Large deviations for the stochastic functional integral equation with nonlocal condition

Gopal Shruthi (Department of Applied Mathematics, Bharathiar University, Coimbatore, India)
Murugan Suvinthra (Department of Applied Mathematics, Bharathiar University, Coimbatore, India)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 8 August 2022

Issue publication date: 23 January 2024

388

Abstract

Purpose

The purpose of this paper is to study large deviations for the solution processes of a stochastic equation incorporated with the effects of nonlocal condition.

Design/methodology/approach

A weak convergence approach is adopted to establish the Laplace principle, which is same as the large deviation principle in a Polish space. The sufficient condition for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis is used in this work.

Findings

Freidlin–Wentzell type large deviation principle holds good for the solution processes of the stochastic functional integral equation with nonlocal condition.

Originality/value

The asymptotic exponential decay rate of the solution processes of the considered equation towards its deterministic counterpart can be estimated using the established results.

Keywords

Citation

Shruthi, G. and Suvinthra, M. (2024), "Large deviations for the stochastic functional integral equation with nonlocal condition", Arab Journal of Mathematical Sciences, Vol. 30 No. 1, pp. 81-94. https://doi.org/10.1108/AJMS-10-2021-0271

Publisher

:

Emerald Publishing Limited

Copyright © 2022, Gopal Shruthi and Murugan Suvinthra

License

Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode


1. Introduction

Differential equations are applied in a wide variety of disciplines, including physics, chemistry, engineering, economics and biology by many researchers. An initial value problem describes the evolution of a physical system. An improvement of initial/local condition is done by imposing a nonlocal condition into the problem. Nonlocal conditions take values/measurements at more places and are more precise than the local condition. That is, if we consider the differential equation

x·(t)=f(t,x(t)),x(t0)=x0,
for t ≥ t0 with x(t)R then the corresponding nonlocal differential equation evolves when the initial condition is replaced by the following nonlocal condition:
x(t0)+g(x())=x0,
where x(⋅) denotes the solution at some specific times t ≥ t0 and g defines a mapping consisting of certain functions on some space. Nonlocal conditions have several applications in real life situations, for example, in the diffusion phenomenon/dripping of a small amount of coloured water in a transparent tube filled with colourless water. Here there will be too little coloured water at the initial time t0, so the measurement x(t0) + g(x(⋅)) will be more precise compared to x(t0). One more example is from the field of kinematics, to determine the location of a physical object with an evolution tx(t) where the initial position holds the nonlocal condition.

Today, nonlocal condition has enormous applications in the field of quantum mechanics, continuum mechanics, damage mechanics, sub-surface flows, image recognition, peri-dynamic models, environment–human coupled systems like AI and decision making problems. Some of the applications are cited in References [1–5]. Recently, in Reference [6], the authors have discussed in detail different mathematical models with non-local initial condition and their applications.

But, since the late 1970s, non-local condition has been studied by many researchers, including Kerefov, who proposed non-local boundary value problems for parabolic equations [7]. Vabishchevich [8] in 1981, studied parabolic problems with non-local condition and problem related to the inverse of heat conduction. Charbowski [9] was the first to propose nonlocal initial-boundary value problems for linear parabolic equations and also investigated their existence and uniqueness. Byszewski, motivated by the physical problems in 1991, investigated nonlocal problems for nonlinear parabolic equations and also found the existence and uniqueness of solutions for non-local hyperbolic equations [10, 11]. Byszewski also found the existence and uniqueness of the solution for semi-linear evolution non-local Cauchy problems [12]. Inspired by the works of Byszewski, Jackson [13] also contributed to the generalization of the classical Cauchy problem to the nonlocal Cauchy problem yielding better results in many physical systems. The research works [14–16] are few more contributions on non-local Cauchy problems. Byszewski and Lakshmikantham were the first to introduce the study of non-local condition in Banach spaces [17], which paved the way for many researchers to study integro-differential equations with non-local condition in Banach spaces (for instance, References [18, 19]). Ntouyas has given a detailed survey about nonlocal initial and boundary problems in Reference [20]. The existence and uniqueness of solutions for nonlocal stochastic differential equations was studied by Lorenz [21]. A Volterra type non-local random integral equation was studied by Abdou et al. [22] by using admissibility of integral operator theory. A study of the Volterra–Itô–Doob type non-local stochastic functional integral equation is developed by Elborai and Youssef [23] using the fixed point technique.

