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

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: $Y_{t} = ξ + \int_{t}^{T} f (s, Y_{s}, Z_{s}) ds - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T,$ Y_{t}=\xi +\int_{t}^{T}f\left(s,Y_{s},Z_{s}\right) ds-\int_{t}^{T}Z_{s}dW_{s},\qquad 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: $Y_{t} = ξ + \int_{t}^{T} f (s, Y_{s}, Z_{s}) ds + \int_{t}^{T} g (s, Y_{s}, Z_{s}) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T,$ Y_{t}=\xi +\int_{t}^{T}f\left(s,Y_{s},Z_{s}\right) ds+\int_{t}^{T}g\left(s,Y_{s},Z_{s}\right) d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\qquad 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 (1) $Y_{t} = ξ + \int_{t}^{T} f (s, Y_{s}, Z_{s}) ds + \int_{t}^{T} g (s, Y_{s}, Z_{s}) d B_{s} + \int_{t}^{T} {dK}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, t \in [0, T],$ Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})dB_{s}+\int_{t}^{T}dK_{s}-\int_{t}^{T}Z_{s}dW_{s},\quad t\in \left[ 0,T\right], 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 $\int_{0}^{T} (Y_{s} - S_{s}) {dK}_{s} = 0 .$ \int_{0}^{T}\left(Y_{s}-S_{s}\right) 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, (2) $\begin{array}{l} Y_{t} = ξ + \int_{t}^{T} E^{'} (f (s, ω, ω^{'}, Y_{s}, {(Y_{s})}^{'}, Z_{s}, {(Z_{s})}^{'})) ds + \int_{t}^{T} {dK}_{s} \\ + \int_{t}^{T} E^{'} (g (s, ω, ω^{'}, Y_{s}, {(Y_{s})}^{'}, Z_{s}, {(Z_{s})}^{'})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T, \end{array}$ \matrix{{Y_{t}=\xi +\int_{t}^{T}E^{^{\prime}}\left(f(s,\omega,\omega^{^{\prime}},Y_{s},\left(Y_{s}\right)^{^{\prime}},Z_{s},\left(Z_{s}\right)^{^{\prime}})\right) ds+\int_{t}^{T}dK_{s}} \hfill &\cr +\int_{t}^{T}E^{^{\prime}}\left(g(s,\omega,\omega^{^{\prime}},Y_{s},\left(Y_{s}\right)^{^{\prime}},Z_{s},\left(Z_{s}\right)^{^{\prime}})\right) d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\quad0\leq t\leq T,\hfill &\cr} 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).

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 $ℱ_{t}^{W} : = σ (W_{s}; 0 \leq s \leq t)$ \mathcal{F}_{t}^{W}:=\sigma (W_{s};0\leq s\leq t) , and $ℱ_{t, T}^{B} : = σ (B_{s} - B_{t}; t \leq s \leq T)$ \mathcal{F}_{t,T}^{B}:=\sigma (B_{s}-B_{t};t\leq s\leq T), , completed with P-null sets. We put, $ℱ_{t} : = ℱ_{t}^{W} \lor ℱ_{t, T}^{B} .$ \mathcal{F}_{t}:=\mathcal{F}_{t}^{W}\vee \mathcal{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 $(\bar{Ω}, \bar{ℱ}, \bar{P}) = (Ω \times Ω, ℱ_{t} \otimes ℱ_{t}, P \otimes P)$ \left(\bar{\Omega},\mathcal{\bar{F}},\bar{P}\right) =\left(\Omega \times \Omega,\mathcal{F}_{t}\mathcal{\otimes F}_{t},P\mathcal{\otimes}P\right) be the (non-completed) product of (Ω ℱ, P) with itself. We denote the filtration of this product space by $\bar{ℱ} = {{\bar{ℱ}}_{t} = ℱ_{t} \otimes ℱ_{t}, 0 \leq t \leq T}$ \mathcal{\bar{F}=}\left\{\mathcal{\bar{F}}_{t}=\mathcal{F}_{t}\mathcal{\otimes F}_{t},0\leq t\leq T\right\} .

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

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

Notice that ${\begin{array}{l} É (θ) = É (θ (\cdot, ω)) \in L^{1} (Ω, ℱ, P) \\ and \\ \bar{E} (θ) = \int_{\bar{Ω}} θ d \bar{P} = \int_{Ω} É (θ (\cdot, ω)) P (d ω) = E (É (θ)) . \end{array}$ \left\{{\matrix{\acute{E}\left(\theta \right) =\acute{E}\left(\theta \left(\cdot,\omega\right) \right) \in L^{1}\left(\Omega,\mathcal{F},P\right) \hfill \\ \text{and} \hfill \\ \bar{E}\left(\theta \right) =\int_{\bar{\Omega}}\theta d\bar{P}=\int_{\Omega}\acute{E}\left(\theta \left(\cdot,\omega \right) \right)P\left(d\omega \right) =E\left(\acute{E}\left(\theta \right) \right).}}\right.

