The convergence and boundedness of solutions to SFDEs with the G-framework

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The convergence and boundedness of solutions to SFDEs with the G-framework

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Submitted:

22 August 2023

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22 August 2023

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Abstract

Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space Cr((−∞,0];Rn). The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both LG2 and exponential estimates.

Keywords:

Subject: Physical Sciences - Other

1. Introduction

Stochastic functional differential equations (SFDEs) find applications in various fields of engineering and science, such as neural networks [15], financial assets [4,26,30], population dynamics [1,20], and gene expression [21]. A vast literature exists on moment estimates, convergence, stability, and existence of solutions for SFDEs [17,18,19,22,23]. The study of SFDEs driven by G-Brownian motion is relatively new, dating back to the invention of G-Brownian theory in 2006 [24]. In [4,25], the classical Lipschitz condition and linear growth condition were used to establish the existence-uniqueness theorem for SFDEs in the space

B C ((- \infty, 0]; R^{n})

. The study of SFDEs under the G-framework with non-Lipschitz conditions and mean square stability was investigated in [8]. The work in [7,9,10,11] provides insights into pth moment estimates, the Cauchy-Maruyama approximation scheme, and exponential estimates for solutions to SFDEs within the framework of G-Brownian motion. Asymptotic estimates were studied in [28], while SFDEs under the G-Lévy processes were investigated in [5]. In this article, we introduce some useful monotone type conditions to study SFDEs under the G-framework within the space

C_{r} ((- \infty, 0]; R^{n})

. Our findings contribute to the growing body of research on SFDEs driven by G-Brownian motion and deepen our understanding of the role of G-framework in stochastic analysis. We study the convergence of solutions for a SFDEs using the framework of G-Brownian motion. Our analysis results in the mean square boundedness of solutions and allows us to compute both

L_{G}^{2}

and exponential estimates. Consider a matrix A; its transpose is denoted by

A^{T}

. Let

C ((- \infty, 0]; R^{n})

denotes the set of continuous mappings from

(- \infty, 0]

R^{n}

. Define the space

C_{r} ((- \infty, 0]; R^{n})

r > 0

C_{r} ((- \infty, 0]; R^{n}) = {α \in C ((- \infty, 0]; R^{n}) : lim_{σ \to - \infty} e^{r σ} α (σ) exists in R^{n}},

Associated with norm

{∥ α ∥}_{r} = {sup}_{- \infty < σ \leq 0} e^{r σ} | α (σ) | < \infty

, the space

C_{r} ((- \infty, 0]; R^{n})

is a Banach space of bounded continuous mappings. For each

0 < r_{1} \leq r_{2} < \infty

C_{r_{1}} \subseteq C_{r_{2}}

[14,29]. Represent the

σ

-algebra of

C_{r}

B (C_{r})

and

C_{r}^{0} = {α \in C_{r} : {lim}_{σ \to - \infty} e^{r σ} α (σ) = 0}

. Let

L^{2} (C_{r})

denote the space of all

F

-measurable stochastic processes

ψ

taking values in

C_{r}

, such that

\hat{E} {| α |}_{r}^{2} < \infty

. Similarly, let

L^{2} (C_{r}^{0})

denote the space of all

F

-measurable stochastic processes

α

taking values in

C_{r}^{0}

, such that

{E | α |}_{r}^{2} < \infty

. Let

(Ω, F, P)

be a complete probability space, where

F

is a sigma-algebra of subsets. The natural filtration

F_{t}

(Ω, F, P)

is defined as the sigma-algebra, denoted by

F_{t} = σ B (v) : 0 \leq v \leq t

, where

B (v)

represents the Borel sigma-algebra of

C_{r}

. We use

P

to represent the set of all probability measures on

(C_{r}, B (C_{r}))

. Additionally,

L_{b} (C_{r})

denotes the collection of continuous bounded functionals on

C_{r}

. Finally, let

Λ_{0}

be the collection of probability measures on

(- \infty, 0]

satisfying

\int_{- \infty}^{0} μ (d σ) = 1

for every

μ \in Λ_{0}

. We define

Λ_{k} = {μ \in Λ_{0} : μ^{(k)} = \int_{- \infty}^{0} e^{- k σ} μ (d σ) < \infty},

(1)

where

Λ_{k_{0}} \subset Λ_{k} \subset Λ_{0}

for any

k \in (0, k_{0})

[29]. Let

κ : C_{r} ((- \infty, 0]; R^{n}) \to R^{n}

η : C_{r} ((- \infty, 0]; R^{n}) \to R^{n}

and

γ : C_{r} ((- \infty, 0]; R^{n}) \to R^{n}

be Borel measurable. Consider the SFDEs driven by G-Brownian motion of the form

d z (t) = κ (z_{t}) d t + η (z_{t}) d 〈 B, B 〉 (t) + γ (z_{t}) d B (t),

(2)

t \geq 0

where

z_{t} = {z (t + θ) : - \infty < θ \leq 0}

. Equation (2) has the starting value

z_{0} = ζ \in C_{r} ((- \infty, 0]; R^{n})

. Let

〈 B, B 〉 (t)

denote the quadratic variation process of the G-Brownian motion

B (t)

defined on a complete probability space

(Ω, F, P)

, where

B (t)

is a one-dimensional process under the filtration

F_{t \geq 0}

satisfying the usual conditions. The remaining paper is arranged as follows. Section 2 presents the basic results. In section 3, some useful lemmas are given. Section 4 investigates the mean square boundedness and convergence of solutions. In section 5, we first study the

L_{G}^{2}

estimate and then derive the exponential estimate. Section 6 contains conclusions.

2. Basic Notions and Results

This section presents some fundamental concepts and results that we utilize in the forthcoming research [2,6,16,27]. Let

H

be a space of real mappings defined on a non-empty set

Ω

Definition 1.

∀

x, y \in H

, a functional

\hat{E} : H \to R

assuring the below given features is called a G-expectation

$\hat{E} [y] \geq \hat{E} [x]$ whenever $y \geq x$ .
$\hat{E} [m_{1}] = m_{1}$ , for any $m_{1} \in R$ .
$\hat{E} [m_{2} y] = m_{2} \hat{E} [y]$ , for any $m_{2} \in R^{+}$ .
$\hat{E} [y + x] \leq \hat{E} [y] + \hat{E} [x]$ .

