Weak order in averaging principle for stochastic differential equations with jumps
Bengong Zhang Hongbo Fu Li Wan Jicheng Liu









 

Bengong Zhang College of Mathematics and Computer Science, Wuhan Textile University 

              
              
              benyan1219@126.com
            Hongbo Fu College of Mathematics and Computer Science, Wuhan Textile University 

              
              
              hbfu@wtu.edu.cn           

           Li Wan College of Mathematics and Computer Science, Wuhan Textile
              University

              wanlinju@aliyun.com
           Jicheng Liu School of Mathematics and Statistics, Huazhong University of Science and Technology 

           jcliu@hust.edu.cn





Weak order in  averaging principle for stochastic differential equations with jumps




    Bengong ZhangHongbo Fu  Li Wan Jicheng Liu
    Received: date / Accepted: date
=======================================================================================




The present article  deals with the  averaging principle  for a
two-time-scale system of jump-diffusion stochastic differential
equation. Under suitable conditions, the weak error is expanded in
powers of timescale parameter. It is proved that the rate of weak
convergence to the averaged  dynamics is of order 1. This reveals
the rate of weak convergence is essentially twice that of strong
convergence.



  60H10 70K70 




§ INTRODUCTION

 We consider a two-time-scale system  of jump-diffusion stochastic
differential equation in form of

    dX^ϵ_t=a(X_t^ϵ,
    Y_t^ϵ)dt+b(X_t^ϵ)d B_t+c(X_t-^ϵ)d P_t,
     X_0^ϵ=x,
          dY^ϵ_t=1/ϵf(X_t^ϵ,
    Y_t^ϵ)dt+1/√(ϵ)g(X_t^ϵ,
    Y_t^ϵ)d W_t+h(X_t-^ϵ,Y_t-^ϵ)d
    N^ϵ_t,  Y_0^ϵ=y,

where  X_t^ϵ∈ℝ^n, Y_t^ϵ∈ℝ^m,
the drift  functions a(x, y)∈ℝ^n, f(x,y)∈ℝ^m, the diffusion functions b(x )∈ℝ^n×
d_1, c(x)∈ℝ^n, g(x,y)∈ℝ^m× d_2 and
h(x,y)∈ℝ^m. B_t and W_t are the vectors of d_1,
d_2-dimensional independent Brownian motions on a complete
stochastic base (Ω, ℱ,ℱ_t, ℙ),
respectively. P_t is a scalar simple Poisson process with
intensity λ_1, and N_t^ϵ is a scalar simple Poisson
process with intensity λ_2/ϵ. The positive
parameter ϵ is small and  describes the ratio of time
scales between X^ϵ_t and Y^ϵ_t. Systems
(<ref>)-(<ref>) with two time scales
occur frequently in applications  including chemical kinetics,
signal processing, complex fluids and  financial engineering.



With the  separation of time scale,  we can view the state variable
of the system as being divided into two parts: the “slow" variable
X^ϵ_t and the “fast" variable Y^ϵ_t. It is often
the case that we are interested only in the dynamics of slow
component. Then a simplified equation, which is independent of fast
variable and possesses the essential features of the system, is
highly desirable. Such a simplified equation is often constructed by
averaging procedure as in <cit.> for
deterministic ordinary differential equations, as well as the
further development
<cit.>
for stochastic differential equations with continuous Gaussian
processes. As far as averaging for stochastic dynamical systems in
infinite dimensional space is concerned, it is worthwhile to quote
the important works of <cit.>
and also the works of <cit.>. For related
works on averaging for multivalued stochastic differential equations
we refer the reader to <cit.>.


In order to derive the averaged dynamics of the system
(<ref>)-(<ref>), we introduce the fast
motion equation with  a frozen slow component x∈ℝ^n in
form of

    dY_t^x= f(x, Y_t^x)dt+g(x, Y_t^x)d W_t+h(x,Y^x_t-)d N_t,
     Y_0^x=y,

whose solution is denoted by Y_t^ϵ(y). Under suitable
conditions on f,g and h, Y_t^ϵ(y) induces a unique
invariant  measure μ^x(dy) on ℝ^m, which   is ergodic
and ensures the averaged equation:

    dX̅_t=a̅(X̅_t)dt+b(X̅_t)dB_t+c(X̅_t-)dP_t,
     X̅_0 =x,

where the averaging nonlinearity is defined by setting

    a̅(x )   =   ∫_ℝ^ma(x,y)μ^x(dy)
       =   lim_t→ +∞𝔼 a(x, Y_t^x(y)).



In <cit.>, it was shown that under the above conditions the
slow motion X^ϵ_t converges strongly to the solution
X̅_t of the above averaged  equation with jumps. The order of
convergence 1/2 in strong sense was provided in
<cit.>.  To our best knowledge, there is no existing
literature to address the weak order in averaging principle for jump
diffusion stochastic differential systems. In fact, it is fair to
say that the weak convergence in stochastic averaging theory of
systems driven by jump noise is not  fully developed yet, although
some strong approximation results on the rate of strong convergence
were obtained <cit.>.

Therefore, we aim to study this problem in this paper. Here we are
interested in  the rate of weak convergence of the averaging
dynamics to the true solution of slow motion X^ϵ_t. In
other word,  we will determine the order, with respect to timescale
parameter ϵ, of weak deviation between original solution to
slow equation and the solution  of the corresponding averaged
equation. The main technique we adapted is to find an expansion
with respect to ϵ of the solutions of the Kolmogorov
equations associated with the jump diffusion system. The solvability
of the Poisson equation associated with the generator of frozen
equation provides an expression for the coefficients of the
expansion. As a result, the boundedness for the coefficients of
expansion can be proved by smoothing effect of the corresponding
transition semigroup in the space of bounded and uniformly
continuous functions, where some regular conditions is needed on
drift and diffusion term.

  Our result shows that the weak
convergence rate to be 1  even when there are jump components in
the system. It is the main contribution of this work. We would like
to stress that asymptotic  method was first applied by Bréhier
<cit.> to an averaging result for stochastic
reaction-diffusion equations in the case of Gaussian noise of
additive type was included only in the fast motion. However, the
extension of this argument  is not straightforward. The method used
in the proof of weak order in <cit.> is strictly related to
the differentiability in time of averaged process. Therefore, once
the noise is introduced in the slow equation, difficulties will
arise and the procedure becomes more complicated. Our result in this
paper bridges such a gap, in which the slow and the fast motions are
both perturbed  by  noise with jumps.