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 $𝔼 \int_{0}^{T} {| φ_{t} |}^{2} dt < \infty$ \mathbb{E}\int_{0}^{T}\left\vert \varphi_{t}\right\vert^{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: ${\begin{matrix} S_{t} \leq Y_{t}, 0 \leq t \leq T, \\ \int_{0}^{T} (Y_{s} - L_{s}) {dK}_{s} = 0 . \end{matrix}$ \left\{\matrix{S_{t}\leq Y_{t},\text{}0\leq t\leq T, \\ \int_{0}^{T}\left(Y_{s}-L_{s}\right) dK_{s}=0.}\right.

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 $\begin{array}{l} 𝔼 ({sup}_{0 \leq t \leq T} {(S_{t}^{+})}^{2}) < + \infty, & where & S_{t}^{+} : = max (S_{t}, 0) \end{array} .$ \mathbb{E}\left({\rm {sup}}_{0\leq t\leq T}\left(S_{t}^{+}\right)^{2}\right)<+\infty, \qquad {\rm where} \qquad S_{t}^{+}:=\max \left(S_{t},0\right).

(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, ${\begin{matrix} f (\cdot, ω, y, y^{'}, z, z^{'}) \in ℳ^{2} (0, T, ℝ^{d}), \\ and \\ g (\cdot, ω, y, y^{'}, z, z^{'}) \in ℳ^{2} (0, T, ℝ^{d}) . \end{matrix}$ \left\{\matrix{f(\cdot,\omega,y,y^{^{\prime}},z,z^{^{\prime}})\in \mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right),\\ \text{and} \\ g(\cdot,\omega,y,y^{^{\prime}},z,z^{^{\prime}})\in \mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right).}\right.

(H.5) There exist constant C ≥ 0 and a constant $0 \leq α \leq \frac{1}{2}$ 0\leq \alpha \leq \frac{1}{2} such that for every (ω, t) ∈ Ω × [0, T ] and (y, y^′) ∈ ℝ², (z, z^′) ∈ ℝ^d × ℝ^d, ${\begin{array}{l} (i) {| f (t, y_{1}, y_{1}^{'}, z_{1}, z_{1}^{'}) - f (t, y_{2}, y_{2}^{'}, z_{2}, z_{2}^{'}) |}^{2} \leq C {{| y_{1} - y_{2} |}^{2} + {| y_{1}^{'} - y_{2}^{'} |}^{2} + {| z_{1} - z_{2} |}^{2} + {| z_{1}^{'} - z_{2}^{'} |}^{2}}, \\ (i i) {| g (t, y_{1}, y_{1}^{'}, z_{1}, z_{1}^{'}) - g (t, y_{2}, y_{2}^{'}, z_{2}, z_{2}^{'}) |}^{2} \leq C {{| y_{1} - y_{2} |}^{2} + {| y_{1}^{'} - y_{2}^{'} |}^{2}} + α {{| z_{1} - z_{2} |}^{2} + {| z_{1}^{'} - z_{2}^{'} |}^{2}} . \end{array}$ \left\{{\matrix{{\left( i \right){{\left| {f(t,{y_1},y_1',{z_1},z_1') - f(t,{y_2},y_2',{z_2},z_2')} \right|}^2} \le C\left\{{{{\left| {{y_1} - {y_2}} \right|}^2} + {{\left| {y_1' - y_2'} \right|}^2} + {{\left| {{z_1} - {z_2}} \right|}^2} + {{\left| {z_1' - z_2'} \right|}^2}} \right\},} \hfill \cr {\left( {ii} \right){{\left| {g(t,{y_1},y_1',{z_1},z_1') - g(t,{y_2},y_2',{z_2},z_2')} \right|}^2} \le C\left\{{{{\left| {{y_1} - {y_2}} \right|}^2} + {{\left| {y_1' - y_2'} \right|}^2}} \right\} + \alpha \left\{{{{\left| {{z_1} - {z_2}} \right|}^2} + {{\left| {z_1' - z_2'} \right|}^2}} \right\}.} \hfill \cr}} \right.

(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 $0 \leq α \leq \frac{1}{2}$ 0\leq \alpha \leq \frac{1}{2} such that for every (ω, t) ∈ Ω × [0, T] and (y, y^′) ∈ ℝ², (z, z^′) ∈ ℝ^d × ℝ^d, ${\begin{array}{l} | f (t, y, y^{'}, z, ź) | \leq C (1 + | y | + | y^{'} | + | z | + | ź |), \\ g satisfies (H . 2) (ii) . \end{array}$ \left\{{\matrix{{\left| {f\left({t,y,{y'},z,z'} \right)} \right| \le C\left({1 + \left| y \right| + \left| {{y'}} \right| + \left| z \right| + \left| {z'} \right|} \right),} \hfill\cr{} \hfill\cr{g\,{\rm{satisfies}}\,\left({H.2} \right)\left({ii} \right).} \hfill\cr}} \right.

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.

