Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

Mostapha Abdelouahab Saouli

doi:10.2478/amns.2020.2.00038

Source Title:: Applied Mathematics and Nonlinear Sciences
Volume/Issue:: Volume 5: Issue 2

Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

Mostapha Abdelouahab Saouli^1
,
2

¹ Laboratory of Applied Mathematics, University of Biskra, Algeria
² Department of Mathematics, University of Kasdi Merbah Ouargla, Algeria

Mostapha Abdelouahab Saouli

Corresponding author
Laboratory of Applied Mathematics, University of Biskra, POB 145, Algeria
Department of Mathematics, University of Kasdi Merbah Ouargla, Algeria
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DOI:: https://doi.org/10.2478/amns.2020.2.00038
Published online:: 28 Oct 2020

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Abstract

In this paper we prove the existence of a solution for mean-field reflected backward doubly stochastic differential equations (MF-RBDSDEs) with one continuous barrier and discontinuous generator (left-continuous). By a comparison theorem establish here for MF-RBDSDEs, we provide a minimal or a maximal solution to MF-RBDSDEs.

Keywords:: Reflected Backward doubly stochastic differential equations; Mean-field; Minimal solution; Comparison theorem; Discontinuous generator; 60H10; 60H05

1 Introduction

The theory of nonlinear backward stochastic differential equations (BSDEs in short) have been first introduced by Pardoux and Peng [6] (1990). They proved the existence and uniqueness of the adapted processes (Y,Z), solution of the following equation:

Y_{t} = ξ + \int_{t}^{T} f (s, Y_{s}, Z_{s}) ds - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T,

where the terminal value ξ is square integrable and the coefficient f is uniformly Lipschitz in (y, z), several authors interested in weakening this assumption; In [5] (1997), the authers prove the existence of a solution for one dimensional backward stochastic differential equations where the coefficient is continuous and it has a linear growth, they also obtain the existence of a minimal solution. In [3] (2008) the author prove the existence of the solution to BSDEs whose coefficient may be discontinuous in y and continuous in z.

A new kind of backward stochastic differential equations was introduced by Pardoux and Peng [7] in (1994) which is a class of backward doubly stochastic differential equation (BDSDE for short) of the form:

Y_{t} = ξ + \int_{t}^{T} f (s, Y_{s}, Z_{s}) ds + \int_{t}^{T} g (s, Y_{s}, Z_{s}) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T,

with two different directions of stochastic integrals, i.e., the equation involves both a standard (forward) stochastic integral dW_t and a backward stochastic integral dB_t and ξ is a random variable termed the terminal condition.

After the authors have proved an existence and unique solution when f and g are uniform Lipschitz, several authors interested to weakening this assumption, see [4]. In [9](2005) the authors obtained the existence of the solution of BDSDE under continuous assumption and gave the comparison theorem for one dimensional BDSDE.

On the other hand Bahlali et al [1] (2009) introduced a special class of reflected BDSDEs (RBDSDEs in short) which is a BDSDE but the solution is forced to stay above a lower barrier. In particular, a solution of such equation is a triplet of processes (Y,Z,K) satisfying

Y_{t} = ξ + \int_{t}^{T} f (s, Y_{s}, Z_{s}) ds + \int_{t}^{T} g (s, Y_{s}, Z_{s}) d B_{s} + \int_{t}^{T} {dK}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, t \in [0, T],

(1)

and Y_t ≥ S_t a.s. for any t ∈ [0, T ]. The role of the nondecreasing continuous process (K_t)_{t∈ [0, T]} is to puch upward the process Y in order to keep it above S, it satisfies the skorohod condition

\int_{0}^{T} (Y_{s} - S_{s}) {dK}_{s} = 0 .

In this paper, motivated by the above results and by the result introduced by Xu, R. (2012) [10], we establish the existence of the a minimal solution to the following reflected MF-BDSDE,

\begin{matrix} Y_{t} = ξ + \int_{t}^{T} E^{'} (f (s, ω, ω^{'}, Y_{s}, {(Y_{s})}^{'}, Z_{s}, {(Z_{s})}^{'})) ds + \int_{t}^{T} {dK}_{s} \\ + \int_{t}^{T} E^{'} (g (s, ω, ω^{'}, Y_{s}, {(Y_{s})}^{'}, Z_{s}, {(Z_{s})}^{'})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T, \end{matrix}

(2)

whose coefficient may be discontinuous in y and continuous in z.

In Section 2, we give some preliminaries about MF-BDSDE with one continuous barrier.
In Section 3, under certain assumptions, we obtain the existence for a minimal solution to the Mean-field backward doubly stochastic differential equation with one continuous barrier and discontinuous generator (left-continuous).

2 Framework

Let (Ω, ℱ, P) be a complete probability space. For T > 0, let {W_t, 0 ≤ t ≤ T} and {B_t, 0 ≤ t ≤ T} be two independent standard Brownian motion defined on (Ω, ℱ, P) with values in ℝ^d and ℝ, respectively.

Let

ℱ_{t}^{W} : = σ (W_{s}; 0 \leq s \leq t)

, and

ℱ_{t, T}^{B} : = σ (B_{s} - B_{t}; t \leq s \leq T)

, completed with P-null sets. We put,

ℱ_{t} : = ℱ_{t}^{W} \lor ℱ_{t, T}^{B} .

It should be noted that (ℱ_t) is not an increasing family of sub σ–fields, and hence it is not a filtration.