The rest of the paper is  structured as follows.  Section 2 is
devoted to  notations, assumptions and summarize preliminary
results. The ergodicity of fast process and the averaged dynamics of
system with jumps is introduced in Section 3. Then the main result
of this article, which is derived via the asymptotic expansions and
uniform error estimates, is presented in Section 4. Finally, we give
the appendix in  section 5.

It should be pointed out that the letter C below with or without
subscripts will denote  generic positive constants independent of
ϵ in the whole paper.




§ ASSUMPTIONS AND PRELIMINARY RESULTS

For any integer d, the scalar product and norm on d-dimensional
Euclidean space ℝ^d are denoted by
(·,·)_ℝ^d and
·_ℝ^d, respectively. For any  integer k, we
denote by C_b^k(ℝ^d,ℝ) the space of all
k-times differentiable functions on ℝ^d, which have
bounded uniformly continuous derivatives up to the k-th order.


In what follows, we shall assume that the drift and diffusion
coefficients arising in the system  fulfill  the following
conditions.

 (A1)
The mappings a(x,y), b(x), c(x), f(x,y), g(x,y) and h(x,y) are
of class C^2 and have bounded first and second derivatives.
Moreover, we assume that a(x,y), b(x) and c(x) are bounded.

(A2) There exists a constant α>0 such that for any
x∈ℝ^n, y∈ℝ^m it holds

    y^Tg(x,y)g^T(x,y)y≥αy_ℝ^m.


(A3) There exists a constant β>0 such that for any
y_1, y_2∈ℝ^m and x∈ℝ^n it holds

    (y_1-y_2,
    f(x,y_1)-f(x,y_2)+λ_2(h(x,y_1)-h(x,y_2)))_ℝ^m
          +g(x,y_1)-g(x,y_2)^2_ℝ^m+λ_2|h(x,y_1)-h(x,y_2)|^2
          ≤ -βy_1-y_2^2_ℝ^m.



Notice that from (A1) it immediately follows that the
following directional derivatives exist and are controlled:

    D_xa(x,y)· k_1_ℝ^n≤
    Lk_1_ℝ^n,
          D_ya(x,y)· l_1_ℝ^n≤
    Ll_1_ℝ^m,
          D_xx^2a(x,y)·(k_1,k_2)_ℝ^n≤
    Lk_1_ℝ^nk_2_ℝ^n,
          D_yy^2a(x,y)·(l_1,l_2)_ℝ^n≤
    Ll_1_ℝ^ml_2_ℝ^m,

where L is a constant independent of x,y,k_1, k_2, l_1 and
l_2. For differentiability of mappings b, c,f, g and h we
possess the analogous  results. For examples,  we have

    D^2_xxb(x)· (k_1,k_2)_ℝ^n≤ L
    k_1_ℝ^nk_2_ℝ^n,  k_1,k_2∈ℝ^n,
          D^2_yyf(x,y)· (l_1,l_2)_ℝ^m≤ L
    l_1_ℝ^ml_2_ℝ^m, l_1,l_2∈ℝ^m.


As far as the assumption (A2) is concerned, it is a sort of
non-degeneracy condition and it is assumed in order to have the
regularizing effect of the Markov transition semigroup associated
with   the fast dynamics. Assumption (A3) is the
dissipative condition   which determines how the fast equation
converges to its equilibrium state.


As assumption (A1) holds,  for any ϵ>0 and any
initial conditions x∈ℝ^n and y∈ℝ^m,
system (<ref>)-(<ref>) admits a unique
solution, which, in order to emphasize the dependence on the initial
data, is denoted by (X_t^ϵ(x,y), Y_t^ϵ(x,y)).
Moreover the following lemma holds (for a proof see e.g.
<cit.>).

Under the assumptions (A1), (A2) and
(A3),
 for any x∈ℝ^n, y∈ℝ^m and ϵ>0
 we have

    𝔼X_t^ϵ(x,y)^2_ℝ^n≤
    C_T(1+x^2_ℝ^n+y^2_ℝ^m),  t∈ [0,
    T]

and

    𝔼Y_t^ϵ(x,y)^2_ℝ^n≤
    C_T(1+x^2_ℝ^m+y^2_ℝ^m),  t∈ [0,
    T].





§ FROZEN EQUATION AND AVERAGED EQUATION 

Fixing ϵ=1, we consider the fast equation with frozen slow
component x∈ℝ^n,

    dY_t^x(y)= f(x, Y_t^x(y))dt+g(x, Y_t^x(y))d W_t+h(x,Y^x_t-(y))d
    N_t, 
    
    Y_0^x=y.

Under assumptions (A1)-(A3), such a problem has a
unique solution, which satisfies <cit.>:

    𝔼Y_t^x(y)^2_ℝ^m≤
    C(1+x^2_ℝ^n+e^-β
    ty^2_ℝ^m), t≥ 0.

Let Y_t^x( y') be the solution of problem (<ref>) with
initial value Y_0^x=y', the Itô formula implies that for any
t≥0,

    𝔼Y_t^x(y)-Y_t^x
    (y')_ℝ^m^2≤y-y'_ℝ^m^2e^-β t.

Moreover, as discussion in <cit.>  and <cit.>,
equation (<ref>) admits a unique ergodic invariant measure
μ^x satisfying

    ∫_ℝ^my_ℝ^m^2μ^x(dy)≤
    C(1+x^2_ℝ^n).

Then, by averaging the coefficient a with respect to the invariant
measure μ^x, we can define an ℝ^n-valued mapping

    a̅(x):=∫_ℝ^ma(x,y)μ^x(dy), x∈ℝ^n.

Due to assumption (A1), it is easily to check that
a̅(x) is 2-times differentiable with bounded derivatives, and
hence it is Lipschitz-continuous such that

    a̅(x_1)-a̅(x_2)_ℝ^n≤
    Cx_1-x_2_ℝ^n, x_1, x_2∈ℝ^n.