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 $\forall (t, y, y^{'}, z) \in [0, T] \times ℝ^{2} \times ℝ^{d}, | f (t, y, y^{'}, z) | \leq f_{t} (ω) + C (| y | + | y^{'} | + | z |) .$ \forall \left(t,y,y^{^{\prime}},z\right) \in \left[ 0,T\right] \times \mathbb{R}^{2}\times \mathbb{R}^{d},\text{}\left\vert f\left(t,y,y^{^{\prime}},z\right) \right\vert \leq f_{t}\left(\omega \right) +C\left(\left\vert y\right\vert +\left\vert y^{^{\prime}}\right\vert +\left\vert z\right\vert \right).

(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, $(y_{1}^{'}, y_{2}^{'}) \in {(ℝ)}^{2}$ \left( y_{1}^{^{\prime}},y_{2}^{^{\prime}}\right) \in \left( \mathbb{R}\right)^{2} , (z₁, z₂) ∈ (ℝ^d)² ${\begin{array}{l} | π (t, y, y^{'}, z) | \leq C (| y | + | y^{'} | + | z |), \\ f (t, ω, y_{1}, y_{1}^{'}, z_{1}) - f (t, ω, y_{2}, y_{2}^{'}, z_{2}) \geq π (t, y_{1} - y_{2}, y_{1}^{'} - y_{2}^{'}, z_{1} - z_{2}) . \end{array}$ \left\{\matrix{\left\vert \pi \left(t,y,y^{^{\prime}},z\right) \right\vert \leq C\left(\left\vert y\right\vert +\left\vert y^{^{\prime}}\right\vert +\left\vert z\right\vert \right), \hfill \\ f\left(t,\omega,y_{1},y_{1}^{^{\prime}},z_{1}\right) -f\left(t,\omega,y_{2},y_{2}^{^{\prime}},z_{2}\right) \geq \pi \left(t,y_{1}-y_{2},y_{1}^{^{\prime}}-y_{2}^{^{\prime}},z_{1}-z_{2}\right).}\right.

(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: (3) $\begin{array}{l} Y_{t} = ξ + \int_{t}^{T} E^{'} (f (s, ω, ω^{'}, Y_{s}, {({\tilde{Y}}_{s})}^{'}, Z_{s})) ds + \int_{t}^{T} {dK}_{s} \\ + \int_{t}^{T} E^{'} (g (s, ω, ω^{'}, Y_{s}, {({\tilde{Y}}_{s})}^{'}, Z_{s})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s} {dW}_{s}, 0 \leq t \leq T . \end{array}$ \matrix{Y_{t}=\xi +\int_{t}^{T}E^{^{\prime}}\left(f(s,\omega,\omega^{^{\prime}},Y_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},Z_{s})\right)ds+\int_{t}^{T}dK_{s} \hfill \\ +\int_{t}^{T}E^{^{\prime}}\left(g(s,\omega,\omega^{^{\prime}},Y_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},Z_{s})\right) d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\ 0\leq t\leq T.}

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 $\tilde{K}$ \tilde{K} belong in 𝒜²we consider the processes $(\tilde{Y}, \tilde{Z}) \in 𝒮^{2} (0, T, ℝ) \times ℳ^{2} (0, T, ℝ^{d})$ \left(\tilde{Y},\tilde{Z}\right) \in \mathcal{S}^{2}\left(0,T,\mathbb{R}\right) \times \mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)such that:(4) ${\begin{array}{l} (i) {\tilde{Y}}_{t} = ξ + \int_{t}^{T} E^{'} (π (s, ω, ω^{'}, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s}) + h (s)) ds + \int_{t}^{T} d {\tilde{K}}_{s} \\ + \int_{t}^{T} E^{'} (g (s, ω, ω^{'}, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} {\tilde{Z}}_{s} {dW}_{s}, 0 \leq t \leq T, \\ (i i) \int_{0}^{T} {\tilde{Y}}_{s}^{-} d {\tilde{K}}_{s} \geq 0 . \end{array}$ \left\{\matrix{\left(i\right) \text{}\tilde{Y}_{t}=\xi +\int_{t}^{T}E^{^{\prime}}\left(\pi \left(s,\omega,\omega^{^{\prime}},\tilde{Y}_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},\tilde{Z}_{s}\right) +h\left(s\right) \right)ds+\int_{t}^{T}d\tilde{K}_{s} \hfill\\ +\int_{t}^{T}E^{^{\prime}}\left(g\left(s,\omega,\omega^{^{\prime}},\tilde{Y}_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},\tilde{Z}_{s}\right)\right) d\overleftarrow{B}_{s}-\int_{t}^{T}\tilde{Z}_{s}dW_{s},\ 0\leq t\leq T, \hfill \\ \left(ii\right) \text{}\int_{0}^{T}\tilde{Y}_{s}^{-}d\tilde{K}_{s}\geq 0. \hfill}\right.Then we have(i)

The MF-RBDSDE (4) has a least one solution $(\tilde{Y}, \tilde{Z}, \tilde{K}) \in 𝒮^{2} (0, T, ℝ^{d}) \times ℳ^{2} (0, T, ℝ^{d}) \times 𝒜^{2}$ \left(\tilde{Y},\tilde{Z},\tilde{K}\right) \in \mathcal{S}^{2}\left(0,T,\mathbb{R}^{d}\right) \times \mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right) \times \mathcal{A}^{2}

(ii)

if h(t) ≥ 0 and ξ ≥ 0, we have ${\tilde{Y}}_{t} \geq 0$ \tilde{Y}_{t}\geq 0 , dℙ × dt – a.s.

