Large deviations for the stochastic functional integral equation with nonlocal condition

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

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.

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

\begin{matrix} \overset{\cdot}{x} (t) & = & f (t, x (t)), \\ x (t_{0}) & = & x_{0}, \end{matrix}

for t ≥ t₀ with $x (t) \in R$ then the corresponding nonlocal differential equation evolves when the initial condition is replaced by the following nonlocal condition:

x (t_{0}) + g (x (\cdot)) = x_{0},

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 t → x(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 Cram $\overset{́}{e}$ r'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:

\begin{matrix} d X (t) = f (t, X (t), A X (t)) d t + g (t, X (t), B (t) X (t)) d W (t), \\ X (0) = X_{0} + h (X (\cdot)), \end{matrix}

(2.1)

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, …, t_p}, where $0 \leq t_{1} < t_{2} < \dots < t_{p} \leq T, p \in N$ ⁠.

Let $(Ω, F, P)$ be a complete probability space with a filtration ${F_{t}}_{t \in J}$ where Ω is a nonempty set known as the sample space, $F = F_{T}$ is a σ - algebra of events of Ω occurring during the time interval J, P is a complete probability measure and ${F_{t}}_{t \in J}$ is an increasing family of sub σ algebras $F_{t} \subset F$ satisfying the usual conditions. Let {X^ϵ, ϵ > 0} be a sequence of random variables taking values in a Polish space $Z$ and defined in $(Ω, F, P)$ ⁠. Also, let $C ≔ C (J, L_{2} (Ω, F, P))$ be the space of all continuous stochastic processes which are adapted to the filtration ${F_{t}}_{t \in J}$ ⁠. The following definitions and results are needed for this work.

Definition 2.1.

[33]: A function $I : Z \to [0, + \infty]$ 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 ${f \in Z : I (f) \leq a}$ is compact.

Definition 2.2.

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

Large deviation upper bound: For each closed subset F of $Z$ ,
$\underset{ϵ \to 0}{lim sup} ϵ \log P (X^{ϵ} \in F) \leq - I (F) .$
Large deviation lower bound: For each open subset G of $Z$ ,
$\underset{ϵ \to 0}{lim inf} ϵ \log P (X^{ϵ} \in G) \geq - I (G) .$

Definition 2.3.

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

\lim_{ϵ \to 0} ϵ \log E \{\exp [- \frac{1}{ϵ} h (X^{ϵ})]\} = - \inf_{f \in Z} {h (f) + I (f)} .

Theorem 2.4.

[31] The family {X^ϵ} satisfies the Laplace principle with good rate function on a Polish space if and only if {X^ϵ} 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:

X (t) = X_{0} + h (X (\cdot)) + \int_{0}^{t} f (τ, X (τ), A X (τ)) d τ + \int_{0}^{t} g (τ, X (τ), B (τ) X (τ)) d W (τ),

(2.2)

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 ${F_{t}}_{t \in J}$ ⁠. 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

| A X (t) | \leq β | X (t) |, t \in J a n d | B (t) X (t) | \leq γ (t) | X (t) |, t \in J,

(2.3)

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 $J \times C \times C$ 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:

$\begin{matrix} | f (t, x, y) | \leq α \sqrt{(1 + | x |^{2} + | y |^{2})}, \\ | g (t, x, y) | \leq α \sqrt{(1 + | x |^{2} + | y |^{2})}, \end{matrix}$

for all t ∈ J, x, $y \in R$ ⁠, where the constant α > 0.

H3.

There exist constants $α_{1} > 0, 0 < α_{2} < \frac{1}{2}$ such that

\begin{matrix} | f (t, x_{2}, y_{2}) - f (t, x_{1}, y_{1}) | \leq α_{1} \sqrt{(| x_{2} - x_{1} |^{2} + | y_{2} - y_{1} |^{2})}, \\ | g (t, x_{2}, y_{2}) - g (t, x_{1}, y_{1}) | \leq α_{1} \sqrt{(| x_{2} - x_{1} |^{2} + | y_{2} - y_{1} |^{2})}, \\ | h (t, x_{2}) - h (t, x_{1}) | \leq α_{2} | x_{2} - x_{1} |, \end{matrix}

for all t ∈ J , x₁, x₂, y₁, $y_{2} \in R$ ⁠.