Suppose that

Ω

be the space of

R^{n}

-valued continuous paths

{(w (t))}_{t \geq 0}

such that

w (0) = 0

associated with the norm

D (w^{1}, w^{2}) = \sum_{j = 1}^{\infty} 2^{- j} (max_{t \in [0, j]} | w^{1} (t) - w^{2} (t) | \land 1) .

Choose

Ω_{T} = {ω \land T : ω \in Ω}

. Assume the canonical process

B (t) = B (t, w)

where

t \geq 0

and

w \in Ω

. Let

λ \in C_{b . L i p} (R^{n \times d})

and

t_{1}, t_{2}, . . ., t_{n} \in [0, T]

then

L_{i p}^{0} (Ω_{T}) = \{λ (B (t_{1}), B (t_{2}), . . ., B (t_{n})) : n \geq 1\} .

Notice that

L_{i p}^{0} (Ω_{t}) \subseteq L_{i p}^{0} (Ω_{T})

L_{i p}^{0} (Ω) = \cup_{m = 1}^{\infty} L_{i p}^{0} (Ω_{m})

and the completion of

L_{i p}^{0} (Ω)

associated with

\hat{E} {[| . |}^{p}]^{\frac{1}{p}}

p \geq 1

L_{G}^{p} (Ω)

. Related to

{B (t)}_{t \geq 0}

, we can express the filtration as

F_{t} = σ {B (u), 0 \leq u \leq t}

where

F = {F_{t}}_{t \geq 0}

. Let

0 \leq t_{0} \leq t_{1} \leq . . . \leq t_{N} < \infty

and

▵_{T} = {t_{0}, t_{1}, . . ., t_{N}}

be a partition of

[0, T] .

Let

p \geq 1

then

M_{G}^{p, 0} (0, T)

is given by

M_{G}^{p, 0} (0, T) = {ρ_{t} (w) = \sum_{o = 0}^{N - 1} ϑ_{j} (w) I_{[t_{o}, t_{o + 1}]} (t) : ϑ_{o} \in L_{G}^{p} (Ω_{t_{o}}), o = 0, 1, . . ., N - 1},

The space

M_{G}^{p} (0, T)

is the completion of

M_{G}^{p, 0} (0, T)

under the norm

∥ ρ ∥ = {\{\int_{0}^{T} E [| ρ_{s} |^{p}] d s\}}^{1 / p},

p \geq 1 .

Definition 2.

Let

ρ_{t} \in M_{G}^{2, 0} (0, T)

. The G-Itô integral

I (ρ)

is defined as the stochastic integral of a function ρ with respect to G-Brownian motion given by

\begin{matrix} I (ρ) = \int_{0}^{T} ρ (s) d B^{a} (s) = \sum_{k = 0}^{N - 1} ξ_{k} (B^{a} (t_{k + 1}) - B^{a} (t_{k})), \end{matrix}

One can extend

I : M_{G}^{2, 0} (0, T) \mapsto L_{G}^{2} (F_{T})

I : M_{G}^{2} (0, T) \mapsto L_{G}^{2} (F_{T})

, where for

ρ \in M_{G}^{2} (0, T)

we have

\begin{matrix} \int_{0}^{T} ρ (s) d B^{a} (s) = I (ρ) . \end{matrix}

Definition 3.

Let

〈 B^{a} 〉 (0) = 0

. The G-quadratic variation process

{〈 B^{a} 〉 (t)}_{t \geq 0}

is given as follows

\begin{matrix} \begin{matrix} 〈 B^{a} 〉 (t) = lim_{N \to \infty} \sum_{j = 0}^{N - 1} {(B^{a} (t_{j + 1}^{N}) - B^{a} (t_{j}^{N}))}^{2} = {B^{a} (t)}^{2} - 2 \int_{0}^{t} B^{a} (s) d B^{a} (s) . \end{matrix} \end{matrix}

Consider a function

K_{0, T} : M_{G}^{0, 1} (0, T) \mapsto L_{G}^{2} (F_{T})

given as

\begin{matrix} K_{0, T} (ρ) = \int_{0}^{T} ρ (s) d 〈 B^{a} 〉 (s) = \sum_{i = 0}^{N - 1} ξ_{i} ({〈 B^{a} 〉}_{(} t_{i + 1}) - 〈 B^{a} 〉 (t_{i})) . \end{matrix}

One can extend

K_{0, T}

M_{G}^{1} (0, T)

. For

ρ \in M_{G}^{1} (0, T)

, it is given by

\begin{matrix} \int_{0}^{T} ρ (s) d 〈 B^{a} 〉 (s) = K_{0, T} (ρ) . \end{matrix}

Lemma 1.

Assume that

γ \in M_{G}^{p} (0, T)

and

p \geq 2

. Then

\hat{E} [sup_{0 \leq t \leq T} | \int_{0}^{t} γ (s) d B (s) |^{p}] \leq a_{3} \hat{E} {[\int_{0}^{t} {| γ (s) |}^{2} d s]}^{\frac{p}{2}},

where

a_{3} \in (0, \infty)

is a p dependent constant.

Lemma 2.

Assume that

γ \in M_{G}^{p} (0, T)

p \geq 1

. Then

\hat{E} [sup_{0 \leq t \leq T} | \int_{0}^{t} γ (s) d 〈 B, B 〉 (s) |^{p}] \leq a_{2} \hat{E} {[\int_{0}^{t} {| γ (s) |}^{2} d s]}^{\frac{p}{2}},

where

a_{2} \in (0, \infty)

depends on p.

Definition 4.

The G-Brownian motion is an n-dimensional stochastic process

{B (t)}_{t \geq 0}

fulfilling the following characteristics

$(i)$: $B (0) = 0 .$
$(ii)$: $B (t + v) - B (t)$ is G-normally distributed and independent of $B (t_{1}), B (t_{2}), . . . . . . . . B (t_{n})$ for any $n \in N$ and $0 \leq t_{1} \leq t_{2} \leq, . . ., \leq t_{n} \leq t$ .

Lemma 3.