Proof

(i) See [2], (2014). (ii) Applying Tanaka's formula to ${| {\tilde{Y}}_{t}^{-} |}^{2}$ \left\vert \tilde{Y}_{t}^{-}\right\vert^{2} , we have $\begin{array}{l} 𝔼 {| {\tilde{Y}}_{t}^{-} |}^{2} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds & = 𝔼 {| ξ^{-} |}^{2} - 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} E^{'} (π (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s}) + h (s)) ds \\ - 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} d {\tilde{K}}_{s} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| | E^{'} (g (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) | |}^{2} ds . \end{array}$ \matrix{\mathbb{E}\left\vert \tilde{Y}_{t}^{-}\right\vert^{2}+\mathbb{E}\int_{t}^{T}1_{\left\{\tilde{Y}_{s}<0\right\}}\left\vert \tilde{Z}_{s}\right\vert^{2}ds &=&\mathbb{E}\left\vert \xi^{-}\right\vert^{2}-2\mathbb{E}\int_{t}^{T}\tilde{Y}_{s}^{-}E^{^{\prime}}\left(\pi (s,\tilde{Y}_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},\tilde{Z}_{s})+h\left(s\right) \right) ds \hfill \\ \hfill &&-2\mathbb{E}\int_{t}^{T}\tilde{Y}_{s}^{-}d\tilde{K}_{s}+\mathbb{E}\int_{t}^{T}1_{\left\{\tilde{Y}_{s}<0\right\}}\left\vert \left\vert E\left(^{^{\prime}}g(s,\tilde{Y}_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},\tilde{Z}_{s})\right) \right\vert \right\vert^{2}ds.}

Since $- 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} d {\tilde{K}}_{s} \leq 0$ -2\mathbb{E}\int_{t}^{T}\tilde{Y}_{s}^{-}d\tilde{K}_{s}\leq 0 , h(s) ≥ 0 and ξ ≥ 0, we get $\begin{array}{l} 𝔼 {| {\tilde{Y}}_{t}^{-} |}^{2} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds \leq & - 2 𝔼 \int_{t}^{T} {\tilde{Y}}_{s}^{-} E^{'} (π (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) ds \\ + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| | E^{'} (g (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s})) | |}^{2} ds \end{array}$ \matrix{\mathbb{E}\left\vert \tilde{Y}_{t}^{-}\right\vert^{2}+\mathbb{E}\int_{t}^{T}1_{\left\{\tilde{Y}_{s}<0\right\}}\left\vert \tilde{Z}_{s}\right\vert^{2}ds & \leq -2\mathbb{E}\int_{t}^{T}\tilde{Y}_{s}^{-}E^{^{\prime}}\left(\pi (s,\tilde{Y}_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},\tilde{Z}_{s})\right) ds \hfill \\& \quad+\mathbb{E}\int_{t}^{T}1_{\left\{\tilde{Y}_{s}<0\right\}}\left\vert \left\vert E^{^{\prime}}\left(g(s,\tilde{Y}_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},\tilde{Z}_{s})\right) \right\vert \right\vert^{2}ds}

By (H.9), we get $| π (s, {\tilde{Y}}_{s}, {({\tilde{Y}}_{s})}^{'}, {\tilde{Z}}_{s}) | \leq C (| {\tilde{Y}}_{s} | + | {({\tilde{Y}}_{s})}^{'} | + | {\tilde{Z}}_{s} |)$ \left\vert \pi \left(s,\tilde{Y}_{s},\left(\tilde{Y}_{s}\right)^{^{\prime}},\tilde{Z}_{s}\right)\right\vert \leq C\left(\left\vert \tilde{Y}_{s}\right\vert +\left\vert\left(\tilde{Y}_{s}\right)^{^{\prime}}\right\vert +\left\vert \tilde{Z}_{s}\right\vert \right) and by assumption (H.11) for g, we have $\begin{array}{l} 𝔼 {| {\tilde{Y}}_{t}^{-} |}^{2} + 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds \\ \leq (4 C^{2} + \frac{C^{2}}{β} + 2 C) 𝔼 \int_{t}^{T} {| {\tilde{Y}}_{s}^{-} |}^{2} ds + (α + β) 𝔼 \int_{t}^{T} 1_{{{\tilde{Y}}_{s} < 0}} {| {\tilde{Z}}_{s} |}^{2} ds . \end{array}$ \matrix{&&\mathbb{E}\left\vert \tilde{Y}_{t}^{-}\right\vert^{2}+\mathbb{E}\int_{t}^{T}1_{\left\{\tilde{Y}_{s}<0\right\}}\left\vert \tilde{Z}_{s}\right\vert^{2}ds \hfill \\ &\leq &\left(4C^{2}+\frac{C^{2}}{\beta}+2C\right) \mathbb{E}\int_{t}^{T}\left\vert \tilde{Y}_{s}^{-}\right\vert^{2}ds+\left(\alpha+\beta \right) \mathbb{E}\int_{t}^{T}1_{\left\{\tilde{Y}_{s}<0\right\}}\left\vert \tilde{Z}_{s}\right\vert^{2}ds.}