According to   invariant property of μ^x,
(<ref>) and assumption (A1), we have

    𝔼a(x, Y_t^x(
    y))-a̅(x)^2_ℝ^n   =   ∫_ℝ^m𝔼(a(x, Y_t^x
    (y))-a(x, Y_t^x(z)))μ^x(dz) ^2_ℝ^n
       ≤   ∫_ℝ^m𝔼Y_t^x
    (y)-Y_t^x(z)_ℝ^m^2μ^x(dz)
       ≤   e^-β t∫_ℝ^my-z^2_ℝ^mμ^x(dz)
       ≤   Ce^-β
    t(1+x^2_ℝ^n+y^2_ℝ^m).

Now we can  introduce the effective dynamical system

    dX̅_t(x)=a̅(X̅_t(x))dt+b(X̅_t(x))dB_t+c(X̅_t-(x))d
    P_t,
    
    X̅_0 =x.

As the coefficients  a̅, b and c are Lipschitz-continuous,
this equation  admits a unique solution such that

    𝔼X̅_t(x)^2_ℝ^n≤
    C_T(1+x^2_ℝ^n),  t∈ [0, T].


 With the above assumptions and notations we have the following   result, which
is a direct consequence of Lemma <ref>, Lemma
<ref> and Lemma <ref>.



Assume that x∈ℝ^n and y∈ℝ^m, Then, under
assumptions (A1), (A2) and (A3), for  any
T>0 and ϕ∈ C_b^3(ℝ^n,ℝ), there exists a
 constant C_T,ϕ,x,y such that
 
    |𝔼ϕ(X^ϵ_T(x,y))-𝔼ϕ(X̅_T(x))|≤
     C_T,ϕ,x,yϵ.

 As a consequence, it can be claimed that the weak order in
 averaging principle for jump-diffusion stochastic  systems is 1.







§ ASYMPTOTIC EXPANSION

Let ϕ∈ C_b^3(ℝ^n, ℝ) and define a function
u^ϵ(t, x,y):[0, T]×ℝ^n×ℝ^m→ℝ by

    u^ϵ(t, x,y)=𝔼ϕ(X_t^ϵ(x,y)).

We are now ready to seek an expansion  formula for u^ϵ(t,
x,y) with respect to ϵ with the form

    u^ϵ(t,x,y)=u_0(t,x,y)+ϵ u_1(t,x,y)+r^ϵ(t,x,y),

where  u_0 and u_1 are smooth functions which will be
constructed below, and r^ϵ is the remainder term. To this
end, let us recall the Kolmogorov operator corresponding to the slow
motion equation, with a frozen fast component y∈ℝ^m,
which is a  second order   operator  taking form

    ℒ_1Φ(x)   =   (a(x,y),D_x
    Φ(x))_ℝ^n+1/2Tr[D_xx^2 Φ(x)
    ·
    b(x ) b^T(x )]
          +λ_1(Φ(x+c(x ))-Φ(x)),  Φ∈ C_b^2(ℝ^n,
    ℝ).

For any frozen slow component x∈ℝ^m, the Kolmogorov
operator for equation (<ref>) is given by

    ℒ_2Ψ(y)   =   (f(x,y),D_y
    Ψ(y))_ℝ^m+1/2Tr[  D_yy^2Ψ(y)
    ·
    g(x,y) g^T(x,y)]
          +λ_2(Ψ(y+h(x,y))-Ψ(y)),  Ψ∈ C_b^2(ℝ^m,
    ℝ).

We  set

    ℒ^ϵ:=ℒ_1+1/ϵℒ_2.

It is known u^ϵ(t,x,y) solves the  equation

    ∂/∂ tu^ϵ(t, x, y)=ℒ^ϵ u^ϵ(t, x, y),
    
    u^ϵ(0, x,y)=ϕ(x),

Also recall the Kolmogorov operator associated with the averaged
equation (<ref>) is defined as

    ℒ̅Φ(x)   =   (a̅(x ),D_x
    Φ(x))_ℝ^n+1/2Tr[  D_xx^2Φ(x)
    ·
    b(x ) b^T(x )]
          +λ_1(Φ(x+c(x))-Φ(x)),  Φ∈ C_b^2(ℝ^n,
    ℝ).

If we set 
    u̅(t, x)=𝔼ϕ(X̅_t(x)),
 we have

    ∂/∂ tu̅(t, x)=ℒ̅u̅(t, x),
    u̅(0, x)=ϕ(x).









 §.§ The leading term

Let us begin with constructing the leading term.
 By substituting expansion (<ref>) into (<ref>), we
see that

    ∂ u_0/∂ t+ϵ∂ u_1/∂
    t+∂ r^ϵ/∂ t   =   ℒ_1u_0+ϵℒ_1u_1+ℒ_1r^ϵ
       +   1/ϵℒ_2u_0+ℒ_2u_1+1/ϵℒ_2r^ϵ.

By  equating powers of      ϵ, we obtain the following
system of equations:

    ℒ_2u_0=0, 
          ∂ u_0/∂ t=ℒ_1u_0+ℒ_2u_1.

According to (<ref>), we can conclude u_0 does not
depend on y, that is 
    u_0(t,x, y)=u_0(t,x).
 We also impose the
initial condition u_0(0,x)=ϕ(x). Note that  ℒ_2 is
the generator of a Markov process defined by equation
(<ref>), which admits  a unique invariant measure μ^x, we
have

    ∫_ℝ^mℒ_2u_1(t,x,y)μ^x(dy)=0.

Thanks to   (<ref>), this yields

    ∂ u_0/∂ t(t,x)   =   ∫_ℝ^m∂ u_0/∂ t(t,x)μ^x(dy)
       =   ∫_ℝ^mℒ_1u_0(t,x)μ^x(dy)
       =   ∫_ℝ^m(a(x,y), D_x
    u_0(t,x))_ℝ^nμ^x(dy)
          +1/2Tr[D_xx^2u_0(t,x)·
    b(x ) b^T(x )]
          +λ_1(u_0(x+c(x ))-u_0(x))
       =   ℒ̅u_0(t,x),

so that u_0 and u̅ are described by the same evolutionary
equation. By  uniqueness argument, we easily have the following
lemma:

Under assumptions (A1), (A2) and (A3),
for any x∈ℝ^n, y∈ℝ^m and T>0, we have
u_0(T,x,y)=u̅(T,x).