Let $(\bar{Ω}, \bar{ℱ}, \bar{P}) = (Ω \times Ω, ℱ_{t} \otimes ℱ_{t}, P \otimes P)$ be the (non-completed) product of (Ω ℱ, P) with itself. We denote the filtration of this product space by $\bar{ℱ} = {{\bar{ℱ}}_{t} = ℱ_{t} \otimes ℱ_{t}, 0 \leq t \leq T}$ .

A random variable ξ ∈ L⁰ (Ω, ℱ, P;ℝⁿ) originally defined on Ω is extended canonically to $\bar{Ω} : \overset{´}{ξ} (\overset{´}{ω}, ω) = ξ (\overset{´}{ω}), (\overset{´}{ω}, ω) \in \bar{Ω} = Ω \times Ω .$ .

For every $θ \in L^{1} (\bar{Ω}, \bar{ℱ}, \bar{P})$ , the variable θ (·, ω) : Ω → ℝ belongs to $L^{1} (\bar{Ω}, \bar{ℱ}, \bar{P})$ , P (dω)−a.s,. We denote its expectation by $É (θ (\cdot, ω)) = \int_{Ω} θ (\overset{´}{ω}, ω) P (d \overset{´}{ω})$

Notice that

{\begin{matrix} É (θ) = É (θ (\cdot, ω)) \in L^{1} (Ω, ℱ, P) \\ and \\ \bar{E} (θ) = \int_{\bar{Ω}} θ d \bar{P} = \int_{Ω} É (θ (\cdot, ω)) P (d ω) = E (É (θ)) . \end{matrix}

We consider the following spaces of processus:

Let ℳ² (0, T, ℝ^d) denote the set of d– dimensional, ℱ_t– progressively measurable processes {φ_t;t ∈ [0, T ]}, such that $𝔼 \int_{0}^{T} {| φ_{t} |}^{2} dt < \infty$ .
We denote by 𝒮² (0, T, ℝ^d), the set of ℱ_t– adapted cádlág processes {φ_t; t ∈ [0, T]}, which satisfy 𝔼(sup_{0 ≤ t ≤ T}|φ_t|²) < ∞.
𝒜² set of continuous, increasing, ℱ_t-adapted process K: [0, T] × Ω → [0, +∞) with K₀ = 0 and 𝔼(K_T)² < +∞.
𝕃² set of ℱ_T- measurable random variables ξ :Ω → ℝ with 𝔼 |ξ|² < +∞.

Definition 1

A solution of equation (2) is a triple (Y, Z, K) which belongs to the space 𝒮² (0, T, ℝ^d) × ℳ² (0, T, ℝ^d) × 𝒜² and satisfies (2) such that:

{\begin{matrix} S_{t} \leq Y_{t}, 0 \leq t \leq T, \\ \int_{0}^{T} (Y_{s} - L_{s}) {dK}_{s} = 0 . \end{matrix}

Remark 1

In the case where S = −∞ (i.e., MF-BDSDEs without lower barrier), the process K has no effect i.e., K ≡ 0.

Remark 2

In the setup of system (2) the process S (·) play the role of reflecting barrier.

Remark 3

The state process Y (·) is forced to stay above the lower barrier S (·), thanks to the action of the increasing reflection process K (·).

The coefficient of mean-field Reflected BDSDE is a function. We assume that f and g satisfy the following assumptions on the data (ξ, f, g, S):

(H.1) The terminal value ξ be a given random variable in 𝕃².
(H.2) (S_t)_{t ≥ 0}, is a continuous progressively measurable real valued process satisfying
$\begin{matrix} 𝔼 ({sup}_{0 \leq t \leq T} {(S_{t}^{+})}^{2}) < + \infty, & where & S_{t}^{+} : = max (S_{t}, 0) \end{matrix} .$
(H.3) For t ∈ [0, T], S_T ≤ ξ, ℙ-almost surely.
(H.4)f : Ω × [0, T] × ℝ × ℝ × ℝ^d × ℝ^d → ℝ; g : Ω × [0, T] × ℝ × ℝ × ℝ^d × ℝ^d → ℝ^k be jointly measurable such that for any (y, y^′, z, z^′) ∈ ℝ × ℝ × ℝ^d × ℝ^d,
${\begin{matrix} f (\cdot, ω, y, y^{'}, z, z^{'}) \in ℳ^{2} (0, T, ℝ^{d}), \\ and \\ g (\cdot, ω, y, y^{'}, z, z^{'}) \in ℳ^{2} (0, T, ℝ^{d}) . \end{matrix}$
(H.5) There exist constant C ≥ 0 and a constant $0 \leq α \leq \frac{1}{2}$ such that for every (ω, t) ∈ Ω × [0, T ] and (y, y^′) ∈ ℝ², (z, z^′) ∈ ℝ^d × ℝ^d,
${\begin{matrix} (i) {| f (t, y_{1}, y_{1}^{'}, z_{1}, z_{1}^{'}) - f (t, y_{2}, y_{2}^{'}, z_{2}, z_{2}^{'}) |}^{2} \leq C {{| y_{1} - y_{2} |}^{2} + {| y_{1}^{'} - y_{2}^{'} |}^{2} + {| z_{1} - z_{2} |}^{2} + {| z_{1}^{'} - z_{2}^{'} |}^{2}}, \\ (i i) {| g (t, y_{1}, y_{1}^{'}, z_{1}, z_{1}^{'}) - g (t, y_{2}, y_{2}^{'}, z_{2}, z_{2}^{'}) |}^{2} \leq C {{| y_{1} - y_{2} |}^{2} + {| y_{1}^{'} - y_{2}^{'} |}^{2}} + α {{| z_{1} - z_{2} |}^{2} + {| z_{1}^{'} - z_{2}^{'} |}^{2}} . \end{matrix}$
(H.6) (i) For a.e (t, ω) the mapping (y, y^′, z, z^′) → f (t, y, y^′, z, z^′) is a cotinuous. (ii) There exist constant C ≥ 0 and a constant $0 \leq α \leq \frac{1}{2}$ such that for every (ω, t) ∈ Ω × [0, T] and (y, y^′) ∈ ℝ², (z, z^′) ∈ ℝ^d × ℝ^d,
${\begin{matrix} | f (t, y, y^{'}, z, ź) | \leq C (1 + | y | + | y^{'} | + | z | + | ź |), \\ g satisfies (H . 2) (ii) . \end{matrix}$

