On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients

0.25 MB·

by  Chaman Kumar

ON MILSTEIN APPROXIMATIONS WITH VARYING
COEFFICIENTS: THE CASE OF SUPER-LINEAR
DIFFUSION COEFFICIENTS
CHAMAN KUMARAND SOTIRIOSSABANIS
6
1 Abstract. AnewclassofexplicitMilsteinschemes,whichapproximate
0
stochasticdifferentialequations(SDEs)withsuperlinearlygrowingdrift
2
and diffusion coefficients, is proposed in this article. It is shown, under

n very mild conditions, that these explicit schemes converge in Lp to the
a solution of thecorresponding SDEswith optimal rate.
J AMS subject classifications: Primary 60H35; secondary 65C30.

1
1

]
R 1. Introduction
P
Followingtheapproachof[8],weextendthetechniquesofconstructingex-
.
h plicit approximations to the solutions of SDEs with super-linear coefficients
t
a in order to develop Milstein-type schemes with optimal rate of (strong) con-
m
vergence.
[ Recent advances in the area of numerical approximations of such non-

1 linear SDEshave producednewEuler-typeschemes, e.g. see[4,7,9,8,3,2],
v which are explicit in nature and hence computationally more efficient than
5
their implicit counterparts. High-order schemes have also been developed in
9
thisdirection. Inparticular,Milstein-type(order1.0)schemesforSDEswith
6
2 super-linear drift coefficients have been studied in [10] and in [5] with the
0 latter article extending the results to include L´evy noise, i.e. discontinuous
.
1 paths. Furthermore, both driftanddiffusion coefficients are allowed to grow
0
super-linearlyin[11]andin[1]. Thelatterreferencehassignificantlyrelaxed
6
theassumptionson theregularity of SDEcoefficients byusingthenotionsof
1
: C-stability and B-consistency. More precisely, the authors in [1] produced
v
optimal rate of convergence results in the case where the drift and diffu-
i
X
sion coefficients are only (once) continuously differentiable functions. Our
r results, which were developed at around the same time as the latter refer-
a
ence by using different methodologies, are obtained under the same relaxed
assumptions with regards to the regularity that is required of the SDE coef-
ficients. Crucially, we relax further the moments bound requirement which
is essential for practical applications.
We illustrate the above statement by considering an example which ap-
pears in [1], namely the one-dimensional SDE given by
dx =x (1−x2)dt+σ(1−x2)dw , ∀t ∈ [0, T],
t t t t t
with initial value x and a positive constant σ. Theorem 1 below yields
0
that for p = 14 (note that ρ = 2) one obtains optimal rate of convergence
0
in L2 (when σ2 ≤ 2 and p > 2 such that σ2(p − 1) ≤ 1) whereas the
13 1 1
corresponding result in [1], Table 1 in Section 8, requires p = 18 for their
0
1
2 C.KUMARANDS.SABANIS
explicit (projective) scheme. The same requirement, i.e. p = 14, as in this
0
article is only achieved by the implicit schemes considered in [1].
Finally, we note that Theorem 1 establishes optimal rate of convergence
results (under suitable assumptions) in Lp for p > 2, which is, to the best
of the authors’ knowledge, the first such results in the case of SDEs with
super-linear coefficients.
We conclude this section by introducingsomenotations which areused in
thisarticle. TheEuclideannormofad-dimensionalvectorbandtheHilbert-
Schmidt norm of a d×m matrix σ are denoted by |b| and |σ| respectively.
The transpose of a matrix σ is denoted by σ∗. The ith element of b is
denoted by bi, whereas σ(i,j) and σ(j) stand for (i,j)-th element and j-th
column of σ respectively for every i = 1,...,d and j = 1,...,m. Further,
xy denotes the inner product of two d-dimensional vectors x and y. The
notation ⌊a⌋ stands for the integer part of a positive real number a. Let
D denote an operator such that for a function f : Rd → Rd, Df(.) gives a
∂fi(.)
d×d matrix whose (i,j)-th entry is for every i,j = 1,...,d. For every
∂xj
j = 1,...,m, let Λj be an operator such that for a function g :Rd → Rd×m,
Λjg(.) gives a matrix of order d×m whose (i,k)-th entry is given by
d ∂g(i,k)(.)