The preceding works in the field of nonlocal condition and its variant applications in real life have motivated us to investigate large deviations for nonlocal stochastic functional differential equations. The subject of the large deviation theory is about controlling the probabilities of atypical events. It is a sub-discipline of probability theory that studies the exponential decline of probability measures of particular kinds of tail events. It has a rich history of development, starting with the works of a Swedish mathematician Cramer in the 1930s for insurance business modelling. In the year 1954, Petrov generalized Cramér's limit theorem. In 1966, Varadhan [24] developed the large deviation principle in a unique manner, making way for many more applications in a more convincing manner, like entropy calculation in statistical mechanics. Using Varadhan's contraction principle, Freidlin and Wentzell [25] developed the large deviation principle (LDP) for differential equations with small stochastic perturbation. Large deviations are established for stochastic differential delay equations by Mo and Luo [26]. Large deviations for stochastic integro-differential equations are studied in Reference [27] and for stochastic functional differential equations with infinite delay in Reference [28]. Large deviations for stochastic partial differential equations driven by a Poisson random measure is worked in Reference [29] and for the mean reflected stochastic differential equation with jumps in Reference [30].

In this paper, the large deviation theory is studied for the stochastic functional integral equation with non-local initial condition by adopting the weak convergence technique, which was formulated by Dupuis and Ellis [31] using Fleming's [32] stochastic control approach. The basic idea behind Dupuis and Ellis's formulation is that, under Polish space, the Laplace principle and the LDP are equivalent. The sufficient conditions for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis [33] are precisely used in this work.

2. Preliminaries

We consider the following nonlocal stochastic functional differential equation:

(2.1)dX(t)=f(t,X(t),AX(t))dt+g(t,X(t),B(t)X(t))dW(t),X(0)=X0+h(X()),
where t ∈ J ≔[0, T], T < , and the nonlocal condition h(X(⋅)) is used in the sense that in the place of “⋅”; we can substitute only elements of the set {t1, t2, …, tp}, where 0t1<t2<<tpT,pN.

Let (Ω,F,P) be a complete probability space with a filtration {Ft}tJ where Ω is a nonempty set known as the sample space, F=FT is a σ - algebra of events of Ω occurring during the time interval J, P is a complete probability measure and {Ft}tJ is an increasing family of sub σ algebras FtF satisfying the usual conditions. Let {Xϵ, ϵ > 0} be a sequence of random variables taking values in a Polish space Z and defined in (Ω,F,P). Also, let CC(J,L2(Ω,F,P)) be the space of all continuous stochastic processes which are adapted to the filtration {Ft}tJ. The following definitions and results are needed for this work.

Definition 2.1.

[33]: A function I:Z[0,+] is called a rate function if I is lower semi-continuous. A rate function I is called a good rate function if for each a < , the level set {fZ:I(f)a} is compact.

Definition 2.2.

[31]: Let I be a rate function on Z. We say the sequence {Xϵ, ϵ > 0} satisfies the LDP with rate function I if the following two conditions hold:

  1. Large deviation upper bound: For each closed subset F of Z,

    lim supϵ0ϵlogP(XϵF)I(F).

  2. Large deviation lower bound: For each open subset G of Z,

    lim infϵ0ϵlogP(XϵG)I(G).

Definition 2.3.