 §.§ Construction of u_1

According to Lemma <ref>, (<ref>) and
(<ref>), we get

    ℒ̅u̅=ℒ_1u̅+ℒ_2u_1,

which means that

    ℒ_2u_1(t,x,y)   =   (a̅(x)-a(x,y),D_xu̅(t,x))_ℝ^n
       :=   -ρ(t,x,y),

where ρ is of class C^2 with respect to y, with uniformly
bounded derivatives. Moreover, for any t≥ 0 and x∈ℝ^n, the equality (<ref>) guarantees that

    ∫_ℝ^mρ(t,x,y)μ^x(dy)=0.

For any y∈ℝ^m and s>0 we have

    ∂/∂ s𝒫_sρ(t,x,y)   =   (f(x,y),D_y [𝒫_sρ(t,x,y)])_ℝ^m
          +1/2Tr[D_yy^2[𝒫_sρ(t,x,y)]·
    g(x,y) g^T(x,y)]
          +λ_2(𝒫_s[ρ(t,x,y+h(x,y))]-𝒫_s[ρ(t,x,y)]),

here 
    𝒫_s[ρ(t,x,y)]:=𝔼ρ(t, x,Y^x_s(y)).

Recalling that   μ^x is the unique invariant measure
corresponding to Markov process Y^x_t(y) defined by  equation
(<ref>), from Lemma <ref> we infer that

    |𝔼ρ(t,
    x,Y^x_s(y))-∫_ℝ^mρ(t,x,z)μ^x(dz)|
          =|∫_ℝ^m𝔼[ρ(t, x,Y^x_s(y))- ρ(t,
    x,Y^x_s(z))]μ^x(dz)|
          ≤∫_ℝ^m|𝔼(a(x,Y^x_s(z))-
    a(x,Y^x_s(y)), D_xu̅(t,x))_ℝ^n|μ^x(dz)
          ≤ C∫_ℝ^m𝔼Y_s^x(z)-
    Y_s^x(y)_ℝ^nμ^x(dz).

Now it follows from   (<ref>) and
(<ref>) that

    |𝔼ρ(t,
    x,Y_s^x(y))-∫_ℝ^mρ(t,x,z)μ^x(dz)|
          ≤
    C(1+x_ℝ^n+y_ℝ^m)e^-β/2s,

which implies

    lim_s→+∞𝔼ρ(t,
    x,Y_s^x(y))=∫_ℝ^mρ(t,x,z)μ^x(dz)=0.

With the aid of the above limit, we can deduce from (<ref>)
that

    (f(x,y),D_y ∫_0^+∞[𝒫_sρ(t,x,y)]ds)_ℝ^m
          +1/2Tr[D_yy^2∫_0^+∞[𝒫_sρ(t,x,y)]·
    g(x,y) g^T(x,y)ds]
          +λ_2(∫_0^+∞𝒫_s[ρ(t,x,y+h(x,y))]ds-∫_0^+∞𝒫_s[ρ(t,x,y)]ds)
          =∫_0^+∞∂/∂ s𝒫_s[ρ(t,x,y)]ds
          =lim_s→+∞𝔼ρ(t,
    x,Y_s^x(y))-ρ(t,x,y)
          =∫_ℝ^mρ(t,x,z)μ^x(dz)-ρ(t,x,y)
          =-ρ(t,x,y),

which implies 
    ℒ_2(∫_0^+∞𝒫_sρ(t,x,y) ds)=-ρ(t,x,y).
 Therefore,

    u_1(t,x,y):=∫_0^+∞𝔼ρ(t,x,Y^x_s(y))ds

is the solution to  equation (<ref>).

Under assumptions (A1), (A2) and (A3),
 for any x∈ℝ^n, y∈ℝ^m and T>0, we have

    |u_1(t,x,y)|≤ C_T(1
    +x_ℝ^n+y_ℝ^m), t∈[0, T].


By (<ref>), we have

    u_1(t,x,y)=∫_0^+∞𝔼(a̅(x)- a(x,Y_s^x(y)), D_xu̅(t,x))_ℝ^n ds,

so that

    |u_1(t,x,y)|   ≤   ∫_0^+∞a̅(x)-
    𝔼[a(x,Y_s^x(y))]_ℝ^n ·D_xu̅(t,x)_ℝ^n  ds.

Therefore, from Lemma <ref> and
(<ref>), we get

    |u_1(t,x,y)|   ≤    C_T(1+x_ℝ^n   +y_ℝ^m )∫_0^+∞e^-β/2 sds
       ≤   C_T(1+x_ℝ^n   +y_ℝ^m ).






 §.§ Determination of remainder r^ϵ

We now turn to  the construction for   remainder term r^ϵ.
It is known that

    (∂_t-ℒ^ϵ)u^ϵ=0,

which, together with (<ref>) and (<ref>),
implies

    (∂_t-ℒ^ϵ)r^ϵ   =   -(∂_t-ℒ^ϵ)
    u_0-ϵ(∂_t-ℒ^ϵ)u_1
       =   -(∂_t-1/ϵℒ_2-ℒ_1)u_0-ϵ(∂_t-1/ϵℒ_2-ℒ_1)u_1
       =   ϵ(ℒ_1u_1-∂_tu_1).

In order to estimate the remainder term r^ϵ  we need the
following   two lemmas.


Under assumptions (A1), (A2) and (A3),
for any x∈ℝ^n, y∈ℝ^m and T>0, we have

    |∂ u_1/∂ t(t,x,y)|≤
    C_T(1+x_ℝ^n+y_ℝ^m).


In view of (<ref>), we get

    ∂ u_1/∂
    t(t,x,y)=∫_0^+∞𝔼(a̅(x)-
    a(x,Y_s^x(y)), ∂/∂
    tD_xu̅(t,x))_ℝ^nds.

By Lemma <ref> introduced in Section <ref>,
we have

    |∂ u_1/∂
    t(t,x,y)|   ≤   ∫_0^+∞𝔼(a̅(x)-
    a(x,Y_s^x(y))_ℝ^n·∂/∂ tD_xu̅(t,x)_ℝ^n)ds
       ≤   C_T∫_0^+∞𝔼a̅(x)-
    a(x,Y_s^x(y))_ℝ^nds,

so that from  (<ref>) we have

    |∂ u_1/∂ t(t,x,y)|≤
     C_T(1+x_ℝ^n+y_ℝ^m).