Therefore, choosing 0 ≤ β ≤ 1 – α and using Gronwall inequality, we have ${\tilde{Y}}_{t}^{-} = 0$ \tilde{Y}_{t}^{-}=0 , ℙ – a.s., ∀t ∈ [0, T], which implies that ${\tilde{Y}}_{t} \geq 0$ \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: (5) ${\begin{array}{l} {\bar{Y}}_{t}^{0} = ξ + \int_{t}^{T} E^{'} (- C (| {\bar{Y}}_{s}^{0} | + {({\bar{Y}}_{s}^{0})}^{'} + | {\bar{Z}}_{s}^{0} |) - f_{s}) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{0} + {({\bar{Y}}_{s}^{0})}^{'} + {\bar{Z}}_{s}^{0})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d {\bar{K}}_{s}^{0} - \int_{t}^{T} {\bar{Z}}_{s}^{0} {dW}_{s}, 0 \leq t \leq T, \\ (i i) {\bar{Y}}_{t}^{0} \geq S_{t}, \\ (i i i) \int_{0}^{T} ({\bar{Y}}_{s}^{0} - S_{s}) d {\bar{K}}_{s}^{0} = 0 . \end{array}$ \left\{\matrix{\bar{Y}_{t}^{0}=\xi +\int_{t}^{T}E^{^{\prime}}\left(-C\left(\left\vert \bar{Y}_{s}^{0}\right\vert +\left(\bar{Y}_{s}^{0}\right)^{^{\prime}}+\left\vert \bar{Z}_{s}^{0}\right\vert \right) -f_{s}\right) ds \hfill \\ +\int_{t}^{T}E^{^{\prime}}\left(g\left(s,\bar{Y}_{s}^{0}+\left(\bar{Y}_{s}^{0}\right)^{^{\prime}}+\bar{Z}_{s}^{0}\right) \right) d\overleftarrow{B}_{s}+\int_{t}^{T}d\bar{K}_{s}^{0}-\int_{t}^{T}\bar{Z}_{s}^{0}dW_{s},\ 0\leq t\leq T, \\ \\ \left(ii\right) \text{}\bar{Y}_{t}^{0}\geq S_{t}, \hfill\\ \\ \left(iii\right) \text{}\int_{0}^{T}\left(\bar{Y}_{s}^{0}-S_{s}\right) d\bar{K}_{s}^{0}=0. \hfill}\right.(6) ${\begin{array}{l} (i) {\bar{Y}}_{t}^{n} = ξ + \int_{t}^{T} E^{'} (f (s, {\bar{Y}}_{s}^{n - 1}, {({\bar{Y}}_{s}^{n - 1})}^{'}, {\bar{Z}}_{s}^{n - 1}) + π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{t}^{n})) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d {\bar{K}}_{s}^{n} - \int_{t}^{T} {\bar{Z}}_{s}^{n} {dW}_{s}, 0 \leq t \leq T, \\ (i i) {\bar{Y}}_{t}^{n} \geq S_{t}, \\ (i i i) \int_{0}^{T} ({\bar{Y}}_{s}^{n} - S_{s}) d {\tilde{k}}_{s}^{n} = 0 . \end{array}$ \left\{\matrix{\left(i\right) \text{}\bar{Y}_{t}^{n}=\xi +\int_{t}^{T}E^{^{\prime}}\left(f\left(s,\bar{Y}_{s}^{n-1},\left(\bar{Y}_{s}^{n-1}\right)^{^{\prime}},\bar{Z}_{s}^{n-1}\right) +\pi \left(s,\delta \bar{Y}_{s}^{n},\delta \left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\delta \bar{Z}_{t}^{n}\right) \right) ds \\ +\int_{t}^{T}E^{^{\prime}}\left(g\left(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n}\right) \right) d\overleftarrow{B}_{s}+\int_{t}^{T}d\bar{K}_{s}^{n}-\int_{t}^{T}\bar{Z}_{s}^{n}dW_{s}, 0\leq t\leq T, \hfill \\ \\ \left(ii\right) \text{}\bar{Y}_{t}^{n}\geq S_{t}, \hfill \\ \\ \left(iii\right) \text{}\int_{0}^{T}\left(\bar{Y}_{s}^{n}-S_{s}\right) d\tilde{k}_{s}^{n}=0. \hfill}\right.(7) ${\begin{array}{l} (i) Y_{t}^{0} = ξ + \int_{t}^{T} E^{'} (C (| Y_{s}^{0} | + {(Y_{s}^{0})}^{'} + | Z_{s}^{0} |) + f_{s}) ds + \int_{t}^{T} d K_{s}^{0} \\ + \int_{t}^{T} E^{'} (g (s, Y_{s}^{0}, {(Y_{s}^{0})}^{'} + Z_{s}^{0})) d {\overset{\leftarrow}{B}}_{s} - \int_{t}^{T} Z_{s}^{0} {dW}_{s}, 0 \leq t \leq T, \\ (i i) Y_{t}^{0} \geq S_{t}, \\ (i i i) \int_{0}^{T} (Y_{s}^{0} - S_{s}) d K_{s}^{0} = 0 . \end{array}$ \left\{\matrix{\left(i\right) \text{}Y_{t}^{0}=\xi +\int_{t}^{T}E^{^{\prime}}\left(C\left(\left\vert Y_{s}^{0}\right\vert +\left(Y_{s}^{0}\right)^{^{\prime}}+\left\vert Z_{s}^{0}\right\vert \right) +f_{s}\right)ds+\int_{t}^{T}dK_{s}^{0} \\ +\int_{t}^{T}E^{^{\prime}}\left(g\left(s,Y_{s}^{0},\left(Y_{s}^{0}\right)^{^{\prime}}+Z_{s}^{0}\right) \right) d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}^{0}dW_{s},\ 0\leq t\leq T, \hfill \\ \\ \left(ii\right) \text{}Y_{t}^{0}\geq S_{t}, \hfill \\ \\ \left(iii\right) \text{}\int_{0}^{T}\left(Y_{s}^{0}-S_{s}\right)dK_{s}^{0}=0. \hfill}\right.