[33]: Let I be a rate function on Z. We say {Xϵ} satisfies the Laplace principle with rate function I if for all real valued bounded continuous functions h defined on Z,

limϵ0ϵlogEexp1ϵh(Xϵ)=inffZ{h(f)+I(f)}.
Theorem 2.4.

[31] The family {Xϵ} satisfies the Laplace principle with good rate function on a Polish space if and only if {Xϵ} satisfies the LDP with the same rate function.

Equation (2.1) is equivalent to the following Volterra–Itô type stochastic functional integral equation with nonlocal condition:
(2.2)X(t)=X0+h(X())+0tf(τ,X(τ),AX(τ))dτ+0tg(τ,X(τ),B(τ)X(τ))dW(τ),
where X0 ∈ R, and the function h(X(⋅)) is random and defined on R with values in the space R.

In the above equation the first integral is a mean square Riemann integral and the second is an Itô integral. W(t) is a real valued Brownian motion adapted to the filtration {Ft}tJ. The operator A is closed, linear and defined on C with values in C. The operators {B(t) : t ∈ J} are linear, bounded and defined on C into C. By the closed graph theorem, we get that

(2.3)|AX(t)|β|X(t)|,tJand|B(t)X(t)|γ(t)|X(t)|,tJ,

in such a way that γ(t) is square integrable on J and β ≥ 0 is a real constant. The functions f and g are real, measurable and defined on J×C×C with values in the space C.

Let |⋅| denote the Euclidean norm. The functions f, g and h will be specified with the conditions below.

H1.

The functions f(t, x, y) and g(t, x, y) are mean square continuous in (x, y) for each t ∈ J.

H2.

f and g have the following restriction on growth:

|f(t,x,y)|α(1+|x|2+|y|2),|g(t,x,y)|α(1+|x|2+|y|2),

for all t ∈ J, x, yR, where the constant α > 0.

H3.

There exist constants α1>0,0<α2<12 such that

|f(t,x2,y2)f(t,x1,y1)|α1(|x2x1|2+|y2y1|2),|g(t,x2,y2)g(t,x1,y1)|α1(|x2x1|2+|y2y1|2),|h(t,x2)h(t,x1)|α2|x2x1|,

for all t ∈ J , x1, x2, y1, y2R.

The existence and uniqueness of solution of equation (2.2) has been established by Elborai and Youseff [23]. In this work we study large deviation principle for Equation (2.2).

Consider Equation (2.2) stochastically perturbed by a small parameter ϵ > 0

(2.4)X(t)=X0+h(X())+0tf(τ,X(τ),AX(τ))dτ+ϵ0tg(τ,X(τ),B(τ)X(τ))dW(τ).

Let {Xϵ} denote the solution of the perturbed Equation (2.4). Since {Xϵ} is a strong solution to Equation (2.4), there exists a Borel-measurable function Gϵ:C(J;R)C(J;R) such that Xϵ(⋅) = Gϵ(W(⋅)) a.s. by the Yamada Watanabe theorem [34].

Let A={v:v is a real valued Ft predictable process and 0T|vτ|2dτ< a.s.}, SN={vL2(J;R):0T|v(τ)|2dτN}, where L2(J;R) is the space of all real valued square integrable functions on J. Then SN endowed with the weak topology in L2(J;R) is a compact Polish space. Let us also define AN={vA:v(ω)SNPa.s}.

The sufficient conditions framed by Budhiraja and Dupuis [33] under which the Laplace principle holds for the family {Xϵ} are the following:

Proposition 2.5.

Suppose that there exists a measurable map G0:C(J;R)C(J;R) such that the following two conditions hold:

  1. Compactness : For each N < , the set KN={G0(0v(τ)dτ):vSN} is a compact subset of C(J;R).

  2. Weak convergence: Let {vϵ:ϵ>0}AN for some N < . If vϵ converge to v in distribution as SN valued random elements, then GϵW()+1ϵ0.vϵ(τ)dτG00.v(τ)dτ in distribution as ϵ → 0.