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

X (t) = X_{0} + h (X (\cdot)) + \int_{0}^{t} f (τ, X (τ), A X (τ)) d τ + \sqrt{ϵ} \int_{0}^{t} g (τ, X (τ), B (τ) X (τ)) d W (τ) .

(2.4)

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

Let $A = {v : v$ is a real valued $F_{t}$ predictable process and $\int_{0}^{T} | v (τ) |^{2} d τ < \infty a . s .},$ $\int_{0}^{T} | v (τ) |^{2} τ < \infty a . s .\},$ $S_{N} = {v \in L_{2} (J; R) : \int_{0}^{T} | v (τ) |^{2} d τ \leq N}$ ⁠, where $L_{2} (J; R)$ is the space of all real valued square integrable functions on J. Then S_N endowed with the weak topology in $L_{2} (J; R)$ is a compact Polish space. Let us also define $A_{N} = {v \in A : v (ω) \in S_{N} P - a . s}$ ⁠.

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

Proposition 2.5.

Suppose that there exists a measurable map $G^{0} : C (J; R) \to C (J; R)$ such that the following two conditions hold:

Compactness : For each N < ∞, the set $K_{N} = {G^{0} (\int_{0}^{\cdot} v (τ) d τ) : v \in S_{N}}$ is a compact subset of $C (J; R)$ ⁠.
Weak convergence: Let ${v^{ϵ} : ϵ > 0} \subset A_{N}$ for some N < ∞. If v^ϵ converge to v in distribution as S_N valued random elements, then $G^{ϵ} (W (\cdot) + \frac{1}{\sqrt{ϵ}} \int_{0}^{.} v^{ϵ} (τ) d τ) \to G^{0} (\int_{0}^{.} v (τ) d τ)$ in distribution as ϵ → 0.

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

I (ϕ) = \inf \{\frac{1}{2} \int_{0}^{T} | v (τ) |^{2} d τ; ϕ = G^{0} (\int_{0}^{.} v (τ) d τ) a n d v \in L_{2} (J; R)\}

for each $ϕ \in C (J; R)$ ⁠.

Theorem 2.6.

[31] Let ${Y_{n}, n \in N}$ be a sequence of real-valued random variables that are defined on a sequence of probability spaces ${(Ω_{n}, F_{n}, P_{n}), n \in N}$ ⁠. If $Y_{n} \overset{D}{\to} Y$ and $f \in C_{b} (X)$ , then $f (Y_{n}) \overset{D}{\to} f (Y)$ ⁠.

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 {X^ϵ} of solutions of Equation (2.4) satisfies the LDP (equivalently, the Laplace principle) in $C (J; R)$ with good rate function

I (ϕ) = \inf \{\frac{1}{2} \int_{0}^{T} | v (τ) |^{2} d τ; X_{v} = ϕ\},

where $v \in L_{2} (J; R)$ andX_vdenotes the solution of the controlled equation

X_{v} (t) = X_{0} + h (X_{v} (\cdot)) + \int_{0}^{t} f (τ, X_{v} (τ), A X_{v} (τ)) d τ + \int_{0}^{t} g (τ, X_{v} (τ), B (τ) X_{v} (τ)) v (τ) d τ

(3.1)

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 $G^{0} : C (J; R) \to C (J; R)$ by

G^{0} (g) ≔ \{\begin{cases} X_{v}, & i f g = \int_{0}^{.} v (τ) d τ f o r s o m e v \in L_{2} (J; R) \\ 0, & o t h e r w i s e, \end{cases}

whereX_vdenotes the solution ofEquation (3.1). Then for eachN < ∞, the set $K_{N} = {G^{0} (\int_{0}^{.} v (τ) d τ) : v \in S_{N}}$ is a compact subset of $C (J; R)$ ⁠.