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

Proof

We will prove ${\bar{Y}}_{t}^{0} \leq {\bar{Y}}_{t}^{n}$ \bar{Y}_{t}^{0}\leq \bar{Y}_{t}^{n} at first. By Eqs. (5), and (6), we have $\begin{array}{l} {\bar{Y}}_{t}^{1} - {\bar{Y}}_{t}^{0} = & \int_{t}^{T} E^{'} (π (s, δ {\bar{Y}}_{s}^{1}, δ {({\bar{Y}}_{s}^{1})}^{'}, δ {\bar{Z}}_{s}^{1}) + Λ_{s}^{1}) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{1}, {({\bar{Y}}_{s}^{1})}^{'}, {\bar{Z}}_{s}^{1}) - g (s, {\bar{Y}}_{s}^{0} + {({\bar{Y}}_{s}^{0})}^{'} + {\bar{Z}}_{s}^{0})) d {\overset{\leftarrow}{B}}_{s} \\ + \int_{t}^{T} (d {\bar{K}}_{s}^{1} - d {\bar{K}}_{s}^{0}) - \int_{t}^{T} δ {\bar{Z}}_{s}^{1} {dW}_{s}, \end{array}$ \matrix{\bar{Y}_{t}^{1}-\bar{Y}_{t}^{0} =&\int_{t}^{T}E^{^{\prime}}\left(\pi\left(s,\delta \bar{Y}_{s}^{1},\delta \left(\bar{Y}_{s}^{1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{1}\right) +\Lambda_{s}^{1}\right) ds \hfill \\&+\int_{t}^{T}E^{^{\prime}}\left(g\left(s,\bar{Y}_{s}^{1},\left(\bar{Y}_{s}^{1}\right)^{^{\prime}},\bar{Z}_{s}^{1}\right) -g\left(s,\bar{Y}_{s}^{0}+\left(\bar{Y}_{s}^{0}\right)^{^{\prime}}+\bar{Z}_{s}^{0}\right)\right) d\overleftarrow{B}_{s} \hfill\\ \hfill &+\int_{t}^{T}\left(d\bar{K}_{s}^{1}-d\bar{K}_{s}^{0}\right)-\int_{t}^{T}\delta \bar{Z}_{s}^{1}dW_{s}, \hfill} where $Λ_{s}^{1} = f (s, {\bar{Y}}_{s}^{0}, {({\bar{Y}}_{s}^{0})}^{'}, {\bar{Z}}_{s}^{0}) + C (| {\bar{Y}}_{s}^{0} | + {({\bar{Y}}_{s}^{0})}^{'} + | {\bar{Z}}_{s}^{0} |) + f_{s}$ \Lambda_{s}^{1}=f\left(s,\bar{Y}_{s}^{0},\left(\bar{Y}_{s}^{0}\right)^{^{\prime}},\bar{Z}_{s}^{0}\right) +C\left(\left\vert \bar{Y}_{s}^{0}\right\vert +\left(\bar{Y}_{s}^{0}\right)^{^{\prime}}+\left\vert \bar{Z}_{s}^{0}\right\vert \right) +f_{s} . By hypothesis (H.7) we have $Λ_{s}^{1} \geq 0$ \Lambda_{s}^{1}\geq 0 , because $({\bar{Y}}_{t}^{0}, {\bar{Z}}_{t}^{0})$ \left(\bar{Y}_{t}^{0},\bar{Z}_{t}^{0}\right) is the solution of Eq. (5), we get $Λ_{s}^{1} \in ℳ^{2} (0, T, ℝ^{d})$ \Lambda_{s}^{1}\in \mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right) . Therefore, from Lemma 3 we get ${\bar{Y}}_{t}^{1} \geq {\bar{Y}}_{t}^{0}$ \bar{Y}_{t}^{1}\geq \bar{Y}_{t}^{0} . Now we want to prove ${\bar{Y}}_{t}^{n} \leq {\bar{Y}}_{t}^{n + 1}$ \bar{Y}_{t}^{n}\leq \bar{Y}_{t}^{n+1} , for any n ≥ 0. We set ${\begin{array}{l} δ ρ_{s}^{n + 1} = ρ_{s}^{n + 1} - ρ_{s}^{n}, \\ Δ ψ^{n + 1} (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1}) \\ = ψ (s, δ {\bar{Y}}_{s}^{n + 1} + {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n + 1})}^{'} + {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n + 1} + {\bar{Z}}_{s}^{n}) - ψ (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) . \end{array}$ \left\{\matrix{\delta \rho_{s}^{n+1}=\rho_{s}^{n+1}-\rho_{s}^{n}, \hfill \\ \Delta \psi^{n+1}\left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right)\hfill \\ =\psi \left(s,\delta \bar{Y}_{s}^{n+1}+\bar{Y}_{s}^{n},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}}+\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}+\bar{Z}_{s}^{n}\right) -\psi \left(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n}\right).}\right.