We recall the following existence results.

Proposition 1

[2] (2014). Under the assumptions (H.1)–(H.5) the reflected BDSDE (2) has a unique solution (Y, Z, K) ∈ 𝒮² (0, T, ℝ^d) × ℳ² (0, T, ℝ^d) × 𝒜².

3 Existence result

In this section we are interested in weakening the conditions on f. We assume that f and g satisfy the following assumptions:

(H.7) Linear growth: There esists a nonnegative process f_t ∈ 𝕄² (0, T, ℝ^d) such that
$\forall (t, y, y^{'}, z) \in [0, T] \times ℝ^{2} \times ℝ^{d}, | f (t, y, y^{'}, z) | \leq f_{t} (ω) + C (| y | + | y^{'} | + | z |) .$
(H.8)f (t, ·, y^′, z): ℝ → ℝ is a left continuous and f (t, y, ·,·) is a cotinuous.
(H.9) There exists a continuous fonction π : [0, T ] × (ℝ)² × ℝ^d satisfying for y₁ ≥ y₂, $(y_{1}^{'}, y_{2}^{'}) \in {(ℝ)}^{2}$ , (z₁, z₂) ∈ (ℝ^d)²
${\begin{matrix} | π (t, y, y^{'}, z) | \leq C (| y | + | y^{'} | + | z |), \\ f (t, ω, y_{1}, y_{1}^{'}, z_{1}) - f (t, ω, y_{2}, y_{2}^{'}, z_{2}) \geq π (t, y_{1} - y_{2}, y_{1}^{'} - y_{2}^{'}, z_{1} - z_{2}) . \end{matrix}$
(H.10) Monotonicity in y^′: ∀ (y, y^′, z), f (t, y, y^′, z) is increasing in y^′.
(H.11)g satisfies (H.5)(ii) and g(t, 0, 0, 0) ≡ 0.

Hence, we only consider the following type of Mean-field reflected BDSDE:

\begin{matrix} Y_{t} = ξ + \int_{t}^{T} E^{'} (f (s, ω, ω^{'}, Y_{s}, {({\tilde{Y}}_{s})}^{'}, Z_{s})) ds + \int_{t}^{T} {dK}_{s} \\ + \int_{t}^{T} E^{'} (g (s, ω, ω^{'}, Y_{s}, {({\tilde{Y}}_{s})}^{'}, Z_{s})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T . \end{matrix}

(3)

Proposition 2

[2] (2014). Under the assumption (H.1)–(H.4) and (H.6), and for any random variable ξ ∈ 𝕃²the mean-field RBDSDE (3) a has an adapted solution (Y, Z, K) ∈ 𝒮² (0, T, ℝ^d) × ℳ² (0, T, ℝ^d) × 𝒜², which is a minimal one, in the sense that, if (Y*, Z*, K*) is any other solution we Y ≤ Y*, P – a.s.

Now we prove a technical Lemma before we introduce the main theorem.

Lemma 3

Let π (t, y, y^′, z) satisfies (H.9), g satisfies (H.11) and h belongs in ℳ² (0, T, ℝ^d). For a continuous function of finite variation $\tilde{K}$ belong in 𝒜²we consider the processes

(\tilde{Y}, \tilde{Z}) \in 𝒮^{2} (0, T, ℝ) \times ℳ^{2} (0, T, ℝ^{d})

such that:

{\begin{matrix} (i) {\tilde{Y}}_{t} = ξ + \int_{t}^{T} E^{'} (π (s, ω, ω^{'}, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s}) + h (s)) ds + \int_{t}^{T} d {\tilde{K}}_{s} \\ + \int_{t}^{T} E^{'} (g (s, ω, ω^{'}, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} {\tilde{Z}}_{s} {dW}_{s}, 0 \leq t \leq T, \\ (i i) \int_{0}^{T} {\tilde{Y}}_{s}^{-} d {\tilde{K}}_{s} \geq 0 . \end{matrix}

(4)

Then we have

(i)The MF-RBDSDE (4) has a least one solution $(\tilde{Y}, \tilde{Z}, \tilde{K}) \in 𝒮^{2} (0, T, ℝ^{d}) \times ℳ^{2} (0, T, ℝ^{d}) \times 𝒜^{2}$
(ii)if h(t) ≥ 0 and ξ ≥ 0, we have ${\tilde{Y}}_{t} \geq 0$ , dℙ × dt – a.s.

