# Generation of travelling sine-Gordon breathers in noisy long Josephson
junctions

Duilio De Santis<EMAIL_ADDRESS>Department of Physics and Chemistry
“E. Segrè”, Group of Interdisciplinary Theoretical Physics, University of
Palermo, I-90128 Palermo, Italy  Claudio Guarcello  Department of Physics “E.
R. Caianiello”, University of Salerno, I-84084 Fisciano, Salerno, Italy
Bernardo Spagnolo  Department of Physics and Chemistry “E. Segrè”, Group of
Interdisciplinary Theoretical Physics, University of Palermo, I-90128 Palermo,
Italy  Radiophysics Department, Lobachevsky State University, 603950 Nizhniy
Novgorod, Russia  Angelo Carollo  Department of Physics and Chemistry “E.
Segrè”, Group of Interdisciplinary Theoretical Physics, University of Palermo,
I-90128 Palermo, Italy  Davide Valenti  Department of Physics and Chemistry
“E. Segrè”, Group of Interdisciplinary Theoretical Physics, University of
Palermo, I-90128 Palermo, Italy

###### Abstract

The generation of travelling sine-Gordon breathers is achieved through the
nonlinear supratransmission effect in a magnetically driven long Josephson
junction, in the presence of losses, a current bias, and a thermal noise
source. We demonstrate how to exclusively induce breather modes by means of
controlled magnetic pulses. A nonmonotonic behavior of the breather-only
generation probability is observed as a function of the noise intensity. An
experimental protocol providing evidence of the Josephson breather’s existence
is proposed.

_Introduction._ —The sine-Gordon (SG) equation is one of the most celebrated
nonlinear wave models, with a domain of application that includes pendula,
Josephson junctions (JJs), gravity, high-energy physics, biophysics, and
seismology [1, 2, 3, 4]. Interestingly, solitonic excitations are sustained in
a SG medium, _kinks_ and _breathers_ being the basic building blocks. Kinks,
i.e., topological solitons, stem from an inherent degeneracy of the system’s
ground state, while breathers are space-localized, time-periodic modes that
can result from the attractive interaction between kinks and antikinks [5, 6].
Indeed, the SG equation provides an ideal environment for the exploration of
soliton dynamics, with subsequent striking experimental realizations, thus it
continues to attract great research interest in many scientific areas.

The electrodynamics of a long Josephson junction (LJJ) can be accurately
described in the SG framework. Here, a kink (or fluxon) represents a quantum
of magnetic flux ${\Phi_{0}}$. Since it can be stored, steered, manipulated,
and made to interact with electronic instrumentation, such a nonlinear wave is
featured in many applications [7, 8, 9, 10, 11, 12]. Notably, the quantum
dynamics of a single fluxon has been experimentally demonstrated [13].

Breathers are far more elusive than kinks, mainly because they radiatively
decay due to dissipation, unless particular forms of driving are employed, and
because their oscillatory nature gives rise to a practically null average
voltage, i.e., beyond sensitivity of the existing high-frequency oscilloscopes
[14]. As a consequence, despite the large variety of numerical and theoretical
studies devoted to them [15, 16, 17], experimental evidence of breather modes
has yet to be found in LJJs.

The interest towards breathers is also motivated by their significant
applicative potential. In fact, such a mode could be effectively used to
develop some novel applications in information transmission [18]. Furthermore,
in contrast to kinks, they possess an internal degree of freedom, i.e., a
proper frequency, which is particularly valuable for quantum computation
purposes. More specifically, breathers behave as macroscopic artificial two-
level atoms in an LJJ with a small capacitance per unit length, so that the
realization of a Josephson breather qubit has been proposed [19, 20]. To these
ends, detailed investigations concerning possible generation and control
mechanisms are crucial.

The excitation and detection of breather-like objects has been deeply studied
in the context of discrete systems as well. In particular, in JJ parallel
arrays, which are modelled by the discrete SG equation (also known as the
Frenkel–Kontorova model), the existence of oscillobreathers has been
predicted, but due to their rapid pulsations, an experimental confirmation is
still missing [21, 1]. Instead, the so-called rotobreather states can be
tracked by measuring the local dc voltages throughout the JJ array, and they
have been successfully observed in JJ ladders [22, 23].

This letter proposes the use of magnetic pulses for the generation of
travelling breather modes into an LJJ by means of the nonlinear
supratransmission (ST) effect. According to the latter phenomenon, a nonlinear
system subjected to a sinusoidal driving with frequency laying in the
forbidden band gap (FBG) can support energy transmission in the form of
solitonic excitations, if the forcing amplitude is strong enough. Initially
discussed in a discrete SG chain [24, 25], the ST mechanism appears to be the
result of a generic nonlinear instability [26], and the corresponding
activation threshold has been found in various situations [27, 28, 29, 30].

Here, by looking at the average voltage drop across the junction, the external
pulse’s frequency/amplitude space is thoroughly analyzed to exploit the ST
process as a controllable source of breathers in LJJs. Interestingly, in the
presence of dissipation, a current bias, and a thermal noise source, vast
regions associated with the exclusive emergence of breather excitations are
observed. In particular, the study reveals a sort of fluctuation-induced
widening of these areas, which is seen to occur in correspondence with a
nonmonotonic behavior of the breather-only generation probability, defined
below, as a function of the noise amplitude. The latter result highlights the
effectiveness of noise as a control parameter in the LJJ device.

Based on these findings, an experimental procedure providing evidence of the
existence of breathers in LJJs is outlined. Moreover, given the degree of
universality of the ST process, and its robustness against discreteness and
finiteness, the application of the proposed approach in a discrete domain
naturally comes to mind. Among the many contexts in which energy localization
and transmission is being actively investigated [31, 32, 33], one could look,
e.g., at oscillobreathers in a JJ parallel array.

_The model._ —An overlap-geometry Josephson tunnel junction is considered,
assuming that the dynamics of the phase difference between the pair wave
functions of the two superconductors ${\varphi(x,t)}$ follows the equation
[34, 35]

$\varphi_{xx}-\varphi_{tt}-\alpha\varphi_{t}=\sin\varphi-\gamma-\gamma_{T}(x,t).$
(1)

The ${x=0}$ end of the junction is subjected to an oscillating magnetic field,
perpendicular to the length of the junction and parallel to the plane of the
barrier, thus the following boundary conditions apply

$\varphi_{x}(0,t)=\widetilde{A}(t)\sin\left(\Omega
t\right),\;\;\;\varphi_{x}(l,t)=0.$ (2)

A subscript notation is used to denote partial differentiation, space is
normalized to the Josephson penetration depth,
${\lambda_{J}=\sqrt{\Phi_{0}/\left(2\pi J_{c}L_{P}\right)}}$, and time is
normalized to the inverse of the Josephson plasma frequency,
${\omega_{p}=\sqrt{2\pi J_{c}/\left(\Phi_{0}C\right)}}$, where $J_{c}$ is the
critical value of the Josephson current density, ${L_{P}}$ is the inductance
per unit length, and ${C}$ is the capacitance per unit length. In Eq. (1),
${\alpha=G/\left(\omega_{p}C\right)}$ is a damping parameter, $G$ is an
effective normal conductance, ${\gamma=J_{b}/J_{c}}$ is the normalized bias
current, and ${\gamma_{T}(x,t)}$ is a Gaussian, zero-average noise source with
the autocorrelation function given by

$\langle\gamma_{T}(x_{1},t_{1})\gamma_{T}(x_{2},t_{2})\rangle=2\alpha\Gamma\delta(x_{1}-x_{2})\delta(t_{1}-t_{2}).$
(3)

Here, ${\Gamma=2ek_{B}T/\left(\hbar J_{c}\lambda_{J}\right)}$ is the noise
amplitude, which is proportional to absolute temperature $T$ ($e$ is the
electron charge, $k_{B}$ is the Boltzmann constant, and $\hbar$ is the reduced
Planck constant). Finally, in Eq. (2), ${\widetilde{A}(t)}$ and ${\Omega}$ are
two dimensionless quantities related, respectively, to the external field’s
amplitude and frequency, and ${l=L/\lambda_{J}}$ is the normalized length of
the junction.

Equation (1) is numerically solved with the initial conditions

$\varphi(x,0)=\arcsin\gamma,\;\;\;\varphi_{t}(x,0)=0;$ (4)

also, in order to access the ST phenomenon, the external field has to
oscillate at a frequency falling into the junction’s FBG, i.e., ${\Omega<1}$,
corresponding to ${\omega<\omega_{p}\sim 100\;\textrm{GHz}-1\;\textrm{THz}}$
[16, 14]. Moreover, to reproduce a meaningful experimental pulse, Gaussian
switching-on/off regimes are chosen as

$\widetilde{A}(t)=\begin{cases}A\exp\left[-\frac{(t-t_{\rm{on}})^{2}}{2\sigma_{\rm{on}}^{2}}\vphantom{\exp\left(-\frac{(t-t_{\rm{off}})^{2}}{2\sigma_{\rm{off}}^{2}}\right)}\right]&t<t_{\rm{on}}\\\
A&t_{\rm{on}}\leq t<t_{\rm{off}}\\\
A\exp\left[-\frac{(t-t_{\rm{off}})^{2}}{2\sigma_{\rm{off}}^{2}}\right]&t\geq
t_{\rm{off}}.\end{cases}$ (5)

In Eq. (5), the Gaussian distribution with standard deviation
$\sigma_{\rm{on}}$ provides a smoothly increasing signal envelope for
${t<t_{\rm{on}}}$ (in practice, ${t_{\rm{on}}=3\sigma_{\rm{on}}}$ is set).
Then, the boundary of the junction is sinusoidally driven until an induced
travelling excitation reaches a selected position 111Since the medium’s FBG is
being considered, one may detect a nonlinear mode’s presence by looking at the
modulus of the phase ${\left\lvert\varphi\right\rvert}$ at the designated
position, which should be taken relatively close to ${x=0}$., i.e., for
${t<t_{\rm{off}}}$. If this event occurs, the driving amplitude is gradually
decreased, with the typical scale $\sigma_{\rm{off}}$. By doing so, one is
mostly able to achieve the controlled generation of single breather modes in
the system.

In what follows, the length of the junction is ${l=100}$, which generally
allows to ignore reflection effects at ${x=l}$, while keeping reasonable
execution times. Also, the damping coefficient is ${\alpha=0.02}$ [38], the
bias current is ${\gamma=0}$, and the width of the increasing (decreasing)
Gaussian tail is ${\sigma_{\rm{on}}=10}$ (${\sigma_{\rm{off}}=2.5}$). A
typical simulation outcome is illustrated in Fig. 1.

Figure 1: Time evolution of an ST-induced travelling breather in the presence
of dissipation. Parameter values: ${t_{\rm{max}}=200}$ (observation time),
${\Omega=0.25}$, and ${A=2.09}$.

_Breather detection._ —As stated above, one of the main aims of the present
work is to identify the regions of the ${(\Omega,A)}$ parameter space in which
the ST generation mechanism leads to breather excitations only. To this end,
the meaning of the expression “breather-only” used throughout the letter shall
now be operatively defined. Essentially, such a term includes all the cases in
which breathers are the only solitons left in the junction from some point
onwards, after a possible transitory phase that could involve kinks and
antikinks. Conversely, the following situations are not considered to belong
to the breather-only class: _i_) at least one bound kink-antikink couple gets
separated by the end of the simulation; _ii_) at least one unpaired kink (or
antikink) is produced; _iii_) the breather state is not observed at all.

Different techniques can be constructed for the classification of the outcomes
of the simulations. For instance, one can take advantage of the practically
null average voltage drop across the junction produced by breathers. By virtue
of the second Josephson relation [34], it is possible to define the
(normalized) time-averaged voltage

$\left\langle
V\right\rangle=\Big{\langle}\int_{0}^{l}\varphi_{t}dx\Big{\rangle}.$ (6)

Through extensive preliminary tests (the values of the simulation parameters
are given above) it was seen that, in the numerical runs where some excitation
is induced, the exclusive emergence of breather modes takes place if the
condition

$\left\lvert\langle V\rangle\right\rvert\lesssim 0.05$ (7)

is verified 222In practice, due to possible reflection effects at the
boundaries of the system, the values ${\left\lvert\varphi(0,t)\right\rvert}$
and ${\left\lvert\varphi(l,t)\right\rvert}$ require additional monitoring.,
otherwise at least one kink (or antikink) is left in the junction at the end
of the simulation, i.e., at ${t=t_{\rm{max}}}$. Equation (7) will be indicated
hereafter as the voltage criterion.