Using Eq. (6), we have $\begin{array}{l} δ {\bar{Y}}_{t}^{n + 1} & = \int_{t}^{T} E^{'} (π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1}) + θ_{s}^{n + 1}) ds - \int_{t}^{T} δ {\bar{Z}}_{s}^{n + 1} {dW}_{s} \\ + \int_{t}^{T} E^{'} (Δ g^{n + 1} (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d (δ {\bar{K}}_{s}^{n + 1}), \end{array}$ \matrix{\delta \bar{Y}_{t}^{n+1} = \hfill &\int_{t}^{T}E^{^{\prime}}\left(\pi \left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right) +\theta_{s}^{n+1}\right)ds-\int_{t}^{T}\delta \bar{Z}_{s}^{n+1}dW_{s} \\&+\int_{t}^{T}E^{^{\prime}}\left(\Delta g^{n+1}(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1})\right) d\overleftarrow{B}_{s}+\int_{t}^{T}d\left(\delta \bar{K}_{s}^{n+1}\right),} where $θ_{s}^{n + 1} = Δ f^{n} (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n}) - π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n})$ \theta_{s}^{n+1}=\Delta f^{n}\left(s,\delta \bar{Y}_{s}^{n},\delta \left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n}\right)-\pi \left(s,\delta \bar{Y}_{s}^{n},\delta \left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n}\right) and $θ_{s}^{0} = Λ_{s}^{1}$ \theta_{s}^{0}=\Lambda_{s}^{1} , ∀n ≥ 0. According to it definition, one cas show that $θ_{s}^{0}$ \theta_{s}^{0} and Δgⁿ⁺¹, ∀n ≥ 0 satisfy all assumption of Lemma 3. Moreover, since ${\bar{K}}_{t}^{n}$ \bar{K}_{t}^{n} is a continuous and increasing process, for all n ≥ 0, $δ {\bar{K}}_{s}^{n + 1}$ \delta \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 $\begin{array}{l} \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d (δ {\bar{K}}_{s}^{n + 1}) & = \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d {\bar{K}}_{s}^{n + 1} - \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d {\bar{K}}_{s}^{n} \\ = \int_{0}^{T} {({\bar{Y}}_{s}^{n + 1} - {\bar{Y}}_{s}^{n})}^{-} d {\bar{K}}_{s}^{n + 1} \geq 0, \end{array}$ \matrix{\int_{0}^{T}\left(\bar{Y}_{s}^{n+1}-\bar{Y}_{s}^{n}\right)^{-}d\left(\delta \bar{K}_{s}^{n+1}\right) & = \hfill \int_{0}^{T}\left(\bar{Y}_{s}^{n+1}-\bar{Y}_{s}^{n}\right)^{-}d\bar{K}_{s}^{n+1}-\int_{0}^{T}\left(\bar{Y}_{s}^{n+1}-\bar{Y}_{s}^{n}\right)^{-}d\bar{K}_{s}^{n} \\ & =\int_{0}^{T}\left(\bar{Y}_{s}^{n+1}-\bar{Y}_{s}^{n}\right)^{-}d\bar{K}_{s}^{n+1}\geq 0,\hfill} by Lemma 3, we deduce that $δ {\bar{Y}}_{t}^{n + 1} \geq 0$ \delta \bar{Y}_{t}^{n+1}\geq 0 , i.e. ${\bar{Y}}_{t}^{n + 1} \geq {\bar{Y}}_{t}^{n}$ \bar{Y}_{t}^{n+1}\geq \bar{Y}_{t}^{n} ∀t ∈ [0, T], we have ${\bar{Y}}_{t}^{n + 1} \geq {\bar{Y}}_{t}^{n} \geq {\bar{Y}}_{t}^{0} .$ \bar{Y}_{t}^{n+1}\geq \bar{Y}_{t}^{n}\geq \bar{Y}_{t}^{0}.