[Λjg(.)] := g(u,j)(.)
(i,k) ∂xu
Xu=1
for every i = 1,...,d, k =1,...,m.
2. Main Results
Suppose (Ω,{F } ,F,P) is a complete filtered probability space sat-
t t≥0
isfying the usual conditions, i.e. the filtration is right continuous and F
0
contains all P-null sets. Let T > 0 beafixed constant and (w ) denote
t t∈[0,T]
an Rm−valued standard Wiener process. Further, suppose that b(x) and
σ(x) are B(Rd)-measurable functions with values in Rd and Rd×m respec-
tively. Moreover, b(x) and σ(x) are continuously differentiable in x ∈ Rd.
For the purpose of this article, the following d-dimensional SDE is consid-
ered,
t t
x =ξ+ b(x )ds+ σ(x )dw , (1)
t s s s
Z Z
0 0
almostsurelyforanyt ∈ [0,T],whereξisanF -measurablerandomvariable
0
in Rd.
Let p ,p ≥ 2 and ρ ≥ 1 (or ρ = 0) are fixed constants. For the purpose
0 1
of this article, the following assumptions are made.
A-1. E|ξ|p0 < ∞.
A-2. There exists a constant L > 0 such that
2xb(x)+(p −1)|σ(x)|2 ≤ L(1+|x|2)
0
for any x ∈Rd.
A-3. There exists a constant L > 0 such that
2(x−x¯)(b(x)−b(x¯))+(p −1)|σ(x)−σ(x¯)|2 ≤ L|x−x¯|2
1
EXPLICIT MILSTEIN-TYPE SCHEME 3
for any x,x¯ ∈ Rd.
A-4. There exists a constant L > 0 such that
|Db(x)−Db(x¯)| ≤ L(1+|x|+|x¯|)ρ−1|x−x¯|
for any x,x¯ ∈ Rd.
A-5. There exists a constant L > 0 such that, for every j = 1,...,m,
|Dσ(j)(x)−Dσ(j)(x¯)| ≤ L(1+|x|+|x¯|)ρ−22|x−x¯|
for any x,x¯ ∈ Rd.
Remark 1. Assumption A-4 means that there is a constant L > 0 such that
∂bi(x)
≤ L(1+|x|)ρ
(cid:12) ∂xj (cid:12)
(cid:12) (cid:12)
for any x ∈ Rd and for ev(cid:12)ery i,j(cid:12)= 1,...,d. As a consequence, one also
obtains that there exists a constant L > 0 such that
|b(x)−b(x¯)| ≤ L(1+|x|+|x¯|)ρ|x−x¯|
for any x,x¯ ∈ Rd. Moreover, this implies that b(x) satisfies,
|b(x)| ≤ L(1+|x|)ρ+1
for any x ∈ Rd. Furthermore, due to Assumption A-5, there exists a con-
stant L > 0 such that
∂σ(i,j)(x) ρ
≤ L(1+|x|)2
∂xk
(cid:12) (cid:12)
(cid:12) (cid:12)
for any x ∈ Rd and for ev(cid:12)ery i,k = 1(cid:12),...,d, j = 1,...,m. Also, Assumption
A-3 implies
ρ
|σ(x)−σ(x¯)| ≤ L(1+|x|+|x¯|)2|x−x¯|
for any x,x¯ ∈ Rd. Moreover, this means σ(x) satisfies,
ρ+2
|σ(x)| ≤ L(1+|x|) 2
for any x ∈Rd. In addition, one notices that
|Λjσ(x)| ≤ L(1+|x|)ρ+1
for any x ∈Rd and for every j = 1,...,m.