Then the family {Xϵ, ϵ > 0} satisfies the Laplace principle in C(J;R) with the rate function I given by

I(ϕ)=inf120T|v(τ)|2dτ;ϕ=G00.v(τ)dτandvL2(J;R)
for each ϕC(J;R).
Theorem 2.6.

[31] Let {Yn,nN} be a sequence of real-valued random variables that are defined on a sequence of probability spaces {(Ωn,Fn,Pn),nN}. If YnDY and fCb(X), then f(Yn)Df(Y).

3. The Large deviation principle

Here we establish the LDP for the family of solution processes of Equation (2.4) by using Proposition 2.5.

Theorem 3.1.

With the assumptions (H1) to (H3), the family {Xϵ} of solutions of Equation (2.4) satisfies the LDP (equivalently, the Laplace principle) in C(J;R) with good rate function

I(ϕ)=inf120T|v(τ)|2dτ;Xv=ϕ,
where vL2(J;R) and Xv denotes the solution of the controlled equation
(3.1)Xv(t)=X0+h(Xv())+0tf(τ,Xv(τ),AXv(τ))dτ+0tg(τ,Xv(τ),B(τ)Xv(τ))v(τ)dτ

with the convention that the infimum of an empty set is infinity.

The above theorem gets proved with the proof of the following two lemmas:

Lemma 3.2.

(Compactness): Define G0:C(J;R)C(J;R) by

G0(g)Xv,ifg=0.v(τ)dτforsomevL2(J;R)0,otherwise,
where Xv denotes the solution of Equation (3.1). Then for each N < , the set KN={G0(0.v(τ)dτ):vSN} is a compact subset of C(J;R).

Proof: Consider a sequence {vn} ∈ SN such that vnv weakly in L2(J;R) as n. Let Xvn denote the solution of the controlled Equation (3.1) with v replaced by vn. That is

(3.2)Xvn(t)=X0+h(Xvn())+0tf(τ,Xvn(τ),AXvn(τ))dτ+0tg(τ,Xvn(τ),B(τ)Xvn(τ))vn(τ)dτ.

From Equations (3.1) and (3.2),

Xvn(t)Xv(t)=h(Xvn())h(Xv())+0tf(τ,Xvn(τ),AXvn(τ))f(τ,Xv(τ),AXv(τ))dτ+0tg(τ,Xvn(τ),B(τ)Xvn(τ))vn(τ)g(τ,Xv(τ),B(τ)Xv(τ))v(τ)dτ.

Taking modulus and by using Hölder's inequality

(3.3)|Xvn(t)Xv(t)||h(Xvn())h(Xv())|+0t|f(τ,Xvn(τ),AXvn(τ))f(τ,Xv(τ),AXv(τ))|2dτ12×T12+0t|g(τ,Xvn(τ),B(τ)Xvn(τ))g(τ,Xv(τ),B(τ)Xv(τ))|2dτ12×0t|vn(τ)|2dτ12+|0tg(τ,Xv(τ),B(τ)Xv(τ))(vn(τ)v(τ))dτ|.

Let ζn(t)=0tg(τ,Xv(τ),B(τ)Xv(τ))(vn(τ)v(τ))dτ. Squaring both sides of Equation (3.3) and using conditions (H2) and (H3),

|XvntXvt|24α22|Xvn()Xv()|2+4α12T0t|XvnτXvτ|2+|AXvnτAXvτ|2dτ+4α12N0t|XvnτXvτ|2+|BτXvnτBτXvτ|2dτ+4|ζn(t)|2.

Using Equation (2.3),

(3.4)|Xvn(t)Xv(t)|24α22|Xvn()Xv()|2+4α12T0t|Xvn(τ)Xv(τ)|2+β2|Xvn(τ)Xv(τ)|2dτ+4α12N0t|Xvn(τ)Xv(τ)|2+γ2(τ)|Xvn(τ)Xv(τ)|2dτ+4|ζn(t)|2.

