \u03f5\to 0, the representation of X(t) is obvious. □