Under assumptions (A1), (A2) and (A3),
for any x∈ℝ^n, y∈ℝ^m and T>0, we have

    |ℒ_1u_1(t,x,y)|≤
    C_T(1+x_ℝ^n+y_ℝ^m),  t∈ [0, T].


Recalling that  u_1(t,x,y) is the solution of equation
(<ref>) and equality (<ref>) holds, we have

    ℒ_1u_1(t,x,y)   =   (a(x,y),D_x
    u_1(t,x,y))_ℝ^n
       +   1/2Tr[D^2_xxu_1(t,x,y)·
    b(x ) b^T(x )]
       +   λ_1[u_1(t,x+c(x ),y)-u_1(t,x,y)],

and then, in order to prove the boundedness of ℒ_1u_1,
we have to estimate the three terms arising in the right hand side
of above  equality.

Step 1:  Estimate of (a(x,y),D_x
u_1(t,x,y))_ℝ^n.

For any k∈ℝ^n, we have

    D_xu_1(t,x,y)· k   =   ∫_0^+∞(D_x(a̅(x)-
    𝔼a(x,Y_s^x(y)))· k,D_xu̅(t,x))_ℝ^n ds
          +∫_0^+∞(a̅(x)-𝔼
    a(x,Y_s^x(y)),D^2_xxu̅(t,x)·
    k)_ℝ^n ds
       =:   I_1(t,x,y,k)+I_2(t,x,y,k).

By Lemma <ref> and <ref>, we infer that

    |I_1(t,x,y,k)|    ≤   D_xu̅(t,x)_ℝ^n∫_0^+∞D_x(a̅(x)-
    𝔼a(x,Y_s^x(y)))· k_ℝ^nds
       ≤   C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m)∫_0^+∞e^-β/2sds
       ≤   C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m).

By Lemma <ref> and inequality
(<ref>), we obtain

    |I_2(t,x,y,k)|    ≤   
    C_Tk_ℝ^n∫_0^+∞a̅(x)-𝔼
    a(x,Y_s^x(y))_ℝ^nds
       ≤   C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m)
    ∫_0^+∞e^-β/2sds
       ≤   C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m).

This, together with (<ref>), implies

    D_xu_1(t,x,y)· k≤
    C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m),

 and then, as a(x,y) is bounded, it follows

    |(a(x,y),D_x u_1(t,x,y))_ℝ^n|   ≤   
    C_Ta(x,y)_ℝ^n(1+x_ℝ^n+y_ℝ^m)
       ≤    C_T
    (1+x_ℝ^n+y_ℝ^m).

Step 2: Estimate of Tr[D^2_xxu_1(t,x,y)· b(x )
b^T(x )].

Since  u_1(t,x,y) is given by the  representation formula
(<ref>),     for any k_1,k_2∈ℝ^n we have

    D^2_xxu_1(t,x,y)· (k_1, k_2)
          =∫_0^+∞𝔼(D^2_xx(a̅(x)-
    a(x,Y_s^x(y)))· (k_1, k_2),D_xu̅(t,x))_ℝ^n ds
          +∫_0^+∞𝔼(D_x(a̅(x)-
    a(x,Y_s^x(y)))· k_1,D^2_xxu̅(t,x)·
    k_2)_ℝ^n ds
          +∫_0^+∞𝔼(D_x(a̅(x)-
    a(x,Y_s^x(y)))· k_2,D^2_xxu̅(t,x)·
    k_1)_ℝ^n ds
          +∫_0^+∞𝔼(a̅(x)- a(x,Y_s^x(y)),
    D^3_xxxu̅(t,x)· (k_1,k_2))_ℝ^n
    ds
          :=∑_i=1^4J_i(t,x,y,k_1,k_2).

Thanks to  Lemma <ref> and Lemma <ref> we
get

    |J_1(t,x,y,k_1,k_2)|
       ≤   ∫_0^+∞|𝔼(D^2_xx(a̅(x)-
    a(x,Y_s^x(y)))·
    (k_1,k_2),D_xu̅(t,x))_ℝ^n | ds
       ≤   
    C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n
    k_2_ℝ^n
    ∫_0^+∞e^-β/2sds
       ≤   C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n
    k_2_ℝ^n .

 By Lemma <ref> and  (<ref>) we infer that

    |J_2(t,x,y,k_1,k_2)|
       ≤   ∫_0^+∞|𝔼(D_x(a̅(x)-
    a(x,Y_s^x(y)))· k_1,D^2_xxu̅(t,x)·
    k_2)_ℝ^n |ds
       ≤   
    C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n
    k_2_ℝ^n
    ∫_0^+∞e^-β/2sds
       ≤   C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n
    k_2_ℝ^n .

With a similar argument we can also show that

    |J_3(t,x,y,k_1,k_2)|
       ≤   C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n
    k_2_ℝ^n .

By making use of Lemma <ref> and
(<ref>), we get

    |J_4(t,x,y,k_1, k_2)|
       ≤   C_Tk_1_ℝ^n·k_2_ℝ^n·(1+x_ℝ^n+y_ℝ^m)∫_0^+∞ e^-β/2sds
       ≤   
    C_Tk_1_ℝ^n·k_2_ℝ^n(1+x_ℝ^n+y_ℝ^m).

In view of the above estimates (<ref>), (<ref>), (<ref>)
and (<ref>), we can conclude that there exists a constant C_T
such that

    |D^2_xxu_1(t,x,y)· (k_1, k_2)|≤ C_T
    k_1_ℝ^n·k_2_ℝ^n(1+x_ℝ^n+y_ℝ^m), t∈[0,T],

which means that for fixed y∈ℝ^m and t∈ [0, T],

    D^2_xxu_1(t,x,y) _L( ℝ^n,ℝ)≤
    C_T(1+x_ℝ^n+y_ℝ^m),

where ·_L(ℝ^n,ℝ) denotes the usual
operator norm on Banach space consisting of  bounded and linear
operators from ℝ^n to ℝ. As the diffusion
function g is bounded, we get

    Tr(D^2_xxu_1(t,x,y)gg^T)
       ≤    C_TD^2_xxu_1(t,x,y)_L( ℝ^n,ℝ)
       ≤   C_T(1+x_ℝ^n+y_ℝ^m).


Step 3: Estimate of λ_1[u_1(t,x+c(x
),y)-u_1(t,x,y)].