Consider

|ζnt|=|0tgτ,Xvτ,BτXvτvnτ-vτdτ|.

Applying Hölder's inequality and taking supremum to the above equation leads to,

(3.5)supτJ|ζn(τ)|α0T1+|Xv|C(J;R)21+γ2(τ)dτ120T|vn(τ)v(τ)|2dτ12αNT+XvC(J;R)2(0T(1+γ2(τ))dτ)12C<,whereCisaconstant.

Observe that {ζn(t)} is a family of linear, continuous, real valued functions mapping SN to C(J;R). By noting that sup0≤τt|ζn(τ)| < C, the constant C not depending on n, we get that the family {ζn} is uniformly bounded by C. It can be concluded that {ζn} is equi-continuous by use of the fact that between Banach spaces the family of point-wise bounded continuous linear functions is equi-continuous. Since vnv in L2(J;R), ζn(t) → 0 point-wise for t ∈ J.

By applying a version of the ArzélaAscoli theorem immediately implies that ζn → 0 uniformly in C(J;R). Hence

(3.6)limnsupτJ|ζn(τ)|=0.

Set κn(t)=sup0τt|Xvn(τ)Xv(τ)|2. Then from Equation (3.4)

κn(t)4α1214α220Tκn(τ)(1+β2)T+(1+γ2(τ))Ndτ+414α22sup0τt|ζn(τ)|2.

Now by using Gronwall's lemma,

κn(t)C1sup0τt|ζn(τ)|2eC2T
where C1 and C2 are constants depending on β, γ(τ), α1, α2, N and T. Hence
|XvnXv|C(J;R)2=supτJ|Xvn(τ)Xv(τ)|2C1suptJ|ζn(t)|2eC2T
and so XvnXv in C(J;R) by Equation (3.6). Since the space SN is compact, it follows the set KN=G00.v(τ)dτ:vSN for N <  is compact.

We consider the following stochastic integral equation for verifying the weak convergence condition of Proposition 2.5.

(3.7)Xvϵϵ(t)=X0+h(Xvϵϵ())+0tf(τ,Xvϵϵ(τ),AXvϵϵ(τ))dτ+0tg(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))vϵ(τ)dτ+ϵ0tg(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))dW(τ).

From Girsanov's theorem, the existence of the above Equation (3.7) follows. So we move on to the weak convergence result.

Lemma 3.3.

(Weak Convergence): Let {vϵ:ϵ>0}AN for some N < . Assume vϵ converges to v in distribution as SN-valued random elements. Then GϵW()+1ϵ0.vϵ(τ)dτG00.v(τ)dτ in distribution as ϵ → 0.

Proof: Applying Itô's formula

(3.8)|Xvϵϵ(t)Xv(t)|2=|h(Xvϵϵ())h(Xv())|2+20t(f(τ,Xvϵϵ(τ),AXvϵϵ(τ))f(τ,Xv(τ),AXv(τ)))×(Xvϵϵ(τ)Xv(τ))dτ+20t(g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))g(τ,Xv(τ),B(τ)Xv(τ)))×(Xvϵϵ(τ)Xv(τ))vϵ(τ)dτ+20t(Xvϵϵ(τ)Xv(τ))g(τ,Xv(τ),B(τ)Xv(τ))(vϵ(τ)v(τ))dτ+2ϵ0t(Xvϵϵ(τ)Xv(τ))g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))dW(τ)+ϵ0t|g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))|2dτ.

Define

(3.9)ζϵ(t)=0tg(τ,Xv(τ),B(τ)Xv(τ))(vϵ(τ)v(τ))dτ.

Taking supremum and expectation we get EsuptJ|ζϵ(t)|C3<, where C3 is a constant.