Proof: Consider a sequence {v_n} ∈ S_N such that v_n → v weakly in $L_{2} (J; R)$ as n → ∞. Let $X_{v_{n}}$ denote the solution of the controlled Equation (3.1) with v replaced by v_n. That is

X_{v_{n}} (t) = X_{0} + h (X_{v_{n}} (\cdot)) + \int_{0}^{t} f (τ, X_{v_{n}} (τ), A X_{v_{n}} (τ)) d τ + \int_{0}^{t} g (τ, X_{v_{n}} (τ), B (τ) X_{v_{n}} (τ)) v_{n} (τ) d τ .

(3.2)

From Equations (3.1) and (3.2),

\begin{array}{l} X_{v_{n}} (t) - X_{v} (t) & = h (X_{v_{n}} (\cdot)) - h (X_{v} (\cdot)) \\ + \int_{0}^{t} f (τ, X_{v_{n}} (τ), A X_{v_{n}} (τ)) - f (τ, X_{v} (τ), A X_{v} (τ)) d τ \\ + \int_{0}^{t} g (τ, X_{v_{n}} (τ), B (τ) X_{v_{n}} (τ)) v_{n} (τ) - g (τ, X_{v} (τ), B (τ) X_{v} (τ)) v (τ) d τ . \end{array}

Taking modulus and by using H $ö$ lder's inequality

| X_{v_{n}} (t) - X_{v} (t) | \leq | h (X_{v_{n}} (\cdot)) - h (X_{v} (\cdot)) | + {(\int_{0}^{t} | f (τ, X_{v_{n}} (τ), A X_{v_{n}} (τ)) - f (τ, X_{v} (τ), A X_{v} (τ)) |^{2} d τ)}^{\frac{1}{2}} \times T^{\frac{1}{2}} + {(\int_{0}^{t} | g (τ, X_{v_{n}} (τ), B (τ) X_{v_{n}} (τ)) - g (τ, X_{v} (τ), B (τ) X_{v} (τ)) |^{2} d τ)}^{\frac{1}{2}} \times {(\int_{0}^{t} | v_{n} (τ) |^{2} d τ)}^{\frac{1}{2}} + | \int_{0}^{t} g (τ, X_{v} (τ), B (τ) X_{v} (τ)) (v_{n} (τ) - v (τ)) d τ | .

(3.3)

Let $ζ^{n} (t) = \int_{0}^{t} g (τ, X_{v} (τ), B (τ) X_{v} (τ)) (v_{n} (τ) - v (τ)) d τ$ ⁠. Squaring both sides of Equation (3.3) and using conditions (H2) and (H3),

| X_{v_{n}} (t) - X_{v} (t) |^{2} \leq 4 α_{2}^{2} | X_{v_{n}} (\cdot) - X_{v} (\cdot) |^{2} + 4 α_{1}^{2} T \int_{0}^{t} (| X_{v_{n}} (τ) - X_{v} (τ) |^{2} + | A X_{v_{n}} (τ) - A X_{v} (τ) |^{2}) d τ + 4 α_{1}^{2} N \int_{0}^{t} (| X_{v_{n}} (τ) - X_{v} (τ) |^{2} + | B (τ) X_{v_{n}} (τ) - B (τ) X_{v} (τ) |^{2}) d τ + 4 | ζ^{n} (t) |^{2} .

Using Equation (2.3),

| X_{v_{n}} (t) - X_{v} (t) |^{2} \leq 4 α_{2}^{2} | X_{v_{n}} (\cdot) - X_{v} (\cdot) |^{2} + 4 α_{1}^{2} T \int_{0}^{t} | X_{v_{n}} (τ) - X_{v} (τ) |^{2} + β^{2} | X_{v_{n}} (τ) - X_{v} (τ) |^{2} d τ + 4 α_{1}^{2} N \int_{0}^{t} (| X_{v_{n}} (τ) - X_{v} (τ) |^{2} + γ^{2} (τ) | X_{v_{n}} (τ) - X_{v} (τ) |^{2}) d τ + 4 | ζ^{n} (t) |^{2} .