_Results and discussion._ —To numerically integrate Eq. (1), a Fortran
implementation of the implicit finite-difference scheme detailed in Ref. [38]
is used, with the noise term approximated according to Ref. [39]. The
discretization steps are ${\Delta x=\Delta t=0.005}$, and the observation time
is usually set to ${t_{\rm{max}}=200}$ in order to follow the entire evolution
of the emitted breathers.

Figure 2: Bifurcation diagram in the ${(\Omega,A)}$ plane. The gray color
indicates ${(\Omega,A)}$ couples for which no excitations of clear solitonic
nature are detected in the medium, the white color corresponds to regions with
at least one kink (or antikink) left in the junction at
${t=t_{\rm{max}}=200}$, and the cyan color is exclusively associated with
breathers according to the voltage criterion [Eq. (7)].

For the first set of simulations, the stochastic term ${\gamma_{T}(x,t)}$ is
not considered. In this case, the ${(\Omega,A)}$ parameter space is explored
by varying the driving frequency $\Omega$ in ${\left[0,1\right]}$ and the
amplitude $A$ in ${\left[0,2.5\right]}$, with increments ${\Delta\Omega=\Delta
A=0.01}$. Figure 2 presents the results obtained through the voltage criterion
[Eq. (7)] in the form of a refined bifurcation diagram. Along with the
essential features of the ST effect, the plot clearly displays the presence of
significant breather-only (cyan) areas, while the gray color indicates
${(\Omega,A)}$ couples for which no excitations of clear solitonic nature are
detected in the medium, and the white color corresponds to regions with at
least one kink (or antikink) left in the junction at ${t=t_{\rm{max}}}$.

Upon changing the values of the parameters ${\alpha}$, ${\gamma}$,
${\sigma_{\rm{on}}}$, and ${\sigma_{\rm{off}}}$, the ranges in which the
overall picture is not substantially altered can be found (not shown here).
Naturally, as the bias current or the duration of the switching-off regime are
increased, the breather-only regions are seen to gradually disappear; in fact,
the ${\gamma}$ perturbation term tends to dissociate a breather into a kink-
antikink pair [14], and additional kinks (and/or antikinks) are more likely
generated if the forcing signal takes a longer time to stop injecting energy
into the medium, i.e., for higher values of ${\sigma_{\rm{off}}}$.
Indicatively, one would choose ${\gamma\lesssim 0.1}$ and
${\sigma_{\rm{off}}\lesssim 10}$. The shape and localization of the zones of
interest may also slightly change if the parameters ${\alpha}$ and
${\sigma_{\rm{on}}}$ are varied, but the main features of Fig. 2 remain.
Moreover, further testing indicates that qualitatively analogous breather-only
areas can be obtained with fixed-duration forcing envelopes as well, i.e.,
with pulses that do not automatically shut down after an externally-induced
excitation is detected.

Figure 3: Breather-only generation probability estimated over ${N=512}$
realizations according to the voltage criterion [Eq. (7)]. Here,
${t_{\rm{max}}=150}$. Moreover, ${\Gamma=0.005}$ in panel (a) and
${\Gamma=0.05}$ in panel (b). The blue (red) circle in panel (a) identifies
the combination ${\Omega=0.55}$ and ${A=2.19}$ (${\Omega=0.52}$ and
${A=2.45}$).

A crucial aspect to discuss is whether the breather-only regions are robust
enough to survive in a noisy environment. Denoting the number of breather-only
cases identified by means of the voltage criterion [Eq. (7)] with
${N_{\rm{breather-only}}}$ and the total number of realizations performed with
$N$, a breather-only generation probability is defined here as
${P_{\rm{breather-only}}=\lim_{N\to\infty}N_{\rm{breather-only}}/N}$.
Estimations of such a quantity over ${N=512}$ realizations are presented in
Fig. 3 for the noise amplitudes ${\Gamma=0.005}$ and ${0.05}$ [8], with
${\Omega\in\left[0.4,0.6\right]}$, ${A\in\left[2,2.5\right]}$, and
${\Delta\Omega=\Delta A=0.01}$. The latter restriction, taken due to the
extent of the computational task, fully contains a single deterministic
breather-only zone, along with the edges of the confining ones, see Fig. 2.

A first remark concerning the panels of Fig. 3 is that the breather-only
regions are still clearly recognizable, since the generation probability
quickly rises in their vicinity, see the black areas. As the value of $\Gamma$
is increased, a progressively smaller core of high probability is expected to
survive because of the larger current fluctuations that tend to break up the
breather state, and this is indeed observed. Nevertheless, a sort of noise-
induced widening of these structures is seen as well, because most of the
neighboring ${(\Omega,A)}$ combinations, that deterministically are not
associated with the formation of breather modes only, acquire nonzero
probability values. Eventually, for ${\Gamma\sim 1}$, different kinds of
excitations (including breathers, kinks, and antikinks) begin to appear in the
junction, in addition to the ST-induced ones, due to the magnitude of the
stochastic perturbation, and the breather-only generation probability falls to
zero in the ${(\Omega,A)}$ parameter space in a uniform way.

Figure 4: Breather-only generation probability estimated over $N=10^{4}$
realizations, according to the voltage criterion [Eq. (7)], as a function of
the noise amplitude $\Gamma$. Here, ${t_{\rm{max}}=200}$. Moreover, the
combination ${\Omega=0.55}$ and ${A=2.19}$ is chosen in panel (a), whereas
${\Omega=0.52}$ and ${A=2.45}$ in panel (b).

To further illustrate its behavior, the quantity ${P_{\rm{breather-
only}}(\Gamma)}$ is calculated for ${N=10^{4}}$ realizations and the two
sample ${(\Omega,A)}$ combinations circled in Fig. 3(a). In the first case
(${\Omega=0.55}$ and ${A=2.19}$), which falls outside of the breather-only
region identified for $\Gamma=0$, a nonmonotonic profile is obtained; as shown
in Fig. 4(a), the probability is practically zero when ${\Gamma\to 0}$ and the
dynamics is closer to the deterministic case, it reaches a peak for an optimal
value of $\Gamma$, and eventually goes back to zero when the stochastic
influence becomes disruptive, i.e., when the fluctuations are sufficiently
strong to both easily break the oscillatory bound state and produce additional
excitations into the junction. The second combination (${\Omega=0.52}$ and
${A=2.45}$), taken as a representative case for the high-probability core,
leads to a decreasing steplike function, see Fig. 4(b). This determines the
temperature values at which breather modes can be safely excited without noise
disturbances.

In view of possible applications, the robustness of the induced breathers
against the stochastic background is also quite relevant. Although not
explicitly shown here, a detailed analysis indicates that there exists a wide
range of noise intensities where their average persistence time above the
level of fluctuations remains close to the quantity ${1/\alpha}$. The latter
is the perturbative prediction on the breather’s radiative decay lifetime for
${\Gamma=0}$ [15, 16], which is seen to hold independently of the position in
the ${(\Omega,A)}$ plane.

Now, based on these results, a test giving experimental evidence of breather
modes in LJJs is outlined. First, scanning a portion of the ${(\Omega,A)}$
space in the presence of a sufficiently high current bias allows for the
characterization of the ST process, if suitable average voltage measurements
can be performed. In fact, the power balance between dissipation and input
from the current bias term leads to well-defined voltage patterns whenever
fluxons emerge (even from a broken up breather state), while very low average
voltages are expected for the ${(\Omega,A)}$ couples that cannot transmit
energy in the medium. Once established that nonlinear (fluxon-based)
excitations are indeed being induced above a certain ${(\Omega,A)}$ threshold,
such a region can be inspected for breather-only ${(\Omega,A)}$ pairs, whose
existence and robustness is demonstrated earlier in the letter. Specifically,
in the unbiased junction a potential breather-only ${(\Omega,A)}$ combination
produces a very low average voltage. If the latter case is encountered, a
sufficiently high current bias ($\gamma\gtrsim 0.1$, see, e.g., Ref. [14])
should be switched on after a time interval much greater than ${1/\alpha}$
from the magnetic pulse’s application. If the low average voltage persists,
breathers only are quite likely being generated, and due to their radiative
decay, no current-driven dissociation into the kink-antikink state can take
place. Conversely, if fluxon-related voltage patterns arise after the current
switch, indicating the presence of at least one kink (or antikink), another
${(\Omega,A)}$ couple can be tried. To summarize, in the case of a persisting
low average voltage, one is observing high-frequency excitations which vanish
within a time interval ${\sim 1/\alpha}$. Linear (plasma) waves can be
reasonably excluded since the junction’s FBG is being considered, and besides
the preliminary ${(\Omega,A)}$ sweep confirms that fluxon-based modes are
being excited. The existence of Josephson breathers would be supported by this
observation.

_Conclusions._ —The present work shows that magnetic pulses can be a
controllable source of travelling breather modes in an LJJ. More precisely, by
looking at the average voltage drop across the junction, refined bifurcation
diagrams are produced in the driving pulse’s ${(\Omega,A)}$ space, in the
presence of dissipation, a current bias, and a thermal noise source. This
allows for the characterization of significant regions where the exclusive
formation of breathers occurs. Moreover, these areas are seen to maintain
their identity in the noisy case, and a sort of noise-induced widening is
found as well. The latter fact happens in correspondence with a nonmonotonic
behavior of the breather-only generation probability, defined above, as a
function of the noise amplitude.

Studies that focus on generation and control techniques are vital for all
breather-related applications, e.g., in information transmission [18] and
quantum computation [19, 20]. Furthermore, given that the detection of
Josephson breathers is a long-standing problem in the realm of mesoscopic
soliton physics, an experimental strategy to confirm their existence is
proposed.

Finally, the degree of universality of the ST process, together with its
robustness against discreteness and finiteness, suggests applying this general
approach to the discrete world. For example, interesting developments can be
expected for oscillobreathers in JJ parallel arrays, but energy localization
and transmission is being actively investigated in many other contexts as of
today [31, 32, 33].

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# Weak convergence of delay SDEs with applications to Carathéodory
approximation

T. C. Son Department of Mathematics, VNU University of Science, Vietnam
National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnam.
N. T. Dung∗ Corresponding author. Email<EMAIL_ADDRESS>N. V. Tan Department
of Foundation, Academy of Cryptography Techniques, 141 Chien Thang, Thanh Tri,
Hanoi, Vietnam T. M. Cuong∗ H. T. P. Thao∗ P. D. Tung∗

(September 4, 2021)

###### Abstract

In this paper, we consider a fundamental class of stochastic differential
equations with time delays. Our aim is to investigate the weak convergence
with respect to delay parameter of the solutions. Based on the techniques of
Malliavin calculus, we obtain an explicit estimate for the rate of
convergence. An application to the Carathéodory approximation scheme of
stochastic differential equations is provided as well.

_Keywords:_ Delay SDEs, Weak convergence, Carathéodory approximation,
Malliavin calculus.
2010 Mathematics Subject Classification: 65C30, 60H10, 60H07.

##  1 Introduction

It is known that the Carathéodory approximation scheme was introduced by
Carathéodory in the early part of 20th century for deterministic differential
equations, see e.g. [5]. In the context of stochastic equations, the first
Carathéodory approximation results were obtained by Bell and Mohammed [3]
(also see Section 2.6 in [14] for a general formulation). Let
$x_{0}\in\mathbb{R},n\geq 1$ and $(B(t))_{t\in[0,T]}$ be a standard Brownian
motion. We consider stochastic differential equation (SDE)

$x(t)=x_{0}+\int_{0}^{t}b(s,x(s))ds+\int_{0}^{t}\sigma(s,x(s))dB(s),\,\,\,t\in[0,T]$
(1.1)

and its Carathéodory approximation

$\left\\{\begin{array}[]{l}x^{n}(t)=x_{0}+\int_{0}^{t}b(s,x^{n}(s-1/n))ds+\int_{0}^{t}\sigma(s,x^{n}(s-1/n))dB(s),\,\,\,t\in[0,T]\\\
\\\ x^{n}(t)=x_{0},\,\,\,t\in[-1/n,0].\end{array}\right.$ (1.2)

It was proved in [3, 14] that, when the coefficients are Lipschitz and have
linear growth, $x^{n}(t)$ strongly converges to $x(t)$ as $n\to\infty.$
Moreover, the following estimate for the strong rate of convergence holds

$E\left[\sup\limits_{0\leq t\leq
T}|x^{n}(t)-x(t)|^{2}\right]\leq\frac{C}{n},\,\,\,n\geq 1,$ (1.3)

where $C$ is a positive constant not depending on $n$

As discussed in Section 2.6 of [14], the advantage of Carathéodory
approximation is that we do not need to compute $x^{1}(t),\cdots,x^{n-1}(t)$
but compute $x^{n}(t)$ directly (this reduces a lot of calculations on
stochastic integrals). In addition, the Carathéodory approximation also works
well for SDEs with non-Lipschitz coefficients. In the last decades, the
Carathéodory approximation has been considered for various stochastic
differential equations. Among others, we mention Turo [19] for stochastic
functional differential equations, Mao [12, 13] and Liu [11] for semilinear
stochastic evolution equations with time delays, Ferrante & Rovira [8] for
delay differential equations driven by fractional Brownian motion, Faizullah
[7] for SDEs under G-Brownian motion, Benabdallah & Bourza [2] for perturbed
SDEs with reflecting boundary, Mao et al. [15] for doubly perturbed SDEs, etc.