Proof

(i) See [2], (2014). (ii) Applying Tanaka's formula to

{| {\tilde{Y}}_{t}^{-} |}^{2}

, we have

\begin{matrix} 𝔼 {| {\tilde{Y}}_{t}^{-} |}^{2} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds & = 𝔼 {| ξ^{-} |}^{2} - 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} E^{'} (π (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s}) + h (s)) ds \\ - 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} d {\tilde{K}}_{s} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| | E^{'} (g (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) | |}^{2} ds . \end{matrix}

Since

- 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} d {\tilde{K}}_{s} \leq 0

, h(s) ≥ 0 and ξ ≥ 0, we get

\begin{matrix} 𝔼 {| {\tilde{Y}}_{t}^{-} |}^{2} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds \leq & - 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} E^{'} (π (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) ds \\ + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| | E^{'} (g (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) | |}^{2} ds \end{matrix}

By (H.9), we get

| π (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s}) | \leq C (| {\tilde{Y}}_{s} | + | {({\tilde{Y}}_{s})}^{'} | + | {\tilde{Z}}_{s} |)

and by assumption (H.11) for g, we have

\begin{matrix} 𝔼 {| {\tilde{Y}}_{t}^{-} |}^{2} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds \\ \leq (4 C^{2} + \frac{C^{2}}{β} + 2 C) 𝔼 \int_{t}^{T} {| {\tilde{Y}}_{s}^{-} |}^{2} ds + (α + β) 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds . \end{matrix}

Therefore, choosing 0 ≤ β ≤ 1 – α and using Gronwall inequality, we have ${\tilde{Y}}_{t}^{-} = 0$ , ℙ – a.s., ∀t ∈ [0, T], which implies that ${\tilde{Y}}_{t} \geq 0$ ℙ – a.s., ∀t ∈ [0, T].

Before we prove the main result, we construct a sequence of MF-RBDSDEs as follows:

{\begin{matrix} {\bar{Y}}_{t}^{0} = ξ + \int_{t}^{T} E^{'} (- C (| {\bar{Y}}_{s}^{0} | + {({\bar{Y}}_{s}^{0})}^{'} + | {\bar{Z}}_{s}^{0} |) - f_{s}) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{0} + {({\bar{Y}}_{s}^{0})}^{'} + {\bar{Z}}_{s}^{0})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d {\bar{K}}_{s}^{0} - \int_{t}^{T} {\bar{Z}}_{s}^{0} {dW}_{s}, 0 \leq t \leq T, \\ (i i) {\bar{Y}}_{t}^{0} \geq S_{t}, \\ (i i i) \int_{0}^{T} ({\bar{Y}}_{s}^{0} - S_{s}) d {\bar{K}}_{s}^{0} = 0 . \end{matrix}

(5)

{\begin{matrix} (i) {\bar{Y}}_{t}^{n} = ξ + \int_{t}^{T} E^{'} (f (s, {\bar{Y}}_{s}^{n - 1}, {({\bar{Y}}_{s}^{n - 1})}^{'}, {\bar{Z}}_{s}^{n - 1}) + π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{t}^{n})) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d {\bar{K}}_{s}^{n} - \int_{t}^{T} {\bar{Z}}_{s}^{n} {dW}_{s}, 0 \leq t \leq T, \\ (i i) {\bar{Y}}_{t}^{n} \geq S_{t}, \\ (i i i) \int_{0}^{T} ({\bar{Y}}_{s}^{n} - S_{s}) d {\tilde{k}}_{s}^{n} = 0 . \end{matrix}

(6)

{\begin{matrix} (i) Y_{t}^{0} = ξ + \int_{t}^{T} E^{'} (C (| Y_{s}^{0} | + {(Y_{s}^{0})}^{'} + | Z_{s}^{0} |) + f_{s}) ds + \int_{t}^{T} d K_{s}^{0} \\ + \int_{t}^{T} E^{'} (g (s, Y_{s}^{0}, {(Y_{s}^{0})}^{'} + Z_{s}^{0})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s}^{0} {dW}_{s}, 0 \leq t \leq T, \\ (i i) Y_{t}^{0} \geq S_{t}, \\ (i i i) \int_{0}^{T} (Y_{s}^{0} - S_{s}) d K_{s}^{0} = 0 . \end{matrix}

(7)

For these solutions above, we get some properties as follows:

Lemma 4

Under the assumptions (H.1) – (H.4) and (H.7) – (H.11), we have for any n ≥ 1 and t ∈ [0, T]

{\bar{Y}}_{t}^{0} \leq {\bar{Y}}_{t}^{n} \leq {\bar{Y}}_{t}^{n + 1} \leq Y_{t}^{0} .