For every n ∈ N and x ∈ Rd, we define the following functions,
b(x)
bn(x) := ,
1+n−θ|x|2ρθ
σ(x)
σn(x) := ,
1+n−θ|x|2ρθ
where θ ≥ 1 and, similarly, for the purposes of establishing a new, explicit
2
Milstein-type scheme, for every j =1,...,m, we define
Λjσ(x)
Λn,jσ(x) := .
1+n−θ|x|2ρθ
4 C.KUMARANDS.SABANIS
Remark 2. The case θ = 1/2 is studied in [8], without the use of Λn,jσ(x),
as the aim is the formulation of a new explicit Euler-type scheme. Through-
out this article, θ is taken to be 1, which corresponds to an order 1.0 Milstein
scheme. By taking different values of θ = 1.5,2,2.5,... and by appropriately
controlling higher order terms, one can obtain optimal rate of convergence
results for higher order schemes by adopting the approach developed in [8]
and in this article.
Moreover, let us also define
m t m
σn(t,x) := Λn,jσ(x)dwj = Λn,jσ(x)(wj −wj )
1 Z r t κ(n,t)
Xj=1 κ(n,t) Xj=1
and hence set
σ˜n(t,x) := σn(x)+σn(t,x)
1
almost surely for any x ∈ Rd, n ∈ N and t ∈ [0,T].
Remark 3. Due to Remark 1, one immediately notices that
|bn(x)| ≤ min(Kn21(1+|x|),|b(x)|)
|σn(x)|2 ≤ min(Kn12(1+|x|2),|σ(x)|2)
|Λn,jσ(x)| ≤ min(Kn12(1+|x|),|Λ(x)|)
for every n∈ N,x ∈Rd and j = 1,...,m.
Letus defineκ(n,t) := ⌊nt⌋/n for any t ∈ [0,T]. We proposebelow anew
variant of the Milstein scheme with coefficients which vary according to the
choice of the time step. The aim is to approximate solutions of non-linear
SDEs such as equation (1). The new explicit scheme is given below
t t
xn =ξ + bn(xn )ds+ σ˜n(s,xn )dw (2)
t Z κ(n,s) Z κ(n,s) s
0 0
almost surely for any t ∈ [0,T].
Remark 4. In the following, K > 0 denotes a generic constant that varies
from place to place, but is always independent of n∈ N.
The main result of this article is stated in the following theorem.
Theorem 1. Let Assumptions A-1 to A-5 be satisfied with p ≥ 2(3ρ+1)
0
and p > 2. Then, the explicit Milstein-type scheme (2) converges in Lp to
1
the true solution of SDE (1) with a rate of convergence equal to 1.0, i.e. for
every n ∈ N
sup E|x −xn|p ≤ Kn−p, (3)
t t
0≤t≤T
when p = 2. Moreover, if p ≥ 4(3ρ+1), then (3) is true for any p ≤ p0
0 3ρ+1
provided that p < p .
1
Remark 5. One observes immediately that for the case ρ = 0, one recov-
ers, due to Assumptions A-1 to A-5 and Theorem 1, the classical Milstein
framework and results (with some improvement perhaps as the coefficients of
(1) are required only to be once continuously differentiable in this article).
EXPLICIT MILSTEIN-TYPE SCHEME 5
Remark 6. In order to ease notation, it is chosen not to explicitly present
the calculations for, and thus it is left as an exercise to the reader, the
case where the drift and/or the diffusion coefficients contain parts which are
Lipschitz continuous and grow at most linearly (in x). In such a case, the
analysis for these parts follows closely the classical approach and the main
theorem/results of this article remain true. Furthermore, note that such a
statement applies also in the case of non-autonomous coefficients in which
typical assumptions for the smoothness of coefficients in t are considered (as,
for example, in [1]).
The details of the proof of the main result, i.e. Theorem 1, and of the
required lemmas are given in the next two sections.
3. Moment Bounds
It is a well-known fact that due to Assumptions A-1 to A-3, the p -th
0
moment of the true solution of (1) is bounded uniformly in time.