(3.4)

Consider

| ζ^{n} (t) | = | \int_{0}^{t} g (τ, X_{v} (τ), B (τ) X_{v} (τ)) (v_{n} (τ) - v (τ)) d τ | .

Applying H $ö$ lder's inequality and taking supremum to the above equation leads to,

\begin{align} \sup_{τ \in J} | ζ^{n} (τ) | \leq & α (\int_{0}^{T} {(1 + | X_{v} |_{C (J; R)}^{2} (1 + γ^{2} (τ)) d τ)}^{\frac{1}{2}} {(\int_{0}^{T} | v_{n} (τ) - v (τ) |^{2} d τ)}^{\frac{1}{2}} \\ \leq α \sqrt{N} {[T + ‖ X_{v} ‖_{C (J; R)}^{2} (\int_{0}^{T} (1 + γ^{2} (τ)) d τ)]}^{\frac{1}{2}} \\ \leq C < \infty, w h e r e C i s a c o n s t a n t . \end{align}

(3.5)

Observe that {ζⁿ(t)} is a family of linear, continuous, real valued functions mapping S_N to $C (J; R)$ ⁠. By noting that sup_0≤τ≤t|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 v_n ⇀ v in $L_{2} (J; R)$ ⁠, ζⁿ(t) → 0 point-wise for t ∈ J.

By applying a version of the $A r z \overset{́}{e} l a - A s c o l i$ theorem immediately implies that ζⁿ → 0 uniformly in $C (J; R)$ ⁠. Hence

\lim_{n \to \infty} \sup_{τ \in J} | ζ^{n} (τ) | = 0 .

(3.6)

Set $κ^{n} (t) = \sup_{0 \leq τ \leq t} | X_{v_{n}} (τ) - X_{v} (τ) |^{2}$ ⁠. Then from Equation (3.4)

κ^{n} (t) \leq \frac{4 α_{1}^{2}}{1 - 4 α_{2}^{2}} \int_{0}^{T} κ^{n} (τ) [(1 + β^{2}) T + (1 + γ^{2} (τ)) N] d τ + \frac{4}{1 - 4 α_{2}^{2}} \sup_{0 \leq τ \leq t} | ζ^{n} (τ) |^{2} .

Now by using Gronwall's lemma,

κ^{n} (t) \leq C_{1} \sup_{0 \leq τ \leq t} | ζ^{n} (τ) |^{2} e^{C_{2} T}

where C₁ and C₂ are constants depending on β, γ(τ), α₁, α₂, N and T. Hence

\begin{matrix} | X_{v_{n}} - X_{v} |_{C (J; R)}^{2} & = \sup_{τ \in J} | X_{v_{n}} (τ) - X_{v} (τ) |^{2} \\ \leq C_{1} \sup_{t \in J} | ζ^{n} (t) |^{2} e^{C_{2} T} \end{matrix}

and so $X_{v_{n}} \to X_{v}$ in $C (J; R)$ by Equation (3.6). Since the space S_N is compact, it follows the set $K_{N} = \{G^{0} (\int_{0}^{.} v (τ) d τ) : v \in S_{N}\}$ for N < ∞ is compact.

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

\begin{align} X_{v^{ϵ}}^{ϵ} (t) & = X_{0} + h (X_{v^{ϵ}}^{ϵ} (\cdot)) + \int_{0}^{t} f (τ, X_{v^{ϵ}}^{ϵ} (τ), A X_{v^{ϵ}}^{ϵ} (τ)) d τ \\ + \int_{0}^{t} g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) v^{ϵ} (τ) d τ \\ + \sqrt{ϵ} \int_{0}^{t} g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) d W (τ) . \end{align}

(3.7)

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 ${v^{ϵ} : ϵ > 0} \subset A_{N}$ for some N < ∞. Assume v^ϵ converges to v in distribution as S_N-valued random elements. Then $G^{ϵ} (W (\cdot) + \frac{1}{\sqrt{ϵ}} \int_{0}^{.} v^{ϵ} (τ) d τ) \to G^{0} (\int_{0}^{.} v (τ) d τ)$ in distribution as ϵ → 0.