Let

p \geq 0

z \in L_{G}^{p}

. For each

m_{1} > 0

\hat{ν} (| z | > m_{1}) \leq \frac{\hat{E} {[| z |}^{p}]}{m_{1}},

where

\hat{E} {| z |}^{p} < \infty

The following two basic lemmas can also be utilized in forthcoming sections of this paper [17].

Lemma 4.

Let

c, l \geq 0

and

δ \in (0, 1) .

Then

\begin{matrix} {(c + l)}^{2} \leq \frac{c^{2}}{δ} + \frac{l^{2}}{1 - δ} . \end{matrix}

Lemma 5.

Let

p \geq 2

and

\hat{δ}, c, l > 0 .

Then

$(i)$: $c^{p - 1} l \leq \frac{(p - 1) \hat{δ} c^{p}}{p} + \frac{l^{p}}{p {\hat{δ}}^{p - 1}} .$
$(ii)$: $c^{p - 2} l^{2} \leq \frac{(p - 2) \hat{δ} c^{p}}{p} + \frac{2 l^{p}}{p {\hat{δ}}^{\frac{p - 2}{2}}} .$

3. Some Useful Results

In this section, we introduce and discuss some important assumptions and establish two lemmas. We consider the following hypotheses

$(H)$: Let $a_{i} > 0$ , $i = 1, 2, . ., 5$ and $α (σ) - β (σ) = Ψ (σ)$ . For any $α, β \in C_{r} ((- \infty, 0]; R^{n})$ and for any probability measure $μ_{1}, μ_{2}, μ_{3} \in Λ_{2 r}$ the following inequalities hold

${[Ψ (0)]}^{T} [κ (α) - κ (β)] \leq - a_{1} {| Ψ (0) |}^{2} + a_{2} \int_{- \infty}^{0} {| Ψ (σ) |}^{2} μ_{1} (d σ),$

(3)

${[Ψ (0)]}^{T} [η (α) - η (β)] \leq - a_{3} {| Ψ (0) |}^{2} + a_{4} \int_{- \infty}^{0} {| Ψ (σ) |}^{2} μ_{2} (d σ),$

(4)

${| γ (α) - γ (β) |}^{2} \leq a_{5} \int_{- \infty}^{0} {| Ψ (σ) |}^{2} μ_{3} (d σ) .$

(5)

Lemma 6.

Let

a < p r

and

p \geq 1

. Then

\begin{matrix} ∥ z_{t} ∥_{r}^{p} & \leq e^{- a t} {∥ ζ ∥}_{r}^{p} + sup_{0 < u \leq t} {| z (u) |}^{p}, \end{matrix}

where

ζ \in C_{r} ((- \infty, 0]; R^{n})

Proof.

Assuming that

p r > a

, we can obtain the following using the definition of the norm

{| \cdot |}_{r}

\begin{matrix} ∥ z_{t} ∥_{r}^{p} & = {[sup_{- \infty < σ \leq 0} e^{r σ} | z (t + σ) |]}^{p} \leq sup_{- \infty < σ \leq 0} e^{a σ} {| z (t + σ) |}^{p} \\ \leq sup_{0 < u \leq t} e^{- a (t - u)} {| z (u) |}^{p} + sup_{- \infty < u \leq 0} e^{- a (t - u)} {| z (u) |}^{p} \\ = e^{- a t} {∥ ζ ∥}_{r}^{p} + e^{- a t} sup_{0 < u \leq t} e^{a s} {| z (u) |}^{p} \\ \leq sup_{0 < u \leq t} {| z (u) |}^{p} + e^{- a t} {∥ ζ ∥}_{r}^{p} . \end{matrix}

The proof stands completed. □

Throughout this paper we let that for any

p \geq 1

a < p r

Lemma 7.

Let

a < p r

p \geq 2

and

μ_{i} \in Λ_{k}

, ∀

i \in N

. Then

\int_{0}^{t} \int_{- \infty}^{0} {| z (σ + s) |}^{p} μ_{i} (d σ) d s \leq \frac{μ_{i}^{(p r)}}{p r} {∥ ζ ∥}_{r}^{p} + \int_{0}^{t} {| z (s) |}^{p} d s,

(6)

\int_{0}^{t} \int_{- \infty}^{0} e^{a s} {| z (σ + s) |}^{p} μ_{i} (d σ) d s \leq \frac{μ_{i}^{(p r)}}{p r - a} {∥ ζ ∥}_{r}^{p} + μ_{i}^{(p r)} \int_{0}^{t} e^{a s} {| z (s) |}^{p} d s,

(7)

where

ζ \in C_{r} ((- \infty, 0]; R^{n})

Proof.

As for each

i \in Z^{+}

μ_{i} \in Λ_{p r}

and

ζ \in C_{r} ((- \infty, 0]; R^{n})

, by using the Fubini theorem and the definition of norm, it follows

\begin{matrix} \int_{0}^{t} \int_{- \infty}^{0} {| z (σ + s) |}^{p} μ_{i} (d σ) d s \\ = \int_{0}^{t} [\int_{- \infty}^{- s} e^{p r (σ + s)} {| z (σ + s) |}^{p} e^{- p r (s + σ)} μ_{i} (d σ) + \int_{- s}^{0} {| z (σ + s) |}^{p} μ_{i} (d σ)] d s \\ \leq {∥ ζ ∥}_{r}^{p} \int_{0}^{t} e^{- p r s} d s \int_{- \infty}^{0} e^{- p r σ} μ_{i} (d σ) + \int_{- \infty}^{0} μ_{i} (d σ) \int_{0}^{t} {| z (s) |}^{p} d s . \end{matrix}

Observing that

\int_{- \infty}^{0} μ_{i} (d σ) = 1

and

\int_{- \infty}^{0} e^{- p r σ} μ_{i} (d σ) = μ_{i}^{(p r)}

i \in N

, it follows

\int_{0}^{t} \int_{- \infty}^{0} {| z (s + σ) |}^{p} μ_{i} (d σ) d s \leq \frac{μ_{i}^{(p r)}}{p r} {∥ ζ ∥}_{r}^{p} + \int_{0}^{t} {| z (s) |}^{p} d s .