By Lemma <ref> and boundedness condition of c(x),
we directly have

    |λ_1[u_1(t,x+c(x ),y)-u_1(t,x,y)]|
          λ_1[|u_1(t,x+c(x ),y)|+|u_1(t,x,y)|]
          ≤ C_T(1
    +x_ℝ^n+y_ℝ^m), t∈ [0,
    T].

Finally, it is now easy to gather all previous estimates  for terms
in (<ref>) and conclude

    |ℒ_1u_1(t,x,y)|≤
    C_T(1+x_ℝ^n+y_ℝ^m),  t∈ [0, T].





Under the conditions of Lemma <ref>, for any T>0,
x∈ℝ^n and y ∈ℝ^m, we have

    |r^ϵ(T,x,y)|≤
    C_Tϵ(1+x_ℝ^n+y_ℝ^m).


By a variation of constant formula, we  write the equation
(<ref>) in its integral form

    r^ϵ(T,x,y)   =   𝔼[r^ϵ(0,X^ϵ_T(x,y),Y^ϵ_T(x, y))]
       +   ϵ[∫_0^T𝔼(ℒ_1u_1-∂
    u_1/∂ s)(s, X^ϵ_T-s(x,y),Y^ϵ_T-s(x,
    y)) ds].

Since u^ϵ and u̅ satisfy the same initial condition,
we have

    |r^ϵ(0, x,y)|   =   |u^ϵ(0,x,y)-u̅(0,x)-ϵ
    u_1(0,x,y)|
       =   ϵ |u_1(0,x,y)|,

so that, thanks to (<ref>), (<ref>) and
(<ref>) we have

    𝔼[r^ϵ(0,X^ϵ_T(x,y),Y^ϵ_T (x,y)]≤
    Cϵ(1+x_ℝ^n+ y_ℝ^m).

Using Lemma <ref> and Lemma <ref> yields

    𝔼[(ℒ_1u_1-∂ u_1/∂ s)(s,
    X^ϵ_T-s(x,y),Y^ϵ_T-s (x,y))]
          ≤ C 𝔼
    (1+X^ϵ_T-s(x,y)+Y^ϵ_T-s(x,y)),

and, according to (<ref>) and (<ref>), this implies
that

    𝔼[∫_0^T(ℒ_1u_1-∂
    u_1/∂ s)(s, X^ϵ_T-s(x,y), Y^ϵ_T-s(x,y))
    ds]
          ≤ C_T (1+x_ℝ^n+ y_ℝ^m).

The last inequality together with (<ref>)  yields

    |r^ϵ(T,x,y)|≤ϵ C_T (1+x_ℝ^n+
    y_ℝ^m).









§ APPENDIX

In this appendix we collect  some technical results to which we
appeal in the proofs of the main results in Section 4 .

For any T>0, there exists a constant C_T>0 such that for any
x,k∈ℝ^n and t∈ [0, T], we have

    |D_xu̅(t,x)· k| ≤ C_Tk_ℝ^n.


Observe that for any   k∈ℝ^n,

    D_xu̅(t,x)·
    k   =   𝔼[Dϕ(X̅_t(x))·η ^k,x_t]
       =   𝔼(ϕ'(X̅_t(x)),η^k,x_t)_ℝ^n,

where η^k,x_t denotes the first mean-square derivative of
X̅_t(x) with respect to x∈ℝ^n along the
direction k∈ℝ^n, then we have

    dη^k,x_t=D_xa̅(X̅_t(x))·η^k,x_tdt+D_xb(X̅_t(x))·η^k,x_td B_t
          +D_xc(X̅_t-(x))·η^k,x_t-d P_t,
    η ^k,x_0=k.

This means that  η ^k, x_t  is the  solution of the integral
equation
 
    η^k, x_t   =   k+∫_0^tD_xa̅(X̅_s(x))·η^k, x_s
    ds+∫_0^tD_xb(X̅_s(x))·η^k, x_sdB_s
          +∫_0^tD_xc(X̅_s-(x))·η_s-^k, xdP_s

and then thanks to  assumption (A1), we get

    𝔼η ^k,x_t^2_ℝ^n≤
    C_Tk^2_ℝ^n +C_T∫_0^t𝔼η
    ^k,x_s^2_ℝ^n ds.

Then by  Gronwall lemma it follows that

    𝔼η ^k,x_t^2_ℝ^n≤ C_Tk
    ^2_ℝ^n,  t∈ [0, T],

so that

    |D_xu̅(t,x)· k|≤  C_T k_ℝ^n.



Next, we introduce an analogous result for the second derivative of
u̅(t,x).

For any T>0, there exists a constant C_T>0 such that for any
x,k_1,k_2∈ℝ^n and t∈ [0, T], we have

    |D^2_xxu̅(t,x)·(k_1,k_2)|≤ C_Tk_1_ℝ^n·k_2_ℝ^n.


For any k_1, k_2 ∈ℝ^n, we have

    D^2_xxu̅(t,x)·(k_1,k_2)   =   𝔼[ϕ”(X̅_t(x))·(η^k_1,x_t,η^k_2,x_t)
       +   ϕ'(X̅_t(x))·ξ^k_1,k_2,x_t],

where ξ^k_1,k_2,x_t is  the solution of     the second
variation equation corresponding to the averaged equation, which may
be rewritten in the following form:

    ξ^k_1,k_2,x_t   =   ∫_0^t[D_xa̅ (X̅_s(x))·ξ^k_1,k_2,x_s+D_xx^2a̅ (X̅_s(x))·(η^k_1,x_s,η^k_2,x_s)]ds
       +   ∫_0^t[D_xx^2b (X̅_s(x))·(η^k_1,x_s,η^k_2,x_s)+D_xb(X̅_s(x))·ξ^k_1,k_2,x_s]dB_s
       +   ∫_0^t[D_xx^2c(X̅_s-(x))·(η^k_1,x_s-,η^k_2,x_s-)+D_xc(X̅_s-(x))·ξ^k_1,k_2,x_s-]dP_s.