Proof: Applying Itô's formula

| X_{v^{ϵ}}^{ϵ} (t) - X_{v} (t) |^{2} = | h (X_{v^{ϵ}}^{ϵ} (\cdot)) - h (X_{v} (\cdot)) |^{2} + 2 \int_{0}^{t} (f (τ, X_{v^{ϵ}}^{ϵ} (τ), A X_{v^{ϵ}}^{ϵ} (τ)) - f (τ, X_{v} (τ), A X_{v} (τ))) \times (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ)) d τ + 2 \int_{0}^{t} (g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) - g (τ, X_{v} (τ), B (τ) X_{v} (τ))) \times (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ)) v^{ϵ} (τ) d τ + 2 \int_{0}^{t} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ)) g (τ, X_{v} (τ), B (τ) X_{v} (τ)) (v^{ϵ} (τ) - v (τ)) d τ + 2 \sqrt{ϵ} \int_{0}^{t} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ)) g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) d W (τ) + ϵ \int_{0}^{t} | g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) |^{2} d τ .

(3.8)

Define

ζ^{ϵ} (t) = \int_{0}^{t} g (τ, X_{v} (τ), B (τ) X_{v} (τ)) (v^{ϵ} (τ) - v (τ)) d τ .

(3.9)

Taking supremum and expectation we get $E (\sup_{t \in J} | ζ^{ϵ} (t) |) \leq C_{3} < \infty$ ⁠, where C₃ is a constant.

By virtue of Itô's formula

\begin{align} 2 \int_{0}^{t} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ)) g (τ, X_{v} (τ), B (τ) X_{v} (τ)) (v^{ϵ} (τ) - v (τ)) d τ \\ = 2 (X_{v^{ϵ}}^{ϵ} (t) - X_{v} (t)) (ζ^{ϵ} (t)) - 2 \int_{0}^{t} (h (X_{v^{ϵ}}^{ϵ} (\cdot)) - h (X_{v} (\cdot))) ζ^{ϵ} (τ) d τ \\ - 2 \int_{0}^{t} (f (τ, X_{v^{ϵ}}^{ϵ} (τ), A X_{v^{ϵ}}^{ϵ} (τ)) - f (τ, X_{v} (τ), A X_{v} (τ))) ζ^{ϵ} (τ) d τ \\ - 2 \int_{0}^{t} (g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) v^{ϵ} (τ) - g (τ, X_{v} (τ), B (τ) X_{v} (τ)) v (τ)) \\ \times ζ^{ϵ} (τ) d τ - 2 \sqrt{ϵ} \int_{0}^{t} g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) ζ^{ϵ} (τ) d W (τ) \\ = I_{1} + I_{2} + I_{3} + I_{4} + I_{5} . \end{align}

(3.10)

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

I_{1} \leq \frac{1}{2} | X_{v^{ϵ}}^{ϵ} (t) - X_{v} (t) |^{2} + 2 | ζ^{ϵ} (t) |^{2} .

(3.11)

I_{2} \leq \frac{1}{2} α_{2}^{2} \int_{0}^{t} | X_{v^{ϵ}}^{ϵ} (\cdot) - X_{v} (\cdot) |^{2} d τ + 2 \int_{0}^{t} | ζ^{ϵ} (τ) |^{2} d τ .

(3.12)

I_{3} \leq \frac{1}{2} α_{1}^{2} \int_{0}^{t} [| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + | A X_{v^{ϵ}}^{ϵ} (τ) - A X_{v} (τ) |^{2}] d τ + 2 \int_{0}^{t} | ζ^{ϵ} (τ) |^{2} d τ .