It is also known that the weak convergence rate of numerical approximations is
very useful in practical applications (see, e.g. [1] for a short discussion).
In fact, for certain numerical schemes such as Milstein scheme, Runge-Kutta
scheme and Euler-Maruyama scheme, etc. many weak convergence results can be
found in the literature, see e.g. [1, 4, 9, 10]. However, to the best of our
knowledge, the weak convergence results of the Carathéodory approximation are
scarce even for the system (1.1)-(1.2). Motivated by this observation, our aim
is to partially fill up this gap.

In this paper, to make the problem more interesting, we consider delay
stochastic differential equations of the form

$\left\\{\begin{array}[]{l}X_{\tau_{1}}(t)=\varphi(0)+\int_{0}^{t}b(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))ds+\int_{0}^{t}\sigma(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))dB(s),\,t\in[0,T]\\\
\\\ X_{\tau_{1}}(t)=\varphi(t),\,\,\,t\in[-\tau_{1},0]\end{array}\right.$
(1.4)

and

$\left\\{\begin{array}[]{l}X_{\tau_{2}}(t)=\varphi(0)+\int_{0}^{t}b(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))ds+\int_{0}^{t}\sigma(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))dB(s),\,t\in[0,T]\\\
\\\ X_{\tau_{2}}(t)=\varphi(t),\,\,\,t\in[-\tau_{2},0],\end{array}\right.$
(1.5)

where $0\leq\tau_{1},\tau_{2}<\infty$ and $\varphi:(-\infty,0]\to\mathbb{R}$
is a bounded deterministic function. Our aim is to study the weak convergence
of $X_{\tau_{2}}(t)$ to $X_{\tau_{1}}(t)$ as $\tau_{2}\to\tau_{1}.$ More
specifically, we will employ the techniques of Malliavin calculus to provide a
quantitative estimate for the quantity

$|Eg(X_{\tau_{2}}(t))-Eg(X_{\tau_{1}}(t))|,$

where $g$ is a bounded and measurable function, see Theorem 3.1. Our method is
different from the existing ones in the literature and based on a new result
established recently in [6]. The restatement of this new result and further
comments will be given in Lemma 2.1 and Remark 2.1.

Our results applied to the Carathéodory approximation system (1.1)-(1.2) yield

$\sup\limits_{0\leq t\leq
T}|Eg(x^{n}(t))-Eg(x(t))|\leq\frac{C}{\sqrt{n}},\,\,\,n\geq 1.$ (1.6)

It should be noted that, when the test function $g$ is Lipschitz continuous,
the strong rate (1.3) implies the weak rate (1.6). However, for bounded and
measurable test functions, there is no such implication and hence, the novelty
of our results lies in the fact that (1.6) holds true for any bounded and
measurable test function. The price to pay is that, besides Lipschitz and
linear growth conditions, we have to impose additional assumptions on the
coefficients, see conditions $(ii)$-$(iii)$ of Corollary 3.3.

The rest of this article is organized as follows. In Section 2, we recall some
concepts of Malliavin calculus and a general result obtained in our recent
work [6]. Our main results are then stated and proved in Section 3. The
conclusion and some remarks are given in Section 4.

##  2 Preliminaries

As we have said in the Introduction, this paper is based on techniques of
Malliavin calculus. For the reader’s convenience, let us recall some elements
of Malliavin calculus (for more details see [16]). We suppose that
$(B(t))_{t\in[0,T]}$ is defined on a complete probability space
$(\Omega,\mathcal{F},\mathbb{F},P)$, where
$\mathbb{F}=(\mathcal{F}_{t})_{t\in[0,T]}$ is a natural filtration generated
by the Brownian motion $B.$ For $h\in L^{2}[0,T],$ we denote by $B(h)$ the
Wiener integral

$B_{h}=\int_{0}^{T}h(t)dB(t).$

Let $\mathcal{S}$ denote the dense subset of $L^{2}(\Omega,\mathcal{F},P)$
consisting of smooth random variables of the form

$F=f(B_{h_{1}},...,B_{h_{n}}),$ (2.1)

where $n\in\mathbb{N},h_{1},...,h_{n}\in L^{2}[0,T]$ and $f$ is an infinitely
differentiable function such that together with all its partial derivatives
has at most polynomial growth order. If $F$ has the form (2.1), we define its
Malliavin derivative as the process $DF:=\\{D_{t}F,t\in[0,T]\\}$ given by

$D_{t}F=\sum\limits_{k=1}^{n}\frac{\partial f}{\partial
x_{k}}(B_{h_{1}},...,B_{h_{n}})h_{k}(t).$

More generally, for each $k\geq 1,$ we can define the iterated derivative
operator by setting $D^{k}_{t_{1},...,t_{k}}F=D_{t_{1}}...D_{t_{k}}F.$ For any
$p,k\geq 1,$ we shall denote by $\mathbb{D}^{k,p}$ the closure of
$\mathcal{S}$ with respect to the norm

$\|F\|^{p}_{k,p}:=E|F|^{p}+E\bigg{[}\int_{0}^{T}|D_{t_{1}}F|^{p}dt_{1}\bigg{]}+...+E\bigg{[}\int_{0}^{T}...\int_{0}^{T}|D^{k}_{t_{1},...,t_{k}}F|^{p}dt_{1}...dt_{k}\bigg{]}.$

A random variable $F$ is said to be Malliavin differentiable if it belongs to
$\mathbb{D}^{1,2}.$ For the convenience of the reader, we recall that the
derivative operator $D$ satisfy the chain rule, i.e.

$D\varphi(F)=\varphi^{\prime}(F)DF.$ (2.2)

Moreover, we have the following relations between Malliavin derivative and the
integrals

$D_{r}\left(\int_{0}^{T}u(s)ds\right)=\int_{r}^{T}D_{r}u(s)ds,$
$D_{r}\left(\int_{0}^{T}u(s)dB(s)\right)=u_{r}+\int_{r}^{T}D_{r}u(s)dB(s)$

for all $0\leq r\leq T,$ where $(u(t))_{t\in[0,T]}$ is an $\mathbb{F}$-adapted
and Malliavin differentiable stochastic process.

An important operator in the Malliavin’s calculus theory is the divergence
operator $\delta$, it is the adjoint of the derivative operator $D$. The
domain of $\delta$ is the set of all functions $u\in L^{2}(\Omega,L^{2}[0,T])$
such that for all $F\in\mathbb{D}^{1,2}$ it holds that

$E|\langle DF,u\rangle_{L^{2}[0,T]}|\leq C(u)\|F\|_{L^{2}[0,T]},$

where $C(u)$ is some positive constant depending on $u$. In particular, if
$u\in dom\delta$, then $\delta(u)$ is characterized by following duality
relationship

$E\langle DF,u\rangle_{L^{2}[0,T]}=E[F\delta(u)]\text{ for any
}F\in\mathbb{D}^{1,2}.$ (2.3)

Let $\mathcal{B}$ denote the space of measurable functions
$g:\mathbb{R}\to\mathbb{R}$ such that
$\|g\|_{\infty}:=\sup\limits_{x\in\mathbb{R}}|g(x)|\leq 1.$ We have the
following.

######  Lemma 2.1.

Let $F_{1}\in\mathbb{D}^{2,2}$ satisfy that
$\|DF_{1}\|_{L^{2}[0,T]}>0\,\,a.s.$ Then, for any random variable
$F_{2}\in\mathbb{D}^{1,2}$ and any $g\in\mathcal{B},$ we have

$\displaystyle|Eg(F_{1})-Eg(F_{2})|$ $\displaystyle\leq
C\left(E\|DF_{1}\|^{-8}_{L^{2}[0,T]}E\left(\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr\right)^{2}+(E\|DF_{1}\|^{-2}_{L^{2}[0,T]})^{2}\right)^{\frac{1}{4}}\|F_{1}-F_{2}\|_{1,2},$

provided that the expectations exist, where $C$ is an absolute constant.

###### Proof.

This lemma comes from Theorem 3.1 in our recent paper [6]. Here, for the
reader’s convenience, we recall its proof. We write $\langle.,.\rangle$
instead of $\langle.,.\rangle_{L^{2}[0,T]}$ and $\|.\|$ instead of
$\|.\|_{L^{2}[0,T]}.$ By the routine approximation argument, we can assume
that $g$ is continuous. Indeed, for example, we can approximate $g$ by bounded
and Lipschitz continuous functions defined by

$g_{\varepsilon}(x)=\int_{-\infty}^{\infty}{\rm
11}_{\\{|y|<\frac{1}{\varepsilon}\\}}g(y)\rho_{\varepsilon}(x-y)dy,\,\,\varepsilon>0,$

where $\rho_{\varepsilon}$ is the standard mollifier:
$\rho_{\varepsilon}(x)=\frac{\rho(x/\varepsilon)}{\varepsilon},$ where
$\rho(x)=C{\rm 11}_{\\{|x|<1\\}}e^{\frac{1}{x^{2}-1}}$ and $C$ is a constant
such that $\int_{-\infty}^{\infty}\rho(x)dx=1.$

The function $\psi(y):=\int_{-\infty}^{y}g(z)dz$ is differentiable with
bounded derivative. Hence, $\psi(F_{i})\in\mathbb{D}^{1,2}$ for $i=1,2.$ We
obtain

$\langle D\psi(F_{i}),DF_{1}\rangle=g(F_{i})\langle
DF_{i},DF_{1}\rangle,\,\,\,i=1,2,$

or equivalently

$\langle D\int_{-\infty}^{F_{i}}g(z)dz,DF_{1}\rangle=g(F_{i})\langle
DF_{i},DF_{1}\rangle,\,\,\,i=1,2.$

As a consequence, we get

$\displaystyle\langle$ $\displaystyle
D\int_{F_{2}}^{F_{1}}g(z)dz,DF_{1}\rangle=g(F_{1})\langle
DF_{1},DF_{1}\rangle-g(F_{2})\langle DF_{2},DF_{1}\rangle$
$\displaystyle=\left(g(F_{1})-g(F_{2})\right)\langle
DF_{1},DF_{1}\rangle+g(F_{2})\langle DF_{1}-DF_{2},DF_{1}\rangle,$

and hence,

$g(F_{1})-g(F_{2})=\dfrac{\langle
D\int_{F_{2}}^{F_{1}}g(z)dz,DF_{1}\rangle}{\|DF_{1}\|^{2}}-\dfrac{g(F_{2})\langle
DF_{1}-DF_{2},DF_{1}\rangle}{\|DF_{1}\|^{2}}.$

Taking the expectation yields

$\displaystyle Eg(F_{1})-Eg(F_{2})$ $\displaystyle=E\left[\dfrac{\langle
D\int_{F_{2}}^{F_{1}}g(z)dz,DF_{1}\rangle}{\|DF_{1}\|^{2}}\right]-E\left[\dfrac{g(F_{2})\langle
DF_{1}-DF_{2},DF_{1}\rangle}{\|DF_{1}\|^{2}}\right].$ (2.4)

Now we consider the stochastic process

$u_{r}:=\dfrac{D_{r}F_{1}}{\|DF_{1}\|^{2}},\ \ 0\leq r\leq T$

and use Proposition 1.3.1 in [16] to get

$\displaystyle E[\delta(u)^{2}]$
$\displaystyle\leq\int_{0}^{T}E|u_{r}|^{2}dr+\int_{0}^{T}\int_{0}^{T}E|D_{\theta}u_{r}|^{2}d\theta
dr$
$\displaystyle=E\|DF_{1}\|^{-2}+\int_{0}^{T}\int_{0}^{T}E|D_{\theta}u_{r}|^{2}d\theta
dr$ (2.5)

By the chain rule (2.2) for Malliavin derivatives, we have

$D_{\theta}u_{r}=\dfrac{D_{\theta}D_{r}F_{1}}{\|DF_{1}\|^{2}}-2\dfrac{D_{r}F_{1}\langle
D_{\theta}DF_{1},DF_{1}\rangle}{\|DF_{1}\|^{4}},\ \ 0\leq\theta,r\leq T.$