Proof

We will prove

{\bar{Y}}_{t}^{0} \leq {\bar{Y}}_{t}^{n}

at first. By Eqs. (5), and (6), we have

\begin{matrix} {\bar{Y}}_{t}^{1} - {\bar{Y}}_{t}^{0} = & \int_{t}^{T} E^{'} (π (s, δ {\bar{Y}}_{s}^{1}, δ {({\bar{Y}}_{s}^{1})}^{'}, δ {\bar{Z}}_{s}^{1}) + Λ_{s}^{1}) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{1}, {({\bar{Y}}_{s}^{1})}^{'}, {\bar{Z}}_{s}^{1}) - g (s, {\bar{Y}}_{s}^{0} + {({\bar{Y}}_{s}^{0})}^{'} + {\bar{Z}}_{s}^{0})) d {\overset{\leftarrow}{B}}_{s} \\ + \int_{t}^{T} (d {\bar{K}}_{s}^{1} - d {\bar{K}}_{s}^{0}) - \int_{t}^{T} δ {\bar{Z}}_{s}^{1} {dW}_{s}, \end{matrix}

where

Λ_{s}^{1} = f (s, {\bar{Y}}_{s}^{0}, {({\bar{Y}}_{s}^{0})}^{'}, {\bar{Z}}_{s}^{0}) + C (| {\bar{Y}}_{s}^{0} | + {({\bar{Y}}_{s}^{0})}^{'} + | {\bar{Z}}_{s}^{0} |) + f_{s}

. By hypothesis (H.7) we have

Λ_{s}^{1} \geq 0

, because

({\bar{Y}}_{t}^{0}, {\bar{Z}}_{t}^{0})

is the solution of Eq. (5), we get

Λ_{s}^{1} \in ℳ^{2} (0, T, ℝ^{d})

. Therefore, from Lemma 3 we get

{\bar{Y}}_{t}^{1} \geq {\bar{Y}}_{t}^{0}

. Now we want to prove

{\bar{Y}}_{t}^{n} \leq {\bar{Y}}_{t}^{n + 1}

, for any n ≥ 0. We set

{\begin{matrix} δ ρ_{s}^{n + 1} = ρ_{s}^{n + 1} - ρ_{s}^{n}, \\ Δ ψ^{n + 1} (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1}) \\ = ψ (s, δ {\bar{Y}}_{s}^{n + 1} + {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n + 1})}^{'} + {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n + 1} + {\bar{Z}}_{s}^{n}) - ψ (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) . \end{matrix}

Using Eq. (6), we have

\begin{matrix} δ {\bar{Y}}_{t}^{n + 1} & = \int_{t}^{T} E^{'} (π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1}) + θ_{s}^{n + 1}) ds - \int_{t}^{T} δ {\bar{Z}}_{s}^{n + 1} {dW}_{s} \\ + \int_{t}^{T} E^{'} (Δ g^{n + 1} (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d (δ {\bar{K}}_{s}^{n + 1}), \end{matrix}

where

θ_{s}^{n + 1} = Δ f^{n} (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n}) - π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n})

and

θ_{s}^{0} = Λ_{s}^{1}

, ∀n ≥ 0. According to it definition, one cas show that

θ_{s}^{0}

and Δgⁿ⁺¹, ∀n ≥ 0 satisfy all assumption of Lemma 3. Moreover, since

{\bar{K}}_{t}^{n}

is a continuous and increasing process, for all n ≥ 0,

δ {\bar{K}}_{s}^{n + 1}

is a contiuous process of finite variation and, using the same argument as one appear in [2], on can show that

\begin{matrix} \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d (δ {\bar{K}}_{s}^{n + 1}) & = \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d {\bar{K}}_{s}^{n + 1} - \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d {\bar{K}}_{s}^{n} \\ = \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d {\bar{K}}_{s}^{n + 1} \geq 0, \end{matrix}

by Lemma 3, we deduce that

δ {\bar{Y}}_{t}^{n + 1} \geq 0

, i.e.

{\bar{Y}}_{t}^{n + 1} \geq {\bar{Y}}_{t}^{n}

∀t ∈ [0, T], we have

{\bar{Y}}_{t}^{n + 1} \geq {\bar{Y}}_{t}^{n} \geq {\bar{Y}}_{t}^{0} .

Now we shall prove that

{\bar{Y}}_{t}^{n + 1} \leq Y_{t}^{0}

∀n ≥ 0, by Eqs.(3) and (7)

\begin{matrix} Y_{t}^{0} - {\bar{Y}}_{t}^{n + 1} & = \int_{t}^{T} E^{'} (- C (| Y_{s}^{0} - {\bar{Y}}_{s}^{+ 1} | + | {(Y_{s}^{0})}^{'} - {({\bar{Y}}_{s}^{n + 1})}^{'} | + | Z_{s}^{0} - {\bar{Z}}_{s}^{n + 1} |) + Λ_{s}^{n + 1}) ds \\ + \int_{t}^{T} E^{'} (g (s, Y_{s}^{0}, {(Y_{s}^{0})}^{'} + Z_{s}^{0}) - g (s, s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})) d {\overset{\leftarrow}{B}}_{s} \\ + \int_{t}^{T} (d K_{s}^{0} - d {\bar{K}}_{s}^{n + 1}) + \int_{t}^{T} (Z_{s}^{0} - {\bar{Z}}_{s}^{n + 1}) {dW}_{s}, \end{matrix}

where

\begin{matrix} Λ_{s}^{n + 1} & = C (| Y_{s}^{0} - {\bar{Y}}_{s}^{+ 1} | + | {(Y_{s}^{0})}^{'} - {({\bar{Y}}_{s}^{n + 1})}^{'} | + | Z_{s}^{0} - {\bar{Z}}_{s}^{n + 1} | + | Y_{s}^{0} | + {(Y_{s}^{0})}^{'} + | Z_{s}^{0} |) \\ + f_{s} - f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1}) . \end{matrix}

By Lemma 3, we deduce that

Y_{t}^{0} - {\bar{Y}}_{t}^{n + 1} \geq 0

, i.e.