The proof of (6) is complete. Using similar arguments as used above we determine

\begin{matrix} \int_{0}^{t} \int_{- \infty}^{0} e^{a s} {| z (σ + s) |}^{p} μ_{i} (d σ) d s \\ = \int_{0}^{t} e^{a s} d s [\int_{- \infty}^{- s} {| z (σ + s) |}^{p} μ_{i} (d σ) + \int_{- s}^{0} {| z (σ + s) |}^{p} μ_{i} (d σ)] \\ = \int_{0}^{t} e^{a s} d s \int_{- \infty}^{- s} {| z (σ + s) |}^{p} μ_{i} (d σ) + \int_{- t}^{0} μ_{i} (d σ) \int_{- σ}^{t} e^{a s} {| z (σ + s) |}^{p} d s \\ \leq \int_{0}^{t} e^{a s} d s \int_{- \infty}^{- s} e^{p r (σ + s)} {| z (s + σ) |}^{p} e^{- p r (σ + s)} μ_{i} (d σ) + \int_{- \infty}^{0} μ_{i} (d σ) \int_{0}^{t} e^{a (s - σ)} {| z (s) |}^{p} d s \\ \leq {∥ ζ ∥}_{r}^{p} \int_{0}^{t} e^{- (p r - a) s} d s \int_{- \infty}^{0} e^{- p r σ} μ_{i} (d σ) + \int_{- \infty}^{0} e^{- a σ} μ_{i} (d σ) \int_{0}^{t} e^{a s} {| z (s) |}^{p} d s . \end{matrix}

With reference to equation (1), and taking note that

p r > a

, we can conclude that

\int_{0}^{t} \int_{- \infty}^{0} e^{a s} {| z (σ + s) |}^{p} μ_{i} (d σ) d s \leq \frac{μ_{i}^{(p r)}}{p r - a} {∥ ζ ∥}_{r}^{p} + μ_{i}^{(p r)} \int_{0}^{t} e^{a s} {| z (s) |}^{p} d s .

The proof of (7) is complete. □

4. Convergence and Mean Square Boundedness

Firstly, let us derive the mean square boundedness for solutions to equation (2).

Theorem 1.

Let the hypothesis H holds. Assume the equation (2) with initial condition

ζ \in C_{r} ((- \infty, 0]; R^{n})

has just one solution

z (t)

. Let

a_{i}

i = 1, 2, . ., 5

assure

2 a_{1} > 2 a_{2} μ_{1}^{(2 r)} + 2 b_{1} a_{4} μ_{2}^{(2 r)} + b_{1} a_{5} μ_{3}^{(2 r)} - 2 b_{1} a_{3}

. Then there is

a \in (0, (2 a_{1} + 2 b_{1} a_{3} - 2 a_{2} μ_{1}^{(2 r)} - 2 b_{1} a_{4} μ_{2}^{(2 r)} - b_{1} a_{5} μ_{3}^{(2 r)}) \land 2 r)

so that

\hat{E} {[| z (t) |}^{2}] \leq c_{1} + c_{2} e^{- a t},

(8)

where

c_{1} = \frac{1}{a} (\frac{1}{δ} {| κ (0) |}^{2} + \frac{b_{1}}{δ_{1}} {| η (0) |}^{2} + \frac{b_{1}}{δ_{2}} {| γ (0) |}^{2})

and

c_{2} = \hat{E} {| z (0) |}^{2} + \frac{2 a_{2} μ_{1}^{(2 r)}}{2 r - a} \hat{E} {∥ ζ ∥}_{r}^{2} + \frac{2 b_{1} a_{4} μ_{2}^{(2 r)}}{2 r - a} \hat{E} {∥ ζ ∥}_{r}^{2} + \frac{b_{1} a_{5} μ_{3}^{(2 r)}}{(2 r - a) (1 - δ_{2})} \hat{E} {∥ ζ ∥}_{r}^{2} .

The values of δ,

δ_{1}

and

δ_{2}

are sufficiently small so that

2 a_{1} - δ - a - b_{1} δ_{1} + 2 b_{1} a_{3} - 2 a_{2} μ_{1}^{(2 r)} - 2 b_{1} a_{4} μ_{2}^{(2 r)} - \frac{b_{1} a_{5}}{1 - δ_{2}} μ_{3}^{(2 r)} > 0 .

Proof.

By using the G-Itô formula, G-Itô integral and Lemma 2, it follows

\begin{matrix} \hat{E} [e^{a t} {| z (t) |}^{2}] & \leq \hat{E} {| z (0) |}^{2} + \hat{E} \int_{0}^{t} e^{a s} [{a | z (s) |}^{2} + 2 z^{T} (s) κ (z_{s})] d s \\ + b_{1} \hat{E} \int_{0}^{t} e^{a s} [2 z^{T} (s) η (z_{s}) + {| γ (z_{s}) |}^{2}] d s . \end{matrix}

(9)

Utilizing (3), (4) and Lemma 5 we determine

\begin{matrix} z^{T} (t) κ (z_{t}) & \leq (\frac{δ}{2} - a_{1}) | z (t) |^{2} + \frac{1}{2 δ} | κ (0) |^{2} + a_{2} \int_{- \infty}^{0} {| z (t + σ) |}^{2} μ_{1} (d σ), \end{matrix}

\begin{matrix} z^{T} (t) η (z_{t}) & \leq (\frac{δ_{1}}{2} - a_{3}) | z (t) |^{2} + \frac{1}{2 δ_{1}} | η (0) |^{2} + a_{4} \int_{- \infty}^{0} {| z (t + σ) |}^{2} μ_{2} (d σ) . \end{matrix}

It follows from Lemma 4 and the condition given in (5) that

\begin{matrix} | γ (z_{t}) |^{2} & \leq \frac{1}{δ_{2}} {| γ (0) |}^{2} + \frac{a_{5}}{1 - δ_{2}} \int_{- \infty}^{0} {| z (t + σ) |}^{2} μ_{3} (d σ) . \end{matrix}