By the Cauchy-Schwarz inequality, we obtain

$\displaystyle\int_{0}^{T}\int_{0}^{T}E|D_{\theta}u_{r}|^{2}d\theta dr$
$\displaystyle\leq
2E\left[\dfrac{\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr}{\|DF_{1}\|^{4}}\right]+8E\int_{0}^{T}\int_{0}^{T}\dfrac{|D_{r}F_{1}|^{2}\|D_{\theta}DF_{1}\|^{2}}{\|DF_{1}\|^{6}}d\theta
dr$
$\displaystyle=2E\left[\dfrac{\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr}{\|DF_{1}\|^{4}}\right]+8E\left[\dfrac{\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr}{\|DF_{1}\|^{4}}\right]$ $\displaystyle\leq
10\left(E\left[\left(\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr\right)^{2}\right]\right)^{\frac{1}{2}}\left(E\left[\|DF_{1}\|^{-8}\right]\right)^{\frac{1}{2}}.$
(2.6)

Inserting (2.6) into (2.5) yields

$E[\delta(u)^{2}]\leq
E\|DF_{1}\|^{-2}+10\left(E\left[\left(\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr\right)^{2}\right]\right)^{\frac{1}{2}}\left(E\left[\|DF_{1}\|^{-8}\right]\right)^{\frac{1}{2}}.$
(2.7)

By using the relation (2.3), the Cauchy-Schwarz inequality and
$\|g\|_{\infty}\leq 1,$ it follows from (2.4) that

$\displaystyle|Eg(F_{1})-Eg(F_{2})|\leq
E\left|\int_{F_{2}}^{F_{1}}g(z)dz\delta\left(u\right)\right|+E\left|\dfrac{g(F_{2})\langle
DF_{1}-DF_{2},DF_{1}\rangle}{\|DF_{1}\|^{2}}\right|$
$\displaystyle\leq\left(E|F_{1}-F_{2}|^{2}\right)^{\frac{1}{2}}\left(E[\delta\left(u\right)^{2}]\right)^{\frac{1}{2}}+E\left|\dfrac{\langle
DF_{1}-DF_{2},DF_{1}\rangle}{\|DF_{1}\|^{2}}\right|$
$\displaystyle\leq\left(E|F_{1}-F_{2}|^{2}\right)^{\frac{1}{2}}\left(E[\delta\left(u\right)^{2}]\right)^{\frac{1}{2}}+E\left[\dfrac{\|DF_{1}-DF_{2}\|}{\|DF_{1}\|}\right]$
$\displaystyle\leq\left(E|F_{1}-F_{2}|^{2}\right)^{\frac{1}{2}}\left(E[\delta\left(u\right)^{2}]\right)^{\frac{1}{2}}+\left(E\|DF_{1}-DF_{2}\|^{2}\right)^{\frac{1}{2}}\left(E\|DF_{1}\|^{-2}\right)^{\frac{1}{2}}.$
(2.8)

So we can obtain the desired conclusion by inserting (2.7) into (2.8) and then
using Cauchy-Schwarz inequality. ∎

######  Remark 2.1.

Lemma 2.1 shows that the weak convergence can be controlled by the norm
$\|.\|_{1,2}$ in the space of Malliavin differentiable random variables. In
its statement, we do not require any special structure of $F_{1}$ and $F_{2}.$
We therefore believe that Lemma 2.1 can be applied to other numerical schemes
such as Euler-Maruyama and Runge-Kutta schemes. In the present paper, the main
reason for choosing Carathéodory scheme is due to the lack of weak convergence
results for this scheme in the literature. It is also worth mentioning that
the proof of Lemma 2.1 heavily relies on dimension one. The generalization to
higher dimensions will be a difficult and interesting problem.

##  3 The main results

In the whole paper, we are going to impose the following assumptions:

######  Assumption 3.1.

The functions $b,\sigma:[0,T]\times\mathbb{R}^{2}\to\mathbb{R}$ are Lipschitz
and have linear growth, i.e. there exists $L>0$ such that

$|b(t,x_{1},y_{1})-b(t,x_{2},y_{2})|+|\sigma(t,x_{1},y_{1})-\sigma(t,x_{2},y_{2})|\leq
L(|x_{1}-x_{2}|+|y_{1}-y_{2}|)$

for all $t\in[0,T],x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}$ and

$|b(t,x,y)|+|\sigma(t,x,y)|\leq
L(|x|+|y|+1),\,\,\forall\,t\in[0,T],x,y\in\mathbb{R}.$

######  Assumption 3.2.

For every $t\in[0,T],$ $b(t,.,.),\sigma(t,.,.)$ are twice differentiable
functions with the partial derivatives bounded by $L$.

Hereafter, we denote by $C$ (with or without an index) a generic constant
which may vary at each appearance. For any function $h$ of $n$ variables, we
denote

$h^{\prime}_{i}(x_{1},\cdots,x_{n}):=\frac{\partial h}{\partial
x_{i}}(x_{1},\cdots,x_{n}),\quad
h^{\prime\prime}_{ij}(x_{1},\cdots,x_{n}):=\frac{\partial h}{\partial
x_{i}\partial x_{j}}(x_{1},\cdots,x_{n}).$

For any $a,b\in\mathbb{R},$ we denote $a\vee b=\max\\{a,b\\}.$ In the proofs,
we frequently use the fundamental inequality

$(a_{1}+\cdots+a_{n})^{p}\leq n^{p-1}(a_{1}^{p}+\cdots+a_{n}^{p})$

for all $a_{1},...,a_{n}\geq 0$ and $p\geq 2.$

###  3.1 Malliavin derivative of delay SDEs

In this Subsection, given $\tau\geq 0,$ we investigate Malliavin derivatives
of the solution to the following delay SDE

$\left\\{\begin{array}[]{l}X_{\tau}(t)=\varphi(0)+\int_{0}^{t}b(s,X_{\tau}(s),X_{\tau}(s-\tau))ds+\int_{0}^{t}\sigma(s,X_{\tau}(s),X_{\tau}(s-\tau))dB(s),\,\,\,t\in[0,T],\\\
\\\ X_{\tau}(t)=\varphi(t),\quad t\in[-\tau,0].\end{array}\right.$ (3.1)

For the Malliavin differentiability of the solutions, we have the following.

######  Proposition 3.1.

Let Assumption 3.1 hold. Then, the equation (3.1) has a unique solution
$(X_{\tau}(t))_{t\in[-\tau,T]}$ and this solution is Malliavin differentiable.
Moreover, the derivative $D_{\theta}X_{\tau}(t)$ satisfies

(i) When $t\in[-\tau,0],$ $D_{\theta}X_{\tau}(t)=0$ for all $0\leq\theta\leq
T,$

(ii) When $t\in(0,T],$ $D_{\theta}X_{\tau}(t)=0$ for $\theta>t$ and

$\displaystyle D_{\theta}X_{\tau}(t)$
$\displaystyle=\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))+\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds+\int_{\theta+\tau}^{t}\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau)ds$
$\displaystyle+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s)+\int_{\theta+\tau}^{t}\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau)dB(s),\,\,\,0\leq\theta\leq
t-\tau,$ (3.2) $\displaystyle
D_{\theta}X_{\tau}(t)=\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))+\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s),\,\,\,(t-\tau)\vee
0<\theta\leq t,$ (3.3)

where $\bar{b}^{\tau}_{2}(s)$, $\bar{b}^{\tau}_{3}(s)$,
$\bar{\sigma}^{\tau}_{2}(s)$, $\bar{\sigma}^{\tau}_{3}(s)$ are adapted
stochastic processes and bounded by the Lipschitz constant $L.$ Here, we use
the convention $[0,t-\tau]=\emptyset$ if $t<\tau.$

###### Proof.

For $t\in[-\tau,0]$, we have $X_{\tau}(t)=\varphi(t)$ is deterministic. Hence,
the Malliavin derivative of $X_{\tau}(t)$ always vanishes. On the other hand,
since the solution $(X_{\tau}(t))_{t\in[0,T]}$ is $\mathbb{F}$-adapted, we
always have $D_{\theta}X_{\tau}(t)=0$ for $\theta>t$. When $\theta\leq t$, by
using the same argument as in the proof of Theorem 2.2.1 in [16], we can show
that the solution $(X_{\tau}(t))_{t\in[0,T]}$ is also Malliavin
differentiable. Applying the operator $D$ to the equation (3.1) we deduce

$\displaystyle D_{\theta}X_{\tau}(t)$
$\displaystyle=\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))$
$\displaystyle+\int_{\theta}^{t}D_{\theta}[b(s,X_{\tau}(s),X_{\tau}(s-\tau))]ds+\int_{\theta}^{t}D_{\theta}[\sigma(s,X_{\tau}(s),X_{\tau}(s-\tau))]dB_{s},\,\,\,t\in(0,T].$
(3.4)

By Proposition 1.2.4 in [16] and Lipchitz property of $b$ and $\sigma$, there
exist adapted processes $\bar{b}^{\tau}_{2}(s)$, $\bar{b}^{\tau}_{3}(s)$,
$\bar{\sigma}^{\tau}_{2}(s)$ and $\bar{\sigma}^{\tau}_{3}(s)$, uniformly
bounded by $L$ such that

$D_{\theta}\left[b(s,X_{\tau}(s),X_{\tau}(s-\tau))\right]=\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)+\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau),$
(3.5)
$D_{\theta}\left[\sigma(s,X_{\tau}(s),X_{\tau}(s-\tau))\right]=\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)+\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau).$
(3.6)

Because $D_{\theta}X_{\tau}(s-\tau)=0$ for $\theta>s-\tau,$ we obtain the
equations (3.2) and (3.3) by inserting (3.5) and (3.6) into (3.4). So the
proof of Proposition is complete. ∎

######  Remark 3.1.

If $b(t,\cdot,\cdot)$ and $\sigma(t,\cdot,\cdot)$ are continuously
differentiable, then
$\bar{b}^{\tau}_{2}(s)=b_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))$,
$\bar{b}^{\tau}_{3}(s)=b_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))$,
$\bar{\sigma}^{\tau}_{2}(s)=\sigma_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))$,
$\bar{\sigma}^{\tau}_{3}(s)=\sigma_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))$.
Consequently, under Assumptions 3.1 and 3.2, we have

$\displaystyle D_{\theta}X_{\tau}(t)$
$\displaystyle=\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))+\int_{\theta}^{t}b_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)ds$
$\displaystyle+\int_{\theta+\tau}^{t}b_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s-\tau)ds+\int_{\theta}^{t}\sigma_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)dB(s)$
$\displaystyle+\int_{\theta+\tau}^{t}\sigma_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s-\tau)dB(s),\,\,0\leq\theta\leq
t-\tau$ (3.7)

and

$\displaystyle D_{\theta}X_{\tau}(t)$
$\displaystyle=\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))+\int_{\theta}^{t}b_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)ds$
$\displaystyle+\int_{\theta}^{t}\sigma_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)dB(s),\,\,\,(t-\tau)\vee
0<\theta\leq t.$ (3.8)

Moreover, $X_{\tau}(t)$ is twice Malliavin differentiable. The proof of this
fact is similar to that of Theorem 2.2.2 in [16], we omit the details. The
second order Malliavin derivative of $X_{\tau}(t)$ will be computed in
Proposition 3.3 below.

We have the following familiar estimates for the moments.

######  Lemma 3.1.

Let Assumption 3.1 hold. Then, for every $p\geq 2,$ we have

$E|X_{\tau}(t)|^{p}\leq C_{p,L,T},\,\,\,t\in[0,T]$ (3.9)

and

$E|X_{\tau}(t)-X_{\tau}(s)|^{p}\leq
C_{p,L,T}|t-s|^{\frac{p}{2}},\,\,\,s,t\in[0,T],$ (3.10)

where $C_{p,L,T}$ is a positive constant.

###### Proof.

The proof is similar to that of Lemmas 6.1 and 6.2 in [14]. So we omit it.
Here, just remark that we use the Burkholder-Davis-Gundy inequality instead of
the Itô isometry. ∎

######  Lemma 3.2.

Let Assumption 3.1 hold. Then, for every $p\geq 2,$ we have

$E|D_{\theta}X_{\tau}(t)|^{p}\leq C_{p,L,T},\,\,\,0\leq\theta\leq t\leq T$
(3.11)

and

$E|D_{\theta}X_{\tau}(t)-D_{\theta}X_{\tau}(s)|^{p}\leq
C_{p,L,T}|t-s|^{p/2},\,\,\,0\leq\theta\leq s\vee t\leq T,$ (3.12)

where $C_{p,L,T}$ is a positive constant.