Lemma 1. Let Assumptions A-1 to A-3 be satisfied. Then, there exists a
unique solution (x ) of SDE (1) and the following holds,
t t∈[0,T]
sup E|x |p0 ≤ K.
t
0≤t≤T
The proof of the above lemma can be found in many textbooks, e.g. see
[6]. The following lemmas are required in order to allow one to obtain
moment bounds for the new explicit scheme (2).
Remark 7. Another useful observations is that for every fixed n ∈ N
and due to Remark 3, the p -th moment of the new Milstein-type scheme
0
(2) is bounded uniformly in time (as in the case of the classical Milstein
scheme/framework withSDEcoefficientswhichgrowatmostlinearly). Clearly,
one cannot claim at this point that such a bound is independent of n. How-
ever, the use of stopping times in the derivation of moment bounds hence-
forth can be avoided.
Lemma 2. Let Assumption A-5 be satisfied. Then,
E|σn(t,xn )|p0 ≤K(1+E|xn |p0)
1 κ(n,t) κ(n,t)
for any t ∈ [0,T] and n∈ N.
Proof. OnusinganelementaryinequalityofstochasticintegralsandH¨older’s
inequality, one obtains
E|σn(t,xn )|p0 = KE m t Λn,jσ(xn )dwj p0
1 κ(n,t) (cid:12)(cid:12)Xj=1Zκ(n,t) κ(n,s) s(cid:12)(cid:12)
(cid:12) (cid:12)
t
≤ Kn−p20+1EZ |Λn,jσ(xnκ(n,s))|p0ds
κ(n,t)
which due to Remark 3 gives
t
E|σ1n(t,xnκ(n,t))|p0 ≤ Kn−p20+1EZ np20(1+|xnκ(n,s)|p0)ds
κ(n,t)
and hence the proof completes. (cid:3)
6 C.KUMARANDS.SABANIS
The following corollary is an immediate consequence of Lemma 2 and
Remark 3.
Corollary 1. Let Assumption A-5 be satisfied. Then
E|σ˜n(t,xnκ(n,t))|p0 ≤ Knp40(1+E|xnκ(n,t)|p0)
for any n ∈ N and t ∈ [0,T].
When p = 2, one proceeds with the following lemma (which is important
0
for the case ρ= 0).
Lemma 3. Let Assumptions A-1 to A-5 be satisfied. Then, the explicit
Milstein-type scheme (2) satisfies the following,
sup sup E|xn|2 ≤ K.
t
n∈N0≤t≤T
Proof. By Itˆo’s formula, one obtains
t t
|xn|2 =|ξ|2 +2 xnbn(xn )ds+2 xnσ˜n(s,xn )dw
t Z s κ(n,s) Z s κ(n,s) s
0 0
t
+ |σ˜n(s,xn )|2ds
Z κ(n,s)
0
for any t ∈ [0,T]. Also, oneuses |z +z |2 = |z |2+2 d m z(i,j)z(i,j)+
1 2 1 i=1 j=1 1 2
|z |2 for any z ,z ∈ Rd×m to estimate the last termPof thePabove equation,
2 1 2
t
E|xn|2 = E|ξ|2 +E 2(xn −xn )bn(xn )ds
t Z s κ(n,s) κ(n,s)
0
t t
+E {2xn bn(xn )+|σn(xn )|2}ds+E |σn(s,xn )|2ds
Z κ(n,s) κ(n,s) κ(n,s) Z 1 κ(n,s)
0 0
d m t
+2E σn,(i,j)(xn )σn,(i,j)(s,xn )ds
Z κ(n,s) 1 κ(n,s)
Xi=1Xj=1 0
which further implies due to Lemma 2 (with p = 2),
0
t s
E|xn|2 ≤ E|ξ|2 +2E bn(xn )drbn(xn )ds
t Z Z κ(n,r) κ(n,s)
0 κ(n,s)
t s
+2E σ˜n(r,xn )dw bn(xn )ds
Z Z κ(n,r) r κ(n,s)
0 κ(n,s)
t 2xn b(xn )+|σ(xn )|2 t
+E κ(n,s) κ(n,s) κ(n,s) ds+KE (1+|xn |2)ds
Z 1+n−1|xn |2ρ+4 Z κ(n,s)
0 κ(n,s) 0
d m t m s
+2E σn,(i,j)(xn ) Λn,kσ(i,j)(xn )dwkds
Z κ(n,s) Z κ(n,r) r
Xi=1Xj=1 0 Xk=1 κ(n,s)
and then on the application of Assumption A-2, Remark 3 (also notice that
third and last terms are zero) gives
t
sup E|xn|2 ≤ E|ξ|2+K +K sup E|xn|2ds < ∞
s r
0≤s≤t Z0 0≤r≤s
for any t ∈ [0,T]. The proof completes on using Gronwall’s lemma. (cid:3)
EXPLICIT MILSTEIN-TYPE SCHEME 7
When p ≥ 4, one proceeds with the following lemma.