(3.13)

I_{4} \leq \frac{1}{2} α_{1}^{2} \int_{0}^{t} [| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + | B (τ) X_{v^{ϵ}}^{ϵ} (τ) - B (τ) X_{v} (τ) |^{2}] d τ

\begin{align} + 2 \sup_{τ \in [0, t]} | ζ^{ϵ} (τ) |^{2} \int_{0}^{t} | v^{ϵ} (τ) |^{2} d τ + 2 {(\sup_{τ \in [0, t]} | ζ^{ϵ} (τ) |)}^{2} \\ \leq \frac{1}{2} α_{1}^{2} \int_{0}^{t} [| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + | B (τ) X_{v^{ϵ}}^{ϵ} (τ) - B (τ) X_{v} (τ) |^{2}] d τ \\ + 2 N \sup_{τ \in [0, t]} | ζ^{ϵ} (τ) |^{2} + 2 {(\sup_{τ \in [0, t]} | ζ^{ϵ} (τ) |)}^{2} . \end{align}

(3.14)

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,

| X_{v^{ϵ}}^{ϵ} (t) - X_{v} (t) |^{2} \leq | h (X_{v^{ϵ}}^{ϵ} (\cdot)) - h (X_{v} (\cdot)) |^{2} + \int_{0}^{t} | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} d τ

\begin{align} + & α_{1}^{2} \int_{0}^{t} [| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + | A X_{v^{ϵ}}^{ϵ} (τ) - A X_{v} (τ) |^{2}] d τ \\ + & \int_{0}^{t} | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} | v^{ϵ} (τ) |^{2} d τ \end{align}

+ α_{1}^{2} \int_{0}^{t} [| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + | B (τ) X_{v^{ϵ}}^{ϵ} (τ) - B (τ) X_{v} (τ) |^{2}] d τ

+ \frac{1}{2} | X_{v^{ϵ}}^{ϵ} (t) - X_{v} (t) |^{2} + 2 | ζ^{ϵ} (t) |^{2} + \frac{1}{2} α_{2}^{2} \int_{0}^{t} | X_{v^{ϵ}}^{ϵ} (\cdot) - X_{v} (\cdot) |^{2} d τ

\begin{align} + & 4 \int_{0}^{t} | ζ^{ϵ} (τ) |^{2} d τ \\ + & \frac{1}{2} α_{1}^{2} \int_{0}^{t} [| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + | A X_{v^{ϵ}}^{ϵ} (τ) - A X_{v} (τ) |^{2}] d τ \end{align}

+ \frac{1}{2} α_{1}^{2} \int_{0}^{t} [| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + | B (τ) X_{v^{ϵ}}^{ϵ} (τ) - B (τ) X_{v} (τ) |^{2}] d τ

\begin{align} + & 2 \sqrt{ϵ} | \int_{0}^{t} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) - ζ^{ϵ} (t)) g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) | d W (τ) | \\ + ϵ \int_{0}^{t} | g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) |^{2} d τ + 2 (N + 1) {(\sup_{τ \in [0, t]} | ζ^{ϵ} (τ) |)}^{2} . \end{align}

(3.15)

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

| X_{v^{ϵ}}^{ϵ} (t) - X_{v} (t) |^{2} \leq \frac{2}{1 - 2 α_{2}^{2}} \int_{0}^{t} | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} d τ

+ \frac{2 α_{1}^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} (| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + β^{2} | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2}) d τ

+ \frac{2}{1 - 2 α_{2}^{2}} \int_{0}^{t} | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} | v^{ϵ} (τ) |^{2} d τ

+ \frac{2 α_{1}^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} (| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + γ^{2} (τ) | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2}) d τ

+ \frac{4}{1 - 2 α_{2}^{2}} | ζ^{ϵ} (t) |^{2} + \frac{α_{2}^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} | X_{v^{ϵ}}^{ϵ} (\cdot) - X_{v} (\cdot) |^{2} d τ

+ \frac{8}{1 - 2 α_{2}^{2}} \int_{0}^{t} | ζ^{ϵ} (τ) |^{2} d τ

+ \frac{α_{1}^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} (| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + β^{2} | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2}) d τ

\begin{array}{l} + \frac{α_{1}^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} (| X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} + γ^{2} (τ) | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2}) d τ \\ + \frac{4}{1 - 2 α_{2}^{2}} (N + 1) {(\sup_{τ \in [0, t]} | ζ^{ϵ} (τ) |)}^{2} \end{array}

\begin{align} + \frac{2 ϵ α^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} [1 + | X_{v^{ϵ}}^{ϵ} (τ) |^{2} + γ^{2} (τ) | X_{v^{ϵ}}^{ϵ} (τ) |^{2}] d τ \\ + \frac{4 \sqrt{ϵ}}{1 - 2 α_{2}^{2}} | \int_{0}^{t} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) - ζ^{ϵ} (τ)) g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) d W (τ) | . \end{align}

(3.16)

Denote

κ^{ϵ} (t) = \sup_{0 \leq τ \leq t} | X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) |^{2} .