Thus, by assumption (A1) and (<ref>)  we have

    𝔼ξ^k_1,k_2,x_t^2_ℝ^n   ≤   
    C_T∫_0^t({𝔼η^k_1,x_s^2_ℝ^n}^1/2{𝔼η^k_2,x_s^2_ℝ^n}^1/2+𝔼ξ^k_1,k_2,x_s^2_ℝ^n)ds
       ≤    C_Tk_1_ℝ^n·k_2_ℝ^n
    +C_T∫_0^t𝔼ξ^k_1,k_2,x_s^2_ℝ^nds.

By applying the Gronwall lemma we have

    𝔼ζ^k_1,k_2,x_t^2_ℝ^n≤
    C_Tk_1_ℝ^n·k_2_ℝ^n.

Returning to (<ref>), we can get

    |D^2_xxu̅(t,x)·(k_1,k_2)|≤ C_T
    h_1_ℝ^n·k_2_ℝ^n.



By using the analogous arguments used before, we can prove the
following estimate for the  third order derivative of u̅(t,x)
with respect to x.

For any T>0, there exists a constant C_T>0 such that for any
x,k_1,k_2,k_3∈ℝ^n and t∈ [0, T], we have

    |D^3_xxxu̅(t,x)·(k_1,k_2,k_3)|≤ C_T k_1_ℝ^n·k_2_ℝ^n·k_3_ℝ^n.



The following lemma states   boundedness for the first derivative of
a̅(x)-𝔼a(x, Y^x_t(y)) with respect to x.

There exists a constant C>0 such that for any x∈ℝ^n,
y∈ℝ^m, k∈ℝ^n and t>0 it holds

    D_x (a̅(x)-𝔼a(x, Y^x_t(y)))·
    k_ℝ^n≤
    Ce^-β/2tk_ℝ^n(1+x_ℝ^n+y_ℝ^m).


 The proof is a modification of the proof of <cit.>. For
any t_0>0, we set

    ã_t_0(x,y,t)=â(x,y,t)-â(x,y,t+t_0),

where

    â(x,y,t):=𝔼a(x, Y^x_t(y)).

Then we have

    lim_t_0→
    +∞ã_t_0(x,y,t)=𝔼a(x, Y^x_t(y))-a̅(x).

By Markov property, we have

    ã_t_0(x,y,t)   =   â(x,y,t)-𝔼a(x,Y_t+t_0^x(y))
       =   â(x,y,t)-𝔼â(x, Y_t_0^x(y),t)

Due to assumption (A1),  for any k∈ℝ^n we
have

    D_xã_t_0(x,y,t)· k   =   D_xâ(x,y,t)·
    k-𝔼D_x(â(x, Y_t_0^x (y),t))· k
       =   â_x'(x,y,t)· k-𝔼â_x'(x,
    Y_t_0^x(y),t)·
    k
          -𝔼â_y'(x,
    Y_t_0^x(y),t)·(D_xY_t_0^x(y)·
    k),

where  the symbols â_x' and â_y'   denote the
directional derivatives with respect to x and y, respectively.
Note that the first derivative ζ_t^x,y,
k=D_xY_t^x(y)· k, at the point x and along the direction
k∈ℝ^n, is the solution of equation

    dζ_t^x, y,k   =   ( f_x'(x, Y_t^x(y))·
    k+f_y'(x, Y_t^x(y))·ζ_t^x, y,k)dt
          +(g_x'(x, Y_t^x(y))· k+g_y'(x,
    Y_t^x(y))·ζ_t^x, y,k)dW_t
          +(h_x'(x, Y_t-^x(y))· k+h'_y(x,
    Y_t-^x(y))·ζ_t-^x, y,k)dN_t

with initial data ζ_0^x,y, k=0. Hence,   by assumption
(A1), it is straightforward to check

    𝔼ζ_t^x,y, k_ℝ^m≤
    Ck_ℝ^n

for any t≥ 0. Note that for any y_1, y_2∈ℝ^m,
we have

    â(x,y_1,
    t)-â(x,y_2,t)_ℝ^n   =   𝔼a(x,
    Y_t^x(y_1))-𝔼a(x,
    Y_t^x(y_2))_ℝ^n
       ≤   C𝔼Y_t^x(y_1)-Y_t^x(y_2)_ℝ^m
       ≤    Ce^-β/2ty_1-y_2_ℝ^m,

where   (<ref>) was used to obtain the last
inequality. This means that

    â_y'(x, y,t)· l_ℝ^m≤
    Ce^-β/2tl_ℝ^m,  l∈ℝ^m.

From  (<ref>) and (<ref>), we obtain

    𝔼[â_y'(x,
    Y_t_0^x(y),t)·(D_xY_t_0^x(y)·
    k)]_ℝ^m
          =𝔼[â_y'(x,
    Y_t_0^x(y),t)· (ζ_t_0^x,y, k )]_ℝ^m
          ≤ C e^-β/2tk_ℝ^n.

Then, by easy calculations, we have

    â_x'(x,y_1,t)· k-â_x'(x,y_2,t)· k
            =𝔼(a_x'(x, Y_t^x(y_1)))·
    k-𝔼(a_x'(x, Y_t^x(y_2)))· k
              +𝔼(a_y'(x, Y_t^x(y_1))·ζ_t^x,y_1,
    k-a_y'(x, Y_t^x(y_2))·ζ_t^x,y_2, k)
            = 𝔼(a_x'(x, Y_t^x(y_1)))·
    k-𝔼(a_x'(x, Y_t^x(y_2)))· k
              +𝔼([a_y'(x, Y_t^x(y_1))-a_y'(x,
    Y_t^x(y_2))]·ζ_t^x,y_1, k)
              +𝔼(a_y'(x,
    Y_t^x(y_2))·(ζ_t^x,y_1, k-ζ_t^x,y_2,
    k))
            := ∑_i=1^3𝒩_i(t,x,y_1,y_2, k).

Now, we estimate the three terms in the right hand side of above
equality. Concerning 𝒩_1(t,x,y_1,y_2, k) we have

    𝒩_1(t,x,y_1,y_2, k)_ℝ^n
          ≤𝔼(a_x'(x, Y_t^x(y_1)))·
    k-(a_x'(x, Y_t^x(y_2)))· k_ℝ^n
          ≤
    C𝔼Y_t^x(y_1)-Y_t^x(y_2)_ℝ^m·k_ℝ^n
          ≤
    Ce^-β/2ty_1-y_2_ℝ^m·k_ℝ^n.