0
Lemma 4. Let Assumptions A-1 to A-5 be satisfied. Then, the explicit
Milstein-type scheme (2) satisfies the following,
sup sup E|xn|p0 ≤ K.
t
n∈N0≤t≤T
Proof. By Itˆo’s formula, one obtains
t
|xn|p0 = |ξ|p0 +p |xn|p0−2xnbn(xn )ds
t 0Z s s κ(n,s)
0
t
+p |xn|p0−2xnσ˜n(s,xn )dw
0Z s s κ(n,s) s
0
p (p −2) t
+ 0 0 |xn|p0−4|σ˜n∗(s,xn )xn|2ds
2 Z s κ(n,s) s
0
p t
+ 0 |xn|p0−2|σ˜n(s,xn )|2ds,
2 Z s κ(n,s)
0
and then on taking expectation along with Schwarz inequality,
t
E|xn|p0 ≤ E|ξ|p0 +p E |xn|p0−2(xn−xn )bn(xn )ds
t 0 Z s s κ(n,s) κ(n,s)
0
t
+p E |xn|p0−2xn bn(xn )ds
0 Z s κ(n,s) κ(n,s)
0
p (p −1) t
+ 0 0 E |xn|p0−2|σ˜n(s,xn )|2ds
2 Z s κ(n,s)
0
foranyt ∈ [0,T]. Then,oneuses|z +z |2 = |z |2+2 d m z(i,j)z(i,j)+
1 2 1 i=1 j=1 1 2
|z |2 for z ,z ∈ Rd×m to obtain the following estimaPtes, P
2 1 2
t
E|xn|p0 ≤ E|ξ|p0 +p E |xn|p0−2(xn −xn )bn(xn )ds
t 0 Z s s κ(n,s) κ(n,s)
0
p t
+ 0E |xn|p0−2{2xn bn(xn )+(p −1)|σn(xn )|2}ds
2 Z s κ(n,s) κ(n,s) 0 κ(n,s)
0
p (p −1) t
+ 0 0 E |xn|p0−2|σn(s,xn )|2ds
2 Z s 1 κ(n,s)
0
t d m
+p (p −1)E |xn|p0−2 σn,(i,j)(xn )σn,(i,j)(s,xn )ds
0 0 Z s κ(n,s) 1 κ(n,s)
0 Xi=1Xj=1
=:C +C +C +C +C . (4)
1 2 3 4 5
Here, C := E|ξ|p0. In order to estimate C , one notices that it can be
1 2
written as
t
C :=p E |xn|p0−2(xn −xn )bn(xn )ds
2 0 Z s s κ(n,s) κ(n,s)
0
t s
=p E |xn|p0−2 bn(xn )drbn(xn )ds
0 Z s Z κ(n,r) κ(n,s)
0 κ(n,s)
t s
+p E |xn|p0−2 σ˜n(r,xn )dw bn(xn )ds
0 Z s Z κ(n,r) r κ(n,s)
0 κ(n,s)
8 C.KUMARANDS.SABANIS
which on the application of Remark 3 and Young’s inequality gives,
t t
C ≤K E|xn|p0ds+K E|xn |p0ds
2 Z s Z κ(n,s)
0 0
t s
+p E |xn |p0−2 σ˜n(r,xn )dw bn(xn )ds
0 Z κ(n,s) Z κ(n,r) r κ(n,s)
0 κ(n,s)
t s
+p E (|xn|p0−2−|xn |p0−2) σ˜n(r,xn )dw bn(xn )ds
0 Z s κ(n,s) Z κ(n,r) r κ(n,s)
0 κ(n,s)
for any t ∈ [0,T]. Further, one observes that the second term of the above