Taking supremum of Equation (3.16)

\begin{array}{l} κ^{ϵ} (t) \leq & M_{1} \int_{0}^{t} (1 + | v^{ϵ} (τ) |^{2}) κ^{ϵ} (τ) d τ + M_{2} \int_{0}^{t} κ^{ϵ} (τ) (1 + γ^{2} (τ)) d τ \\ + M_{3} {(\sup_{τ \in [0, t]} | (ζ^{ϵ} (τ)) |)}^{2} + \frac{2 ϵ α^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} [1 + | X_{v^{ϵ}}^{ϵ} (τ) |^{2} (1 + γ^{2} (τ))] d τ \\ + \frac{4 \sqrt{ϵ}}{1 - 2 α_{2}^{2}} \sup_{s \in [0, t]} | \int_{0}^{s} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) - ζ^{ϵ} (τ)) g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) d W (τ) | . \end{array}

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

\begin{array}{l} κ^{ϵ} (t) & \leq (M_{3} {(\sup_{τ \in [0, t]} | ζ^{ϵ} (τ) |)}^{2} + \frac{2 ϵ α^{2}}{1 - 2 α_{2}^{2}} \int_{0}^{t} [1 + | X_{v^{ϵ}}^{ϵ} (τ) |^{2} (1 + γ^{2} (τ))] d τ \\ + \frac{4 \sqrt{ϵ}}{1 - 2 α_{2}^{2}} \sup_{s \in [0, t]} | \int_{0}^{s} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) - ζ^{ϵ} (τ)) g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) d W (τ) |) \\ \times \exp (M_{1} \int_{0}^{t} (1 + | v^{ϵ} (τ) |^{2}) d τ + M_{2} \int_{0}^{t} (1 + γ^{2} (τ)) d τ |) . \end{array}

Simplifying further

\begin{align} κ^{ϵ} (T) & \leq C (M_{3} {(\sup_{τ \in J} | ζ^{ϵ} (τ) |)}^{2} + \frac{2 ϵ α^{2}}{1 - 2 α_{2}^{2}} [T + \sup_{τ \in J} | X_{v^{ϵ}}^{ϵ} (τ) |^{2} \int_{0}^{T} (1 + γ^{2} (τ)) d τ] \\ + \frac{4 \sqrt{ϵ}}{1 - 2 α_{2}^{2}} \sup_{s \in J} | \int_{0}^{s} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) - ζ^{ϵ} (τ)) g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) d W (τ) |) . \end{align}

(3.17)

For showing convergence of ζ^ϵ(τ), we define

F (u) = \int_{0}^{t} g (τ, X_{v} (τ), B (τ) X_{v} (τ)) u (τ) d τ, u \in A_{N} .

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

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

E (\sup_{s \in J} | \int_{0}^{s} (X_{v^{ϵ}}^{ϵ} (τ) - X_{v} (τ) - ζ^{ϵ} (τ)) g (τ, X_{v^{ϵ}}^{ϵ} (τ), B (τ) X_{v^{ϵ}}^{ϵ} (τ)) d W (τ) |) \leq C .

With the above bound and with the distributional convergence of ζ^ϵ to 0, it follows from (3.17) that κ^ϵ(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.

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.

References

Bruno

Tosin

Tricerri

Venuti

Non-local first-order modelling of crowd dynamics: a multidimensional framework with applications

Appl Math Model

2011

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426

Large deviations for the stochastic functional integral equation with nonlocal condition Open Access

1. Introduction

2. Preliminaries

3. The Large deviation principle

4. Conclusion

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Large deviations for the stochastic functional integral equation with nonlocal condition