Y_{t}^{0} \geq {\bar{Y}}_{t}^{n + 1}

, for all t ∈ [0, T]. Thus we have for all n ≥ 0

Y_{t}^{0} \geq {\bar{Y}}_{t}^{n + 1} \geq {\bar{Y}}_{t}^{n} \geq {\bar{Y}}_{t}^{0}, d \bar{ℙ} \times d t - a . s . \forall t \in [0, T] .

The proof of Lemma 4 is complete.

Theorem 5

Let ξ ∈ 𝕃² (ℱ_T, ℝ) and t ∈ [0, T]. Under assumption (H.1) – (H.4) and (H.7) – (H.11), the reflected MF-BDSDEs (2) has a minimal solution

{(Y_{t}, Z_{t}, K_{t})}_{0 \leq t \leq T} \in 𝒮^{2} (0, T, ℝ) \times ℳ^{2} (0, T, ℝ^{d}) \times 𝒜^{2} .

Proof

From Lemma 4, we know

{({\bar{Y}}_{t}^{n})}_{n \geq 0}

is increasing and bounded in ℳ² (0, T, ℝ^d). Since

| {\tilde{Y}}_{t}^{n} | \leq max ({\tilde{Y}}_{t}^{0}, Y_{t}^{0}) \leq | {\tilde{Y}}_{t}^{0} | + | Y_{t}^{0} |

for all t ∈ [0, T], we have

sup_{n} 𝔼 (sup_{0 \leq t \leq T} {| {\bar{Y}}_{t}^{n} |}^{2}) \leq 𝔼 (sup_{0 \leq t \leq T} {| {\bar{Y}}_{t}^{0} |}^{2}) + 𝔼 (sup_{0 \leq t \leq T} {| Y_{t}^{0} |}^{2}) < \infty,

then according to the Lebesgue's dominated convergence theorem, we deduce that

{({\bar{Y}}_{t}^{n})}_{n \geq 0}

converges in 𝒮² (0, T, ℝ). We denote by

\bar{Y}

the limit of

{({\bar{Y}}_{t}^{n})}_{n \geq 0}

On the other hand from Eq. (6), we deduce that

\begin{matrix} {\bar{Y}}_{0}^{n + 1} & = {\bar{Y}}_{T}^{n + 1} + \int_{0}^{T} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d {\bar{K}}_{s}^{n + 1} - \int_{t}^{T} {\bar{Z}}_{s}^{n + 1} {dW}_{s} . \end{matrix}

Applying Itô's formula, we obtain

\begin{matrix} 𝔼 {| {\bar{Y}}_{0}^{n + 1} |}^{2} + 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq & 𝔼 {| {\bar{Y}}_{T}^{n + 1} |}^{2} + 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} d {\bar{K}}_{s}^{n + 1} + 𝔼 \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) | |}^{2} ds \\ + 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds . \end{matrix}

From assumption (H.7) and Yöung's inequality, we get

\begin{matrix} 2 𝔼 \int_{0}^{T} {\bar{Y}}_{t}^{n + 1} E^{'} [f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})] ds \\ \leq 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} [f_{s} (ω) + C (1 + | {\bar{Y}}_{s}^{n} | + | {({\bar{Y}}_{s}^{n})}^{'} | + | {\bar{Z}}_{s}^{n} |)] ds, \\ \leq 4 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + (4 C^{2} + 1) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds \\ + 16 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds, \\ = (24 C^{2} + 1) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds, \end{matrix}

(8)

and from hypothesis(H.9) we get

\begin{matrix} 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ \leq 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (C (| δ {\bar{Y}}_{s}^{n + 1} | + | {(δ {\bar{Y}}_{s}^{n + 1})}^{'} | + | δ {\bar{Z}}_{s}^{n + 1} |)) ds, \\ \leq 4 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 4 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds + 8 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds \\ + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + 16 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds, \\ = (4 C + 28 C^{2}) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds . \end{matrix}

(9)

Using the two inequalities (8) and (9), we obtain

\begin{matrix} 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ \leq (52 C^{2} + 4 C + 1) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 2 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds \\ + \frac{1}{8} 𝔼 \int_{0}^{T} ({| {\bar{Z}}_{s}^{n + 1} |}^{2} + {| {\bar{Z}}_{s}^{n} |}^{2}) ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds . \end{matrix}

Then, we get

\begin{matrix} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq & 𝔼 {| ξ |}^{2} + 𝔼 \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) | |}^{2} ds \\ + C + 2 𝔼 \int_{0}^{T} 〈 {\bar{Y}}_{s}^{n + 1}, d {\bar{K}}_{s}^{n + 1} 〉 + \frac{1}{8} 𝔼 \int_{0}^{T} ({| {\bar{Z}}_{s}^{n + 1} |}^{2} + {| {\bar{Z}}_{s}^{n} |}^{2}) ds, \end{matrix}

where

C = 2 𝔼 \int_{0}^{T} | {\bar{Y}}_{s}^{n} | ds + (52 C + 4 C + 1) \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds

Applying hypothesis (H. 11), we have

\begin{matrix} 𝔼 \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) | |}^{2} ds \\ \leq 4 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 2 α 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + 2 𝔼 \int_{0}^{T} {| | g (s, 0, 0, 0) | |}^{2} ds . \end{matrix}

Using Yöung's inequality, we obtain

2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} d {\bar{K}}_{s}^{n + 1} \leq 2 𝔼 \int_{0}^{T} S_{s} d {\bar{K}}_{s}^{n + 1} \leq \frac{1}{θ} 𝔼 (sup_{0 \leq t \leq T} {| S_{t} |}^{2}) + θ 𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} .