###### Proof.

By the linear growth property of $\sigma$ and the estimate (3.9) we have

$E|\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))|^{p}\leq
C_{p,L,T},\,\,\forall\,\,0\leq\theta\leq T.$ (3.13)

It follows from equations (3.2) and (3.3) that the Malliavin derivative
$D_{\theta}X_{\tau}(t)$ satisfies

$\displaystyle D_{\theta}X_{\tau}(t)$
$\displaystyle=\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))+\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds+\int_{\theta}^{t}\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds$
$\displaystyle+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s)+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s),\,\,\,0\leq\theta\leq t\leq T.$ (3.14)

We therefore get

$\displaystyle E$ $\displaystyle|D_{\theta}X_{\tau}(t)|^{p}$
$\displaystyle\leq
5^{p-1}\bigg{(}E|\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))|^{p}+E\left|\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds\right|^{p}+E\left|\int_{\theta}^{t}\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds\right|^{p}$
$\displaystyle\quad+E\left|\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s)\right|^{p}+E\left|\int_{\theta}^{t}\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s)\right|^{p}\bigg{)},\,\,\,0\leq\theta\leq t\leq
T.$

By the boundedness of $\bar{b}^{\tau}_{2}(s)$, $\bar{b}^{\tau}_{3}(s)$,
$\bar{\sigma}^{\tau}_{2}(s)$, $\bar{\sigma}^{\tau}_{3}(s),$ the estimate
(3.13) and the Hölder and Burkholder-Davis-Gundy inequalities, it is easy to
see that

$\displaystyle E|D_{\theta}X_{\tau}(t)|^{p}$ $\displaystyle\leq
C_{p,L,T}+C_{p,L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau}(s)|^{p}ds+C_{p,L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau}(s-\tau)|^{p}ds$
$\displaystyle\leq
C_{p,L,T}+C_{p,L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau}(s)|^{p}ds,\,\,\,0\leq\theta\leq
t\leq T,$

where $C_{p,L,T}$ is some positive constant. So, we can obtain (3.11) by using
Gronwall’s lemma.

In order to prove (3.12), we assume, without the loss of generally, that
$s<t$. We consider three cases separately.

Case 1: $0\leq\theta\leq s-\tau$. From the equation (3.2), we have

$\displaystyle D_{\theta}X_{\tau}(t)-D_{\theta}X_{\tau}(s)$
$\displaystyle=\int_{s}^{t}\bar{b}_{2}^{\tau}(u)D_{\theta}X_{\tau}(u)du+\int_{s}^{t}\bar{b}_{3}^{\tau}(u)D_{\theta}X_{\tau}(u-\tau)du$
$\displaystyle+\int_{s}^{t}\bar{\sigma}_{2}^{\tau}(u)D_{\theta}X_{\tau}(u)dB_{u}+\int_{s}^{t}\bar{\sigma}_{3}^{\tau}(u)D_{\theta}X_{\tau}(u-\tau)du.$

Using the Hölder and Burkholder-Davis-Gundy inequalities and (3.11), we have

$\displaystyle E|D_{\theta}X_{\tau}(t)-D_{\theta}X_{\tau}(s)|^{p}$
$\displaystyle\leq
C_{p,L,T}\left((t-s)^{p-1}\int_{s}^{t}E|D_{\theta}X_{\tau}(u)|^{p}du+(t-s)^{\frac{p}{2}-1}\int_{s}^{t}E|D_{\theta}X_{\tau}(u)|^{p}du\right)$
$\displaystyle\leq C_{p,L,T}|t-s|^{p/2}.$

Case 2: $s-\tau<\theta\leq t-\tau$. From the equations (3.2) and (3.3) we have

$\displaystyle D_{\theta}X_{\tau}(t)-D_{\theta}X_{\tau}(s)$
$\displaystyle=\int_{s}^{t}\bar{b}_{2}^{\tau}(u)D_{\theta}X_{\tau}(u)du+\int_{\theta+\tau}^{t}\bar{b}_{3}^{\tau}(u)D_{\theta}X_{\tau}(u-\tau)du$
$\displaystyle+\int_{s}^{t}\bar{\sigma}_{2}^{\tau}(u)D_{\theta}X_{\tau}(u)dB_{u}+\int_{\theta+\tau}^{t}\bar{\sigma}_{3}^{\tau}(u)D_{\theta}X_{\tau}(u-\tau)du.$

Using the same arguments as in the proof of Case 1, we obtain

$\displaystyle E|D_{\theta}X_{\tau}(t)-D_{\theta}X_{\tau}(s)|^{p}$
$\displaystyle\leq C_{p,L,T}\left(|t-s|^{p/2}+|t-\theta-\tau|^{p/2}\right)$
$\displaystyle\leq C_{p,L,T}|t-s|^{p/2}.$

Case 3: $t-\tau<\theta\leq t$. From $\eqref{mall2}$ we have

$\displaystyle
D_{\theta}X_{\tau}(t)-D_{\theta}X_{\tau}(s)=\int_{s}^{t}\bar{b}_{2}^{\tau}(u)D_{\theta}X_{\tau}(u)du+\int_{s}^{t}\bar{\sigma}_{2}^{\tau}(u)D_{\theta}X_{\tau}(u)dB_{u},$

and hence, we also have

$E|D_{\theta}X_{\tau}(t)-D_{\theta}X_{\tau}(s)|^{p}\leq C_{p,L,T}|t-s|^{p/2}.$

This finishes the proof of Proposition. ∎

######  Proposition 3.2.

Let Assumptions 3.1 hold and, in addition, we assume that

$|\sigma(t,x,y)|\geq\sigma_{0}>0,\,\,\forall\,t\in[0,T],x,y\in\mathbb{R}.$

Then, for every $p\geq 1$ and for all $0<t\leq T,$ we have

$E\left[\frac{1}{\|DX_{\tau}(t)\|^{2p}_{L^{2}[0,T]}}\right]\leq
C_{p,L,T}\,t^{-p},$ (3.15)

where $C_{p,L,T}$ is a positive constant.

###### Proof.

Fixed $t\in(0,T].$ By using the fundamental inequality
$(a+b+c)^{2}\geq\frac{a^{2}}{2}-2(b^{2}+c^{2}),$ we obtain from the equation
(3.14) that

$\displaystyle|D_{\theta}X_{\tau}(t)|^{2}$
$\displaystyle\geq\frac{1}{2}\sigma^{2}(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))-2\bigg{(}\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds+\int_{\theta}^{t}\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds\bigg{)}^{2}$
$\displaystyle-2\bigg{(}\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s)+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s)\bigg{)}^{2},\,\,\,0\leq\theta\leq t\leq T.$

For each $y\geq y_{0}:=\frac{4}{t\sigma_{0}^{2}},$ the real number
$\varepsilon:=\frac{4}{y\sigma_{0}^{2}t}$ belongs to $(0,1].$ Hence,

$\displaystyle\|DX_{\tau}(t)\|^{2}_{L^{2}[0,T]}$
$\displaystyle\geq\int_{t(1-\varepsilon)}^{t}|D_{\theta}X_{\tau}(t)|^{2}d\theta\geq\int_{t(1-\varepsilon)}^{t}\frac{\sigma^{2}(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))}{2}d\theta$
$\displaystyle-2\int_{t(1-\varepsilon)}^{t}\bigg{(}\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds+\int_{\theta}^{t}\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds\bigg{)}^{2}d\theta$
$\displaystyle-2\int_{t(1-\varepsilon)}^{t}\bigg{(}\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s)+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s)\bigg{)}^{2}d\theta$
$\displaystyle\geq\frac{\sigma_{0}^{2}t\varepsilon}{2}-I_{y}(t)=\frac{2}{y}-I_{y}(t),$

where

$\displaystyle I_{y}(t)$
$\displaystyle:=2\int_{t(1-\varepsilon)}^{t}\bigg{(}\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds+\int_{\theta}^{t}\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds\bigg{)}^{2}d\theta$
$\displaystyle+2\int_{t(1-\varepsilon)}^{t}\bigg{(}\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s)+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s)\bigg{)}^{2}d\theta.$

Then, by Markov inequality, we obtain

$\displaystyle
P\left(\|DX_{\tau}(t)\|^{2}_{L^{2}[0,T]}\leq\frac{1}{y}\right)\leq
P\left(\frac{2}{y}-I_{y}(t)\leq\frac{1}{y}\right)=P\left(I_{y}(t)\geq\frac{1}{y}\right)\leq
y^{q/2}E\left(|I_{y}(t)|^{q/2}\right)\,\,\,\forall\,\,q\geq 2.$ (3.16)

By the inequality $(|a|+|b|)^{q/2}\leq 2^{q/2-1}(|a|^{q/2}+(|b|^{q/2}),$ we
get

$\displaystyle E|I_{y}(t)|^{q/2}$ $\displaystyle\leq
2^{q-1}\bigg{[}E\left(\bigg{(}\int_{\theta}^{t}\bar{b}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)ds+\int_{\theta}^{t}\bar{b}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds\bigg{)}^{2}d\theta\right)^{q/2}$
$\displaystyle+E\left(\int_{t(1-\varepsilon)}^{t}\bigg{(}\int_{\theta}^{t}\bar{\sigma}^{\tau}_{2}(s)D_{\theta}X_{\tau}(s)dB(s)+\int_{\theta}^{t}\bar{\sigma}^{\tau}_{3}(s)D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s)\bigg{)}^{2}d\theta\right)^{q/2}\bigg{]}.$

By using the Hölder and Burkholder-Davis-Gundy inequalities, it follows from
(3.11) that

$\displaystyle E|I_{y}(t)|^{q/2}$ $\displaystyle\leq
C_{q,L,T}(t\varepsilon)^{\frac{q-2}{2}}\left(\int_{t(1-\varepsilon)}^{t}(t-\theta)^{\frac{q-2}{2}}\int_{\theta}^{t}E|D_{\theta}X_{\tau}(s)|^{p}dsd\theta+\int_{t(1-\varepsilon)}^{t}\left(\int_{\theta}^{t}E|D_{\theta}X_{\tau}(s)|^{2}ds\right)^{q/2}d\theta\right)$
$\displaystyle\leq
C_{q,L,T}(t\varepsilon)^{\frac{q-2}{2}}\left(\int_{t(1-\varepsilon)}^{t}(t-\theta)^{\frac{q}{2}}d\theta+\int_{t(1-\varepsilon)}^{t}\left(t-\theta\right)^{q/2}d\theta\right)$
$\displaystyle\leq
C_{q,L,T}(t\varepsilon)^{\frac{q-2}{2}}(t\varepsilon)^{\frac{q}{2}+1}$
$\displaystyle=C_{q,L,T}\left(\frac{4}{y\sigma_{0}^{2}}\right)^{q},$ (3.17)

where $C_{q,L,T}$ is a positive constant. Combining $\eqref{3.13}$ and
$\eqref{3.14}$ we deduce

$\displaystyle
P\left(\|DX_{\tau}(t)\|^{2}_{L^{2}[0,T]}\leq\frac{1}{y}\right)\leq
C_{q,L,T}y^{p/2}\left(\frac{4}{y\sigma_{0}^{2}}\right)^{p}\,\,\,\forall\,\,p\geq
2,y\geq y_{0}.$

For any $p\geq 2$ and $q=2p+1$, we obtain the following estimates

$\displaystyle E\left(\|DX_{\tau}(t)\|^{-2p}_{L^{2}[0,T]}\right)$
$\displaystyle=\int_{0}^{\infty}py^{p-1}P\left(\|DX_{\tau}(t)\|^{-2}_{L^{2}[0,T]}>y\right)dy$
$\displaystyle\leq\int_{0}^{y_{0}}py^{p-1}dy+\int_{y_{0}}^{\infty}py^{p-1}P\left(\|DX_{\tau}(t)\|^{2}_{L^{2}[0,T]}<\frac{1}{y}\right)dy$
$\displaystyle\leq
y_{0}^{p}+pC_{p,L,T}\int_{y_{0}}^{\infty}y^{p-1}y^{q/2}\left(\frac{4}{y\sigma_{0}^{2}}\right)^{q}dy$
$\displaystyle=y_{0}^{p}+2pC_{p,L,T}\left(\frac{4}{\sigma_{0}^{2}}\right)^{2p+1}y_{0}^{-\frac{1}{2}}.$
(3.18)

We recall here that $y_{0}=\frac{4}{t\sigma_{0}^{2}}.$ So (3.15) follows from
(3.18). ∎

######  Proposition 3.3.

Let Assumptions 3.1 and 3.2 hold. It holds that

$E|D_{r}D_{\theta}X_{\tau}(t)|^{4}\leq C_{T,L}\,\,\forall\,0\leq\theta,r\leq
t\leq T,$

where $C_{L,T}$ is a positive constant.