Using the above inequalities, (9) becomes

\begin{matrix} \hat{E} [e^{a t} {| z (t) |}^{2}] & \leq \hat{E} {| z (0) |}^{2} + \frac{1}{a} (\frac{1}{δ} {| κ (0) |}^{2} + \frac{b_{1}}{δ_{1}} {| η (0) |}^{2} + \frac{b_{1}}{δ_{2}} {| γ (0) |}^{2}) (e^{a t} - 1) \\ + (δ + a - 2 a_{1} + b_{1} δ_{1} - 2 b_{1} a_{3}) \hat{E} \int_{0}^{t} e^{a s} {| z (s) |}^{2} d s \\ + 2 a_{2} \hat{E} \int_{0}^{t} e^{a s} \int_{- \infty}^{0} {| z (s + σ) |}^{2} μ_{1} (d σ) d s \\ + 2 b_{1} a_{4} \hat{E} \int_{0}^{t} e^{a s} \int_{- \infty}^{0} {| z (s + σ) |}^{2} μ_{2} (d σ) d s \\ + b_{1} \frac{a_{5}}{1 - δ_{2}} \hat{E} \int_{0}^{t} e^{a s} \int_{- \infty}^{0} {| z (s + σ) |}^{2} μ_{3} (d σ) d s \end{matrix}

(10)

In view of Lemma 7, it follows

\int_{0}^{t} \int_{- \infty}^{0} e^{a s} {| z (s + σ) |}^{2} μ_{i} (d σ) d s \leq \frac{1}{2 r - a} {∥ ζ ∥}_{r}^{2} μ_{i}^{(2 r)} + μ_{i}^{(2 r)} \int_{0}^{t} e^{a s} {| z (s) |}^{2} d s .

(11)

Substituting (11) in (10) we derive

\begin{matrix} \hat{E} [e^{a t} {| z (t) |}^{2}] \leq \hat{E} {| z (0) |}^{2} + \frac{2 a_{2} μ_{1}^{(2 r)}}{2 r - a} \hat{E} {∥ ζ ∥}_{r}^{2} + \frac{2 b_{1} a_{4} μ_{2}^{(2 r)}}{2 - r a} \hat{E} {∥ ζ ∥}_{r}^{2} + \frac{b_{1} a_{5} μ_{3}^{(2 r)}}{(2 r - a) (1 - δ_{2})} \hat{E} {∥ ζ ∥}_{r}^{2} \\ + \frac{1}{λ} (\frac{1}{δ} {| κ (0) |}^{2} + \frac{b_{1}}{δ_{1}} {| η (0) |}^{2} + \frac{b_{1}}{δ_{2}} {| γ (0) |}^{2}) (e^{a t} - 1) \\ - (2 a_{1} - δ - a - b_{1} δ_{1} + 2 b_{1} a_{3} - 2 a_{2} μ_{1}^{(2 r)} - 2 b_{1} a_{4} μ_{2}^{(2 r)} - \frac{b_{1} a_{5}}{1 - δ_{2}} μ_{3}^{(2 r)}) \hat{E} \int_{0}^{t} e^{a s} {| z (s) |}^{2} d s . \end{matrix}

2 a_{1} > 2 a_{2} μ_{1}^{(2 r)} + 2 b_{1} a_{4} μ_{2}^{(2 r)} + b_{1} a_{5} μ_{3}^{(2 r)} - 2 b_{1} a_{3}

and

a \in (0, (2 a_{1} + 2 b_{1} a_{3} - 2 a_{2} μ_{1}^{(2 r)} - 2 b_{1} a_{4} μ_{2}^{(2 r)} - b_{1} a_{5} μ_{3}^{(2 r)}) \land 2 r)

. Selecting

δ

δ_{1}

and

δ_{2}

sufficiently small so that

2 a_{1} - δ - a - b_{1} δ_{1} + 2 b_{1} a_{3} - 2 a_{2} μ_{1}^{(2 r)} - 2 b_{1} a_{4} μ_{2}^{(2 r)} - \frac{b_{1} a_{5}}{1 - δ_{2}} μ_{3}^{(2 r)} > 0,

we obtain the desired result

\hat{E} {[| z (t) |}^{2}] \leq c_{1} + c_{2} e^{- a t},

where

c_{1} = \frac{1}{a} (\frac{1}{δ} {| κ (0) |}^{2} + \frac{b_{1}}{δ_{1}} {| η (0) |}^{2} + \frac{b_{1}}{δ_{2}} {| γ (0) |}^{2})

and

c_{2} = \hat{E} | z_{0} |^{2} + \frac{2 a_{2} μ_{1}^{(2 q)}}{2 r - a} \hat{E} {∥ ζ ∥}_{r}^{2} + \frac{2 b_{1} a_{4} μ_{2}^{(2 r)}}{2 r - a} \hat{E} {∥ ζ ∥}_{r}^{2} + \frac{b_{1} a_{5} μ_{3}^{(2 r)}}{(2 r - a) (1 - δ_{2})} \hat{E} {∥ ζ ∥}_{r}^{2} .

□

Theorem 1 describes that equation (2) has a mean square bounded solution. The following Theorem 2 expresses that any two distinct solutions of equation (2) are convergent.

Theorem 2.

Assuming that all hypotheses of Theorem 1 are satisfied, let

z (t)

and

y (t)

be two solutions of equation (2) associated with initial values ζ and ξ, respectively. Then, we have:

\hat{E} {[| z (t) - y (t) |}^{2}] \leq c_{3} \hat{E} {| ξ - ζ |}_{r}^{2} e^{- a t},

(12)

where

c_{3} = 1 + \frac{1}{2 r - a} (2 a_{2} μ_{1}^{(2 r)} + 2 b_{1} a_{4} μ_{2}^{(2 r)} + b_{1} a_{5} μ_{3}^{(2 r)})

Proof.