equation is zero and the third term can be estimated by the application of
Itˆo’s formula as below,
t t s
C ≤ K sup E|xn|p0ds+KE |xn|p0−4xnbn(xn )dr
2 Z0 0≤r≤s r Z0 Zκ(n,s) r r κ(n,r)
s
× σ˜n(r,xn )dw bn(xn )ds
Z κ(n,r) r κ(n,s)
κ(n,s)
t s
+KE |xn|p0−4xnσ˜n(r,xn )dw
Z Z r r κ(n,r) r
0 κ(n,s)
s
× σ˜n(r,xn )dw bn(xn )ds
Z κ(n,r) r κ(n,s)
κ(n,s)
t s
+KE |xn|p0−4|σ˜n(r,xn )|2dr
Z Z r κ(n,r)
0 κ(n,s)
s
× | σ˜n(r,xn )dw || bn(xn )| ds
Z κ(n,r) r κ(n,s)
κ(n,s)
for any t ∈ [0,T]. Due to Remark 3 along with an elementary inequality of
stochastic integrals, the following estimates can be obtained,
t
C ≤ K sup E|xn|p0ds
2 r
Z0 0≤r≤s
t s s
+KnE (1+|xn |2)|xn|p0−3dr σ˜n(r,xn )dw ds
Z0 Zκ(n,s) κ(n,s) r (cid:12)Zκ(n,s) κ(n,r) r(cid:12)
(cid:12) (cid:12)
t s (cid:12) (cid:12)
+Kn12EZ Z (1+|xnκ(n,s)|)|xnr|p0−3|σ˜n(r,xnκ(n,r))|2drds
0 κ(n,s)
t s
+Kn12EZ Z (1+|xnκ(n,s)|)|xnr|p0−4|σ˜n(r,xnκ(n,r))|2dr
0 κ(n,s)
s
× σ˜n(r,xn )dw ds
(cid:12)Zκ(n,s) κ(n,r) r(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
EXPLICIT MILSTEIN-TYPE SCHEME 9
which can also be estimated as,
t
C ≤ K sup E|xn|p0ds
2 r
Z0 0≤r≤s
t s s
+KEZ0 n43 Zκ(n,s)(1+|xnκ(n,s)|2)|xnr|p0−3drn41(cid:12)Zκ(n,s)σ˜n(r,xnκ(n,r))dwr(cid:12)ds
(cid:12) (cid:12)
t s (cid:12) (cid:12)
+KEZ Z n1−p20(1+|xnκ(n,s)|)|xnr|p0−3n−21+p20|σ˜n(r,xnκ(n,r))|2drds
0 κ(n,s)
t s
+KEZ n41 Z (1+|xnκ(n,s)|)|xnr|p0−4|σ˜n(r,xnκ(n,r))|2dr
0 κ(n,s)
s
×n14(cid:12)Zκ(n,s)σ˜n(r,xnκ(n,r))dwr(cid:12)ds
(cid:12) (cid:12)
(cid:12) (cid:12)
and then one uses Young’s inequality to obtain the following estimates,
t
C ≤ K sup E|xn|p0ds
2 r
Z0 0≤r≤s
+Kn4(p30p−01)EZ0t(cid:16)Zκs(n,s)(1+|xnκ(n,s)|2)|xnr|p0−3dr(cid:17)p0p−01ds
+KnEZ tZ s (1+|xnκ(n,s)|)|xnr|p0−3 p0p−02drds
0 κ(n,s)(cid:0) (cid:1)
+Kn4(pp00−1)EZ0t(cid:16)Zκs(n,s)(1+|xnκ(n,s)|)|xnr|p0−4 |σ˜n(r,xnκ(n,r))|2 dr(cid:17)p0p−01ds
+Knp40EZ0t(cid:12)Zκs(n,s)σ˜n(r,xnκ(n,r))dwr(cid:12)p0ds