###### Proof.

We rewrite the equations (3.1) and (3.1) as follows

$\displaystyle D_{\theta}$ $\displaystyle
X_{\tau}(t)=\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))+\int_{\theta}^{t}b_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)ds$
$\displaystyle+\int_{\theta}^{t}b_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds+\int_{\theta}^{t}\sigma_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)dB(s)$
$\displaystyle+\int_{\theta}^{t}\sigma_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s),\,\,0\leq\theta\leq t\leq T.$

Hence, the second order Malliavin derivative of $X_{\tau}(t)$ can be computed
by

$\displaystyle
D_{r}D_{\theta}X_{\tau}(t)=D_{r}[\sigma(\theta,X_{\tau}(\theta),X_{\tau}(\theta-\tau))]+\sigma_{2}^{\prime}(r,X_{\tau}(r),X_{\tau}(r-\tau))D_{\theta}X_{\tau}(r)$
$\displaystyle+\sigma_{3}^{\prime}(r,X_{\tau}(r),X_{\tau}(r-\tau))D_{\theta}X_{\tau}(r-\tau){\rm
11}_{[\theta+\tau,t]}(r)+\int_{\theta\vee
r}^{t}D_{r}[b_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))]D_{\theta}X_{\tau}(s)ds$
$\displaystyle+\int_{\theta\vee
r}^{t}b_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{r}D_{\theta}X_{\tau}(s)ds+\int_{\theta\vee
r}^{t}D_{r}[b_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))]D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds$ $\displaystyle+\int_{\theta\vee
r}^{t}b_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{r}D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)ds+\int_{\theta\vee
r}^{t}D_{r}[\sigma_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))]D_{\theta}X_{\tau}(s)dB(s)$
$\displaystyle+\int_{\theta\vee
r}^{t}\sigma_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{r}D_{\theta}X_{\tau}(s)dB(s)+\int_{\theta\vee
r}^{t}D_{r}[\sigma_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))]D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s)$ $\displaystyle+\int_{\theta\vee
r}^{t}\sigma_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{r}D_{\theta}X_{\tau}(s-\tau){\rm
11}_{[\theta+\tau,t]}(s)dB(s),\,\,0\leq r,\theta\leq t\leq T,$

where, for
$h\in\\{b_{2}^{\prime},b_{3}^{\prime},\sigma_{2}^{\prime},\sigma_{3}^{\prime}\\},$
we have

$D_{r}\left[h(s,X_{\tau}(s),X_{\tau}(s-\tau))\right]=h_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{r}X_{\tau}(s)+h_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{r}X_{\tau}(s-\tau).$

Note that the partial derivatives of $b$ and $\sigma$ are bounded. By using
(3.11) and the Hölder and Burkholder-Davis-Gundy inequalities, we verify that

$\displaystyle E|D_{r}D_{\theta}X_{\tau}(t)|^{4}$ $\displaystyle\leq
C_{T,L}+C_{T,L}\int_{\theta\vee
r}^{t}E|D_{r}D_{\theta}X_{\tau}(s)|^{4}ds+C_{T,L}\int_{\theta\vee
r}^{t}E|D_{r}D_{\theta}X_{\tau}(s-\tau)|^{4}ds$ $\displaystyle\leq
C_{T,L}+C_{T,L}\int_{\theta\vee
r}^{t}E|D_{r}D_{\theta}X_{\tau}(s)|^{4}ds,\,\,0\leq r,\theta\leq t\leq T,$

which, by Gronwall’s lemma, gives us the desired conclusion. ∎

###  3.2 $L^{p}$-distances and weak convergence

In this Subsection, we consider the equations (1.4) and (1.5). We first bound
$L^{p}$-distances between the solutions and between their Malliavin
derivatives. Then, we use Lemma 2.1 to estimate the weak rate of convergence.
In the whole subsection, we assume without loss of generality that
$\tau_{1}<\tau_{2}.$

######  Lemma 3.3.

Let Assumptions 3.1 hold and let $h:[0,T]\times\mathbb{R}^{2}\to\mathbb{R}$ be
such that

$|h(t,x_{1},y_{1})-h(t,x_{2},y_{2})|\leq L(|x_{1}-x_{2}|+|y_{1}-y_{2}|)$

for all $t\in[0,T],x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}.$ Then, for every
$p\geq 2$ and for all $t\in[0,T],$

$\displaystyle\int_{0}^{t}E|h(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-h(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{p}ds$
$\displaystyle\leq
C_{p,L,T}\bigg{(}t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds$
$\displaystyle\hskip
250.38434pt+\int_{0}^{t}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds\bigg{)},$
(3.19)

where $C_{p,L,T}$ is a positive constant.

###### Proof.

We have

$\displaystyle\int_{0}^{t}E|h(s,$ $\displaystyle
X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-h(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{p}ds$
$\displaystyle\leq
L^{p}\int_{0}^{t}E(|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|+|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|)^{p}ds$
$\displaystyle\leq
L^{p}2^{p-1}\int_{0}^{t}(E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}+E|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p})ds$
$\displaystyle=L^{p}2^{p-1}M_{p}(t)+L^{p}2^{p-1}\int_{0}^{t}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds,\,\,\,t\in[0,T],$
(3.20)

where

$\displaystyle M_{p}(t)$
$\displaystyle:=\int_{0}^{t}E|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p}ds$
$\displaystyle\leq
2^{p-1}\int_{0}^{t}(E|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{1})|^{p}+E|X_{\tau_{2}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p})ds.$

Since $X_{\tau_{1}}(s)=X_{\tau_{2}}(s)=\varphi(s),s\in[-\tau_{1},0],$ it holds
that

$\displaystyle\int_{0}^{t}$ $\displaystyle
E|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{1})|^{p}ds$
$\displaystyle=\int_{0}^{t\wedge\tau_{1}}E|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{1})|^{p}ds+\int_{t\wedge\tau_{1}}^{t}E|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{1})|^{p}ds$
$\displaystyle=\int_{t\wedge\tau_{1}}^{t}E|X_{\tau_{1}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{1})|^{p}ds=\int_{0}^{t-\tau_{1}}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds$
$\displaystyle\leq\int_{0}^{t}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds,\,\,\,t\in[0,T].$

On the other hand, we have

$\displaystyle\int_{0}^{t}E|X_{\tau_{2}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p}ds=\int_{0}^{t\wedge\tau_{1}}E|X_{\tau_{2}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p}ds$
$\displaystyle+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}E|X_{\tau_{2}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{2}}^{t}E|X_{\tau_{2}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p}ds,\,\,\,t\in[0,T].$

Then, recalling (3.10), we obtain

$\displaystyle\int_{0}^{t}E|X_{\tau_{2}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p}ds$
$\displaystyle\leq\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+2^{p-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}E|X_{\tau_{2}}(s-\tau_{1})-\varphi(0)|^{p}ds$
$\displaystyle+2^{p-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-X_{\tau_{2}}(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{2}}^{t}E|X_{\tau_{2}}(s-\tau_{1})-X_{\tau_{2}}(s-\tau_{2})|^{p}ds$
$\displaystyle\leq\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+C_{p,L,T}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}(s-\tau_{1})^{p/2}ds$
$\displaystyle+2^{p-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds+C_{p,L,T}\int_{t\wedge\tau_{2}}^{t}(\tau_{2}-\tau_{1})^{p/2}ds$
$\displaystyle\leq
C_{p,L,T}\bigg{(}t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\bigg{)}$

Hence,

$\displaystyle M_{p}(t)$ $\displaystyle\leq
C_{p,L,T}\bigg{(}t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds$
$\displaystyle\hskip
170.71652pt+\int_{0}^{t}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds\bigg{)},\,\,\,t\in[0,T].$
(3.21)

As a consequence, we obtain (3.19) by inserting (3.21) into (3.20). The proof
of the lemma is complete. ∎

######  Proposition 3.4.

Let Assumption 3.1 hold. Then, for every $p\geq 2$ and for all $t\in[0,T],$
there exists a positive constants $C_{p,L,T}$ such that

$\displaystyle E|X_{\tau_{1}}(t)-X_{\tau_{2}}(t)|^{p}$ $\displaystyle\leq
C_{p,L,T}t^{\frac{p}{2}-1}\bigg{(}t|\tau_{1}-\tau_{2}|^{p/2}$
$\displaystyle+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\bigg{)}.$
(3.22)

###### Proof.

We write

$\displaystyle X_{\tau_{1}}(t)-X_{\tau_{2}}(t)=I_{1}(t)+I_{2}(t),\,\,\,0\leq
t\leq T,$ (3.23)

where the terms $I_{1}(t),I_{2}(t)$ are defined by

$\displaystyle
I_{1}(t):=\int_{0}^{t}[b(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))]ds,$
$\displaystyle
I_{2}(t):=\int_{0}^{t}[\sigma(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-\sigma(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))]dB(s).$

Using the Hölder inequality and Lemma 3.3, we deduce

$\displaystyle E|I_{1}(t)|^{p}$ $\displaystyle\leq
t^{p-1}\int_{0}^{t}E|b(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{p}ds$
$\displaystyle\leq
C_{p,L,T}t^{p-1}\left(t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\right)$
$\displaystyle\ \
+C_{p,L,T}\int_{0}^{t}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds,\,\,\,0\leq
t\leq T.$ (3.24)

Similarly, we also have

$\displaystyle E|I_{2}(t)|^{p}$ $\displaystyle\leq
C_{p,L,T}E\left(\int_{0}^{t}|\sigma(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-\sigma(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{2}ds\right)^{p/2}$
$\displaystyle\leq
C_{p,L,T}t^{\frac{p}{2}-1}\int_{0}^{t}E|\sigma(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-\sigma(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{p}ds$
$\displaystyle\leq
C_{p,L,T}t^{\frac{p}{2}-1}\left(t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\right)$
$\displaystyle\ \
+C_{p,L,T}\int_{0}^{t}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds,\,\,\,0\leq
t\leq T.$ (3.25)

Combining (3.23), (3.24) and (3.2), we obtain

$\displaystyle E|X_{\tau_{1}}(t)$ $\displaystyle-X_{\tau_{2}}(t)|^{p}$
$\displaystyle\leq
C_{p,L,T}t^{\frac{p}{2}-1}\left(t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\right)$
$\displaystyle\ \
+C_{p,L,T}\int_{0}^{t}E|X_{\tau_{1}}(s)-X_{\tau_{2}}(s)|^{p}ds,\,\,\,0\leq
t\leq T.$

Since the function $t\mapsto
t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds$
is non-decreasing, we can use the Gronwall-type lemma (see, Theorem 1.4.2 in
[17]) to get

$\displaystyle E|X_{\tau_{1}}(t)-X_{\tau_{2}}(t)|^{p}$ $\displaystyle\leq
C_{p,L,T}t^{\frac{p}{2}-1}\left(t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\right)e^{C_{p,L,T}t}$
$\displaystyle\leq
C_{p,L,T}t^{\frac{p}{2}-1}\left(t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\right).$

The proof of Proposition is complete. ∎

######  Corollary 3.1.

Under the assumptions of Lemma 3.3, we have

$\displaystyle\int_{0}^{t}E|h(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-h(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{p}ds$
$\displaystyle\leq
C_{p,L,T}\bigg{(}t|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\bigg{)}$
$\displaystyle\leq
C_{p,L,T}\bigg{(}|\tau_{1}-\tau_{2}|^{p/2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{p}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{p}ds\bigg{)}$

for every $p\geq 2$ and for all $t\in[0,T],$ where $C_{p,L,T}$ is a positive
constant.

###### Proof.

Follows directly from Lemma 3.3 and Proposition 3.4. ∎

Next, we estimate the distance between the Malliavin derivatives. For this
purpose, we set

$\displaystyle
I_{1,\tau}(\theta,t):=\int_{\theta}^{t}b_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)ds,$
$\displaystyle
I_{2,\tau}(\theta,t):=\int_{\theta+\tau}^{t}b_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s-\tau)ds,$
$\displaystyle
J_{1,\tau}(\theta,t):=\int_{\theta}^{t}\sigma_{2}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s)dB(s),$
$\displaystyle
J_{2,\tau}(\theta,t):=\int_{\theta+\tau}^{t}\sigma_{3}^{\prime}(s,X_{\tau}(s),X_{\tau}(s-\tau))D_{\theta}X_{\tau}(s-\tau)dB(s),$

where $(X_{\tau}(t))_{t\in[-\tau,T]}$ is the solution to the equation (3.1).