Define

ϕ (t) = z (t) - y (t)

\hat{γ} (t) = γ (z_{t}) - γ (y_{t})

\hat{κ} (t) = κ (z_{t}) - κ (y_{t})

, and

\hat{η} (t) = η (z_{t}) - η (y_{t})

. Utilizing the G-Itô integral, Lemma 2, and G-Itô formula, it follows

\begin{matrix} \hat{E} [e^{a t} {| ϕ (t) |}^{2}] & \leq \hat{E} {| ξ (0) - ζ (0) |}^{2} + \hat{E} \int_{0}^{t} e^{a s} {[a | ϕ (s) |}^{2} + 2 ϕ^{T} (s) \hat{κ} (s)] d s \\ + b_{1} \hat{E} \int_{0}^{t} e^{ϕ s} [2 ϕ^{T} (s) \hat{η} (s) + | \hat{γ} (s) |^{2}] d s . \end{matrix}

(13)

From hypothesis H, we derive

\begin{matrix} ϕ^{T} (t) \hat{κ} (t) \leq - a_{1} {| ϕ (t) |}^{2} + a_{2} \int_{- \infty}^{0} ϕ (σ + t) μ_{1} (d σ), \\ ϕ^{T} (t) \hat{η} (t) \leq - a_{3} {| ϕ (t) |}^{2} + a_{4} \int_{- \infty}^{0} ϕ (σ + t) μ_{2} (d σ) \end{matrix}

and

| \hat{γ} {(t) |}^{2} \leq a_{5} \int_{- \infty}^{0} ϕ (t + σ) μ_{3} (d σ) .

Utilizing the above inequalities, (13) becomes

\begin{matrix} \hat{E} [e^{a t} {| ϕ (t) |}^{2}] & \leq \hat{E} {| ξ (0) - ζ (0) |}^{2} + (a - 2 a_{1} - 2 b_{1} a_{3}) \hat{E} \int_{0}^{t} e^{a s} {| ϕ (s) |}^{2} d s \\ + 2 a_{2} \hat{E} \int_{0}^{t} \int_{- \infty}^{0} e^{a s} ϕ (s + σ) μ_{1} (d σ) d s + 2 b_{1} a_{4} \hat{E} \int_{0}^{t} \int_{- \infty}^{0} e^{a s} ϕ (s + σ) μ_{2} (d σ) d s \\ + k_{1} a_{5} \hat{E} \int_{0}^{t} \int_{- \infty}^{0} e^{a s} ϕ (s + σ) μ_{3} (d σ) d s . \end{matrix}

(14)

From Lemma 7 for

i = 1, 2, 3

it follows

\begin{matrix} \int_{0}^{t} \int_{- \infty}^{0} e^{a s} {| ϕ (s + σ) |}^{2} μ_{i} (d σ) d s \leq \frac{1}{2 q - a} {∥ ζ - ξ ∥}_{r}^{2} μ_{i}^{(2 r)} + μ_{i}^{(2 r)} \int_{0}^{t} e^{a s} {| ϕ (s) |}^{2} d s \end{matrix}

(15)

Plugging (15) in (14) we determine

\begin{matrix} e^{a t} \hat{E} {| ϕ (t) |}^{2} & \leq \hat{E} {| ξ (0) - ζ (0) |}^{2} + \frac{1}{2 r - a} [2 a_{2} μ_{1}^{(2 r)} + 2 b_{1} a_{4} μ_{2}^{(2 r)} + b_{1} a_{5} μ_{3}^{(2 r)}] \hat{E} {∥ ζ - ξ ∥}_{r}^{2} \\ - (2 a_{1} + 2 b_{1} a_{3} - a - 2 a_{2} μ_{1}^{(2 r)} - 2 k_{1} a_{4} μ_{2}^{(2 r)} - b_{1} a_{5} μ_{3}^{(2 r)}) \hat{E} \int_{0}^{t} e^{a s} {| ϕ (s) |}^{2} d s . \end{matrix}

2 a_{1} > 2 a_{2} μ_{1}^{(2 r)} + 2 b_{1} a_{4} μ_{2}^{(2 r)} + b_{1} a_{5} μ_{3}^{(2 r)} - 2 b_{1} a_{3}

and

a \in (0, (2 a_{1} + 2 b_{1} a_{3} - 2 a_{2} μ_{1}^{(2 r)} - 2 b_{1} a_{4} μ_{2}^{(2 r)} - b_{1} a_{5} μ_{3}^{(2 r)}) \land 2 r)

it follows

\begin{matrix} \hat{E} {| ϕ (t) |}^{2} & \leq [1 + \frac{1}{2 r - a} (2 a_{2} μ_{1}^{(2 r)} + 2 b_{1} a_{4} μ_{2}^{(2 r)} + b_{1} a_{5} μ_{3}^{(2 r)})] \hat{E} {∥ ζ - ξ ∥}_{r}^{2} e^{- a t}, \end{matrix}

consequently, we derive the following required result

\begin{matrix} \hat{E} {| z (t) - y (t) |}^{2} & \leq c_{3} \hat{E} {∥ ζ - ξ ∥}_{r}^{2} e^{- a t}, \end{matrix}

where

c_{3} = 1 + \frac{1}{2 r - a} (2 a_{2} μ_{1}^{(2 r)} + 2 b_{1} a_{4} μ_{2}^{(2 r)} + b_{1} a_{5} μ_{3}^{(2 r)})

. □

κ (0) = η (0) = γ (0) = 0

, then from Theorem 2 we can obtain that the trivial solution of equation (2) is mean square exponentially stable.

Example 1.

Consider

z (t)

and

y (t)

be two solutions of the equation

d z (t) = z_{t} d t + s i n (z_{t}) d 〈 B 〉 (t) + z_{t} d B (t)

with initial values ζ and ξ respectively. Define

ϕ (t) = z (t) - y (t)

\hat{η} (t) = \hat{γ} (t) = z_{t} - y_{t}

and

\hat{κ} (t) = s i n (z_{t}) - s i n (y_{t})

. Under the given hypothesis one can easily derive that

y (t)

is mean square convergent to

z (t)

5. The Exponential Estimate

Firstly, let us determine the

L_{G}^{2}

estimates. Let equation (2) with initial condition

ζ \in C_{r} ((- \infty, 0]; R^{n})

has just one solution

z (t)

t \in [0, \infty)

Theorem 3.