Therefore, there exists a constant C^θ depending on α, ξ, C and θ, we derive

𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq C^{θ} + (\frac{1}{8} + 2 α) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + θ 𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2},

(10)

where

C^{θ} = C + 𝔼 {| ξ |}^{2} + 4 C \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + \frac{1}{θ} 𝔼 ({sup}_{0 \leq t \leq T} {| S_{t} |}^{2}) + 2 𝔼 \int_{0}^{T} {| | g (s, 0, 0, 0) | |}^{2} ds

Chossing α such that

0 < \frac{1}{8} + 2 α < 1

, we obtain

𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq C^{θ} + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + θ 𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} .

Moreover, since

\begin{matrix} {\bar{K}}_{T}^{n + 1} = & {\bar{Y}}_{0}^{n + 1} - ξ - \int_{0}^{T} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ - \int_{0}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} {\bar{Z}}_{s}^{n + 1} {dW}_{s}, \end{matrix}

by the Hölder inequality and B-D-G inequality, 𝔼 (X)² ≤ 𝔼 (X²) and the properties on f, g, π that there exists two constants C₁ and C₂ depending on α, ξ and C of n such that

𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} \leq C_{1} + C_{2} (𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} + {| {\bar{Z}}_{s}^{n} |}^{2} ds) .

Return to inequality (10), we get

𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq C^{θ} + θ C_{1} + (\frac{1}{8} + θ C_{2}) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + θ C_{2} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds,

we chosing θ, such that θC₂ ≤ 1, we have

\begin{matrix} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds & \leq C^{θ} + θ C_{1} + (\frac{1}{8} + θ C_{2}) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds \\ \leq (C^{θ} + θ C_{1}) \sum_{i = 0}^{i = n - 1} {(\frac{1}{8} + θ C_{2})}^{i} + {(\frac{1}{8} + θ C_{2})}^{n} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{0} |}^{2} ds . \end{matrix}

Now chossing θ such that

\frac{1}{8} + θ C_{2} < 1

and notting

𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{0} |}^{2} ds < \infty

. Obtain

sup_{n \in ℕ} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds < \infty,

consequently, we deduce

𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} < \infty .

Now we shall prove that

({\bar{Z}}^{n}, {\bar{K}}^{n})

is a Cauchy sequence in ℳ² (0, T, ℝ^d) × 𝒜².

Applying Itô's formula to

{| δ {\tilde{Y}}_{s}^{n, m} |}^{2} = {| {\tilde{Y}}_{s}^{n} - {\tilde{Y}}_{s}^{m} |}^{2}

, we have

\begin{matrix} 𝔼 {| {\bar{Y}}_{t}^{n} - {\bar{Y}}_{t}^{m} |}^{2} + 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds = & 2 𝔼 \int_{0}^{T} ({\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m}) E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) ds + 2 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} (d {\bar{K}}_{s}^{n} - d {\bar{K}}_{s}^{m}) \\ + \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) - g (s, {\bar{Y}}_{s}^{m}, {({\bar{Y}}_{s}^{m})}^{'}, {\bar{Z}}_{s}^{m})) | |}^{2} ds . \end{matrix}

where

Γ_{s}^{n} = f (s, {\bar{Y}}_{s}^{n - 1}, {({\bar{Y}}_{s}^{n - 1})}^{'}, {\bar{Z}}_{s}^{n - 1}) + π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n})

. Since

\int_{0}^{T} {\bar{Y}}_{s}^{n + 1} (d {\bar{K}}_{s}^{n} - d {\bar{K}}_{s}^{m}) \leq 0

, we obtain

\begin{matrix} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds & \leq 2 𝔼 \int_{0}^{T} ({\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m}) E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) ds \\ + \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) - g (s, {\bar{Y}}_{s}^{m}, {({\bar{Y}}_{s}^{m})}^{'}, {\bar{Z}}_{s}^{m})) | |}^{2} ds . \end{matrix}

By the Hölder inequality and hypothesis (H.11), we deduce that

\begin{matrix} (1 - α) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds \leq 2 𝔼 {(\int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds)}^{\frac{1}{2}} 𝔼 {(\int_{0}^{T} {| E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) |}^{2} ds)}^{\frac{1}{2}} \\ + 2 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds . \end{matrix}

The boundedness of the sequence

({\bar{Y}}^{n}, {\bar{Z}}^{n}, {\bar{K}}^{n})

, we deduce that the

Λ = {sup}_{n \in ℕ} [𝔼 \int_{0}^{T} E^{'} {| Γ_{s}^{n} |}^{2} ds] < \infty

, this yields that

(1 - α) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds \leq 4 Λ 𝔼 {(\int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds)}^{\frac{1}{2}} + 2 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds,

which yields that

{({\bar{Z}}^{n})}_{n \geq 0}

is a Cauchy sequence in ℳ² (0, T, ℝ^d). Then there exists Z ∈ ℳ² (ℝ^d) such that

𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - Z_{s} |}^{2} ds \to 0 as n \to \infty .