######  Lemma 3.4.

Let Assumptions 3.1 and 3.2 hold. We have

$\displaystyle
E|I_{1,\tau_{1}}(\theta,t)-I_{1,\tau_{2}}(\theta,t)|^{2}+E|J_{1,\tau_{1}}(\theta,t)-J_{1,\tau_{2}}(\theta,t)|^{2}$
$\displaystyle\leq
C_{L,T}\left(|\tau_{1}-\tau_{2}|^{2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}$
$\displaystyle+C_{L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}ds,\,\,0\leq\theta\leq
t\leq T,$ (3.26)

where $C_{L,T}$ is a positive constant.

###### Proof.

We have

$\displaystyle I_{1,\tau_{1}}$
$\displaystyle(\theta,t)-I_{1,\tau_{2}}(\theta,t)$
$\displaystyle=\int_{\theta}^{t}[b_{2}^{\prime}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))]D_{\theta}X_{\tau_{1}}(s)ds$
$\displaystyle+\int_{\theta}^{t}b_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))[D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)]ds,\,\,0\leq\theta\leq
t.$

Then, by the Hölder inequality, we obtain

$\displaystyle E|I_{1,\tau_{1}}(\theta,t)-I_{1,\tau_{2}}(\theta,t)|^{2}$
$\displaystyle\leq
2(t-\theta)\int_{\theta}^{t}E|[b_{2}^{\prime}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))]D_{\theta}X_{\tau_{1}}(s)|^{2}ds$
$\displaystyle+2(t-\theta)\int_{\theta}^{t}E|b_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))[D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)]|^{2}ds$
$\displaystyle\leq
C_{L,T}\left(\int_{\theta}^{t}E|b_{2}^{\prime}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{4}ds\right)^{1/2}\left(\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)|^{4}ds\right)^{1/2}$
$\displaystyle+C_{L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}ds,\,\,0\leq\theta\leq
t.$ (3.27)

Note that, by the estimate (3.11), we have
$\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)|^{4}ds\leq C_{L,T}.$
Furthermore, an application of Corollary 3.1 to $h=b_{2}^{\prime}$ yields

$\displaystyle\int_{\theta}^{t}E|b_{2}^{\prime}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{4}ds$
$\displaystyle\leq
C_{L,T}\left(|\tau_{1}-\tau_{2}|^{2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right),\,\,0\leq\theta\leq
t.$

Hence, we get

$\displaystyle E|I_{1,\tau_{1}}(\theta,t)-I_{1,\tau_{2}}(\theta,t)|^{2}$
$\displaystyle\leq
C_{L,T}\left(|\tau_{1}-\tau_{2}|^{2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}$
$\displaystyle+C_{L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}ds,\,\,0\leq\theta\leq
t.$

Thus (3.26) is verified for
$E|I_{1,\tau_{1}}(\theta,t)-I_{1,\tau_{2}}(\theta,t)|^{2}.$ Similarly, we have

$\displaystyle E|J_{1,\tau_{1}}(\theta,t)-J_{1,\tau_{2}}(\theta,t)|^{2}\leq
2\int_{\theta}^{t}E|[\sigma_{2}^{\prime}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-\sigma_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))]D_{\theta}X_{\tau_{1}}(s)|^{2}ds$
$\displaystyle+2\int_{\theta}^{t}E|\sigma_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))[D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)]|^{2}ds$
$\displaystyle\leq
C\left(\int_{\theta}^{t}E|\sigma_{2}^{\prime}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-\sigma_{2}^{\prime}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))|^{4}ds\right)^{1/2}\left(\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)|^{4}ds\right)^{1/2}$
$\displaystyle+C\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}ds,\,\,0\leq\theta\leq
t.$ (3.28)

The right hand side of (3.28) has the same form as that of (3.27). We
therefore can conclude that (3.26) also holds true for
$E|J_{1,\tau_{1}}(\theta,t)-J_{1,\tau_{2}}(\theta,t)|^{2}.$ The proof of the
lemma is complete. ∎

######  Lemma 3.5.

Let Assumptions 3.1 and 3.2 hold. Then, we have for $t>\tau_{2},$

$\displaystyle
E|I_{2,\tau_{1}}(\theta,t)-I_{2,\tau_{2}}(\theta,t)|^{2}+E|J_{2,\tau_{1}}(\theta,t)-J_{2,\tau_{2}}(\theta,t)|^{2}$
$\displaystyle\leq
C_{L,T}\left(|\tau_{1}-\tau_{2}|^{2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}$
$\displaystyle+C_{L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}ds,\,\,0\leq\theta\leq
t-\tau_{2},$ (3.29)

where $C_{L,T}$ is a positive constant.

###### Proof.

When $t>\tau_{2}$ and $0\leq\theta\leq t-\tau_{2}$, we have

$\displaystyle I_{2,\tau_{1}}(\theta,t)-I_{2,\tau_{2}}(\theta,t)$
$\displaystyle=\int_{\theta+\tau_{1}}^{t}b^{\prime}_{3}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))D_{\theta}X_{\tau_{1}}(s-\tau_{1})ds-\int_{\theta+\tau_{2}}^{t}b^{\prime}_{3}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))D_{\theta}X_{\tau_{2}}(s-\tau_{2})ds$
$\displaystyle=\int_{\theta+\tau_{2}}^{t}[b^{\prime}_{3}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b^{\prime}_{3}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))]D_{\theta}X_{\tau_{1}}(s-\tau_{1})ds$
$\displaystyle+\int_{\theta+\tau_{1}}^{\theta+\tau_{2}}b^{\prime}_{3}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))D_{\theta}X_{\tau_{1}}(s-\tau_{1})ds$
$\displaystyle+\int_{\theta+\tau_{2}}^{t}b^{\prime}_{3}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))[D_{\theta}X_{\tau_{1}}(s-\tau_{1})-D_{\theta}X_{\tau_{2}}(s-\tau_{2})]ds.$
(3.30)

We observe that the first addend in the right hand side of (3.30) can be
estimated as in the proof of Lemma 3.4 and we obtain

$\displaystyle E\big{|}$
$\displaystyle\int_{\theta+\tau_{2}}^{t}[b^{\prime}_{3}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))-b^{\prime}_{3}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))]D_{\theta}X_{\tau_{1}}(s-\tau_{1})ds\big{|}^{2}$
$\displaystyle\leq
C_{L,T}\left(|\tau_{1}-\tau_{2}|^{2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}.$

For the second addend, it follows from (3.11) that

$\displaystyle E\big{|}$
$\displaystyle\int_{\theta+\tau_{1}}^{\theta+\tau_{2}}b^{\prime}_{3}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))D_{\theta}X_{\tau_{1}}(s-\tau_{1})ds\big{|}^{2}$
$\displaystyle\leq|\tau_{1}-\tau_{2}|\int_{\theta+\tau_{1}}^{\theta+\tau_{2}}E|b^{\prime}_{3}(s,X_{\tau_{1}}(s),X_{\tau_{1}}(s-\tau_{1}))D_{\theta}X_{\tau_{1}}(s-\tau_{1})|^{2}ds$
$\displaystyle\leq C_{L,T}|\tau_{1}-\tau_{2}|^{2}\leq
C_{L,T}|\tau_{1}-\tau_{2}|.$

For the third addend, we use the Hölder inequality and (3.12) to get

$\displaystyle E\big{|}$
$\displaystyle\int_{\theta+\tau_{2}}^{t}b^{\prime}_{3}(s,X_{\tau_{2}}(s),X_{\tau_{2}}(s-\tau_{2}))[D_{\theta}X_{\tau_{1}}(s-\tau_{1})-D_{\theta}X_{\tau_{2}}(s-\tau_{2})]ds\big{|}^{2}$
$\displaystyle\leq
C_{L,T}\int_{\theta+\tau_{2}}^{t}E|D_{\theta}X_{\tau_{1}}(s-\tau_{1})-D_{\theta}X_{\tau_{2}}(s-\tau_{2})|^{2}ds$
$\displaystyle\leq
C_{L,T}\int_{\theta+\tau_{2}}^{t}E|D_{\theta}X_{\tau_{1}}(s-\tau_{1})-D_{\theta}X_{\tau_{1}}(s-\tau_{2})|^{2}ds+C_{L,T}\int_{\theta+\tau_{2}}^{t}E|D_{\theta}X_{\tau_{1}}(s-\tau_{2})-D_{\theta}X_{\tau_{2}}(s-\tau_{2})|^{2}ds$
$\displaystyle\leq
C_{L,T}\int_{\theta+\tau_{2}}^{t}|\tau_{1}-\tau_{2}|ds+C_{L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}ds$
$\displaystyle\leq
C_{L,T}|\tau_{1}-\tau_{2}|+C_{L,T}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}ds.$

Hence, we can obtain (3.29) for
$E|I_{2,\tau_{1}}(\theta,t)-I_{2,\tau_{2}}(\theta,t)|^{2}.$ This finishes the
proof of Lemma because the estimate for
$E|J_{2,\tau_{1}}(\theta,t)-J_{2,\tau_{2}}(\theta,t)|^{2}$ can be treated
similarly. ∎

######  Proposition 3.5.

Let Assumptions 3.1 and 3.2 hold. Then, there exists a positive constants
$C_{L,T}$ such that, for all $t\in[0,T],$

$\displaystyle E\|DX_{\tau_{1}}(t)$ $\displaystyle-
DX_{\tau_{2}}(t)\|^{2}_{L^{2}[0,T]}\leq
C_{T,L}\bigg{(}t^{2}|\tau_{1}-\tau_{2}|^{2}$
$\displaystyle+t\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+t\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\bigg{)}^{1/2},$
(3.31)

where $C_{L,T}$ is a positive constant.

###### Proof.

We consider the following cases.

Case 1: $t>\tau_{2}$. In this case, we write

$\displaystyle E\|DX_{\tau_{1}}(t))-DX_{\tau_{2}}(t)\|^{2}_{L^{2}[0,T]}$
$\displaystyle=\int_{0}^{t-\tau_{2}}E|D_{\theta}X_{\tau_{1}}(t))-D_{\theta}X_{\tau_{2}}(t)|^{2}d\theta+\int_{t-\tau_{2}}^{t-\tau_{1}}E|D_{\theta}X_{\tau_{1}}(t))-D_{\theta}X_{\tau_{2}}(t)|^{2}d\theta$
$\displaystyle+\int_{t-\tau_{1}}^{t}E|D_{\theta}X_{\tau_{1}}(t))-D_{\theta}X_{\tau_{2}}(t)|^{2}d\theta.$

We observe from the equations (3.1) and (3.1) that, when $0\leq\theta\leq
t-\tau_{2},$ we have

$\displaystyle D_{\theta}X_{\tau_{1}}(t))-D_{\theta}X_{\tau_{2}}(t)$
$\displaystyle=\sigma(\theta,X_{\tau_{1}}(\theta),X_{\tau_{1}}(\theta-\tau_{1}))-\sigma(\theta,X_{\tau_{2}}(\theta),X_{\tau_{2}}(\theta-\tau_{2}))$
$\displaystyle+I_{1,\tau_{1}}(\theta,t)-I_{1,\tau_{2}}(\theta,t)+I_{2,\tau_{1}}(\theta,t)-I_{2,\tau_{2}}(\theta,t)$
$\displaystyle+J_{1,\tau_{1}}(\theta,t)-J_{1,\tau_{2}}(\theta,t)+J_{2,\tau_{1}}(\theta,t)-J_{2,\tau_{2}}(\theta,t).$

When $t-\tau_{2}<\theta\leq t-\tau_{1},$ we have

$\displaystyle D_{\theta}X_{\tau_{1}}(t))-D_{\theta}X_{\tau_{2}}(t)$
$\displaystyle=\sigma(\theta,X_{\tau_{1}}(\theta),X_{\tau_{1}}(\theta-\tau_{1}))-\sigma(\theta,X_{\tau_{2}}(\theta),X_{\tau_{2}}(\theta-\tau_{2}))$
$\displaystyle+I_{1,\tau_{1}}(\theta,t)-I_{1,\tau_{2}}(\theta,t)+I_{2,\tau_{1}}(\theta,t)$
$\displaystyle+J_{1,\tau_{1}}(\theta,t)-J_{1,\tau_{2}}(\theta,t)+J_{2,\tau_{1}}(\theta,t).$