Next, by  assumption (A1)  we get

    𝒩_2(t,x,y_1,y_2, k)_ℝ^n
          ≤𝔼[a_y'(x, Y_t^x(y_1))-a_y'(x,
    Y_t^x(y_2))]·ζ_t^x,y_1, k_ℝ^n
          ≤ C{𝔼ζ_t^x,y_1,
    k^2_ℝ^m}^1/2·{𝔼Y_t^x(y_1)-Y_t^x(y_2)^2_ℝ^m}^1/2
          ≤ C
    e^-β/2tk_ℝ^n·y_1-y_2_ℝ^m.

For the third term, by making use of assumption (A1) again,
we can infer that

    𝒩_3(t,x,y_1,y_2, k)_ℝ^n
          ≤𝔼a_y'(x, Y_t^x(y_2))·(ζ_t^x,y_1, k-ζ_t^x,y_2, k)_ℝ^n
          ≤ C𝔼ζ_t^x,y_1, k-ζ_t^x,y_2, k_ℝ^m
          ≤ C
    e^-β/2ty_1-y_2_ℝ^m·k_ℝ^n.

Now, returning to (<ref>) and  taking into account of
(<ref>), (<ref>) and (<ref>), we get

    â_x'(x,y_1,t)· k-â_x'(x,y_2,t)· k
          ≤ C
    e^-β/2ty_1-y_2_ℝ^m·k_ℝ^n,

which leads to

    â_x'(x,y,t)· h- 𝔼â_x'(x,Y_t_0^x(y),t)· k_ℝ^n
          ≤ C
    e^-β/2t(1+y_ℝ^m+Y_t_0^x(y)_ℝ^m)·k_ℝ^n
          ≤
    e^-β/2t(1+x_ℝ^n+y_ℝ^m)·k_ℝ^n,

where we used the inequality (<ref>). Returning to
(<ref>), by (<ref>) and (<ref>) we conclude that

    D_xã_t_0(x,y,t)· k_ℝ^n≤
    Ce^-β/2t(1+x_ℝ^n+y_ℝ^m)k_ℝ^n.

Taking the limit as t_0→ +∞  we obtain

    D_x (a̅(x)-𝔼a(x, Y^x_t(y)))_ℝ^n≤
    Ce^-β/2tk_ℝ^n(1+x_ℝ^n+y_ℝ^m).




 Proceeding with similar arguments above we obtain the following higher order
 differentiability.

There exists a constant C>0 such that for any x, k_1, k_2∈ℝ^n, y∈ℝ^m and t>0 it holds

    D^2_xx(a̅(x)-𝔼a(x,
    Y^x_t(y)))(k_1,k_2)_ℝ^n
          ≤
    Ce^-β/2tk_1_ℝ^nk_2_ℝ^n(1+x_ℝ^n+y_ℝ^m).





Finally, we introduce the following auxiliary result.

There exists a constant C>0 such that for any x, k∈ℝ^n, y∈ℝ^m and t>0 it holds

    ∂/∂ tD_xu̅(t,x)· k_ℝ^n≤ Ck_ℝ^n.


For simplicity of presentation, we will prove it for the
1-dimensional case. The multi-dimensional situation can be treated
similarly, only notations are somewhat involved.  In this case we
only need to show

    |∂/∂ t∂/∂
    xu̅(t,x)|≤ C.

Actually, for any ϕ∈ C_b^3(ℝ,ℝ) we have

    ∂/∂ xu̅(t,x)=∂/∂
    x𝔼ϕ(X̅_t(x))=𝔼(ϕ'(X̅_t(x))·∂/∂
    xX̅_t(x)).

If we define

    ς^x_t:=∂/∂ xX̅_t(x),

we have

    ς^x_t   =   1+∫_0^ta̅'(X̅_s(x))·ς^x_s
    ds+∫_0^tb'(X̅_s(x))·ς^x_sdB_s
          +∫_0^tc'(X̅_s-(x))·ς_s-^xdP_s.

The boundedness of a̅', b' and c'   guarantees

    𝔼|ς^x_t|^2≤ C_T,  t∈ [0,
    T].

 By    using Itô formula we have

    𝔼[ϕ'(X̅_t(x))·ς_t^x]
          =ϕ'(x)+𝔼∫_0^t[ϕ'(X̅_s(x))a̅'(X̅_s(x))ς_s^x
    +ς_s^xϕ”(X̅_s(x))a̅(X̅_s(x))]ds
          +𝔼∫_0^tb'(X̅_s(x)))ς_s^xϕ”(X̅_s(x))b(X̅_s(x))ds
          +1/2𝔼∫_0^tς_s^xϕ”'(X̅_s(x))b^2(X̅_s(x))ds
          +λ_1𝔼∫_0^tϕ'(X̅_s(x))c'(X̅_s-(x))ς_s^xds
          +λ_1𝔼∫_0^tς_s^x[ϕ'(X̅_s-(x)+c(X̅_s-(x)))-ϕ'(X̅_s-(x))]ds
          +λ_1𝔼∫_0^tc'(X̅_s-(x))ς_s^x[ϕ'(X̅_s-(x)+c(X̅_s-(x)))-ϕ'(X̅_s-(x))]ds.

 Since ϕ belongs to C_b^3(ℝ, ℝ), from
 the assumption (A1) it follows that for any t∈ [0, T],

    |∂/∂ t[∂/∂
    xu̅(t,x)]|   =   |∂/∂ t𝔼[ϕ'(X̅_t(x))·ς_t^x]|
       ≤    C|𝔼ς_t^x|,

then, by taking (<ref>) into account, one would
easily arrive at (<ref>).











§ ACKNOWLEDGMENTS

We would like to thank Professor Jinqiao Duan for helpful
discussions and comments. Hongbo Fu is supported by Natural Science
Foundation of Hubei Province (No. 2018CFB688), NSF of China (No.
11301403) and  Chinese Scholarship Council (No. [2015]5104). Bengong
Zhang is supported by NSF of China (No. 11401448).  Li Wan is
supported by NSF  of China (No. 61573011). Jicheng Liu is supported
by NSF  of China (No. 11271013).



§ COMPETING INTERESTS

The authors declare  that they have no competing interests.



§ AUTHORS CONTRIBUTIONS

The authors declare that the work was realized in collaboration with
the same responsibility. All authors read and approved the final
manuscript.

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