On the other hand, by Burkhölder-Davis-Gundy inequality, we get

{\begin{matrix} 𝔼 sup_{0 \leq t \leq T} {| \int_{t}^{T} {\bar{Z}}_{s}^{n} {dW}_{s} - \int_{t}^{T} Z_{s} {dW}_{s} |}^{2} \leq 𝔼 \int_{t}^{T} {| {\bar{Z}}_{s}^{n} - Z_{s} |}^{2} ds \to 0, as n \to \infty, \\ 𝔼 sup_{0 \leq t \leq T} {| \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})) - E^{'} (g (s, Y_{s}, {(Y_{s})}^{'}, Z_{s})) |}^{2} \\ \leq 2 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} - Y_{s} |}^{2} ds + α 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - Z_{s} |}^{2} ds \to 0, as n \to \infty . \end{matrix}

Therefore, from the properieties of f and π

Γ_{s}^{n} = f (s, {\bar{Y}}_{s}^{n - 1}, {({\bar{Y}}_{s}^{n - 1})}^{'}, {\bar{Z}}_{s}^{n - 1}) + π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n}) \to f (s, Y_{s}, {(Y_{s})}^{'}, Z_{s}),

\bar{ℙ}

– a.s., for all t ∈ [0, T] as n → ∞. Then follows by Lebesgue's dominated convergence theorem that

𝔼 \int_{0}^{T} {| E^{'} (Γ_{s}^{n} - f (s, Y_{s}, {(Y_{s})}^{'}, Z_{s})) |}^{2} ds \to 0, n \to \infty

Since

({\tilde{Y}}_{s}, {\tilde{Z}}_{s}, Γ_{s}^{n})

converges in 𝒮² (0, T, ℝ) × ℳ² (0, T, ℝ^d) × ℳ² (0, T, ℝ²) and

\begin{matrix} 𝔼 (sup_{0 \leq t \leq T} {| {\bar{K}}_{t}^{n} - {\bar{K}}_{t}^{m} |}^{2}) \leq & 𝔼 {| {\bar{Y}}_{0}^{n} - {\bar{Y}}_{0}^{m} |}^{2} + 𝔼 sup_{0 \leq t \leq T} {| {\bar{Y}}_{t}^{n} - {\bar{Y}}_{t}^{m} |}^{2} + 𝔼 \int_{0}^{T} {| E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) |}^{2} ds \\ + 𝔼 sup_{0 \leq t \leq T} {| \int_{0}^{t} E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) - g (s, {\bar{Y}}_{s}^{m}, {({\bar{Y}}_{s}^{m})}^{'}, {\bar{Z}}_{s}^{m})) d \overset{\leftarrow}{B_{s}} |}^{2} \\ + 𝔼 sup_{0 \leq t \leq T} {| \int_{0}^{t} ({\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m}) {dW}_{s} |}^{2} \end{matrix}

for any n ≥ 0, we deduce from Bukhölder-Davis-Gundy inequality that

𝔼 (sup_{0 \leq t \leq T} {| {\bar{K}}_{t}^{n} - {\bar{K}}_{t}^{m} |}^{2}) \to 0,

as n → ∞. Consequently, there exists a ℱ_t–mesurable process K wich value in ℝ such that

𝔼 (sup_{0 \leq t \leq T} {| {\bar{K}}_{t}^{n} - K_{t} |}^{2}) \to 0,

as n → ∞. Obviously, K₀ = 0 and {K_t; 0 ≤ t ≤ T} is a increasing and continuous process. From Eq. (6), we have for all n ≥ 0,

{\bar{Y}}_{t}^{n} \geq S_{t}

, ∀t ∈ [0, T], then Y_t ≥ S_t, ∀t ∈ [0, T]. On the other hand, from the result of Saisho [8] (in 1987, p. 465), we have

\int_{0}^{T} ({\bar{Y}}_{s}^{n} - S_{s}) d {\bar{K}}_{s}^{n} \to \int_{0}^{T} (Y_{s} - S_{s}) {dK}_{s},

$\bar{ℙ}$ – a.s., as n → ∞. Using the identite $\int_{0}^{T} ({\bar{Y}}_{s}^{n} - S_{s}) d {\bar{K}}_{s}^{n} = 0$ , for all n ≥ 0 we conclude that $\int_{0}^{T} (Y_{s} - S_{s}) {dK}_{s} \geq 0$ . Letting n → +∞ in Eq. (3), we prove that (Y, Z, K) is solution to Eq. (3). Let (Y*, Z*, K*) be any solution of the MF-RBDSDE (3), we have ${\bar{Y}}_{\cdot}^{n} \leq Y_{\cdot}^{*}$ , for all n ≥ 0 and therefore, Y. ≤ Y* i.e., Y is the minimal solution.

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Received:: 2019-05-14

Accepted:: 2020-03-30

Published Online:: 2020-10-28

Citation Information:: Applied Mathematics and Nonlinear Sciences, Volume 5, Issue 2, Pages 205–216, eISSN 2444-8656, DOI: https://doi.org/10.2478/amns.2020.2.00038.

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Applied Mathematics and Nonlinear Sciences

Online ISSN:: 2444-8656

First Published:: 01 Jan 2016

Language:: English

Publisher:: Sciendo

Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

Abstract

1 Introduction

2 Framework

3 Existence result

References

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Journal + Issues

Applied Mathematics and Nonlinear Sciences

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