When $t-\tau_{1}<\theta\leq t,$ we have

$\displaystyle D_{\theta}X_{\tau_{1}}(t)-D_{\theta}X_{\tau_{2}}(t)$
$\displaystyle=\sigma(\theta,X_{\tau_{1}}(\theta),X_{\tau_{1}}(\theta-\tau_{1}))-\sigma(\theta,X_{\tau_{2}}(\theta),X_{\tau_{2}}(\theta-\tau_{2}))$
$\displaystyle+I_{1,\tau_{1}}(\theta,t)-I_{1,\tau_{2}}(\theta,t)+J_{1,\tau_{1}}(\theta,t)-J_{1,\tau_{2}}(\theta,t).$

Hence, using the estimates established in Lemmas 3.4 and 3.5, we deduce

$\displaystyle E\|D$ $\displaystyle
X_{\tau_{1}}(t))-DX_{\tau_{2}}(t)\|^{2}_{L^{2}[0,T]}$
$\displaystyle\leq\int_{0}^{t}E|\sigma(\theta,X_{\tau_{1}}(\theta),X_{\tau_{1}}(\theta-\tau_{1}))-\sigma(\theta,X_{\tau_{2}}(\theta),X_{\tau_{2}}(\theta-\tau_{2}))|^{2}d\theta$
$\displaystyle+\int_{t-\tau_{2}}^{t-\tau_{1}}[E|I_{2,\tau_{1}}(\theta,t)|^{2}+E|J_{2,\tau_{1}}(\theta,t)|^{2}]d\theta$
$\displaystyle+C_{L,T}\,t\left(|\tau_{1}-\tau_{2}|^{2}+\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}$
$\displaystyle+C_{L,T}\int_{0}^{t}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}dsd\theta,\,\,\,t>\tau_{2}.$

By the Hölder inequality and Corollary 3.1

$\displaystyle\int_{0}^{t}E|\sigma(\theta,X_{\tau_{1}}(\theta),X_{\tau_{1}}(\theta-\tau_{1}))-\sigma(\theta,X_{\tau_{2}}(\theta),X_{\tau_{2}}(\theta-\tau_{2}))|^{2}d\theta$
$\displaystyle\leq\sqrt{t}\left(\int_{0}^{t}E|\sigma(\theta,X_{\tau_{1}}(\theta),X_{\tau_{1}}(s-\theta))-\sigma(\theta,X_{\tau_{2}}(\theta),X_{\tau_{2}}(s-\theta))|^{4}d\theta\right)^{1/2}$
$\displaystyle\leq
C_{T,L}\left(t^{2}|\tau_{1}-\tau_{2}|^{2}+t\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+t\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}.$

On the other hand, it is easy to see from the boundedness of
$b_{3}^{\prime},\sigma_{3}^{\prime}$ and (3.11) that
$E|I_{2,\tau_{1}}(\theta,t)|^{2}+E|J_{2,\tau_{1}}(\theta,t)|^{2}\leq
C_{L,T}(t-\theta-\tau_{1})$ and hence,

$\int_{t-\tau_{2}}^{t-\tau_{1}}[E|I_{2,\tau_{1}}(\theta,t)|^{2}+E|J_{2,\tau_{1}}(\theta,t)|^{2}]d\theta\leq
C_{L,T}|\tau_{1}-\tau_{2}|^{2}\leq C_{L,T}\,t|\tau_{1}-\tau_{2}|.$

Combining the above estimates yields

$\displaystyle E\|D$ $\displaystyle
X_{\tau_{1}}(t)-DX_{\tau_{2}}(t)\|^{2}_{L^{2}[0,T]}$ $\displaystyle\leq
C_{L,T}\left(t^{2}|\tau_{1}-\tau_{2}|^{2}+t\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+t\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}$
$\displaystyle+C_{L,T}\int_{0}^{t}\int_{\theta}^{t}E|D_{\theta}X_{\tau_{1}}(s)-D_{\theta}X_{\tau_{2}}(s)|^{2}dsd\theta$
$\displaystyle=C_{L,T}\left(t^{2}|\tau_{1}-\tau_{2}|^{2}+t\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+t\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/2}$
$\displaystyle+C_{L,T}\int_{0}^{t}E\|DX_{\tau_{1}}(s)-DX_{\tau_{2}}(s)\|^{2}_{L^{2}[0,T]}ds,\,\,\,t>\tau_{2},$

which, by the Gronwall-type lemma (see, Theorem 1.4.2 in [17]), gives us
(3.31) when $t>\tau_{2}.$

We omit the detailed proof of Case 2: $\tau_{1}<t\leq\tau_{2}$ and Case 3:
$t\leq\tau_{1}$ because the arguments of Case 1 can be replicated.

The proof of Proposition is complete. ∎

We now are ready to state and prove the main result of this paper.

######  Theorem 3.1.

Let Assumption 3.1 and 3.2 hold. We assume, in addition, that

$|\sigma(t,x,y)|\geq\sigma_{0}>0,\,\,\forall\,t\in[0,T],x,y\in\mathbb{R}.$

Then, for any $g\in\mathcal{B}$ and $t\in(0,T],$ we have

$\displaystyle|Eg(X_{\tau_{1}}(t))$ $\displaystyle-Eg(X_{\tau_{2}}(t))|\leq
C_{T,L}\bigg{(}|\tau_{1}-\tau_{2}|^{2}$
$\displaystyle+t^{-1}\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+t^{-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\bigg{)}^{1/4},$
(3.32)

where $C_{L,T}$ is a positive constant.

###### Proof.

Fixed $t\in(0,T],$ we consider the random variables $F_{1}=X_{\tau_{1}}(t)$
and $F_{2}=X_{\tau_{2}}(t).$ Thanks to Propositions 3.2, 3.3, 3.4 and 3.5 we
have the following

$E\|DF_{1}\|^{-8}_{L^{2}[0,T]}\leq
C_{T,L}\,t^{-4},\,\,\,(E\|DF_{1}\|^{-2}_{L^{2}[0,T]})^{2}\leq
C_{T,L}\,t^{-2},$ $\displaystyle
E\left(\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr\right)^{2}$
$\displaystyle=E\left(\int_{0}^{t}\int_{0}^{t}|D_{\theta}D_{r}E|X_{\tau_{1}}(t)|^{2}d\theta
dr\right)^{2}$ $\displaystyle\leq
t^{2}\int_{0}^{t}\int_{0}^{t}E|D_{\theta}D_{r}E|X_{\tau_{1}}(t)|^{4}d\theta
dr$ $\displaystyle\leq C_{L,T}\,t^{4},$ (3.33)

and

$\displaystyle\|F_{1}-F_{2}\|_{1,2}=\left(E|X_{\tau_{1}}(t)-X_{\tau_{2}}(t)|^{2}+E\|DX_{\tau_{1}}(t)-DX_{\tau_{2}}(t)\|^{2}_{L^{2}[0,T]}\right)^{1/2}$
$\displaystyle\leq
C_{T,L}\left(t^{2}|\tau_{1}-\tau_{2}|^{2}+t\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+t\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/4}.$

Consequently, in view of Lemma 2.1, we obtain

$\displaystyle|Eg(X_{\tau_{1}}(t))-Eg(X_{\tau_{2}}(t))|=|Eg(F_{1})-Eg(F_{2})|$
$\displaystyle\leq
C\left(E\|DF_{1}\|^{-8}_{L^{2}[0,T]}E\left(\int_{0}^{T}\int_{0}^{T}|D_{\theta}D_{r}F_{1}|^{2}d\theta
dr\right)^{2}+(E\|DF_{1}\|^{-2}_{L^{2}[0,T]})^{2}\right)^{\frac{1}{4}}\|F_{1}-F_{2}\|_{1,2}$
$\displaystyle\leq
C_{T,L}\left(|\tau_{1}-\tau_{2}|^{2}+t^{-1}\int_{0}^{t\wedge\tau_{1}}|\varphi(s-\tau_{1})-\varphi(s-\tau_{2})|^{4}ds+t^{-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}|\varphi(0)-\varphi(s-\tau_{2})|^{4}ds\right)^{1/4}.$

This completes the proof. ∎

Clearly, when the initial data $\varphi$ is a continuous function, the
estimate (3.32) implies that $X_{\tau_{2}}(t)$ weakly converges to
$X_{\tau_{1}}(t)$ as $\tau_{2}\to\tau_{1}.$ Moreover, when $\varphi$ is Hölder
continuous, we have the following Hölder continuity of solutions with respect
to delay parameter.

######  Corollary 3.2.

Suppose the assumptions of Theorem 3.1. In addition, we assume that the
initial data $\varphi$ is Hölder continuous with exponent $\beta\in(0,1).$
Then, for any $g\in\mathcal{B},$ we have

$\sup\limits_{0\leq t\leq T}|Eg(X_{\tau_{1}}(t))-Eg(X_{\tau_{2}}(t))|\leq
C_{T,L}\,|\tau_{1}-\tau_{2}|^{\beta\wedge\frac{1}{2}},$ (3.34)

where $C_{L,T}$ is a positive constant.

###### Proof.

We have

$t^{-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}(\tau_{2}-s)^{4\beta}ds\leq
t^{-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}(\tau_{2}-\tau_{1})^{4\beta}ds\leq(\tau_{2}-\tau_{1})^{4\beta}.$

Hence, it follows from (3.32) that

$\displaystyle|Eg(X_{\tau_{1}}(t))-Eg(X_{\tau_{2}}(t))|$ $\displaystyle\leq
C_{T,L}\left(|\tau_{1}-\tau_{2}|^{2}+t^{-1}\int_{0}^{t\wedge\tau_{1}}|\tau_{1}-\tau_{2}|^{4\beta}ds+t^{-1}\int_{t\wedge\tau_{1}}^{t\wedge\tau_{2}}(\tau_{2}-s)^{4\beta}ds\right)^{1/4}$
$\displaystyle\leq
C_{T,L}\,|\tau_{1}-\tau_{2}|^{\beta\wedge\frac{1}{2}},\,\,\,0<t\leq T.$

This, together with the fact that
$|Eg(X_{\tau_{1}}(t))-Eg(X_{\tau_{2}}(t))|=0$ when $t=0,$ gives us (3.34). ∎

When the coefficients $b(t,x,y),\sigma(t,x,y)$ do not depend on $x,$ the
delays $\tau_{1}=0,\tau_{2}=1/n$ and the initial data
$\varphi(t)=x_{0},\,t\in[-1/n,0],$ we obtain the following weak rate of
convergence for the Carathéodory approximation.

######  Corollary 3.3.

Consider the Carathéodory approximation system (1.1)-(1.2). Suppose that

(i) $b(t,x)$ and $\sigma(t,x)$ are Lipschitz in $x$ and satisfy linear growth,

(ii) $b(t,\cdot)$ and $\sigma(t,\cdot)$ are twice differentiable with bounded
derivatives,

(iii) $|\sigma(t,x)|\geq\sigma_{0}>0$ for all $t\in[0,T]$ and
$x\in\mathbb{R}.$

Then, for any $g\in\mathcal{B},$ we have

$\sup\limits_{0\leq t\leq
T}|Eg(x^{n}(t))-Eg(x(t))|\leq\frac{C_{L,T}}{\sqrt{n}},\,\,\,n\geq 1,$

where $C_{L,T}$ is a positive constant.

##  4 Conclusion

In this paper, we employed the technique of Malliavin calculus to study the
weak convergence of delay SDEs. The interesting point of our results lies in
the fact that we are able to provide an explicit estimate for the rate of
convergence and we only require the test function $g$ to be bounded.
Furthermore, the method introduced in the paper can be used to investigate the
weak convergence and the Carathéodory approximation for more general equations
such as SDEs with multiple delays

$dX(t)=b(t,X(t),X(t-\tau_{1}),\cdots,X(t-\tau_{k}))dt+\sigma(t,X(t),X(t-\tau_{1}),\cdots,X(t-\tau_{k}))dB(t),\,\,\,t\in[0,T]$

or SDEs with variable delays

$dX(t)=b(t,X(t),X(t-\tau(t)))dt+\sigma(t,X(t),X(t-\tau(t)))dB(t),\,\,\,t\in[0,T].$

However, the computations will be more complex. Hence, in the present paper,
we have chosen the simplest class of delay SDEs to illustrate the main
features of the method rather than getting bogged down with complex notations.

Acknowledgments. The authors would like to thank the anonymous referees for
their valuable comments for improving the paper. N. T. Dung and H. T. P. Thao
are supported by the Vietnam National University, Hanoi under grant number
QG.20.21. T. C. Son, N. V. Tan, T. M. Cuong and P. D. Tung are supported by
Viet Nam National Foundation for Science and Technology Development (NAFOSTED)
under grant number 101.03-2019.08. A part of this paper was done while the
authors were visiting the Vietnam Institute for Advanced Study in Mathematics
(VIASM). The authors would like to thank the VIASM for financial support and
hospitality.

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