Assume that the hypothesis H holds and

{E ∥ ζ ∥}_{r}^{2} < \infty

. For every

t \geq 0

\hat{E} [sup_{- \infty < s \leq t} {| z (t) |}^{2}] \leq [\hat{E} {∥ ζ ∥}_{r}^{2} + m_{1}] e^{m_{2} t},

where

m_{1} = c + \frac{2}{r} [r + a_{2} μ_{1}^{(2 r)} + b_{1} (a_{5} μ_{3}^{(2 r)} + μ_{2}^{(2 r)}) + 2 b_{3} a_{5} μ_{3}^{(2 r)}] \hat{E} {∥ ζ ∥}_{r}^{2}

c = {2 [| κ (0) |}^{2} + b_{1} {(| η (0) |}^{2} + {2 | γ (0) |}^{2}) + 4 b_{3} {| γ (0) |}^{2}] T

and

m_{2} = 2 [2 a_{2} - 2 a_{1} + 1 + b_{1} (2 a_{5} - 2 a_{3} + 3) + 4 b_{3} a_{5}]

Proof.

Using the G-It

\hat{o}

formula and properties of the G-expectation, it follows

\begin{matrix} \hat{E} [sup_{0 \leq s \leq t} {| z (t) |}^{2}] & \leq \hat{E} {| z (0) |}^{2} + 2 \hat{E} [sup_{0 \leq s \leq t} \int_{0}^{t} z^{T} (s) κ (z_{s}) d s] \\ + \hat{E} [sup_{0 \leq s \leq t} \int_{0}^{t} (2 z^{T} (s) η (z_{s}) + | γ (z_{s}) |^{2}) d 〈 B, B 〉 (s)] \\ + 2 \hat{E} [sup_{0 \leq s \leq t} \int_{0}^{t} z^{T} (s) γ (z_{s}) d B (s)] \end{matrix}

(16)

From our assumption H,

2 a_{1} a_{2} \leq \sum_{i = 1}^{2} a_{i}^{2}

and

{(\sum_{i = 1}^{2} a_{i})}^{2} \leq 2 \sum_{i = 1}^{2} a_{i}^{2}

, it follows

\begin{matrix} z^{T} (t) κ (z_{t}) & \leq - (a_{1} - \frac{1}{2}) | z (t) |^{2} + \frac{1}{2} | κ (0) |^{2} + a_{2} \int_{- \infty}^{0} {| z (t + σ) |}^{2} μ_{1} (d σ), \end{matrix}

(17)

\begin{matrix} z^{T} (t) η (z_{t}) & \leq - (a_{3} - \frac{1}{2}) | z (t) |^{2} + \frac{1}{2} | η (0) |^{2} + a_{4} \int_{- \infty}^{0} {| z (t + σ) |}^{2} μ_{2} (d σ), \end{matrix}

(18)

\begin{matrix} | γ (z_{t}) |^{2} & \leq {2 | γ (0) |}^{2} + 2 a_{5} \int_{- \infty}^{0} {| z (t + σ) |}^{2} μ_{3} (d σ) . \end{matrix}

(19)

In view of (17) and (11), we determine

\begin{matrix} 2 \hat{E} [sup_{0 < s \leq t} \int_{0}^{t} z^{T} (s) κ (z_{s}) d s] \leq {| κ (0) |}^{2} T + \frac{a_{2}}{r} \hat{E} {∥ ζ ∥}_{r}^{2} μ_{1}^{(2 r)} + (2 a_{2} - 2 a_{1} + 1) \hat{E} \int_{0}^{t} {| z (s) |}^{2} d s . \end{matrix}

Utilizing (18), (19), (11) and Lemma 2, it follows

\begin{matrix} \hat{E} [sup_{0 \leq s \leq t} \int_{0}^{t} (2 z^{T} (s) η (z_{s}) + | γ (z_{s}) |^{2}) d 〈 B, B 〉 (s)] \leq b_{1} \hat{E} [\int_{0}^{t} (2 z^{T} (s) η (z_{s}) + | γ (z_{s}) |^{2})] d s \\ \leq b_{1} {[| η (0) |}^{2} + {2 | γ (0) |}^{2}] T + b_{1} \frac{1}{r} (a_{5} μ_{3}^{(2 r)} + μ_{2}^{(2 r)}) \hat{E} {∥ ζ ∥}_{r}^{2} \\ + b_{1} (2 a_{5} - 2 a_{3} + 3) \hat{E} \int_{0}^{t} {| z (s) |}^{2} d s . \end{matrix}

The inequality

a_{1} a_{2} \leq \frac{1}{2} \sum_{i = 1}^{2} a_{i}

, Lemma 1 and (19) give

\begin{matrix} 2 \hat{E} [sup_{0 < s \leq t} \int_{0}^{t} z^{T} (s) γ (z_{s}) d B (t)] \leq 2 b_{2} \hat{E} {[\int_{0}^{t} {| z^{T} (s) γ (z_{s}) |}^{2} d s]}^{\frac{1}{2}} \\ \leq \frac{1}{2} \hat{E} [sup_{0 < s \leq t} {| z (s) |}^{2}] + 4 b_{2}^{2} {| γ (0) |}^{2} T + 4 b_{2}^{2} a_{5} \hat{E} \int_{0}^{t} \int_{- \infty}^{0} {| z (s + σ) |}^{2} μ_{3} (d σ) d s . \end{matrix}

Using Lemma 7, we derive

\begin{matrix} 2 \hat{E} [sup_{0 < s \leq t} \int_{0}^{t} z^{T} (s) γ (z_{s}) d B (s)] \\ \leq \frac{1}{2} \hat{E} [sup_{0 < s \leq t} {| z (s) |}^{2}] + 4 b_{3} {| γ (0) |}^{2} T + \frac{4 b_{3} a_{5}}{2 r} μ_{3}^{(2 r)} \hat{E} {∥ ζ ∥}_{r}^{2} + 4 b_{3} a_{5} \hat{E} \int_{0}^{t} {| z (s) |}^{2} d s, \end{matrix}

(20)

where

b_{3} = b_{2}^{2}

. By substituting the aforementioned inequalities into equation (16), and letting

m_{1} = c + \frac{2}{r} [r + a_{2} μ_{1}^{(2 r)} + b_{1} (a_{5} μ_{3}^{(2 r)} + μ_{2}^{(2 r)}) + 2 b_{3} a_{5} μ_{3}^{(2 r)}] E {∥ ζ ∥}_{r}^{2}

m_{2} = 2 [2 a_{2} - 2 a_{1} + 1 + b_{1} (2 a_{5} - 2 a_{3} + 3) + 4 b_{3} a_{5}]