(cid:12) (cid:12)
(cid:12) t s (cid:12)
+Kn−p40+1EZ Z |σ˜n(r,xnκ(n,r))|p0drds
0 κ(n,s)
for any t ∈ [0,T]. Further, by the application of H¨older’s inequality and
an elementary inequality of stochastic integrals, one obtains the following
estimates,
t
C ≤ K sup E|xn|p0ds
2 r
Z0 0≤r≤s
+Kn4(p30p−01)−p0p−01+1EZ tZ s (1+|xnκ(n,s)|p20p−01)|xnr|p0p(0p0−−13)drds
0 κ(n,s)
+KnEZ tZ s (1+|xnκ(n,s)|p0p−02)|xnr|(p0p0−−3)2p0drds
0 κ(n,s)
+Kn4(pp00−1)−p0p−01+1EZ tZ s (1+|xnκ(n,s)|p0p−01)|xnr|p0p(0p0−−14)
0 κ(n,s)
× |σ˜n(r,xnκ(n,r))|p20p−01 drds
t s
+Kn−p40+1EZ Z |σ˜n(r,xnκ(n,r))|p0drds
0 κ(n,s)
10 C.KUMARANDS.SABANIS
which due to Corollary 1 yields
t
C ≤ K sup E|xn|p0ds
2 r
Z0 0≤r≤s
+KEZ t(1+|xnκ(n,s)|p20p−01)n−4(pp00−1)+1Z s |xnr|p0p(0p0−−13)drds
0 κ(n,s)
+KEZ t(1+|xnκ(n,s)|p0p−02)nZ s |xnr|(p0p0−−3)2p0drds
0 κ(n,s)
+KEZ t(1+|xnκ(n,s)|p0p−01)n−4(p30p−01)+1Z s |xnr|p0p(0p0−−14)
0 κ(n,s)
× |σ˜n(r,xnκ(n,r))|p20p−01 drds
t s
+Kn−p40+1Z Z np40(1+E|xnκ(n,r)|p0)drds
0 κ(n,s)
and then on further application of Young’s inequality, following estimates
are obtained
t t
C ≤ K +K sup E|xn|p0ds+KE (1+|xn |p0)ds
2 Z0 0≤r≤s r Z0 κ(n,s)
+KE tn−4(pp00−3)+pp00−−13 s |xnr|p0p(0p0−−13)dr pp00−−31ds
Z0 (cid:16)Zκ(n,s) (cid:17)
+KE tnpp00−−23 s |xnr|(p0p0−−3)2p0dr pp00−−23ds
Z0 (cid:16)Zκ(n,s) (cid:17)
+KEZ0tn−4(p30p−02)+pp00−−12(cid:16)Zκs(n,s)|xnr|p0p(0p0−−14) | σ˜n(r,xnκ(n,r)) |p20p−01 dr(cid:17)pp00−−21ds
for any t ∈ [0,T]. Thus, on using H¨older’s inequality, one obtains
C2 ≤ K +K t sup E|xnr|p0ds+KE tn−4(pp00−3)+1 s |xnr|p0drds
Z0 0≤r≤s Z0 Zκ(n,s)
t s
+KE n |xn|p0drds
r
Z Z
0 κ(n,s)
+KEZ tn−4(p30p−02)+1Z s |xnr|p0p(0p0−−24) | σ˜n(r,xnκ(n,r)) |p20p−02 drds
0 κ(n,s)
for any t ∈ [0,T]. Also, one can write above inequality as
t
C ≤ K +K sup E|xn|p0ds
2 r
Z0 0≤r≤s
+KEZ tZ s npp00−−42|xnr|p0p(0p0−−24)n−4(7pp00−+21)6+1 | σ˜n(r,xnκ(n,r)) |p20p−02 drds
0 κ(n,s)