By virtue of Itô's formula

(3.10)20tXvϵϵ(τ)Xv(τ)g(τ,Xv(τ),B(τ)Xv(τ))(vϵ(τ)v(τ))dτ=2(Xvϵϵ(t)Xv(t))(ζϵ(t))20th(Xvϵϵ())h(Xv())ζϵ(τ)dτ20tf(τ,Xvϵϵ(τ),AXvϵϵ(τ))f(τ,Xv(τ),AXv(τ))ζϵ(τ)dτ20tg(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))vϵ(τ)g(τ,Xv(τ),B(τ)Xv(τ))v(τ)×ζϵ(τ)dτ2ϵ0tg(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))ζϵ(τ)dW(τ)=I1+I2+I3+I4+I5.

By the help of Young's inequality and (H3), we get

(3.11)I112|Xvϵϵ(t)Xv(t)|2+2|ζϵ(t)|2.
(3.12)I212α220t|Xvϵϵ()Xv()|2dτ+20t|ζϵ(τ)|2dτ.
(3.13)I312α120t|Xvϵϵ(τ)Xv(τ)|2+|AXvϵϵ(τ)AXv(τ)|2dτ+20t|ζϵ(τ)|2dτ.
I412α120t|Xvϵϵ(τ)Xv(τ)|2+|B(τ)Xvϵϵ(τ)B(τ)Xv(τ)|2dτ
(3.14)+2supτ[0,t]|ζϵ(τ)|20t|vϵ(τ)|2dτ+2supτ[0,t]|ζϵ(τ)|212α120t|Xvϵϵ(τ)Xv(τ)|2+|B(τ)Xvϵϵ(τ)B(τ)Xv(τ)|2dτ+2Nsupτ[0,t]|ζϵ(τ)|2+2supτ[0,t]|ζϵ(τ)|2.

Using Young's inequality, condition (H3) and substituting the estimates (3.11)–(3.14) of I1, I2, I3 and I4 in Equation (3.8), we get,

|Xvϵϵ(t)Xv(t)|2|h(Xvϵϵ())h(Xv())|2+0t|Xvϵϵ(τ)Xv(τ)|2dτ
+α120t|Xvϵϵ(τ)Xv(τ)|2+|AXvϵϵ(τ)AXv(τ)|2dτ+0t|Xvϵϵ(τ)Xv(τ)|2|vϵ(τ)|2dτ
+α120t|Xvϵϵ(τ)Xv(τ)|2+|B(τ)Xvϵϵ(τ)B(τ)Xv(τ)|2dτ
+12|Xvϵϵ(t)Xv(t)|2+2|ζϵ(t)|2+12α220t|Xvϵϵ()Xv()|2dτ
+40t|ζϵ(τ)|2dτ+12α120t|Xvϵϵ(τ)Xv(τ)|2+|AXvϵϵ(τ)AXv(τ)|2dτ
+12α120t|Xvϵϵ(τ)Xv(τ)|2+|B(τ)Xvϵϵ(τ)B(τ)Xv(τ)|2dτ
(3.15)+2ϵ|0t(Xvϵϵ(τ)Xv(τ)ζϵ(t))g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))|dW(τ)|+ϵ0t|g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))|2dτ+2(N+1)supτ[0,t]|ζϵ(τ)|2.

Using conditions (H2) and (H3) and simplifying, we get

|Xvϵϵ(t)Xv(t)|2212α220t|Xvϵϵ(τ)Xv(τ)|2dτ
+2α1212α220t|Xvϵϵ(τ)Xv(τ)|2+β2|Xvϵϵ(τ)Xv(τ)|2dτ
+212α220t|Xvϵϵ(τ)Xv(τ)|2|vϵ(τ)|2dτ
+2α1212α220t|Xvϵϵ(τ)Xv(τ)|2+γ2(τ)|Xvϵϵ(τ)Xv(τ)|2dτ
+412α22|ζϵ(t)|2+α2212α220t|Xvϵϵ()Xv()|2dτ
+812α220t|ζϵ(τ)|2dτ
+α1212α220t|Xvϵϵ(τ)Xv(τ)|2+β2|Xvϵϵ(τ)Xv(τ)|2dτ
+α1212α220t|Xvϵϵ(τ)Xv(τ)|2+γ2(τ)|Xvϵϵ(τ)Xv(τ)|2dτ+412α22(N+1)supτ[0,t]|ζϵ(τ)|2
(3.16)+2ϵα212α220t1+|Xvϵϵ(τ)|2+γ2(τ)|Xvϵϵ(τ)|2dτ+4ϵ12α22|0t(Xvϵϵ(τ)Xv(τ)ζϵ(τ))g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))dW(τ)|.