, we can evaluate the result

\begin{matrix} \hat{E} [sup_{0 \leq s \leq t} {| z (t) |}^{2}] & \leq m_{1} + m_{2} \int_{0}^{t} \hat{E} [sup_{0 \leq s \leq t} {| z (s) |}^{2}] d s, \end{matrix}

(21)

where

c = 2 [{| κ (0) |}^{2} + b_{1} {(| η (0) |}^{2} + {2 | γ (0) |}^{2}) + 4 b_{3} {| γ (0) |}^{2}] T

. By observing that

\hat{E} [sup_{- \infty < s \leq t} {| z (s) |}^{2}] \leq \hat{E} {∥ ζ ∥}_{r}^{2} + \hat{E} [sup_{0 \leq s \leq t} {| z (s) |}^{2}],

it follows

\begin{matrix} \hat{E} [sup_{- \infty < s \leq t} {| z (s) |}^{2}] & \leq \hat{E} {∥ ζ ∥}_{r}^{2} + m_{1} + m_{2} \int_{0}^{t} \hat{E} [sup_{0 \leq s \leq t} {| z (s) |}^{2}] d s \\ \leq \hat{E} {∥ ζ ∥}_{r}^{2} + m_{1} + m_{2} \int_{0}^{t} \hat{E} [sup_{- \infty < s \leq t} {| z (s) |}^{2}] d s . \end{matrix}

Finally, the required result is obtained by using the Grownwall inequality. □

Theorem 4.

Under the conditions of Theorem 3, it follows

lim_{t \to \infty} sup \frac{1}{t} l o g | z (t) | \leq α,

and

α = 2 a_{2} - 2 a_{1} + 1 + b_{1} (2 a_{5} - 2 a_{3} + 3) + 4 b_{3} a_{5}

Proof.

Assuming that

m_{1} = m + \frac{2}{r} [r + a_{2} μ_{1}^{(2 r)} + b_{1} (a_{5} μ_{3}^{(2 r)} + μ_{2}^{(2 r)}) + 2 b_{3} a_{5} μ_{3}^{(2 r)}] E {∥ ζ ∥}_{r}^{2}

and

m_{2} = 2 [2 a_{2} - 2 a_{1} + 1 + b_{1} (2 a_{5} - 2 a_{3} + 3) + 4 b_{3} a_{5}]

then from the inequality (21), we can conclude that:

\hat{E} [sup_{0 \leq s \leq t} {| z (s) |}^{2}] \leq m_{1} e^{m_{2} t},

(22)

where

m = 2 [{| κ (0) |}^{2} + b_{1} {(| η (0) |}^{2} + {2 | γ (0) |}^{2}) + 4 b_{3} {| γ (0) |}^{2}] T

. For each

q = 1, 2, 3, . . .,

from (22) it follows

\hat{E} [sup_{q - 1 \leq t \leq q} {| z (t) |}^{2}] \leq m_{1} e^{m_{2} q} .

By utilizing Lemma 3 for every given

δ > 0

, we obtain:

\begin{matrix} ν \{w : sup_{q - 1 \leq t \leq q} {| z (t) |}^{2} > e^{(m_{2} + δ) q}\} & \leq \frac{\hat{E} [{sup}_{q - 1 \leq t \leq q} {| z (t) |}^{2}]}{e^{(m_{2} + δ) q}} \\ \leq \frac{m_{1} e^{m_{2} q}}{e^{(m_{2} + δ) q}} \\ = m_{1} e^{- δ q} . \end{matrix}

But the Borel-Cantelli lemma gives that for almost every

w \in Ω

there is a random number

q_{0} = q_{0} (w) \in Z

in a manner that when

q \geq q_{0},

then

sup_{q - 1 \leq t \leq q} {| z (t) |}^{2} \leq e^{(m_{2} + δ) q},

which implies

\begin{matrix} lim_{t \to \infty} sup \frac{1}{t} l o g | z (t) | & \leq \frac{m_{2} + δ}{2} \\ = 2 a_{2} - 2 a_{1} + 1 + b_{1} (2 a_{5} - 2 a_{3} + 3) + 4 b_{3} a_{5} + \frac{δ}{2}, \end{matrix}

δ

is arbitrary and letting

α = 2 a_{2} - 2 a_{1} + 1 + b_{1} (2 a_{5} - 2 a_{3} + 3) + 4 b_{3} a_{5}

, we can conclude that

α \geq lim_{t \to \infty} sup \frac{1}{t} l o g | z (t) | .

The proof stands completed. □

The lemma above expresses that the second moment of the Lyapunov exponent, as defined in [13] as

{lim}_{t \to \infty} sup \frac{1}{t} log | z (t) |

, is bounded above by M.

6. Conclusion

Several stochastic functional differential equations (SFDEs) in financial mathematics do not hold the standard Lipschitz assumption such as the Cox-Ingersoll-Ross, Heston and Ait-Sahalia models. In this article some useful monotone type conditions have been introduced. We have proved that any two solutions of SFDEs in the G-framework under distinct initial conditions are convergent. The solutions are mean square bounded. The

L_{G}^{2}

and exponential estimates have been calculated. We anticipate that the findings presented in this article will offer valuable insights into the analysis of equations, even when not under the constraints of standard assumptions. This contribution is poised to have a substantial positive impact on the examination of various unresolved inquiries, including the investigation into the existence, uniqueness, convergence and stability of solutions for backward and forward stochastic dynamic systems driven by G-Brownian motion with conditions of a monotone nature.

Author Contributions

”Conceptualization, F.F.; methodology, R.U.; formal analysis, Q.C.; investigation, F.F. and R.U.; All authors have read and agreed to the published version of the manuscript.”,

Funding

“This research was funded by NUST Pakistan.”

Acknowledgments

“The financial support of NUST Pakistan is acknowledged with thanks.”

Conflicts of Interest

“All authors have no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results”.

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The convergence and boundedness of solutions to SFDEs with the G-framework

Abstract

1. Introduction

2. Basic Notions and Results

3. Some Useful Results

4. Convergence and Mean Square Boundedness

5. The Exponential Estimate

6. Conclusion

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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