Denote

κϵ(t)=sup0τt|Xvϵϵ(τ)Xv(τ)|2.

Taking supremum of Equation (3.16)

κϵ(t)M10t(1+|vϵ(τ)|2)κϵ(τ)dτ+M20tκϵ(τ)(1+γ2(τ))dτ+M3supτ[0,t]|(ζϵ(τ))|2+2ϵα212α220t1+|Xvϵϵ(τ)|2(1+γ2(τ))dτ+4ϵ12α22sups[0,t]|0s(Xvϵϵ(τ)Xv(τ)ζϵ(τ))g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))dW(τ)|.

Here M1, M2 and M3 are constants depending on β, N, α, α1, α2 and T. By using Gronwall's lemma

κϵ(t)(M3supτ[0,t]|ζϵ(τ)|2+2ϵα212α220t[1+|Xvϵϵ(τ)|2(1+γ2(τ))]dτ+4ϵ12α22sups[0,t]|0s(Xvϵϵ(τ)Xv(τ)ζϵ(τ))g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))dW(τ)|)×expM10t(1+|vϵ(τ)|2)dτ+M20t(1+γ2(τ))dτ|.

Simplifying further

(3.17)κϵ(T)CM3(supτJ|ζϵ(τ)|)2+2ϵα212α22T+supτJ|Xvϵϵ(τ)|20T(1+γ2(τ))dτ+4ϵ12α22supsJ|0s(Xvϵϵ(τ)Xv(τ)ζϵ(τ))g(τ,Xvϵϵ(τ),B(τ)Xvϵϵ(τ))dW(τ)|.

For showing convergence of ζϵ(τ), we define

F(u)=0tg(τ,Xv(τ),B(τ)Xv(τ))u(τ)dτ,uAN.

Recall that vϵ converges to v in distribution as SN valued random elements which is endowed with the weak topology. By linear growth of g, we could observe that F as a mapping from SN to C(J;R) is bounded and continuous. Now applying Theorem 2.6, we get ζϵ → 0 in distribution as ϵ → 0.

Using Burkholder–Davis–Gundy inequality, it can be easily verified that

EsupsJ|0sXvϵϵτ-Xvτ-ζϵτgτ,Xvϵϵτ,BτXvϵϵτdWτ|C.

With the above bound and with the distributional convergence of ζϵ to 0, it follows from (3.17) that κϵ(T) → 0 in distribution as ϵ → 0. Thus the lemma is established, thereby proving the main theorem.

4. Conclusion

Here, a control system for the corresponding original system is taken, and the compactness of the solution is proved. And with the help of the estimates of the solution, it is proved successfully that, for a controlled deterministic system, a controlled stochastic system converges weakly. Thus, the stochastic functional integral equation with a nonlocal condition under Polish space satisfies the Laplace principle, and thereby the large deviation principle is also proved. In the future, one can move to study exit time problems, the uniform large deviation principle, and the moderate deviation principle for the equation considered in this work, and also study the LDP and moderate deviation principle for coupled stochastic integral equations with nonlocal condition.

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Acknowledgements

The authors are thankful to the referee for the valuable comments and suggestions given to improve the quality of this work.

Conflict of interest: The authors declare that there is no conflict of interest.

Corresponding author

Gopal Shruthi can be contacted at: shruthigopal89@gmail.com

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