Now we shall prove that ${\bar{Y}}_{t}^{n + 1} \leq Y_{t}^{0}$ \bar{Y}_{t}^{n+1}\leq Y_{t}^{0} ∀n ≥ 0, by Eqs.(3) and (7) $\begin{array}{l} Y_{t}^{0} - {\bar{Y}}_{t}^{n + 1} & = \int_{t}^{T} E^{'} (- C (| Y_{s}^{0} - {\bar{Y}}_{s}^{+ 1} | + | {(Y_{s}^{0})}^{'} - {({\bar{Y}}_{s}^{n + 1})}^{'} | + | Z_{s}^{0} - {\bar{Z}}_{s}^{n + 1} |) + Λ_{s}^{n + 1}) ds \\ + \int_{t}^{T} E^{'} (g (s, Y_{s}^{0}, {(Y_{s}^{0})}^{'} + Z_{s}^{0}) - g (s, s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})) d {\overset{\leftarrow}{B}}_{s} \\ + \int_{t}^{T} (d K_{s}^{0} - d {\bar{K}}_{s}^{n + 1}) + \int_{t}^{T} (Z_{s}^{0} - {\bar{Z}}_{s}^{n + 1}) {dW}_{s}, \end{array}$ \matrix{Y_{t}^{0}-\bar{Y}_{t}^{n+1} =&\int_{t}^{T}E^{^{\prime}}\left(-C\left(\left\vert Y_{s}^{0}-\bar{Y}_{s}^{+1}\right\vert +\left\vert \left(Y_{s}^{0}\right)^{^{\prime}}-\left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}}\right\vert +\left\vert Z_{s}^{0}-\bar{Z}_{s}^{n+1}\right\vert \right)+\Lambda_{s}^{n+1}\right) ds\\&+\int_{t}^{T}E^{^{\prime}}\left(g(s,Y_{s}^{0},\left(Y_{s}^{0}\right)^{^{\prime}}+Z_{s}^{0})-g(s,s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n})\right) d\overleftarrow{B}_{s}\hfill \\&+\int_{t}^{T}\left(dK_{s}^{0}-d\bar{K}_{s}^{n+1}\right)+\int_{t}^{T}\left(Z_{s}^{0}-\bar{Z}_{s}^{n+1}\right) dW_{s},\hfill} where $\begin{array}{l} Λ_{s}^{n + 1} & = C (| Y_{s}^{0} - {\bar{Y}}_{s}^{+ 1} | + | {(Y_{s}^{0})}^{'} - {({\bar{Y}}_{s}^{n + 1})}^{'} | + | Z_{s}^{0} - {\bar{Z}}_{s}^{n + 1} | + | Y_{s}^{0} | + {(Y_{s}^{0})}^{'} + | Z_{s}^{0} |) \\ + f_{s} - f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1}) . \end{array}$ \matrix{\Lambda_{s}^{n+1} &=& C\left(\left\vert Y_{s}^{0}-\bar{Y}_{s}^{+1}\right\vert +\left\vert \left(Y_{s}^{0}\right)^{^{\prime}}-\left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}}\right\vert +\left\vert Z_{s}^{0}-\bar{Z}_{s}^{n+1}\right\vert +\left\vert Y_{s}^{0}\right\vert+\left(Y_{s}^{0}\right)^{^{\prime}}+\left\vert Z_{s}^{0}\right\vert \right) \\ &&+f_{s}-f(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n})+\pi \left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right).}

By Lemma 3, we deduce that $Y_{t}^{0} - {\bar{Y}}_{t}^{n + 1} \geq 0$ Y_{t}^{0}-\bar{Y}_{t}^{n+1}\geq 0 , i.e. $Y_{t}^{0} \geq {\bar{Y}}_{t}^{n + 1}$ Y_{t}^{0}\geq \bar{Y}_{t}^{n+1} , for all t ∈ [0, T]. Thus we have for all n ≥ 0 $Y_{t}^{0} \geq {\bar{Y}}_{t}^{n + 1} \geq {\bar{Y}}_{t}^{n} \geq {\bar{Y}}_{t}^{0}, d \bar{ℙ} \times d t - a . s . \forall t \in [0, T] .$ Y_{t}^{0}\geq \bar{Y}_{t}^{n+1}\geq \bar{Y}_{t}^{n}\geq \bar{Y}_{t}^{0},\text{}d\mathbb{\bar{P}}\times dt-a.s.\text{}\forall t\in \left[ 0,T\right].

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

Proof

From Lemma 4, we know ${({\bar{Y}}_{t}^{n})}_{n \geq 0}$ \left(\bar{Y}_{t}^{n}\right)_{n\geq 0} is increasing and bounded in ℳ² (0, T, ℝ^d). Since $| {\tilde{Y}}_{t}^{n} | \leq max ({\tilde{Y}}_{t}^{0}, Y_{t}^{0}) \leq | {\tilde{Y}}_{t}^{0} | + | Y_{t}^{0} |$ \left\vert \tilde{Y}_{t}^{n}\right\vert \leq \max\left(\tilde{Y}_{t}^{0},Y_{t}^{0}\right) \leq \left\vert \tilde{Y}_{t}^{0}\right\vert +\left\vert Y_{t}^{0}\right\vert for all t ∈ [0, T], we have $sup_{n} 𝔼 (sup_{0 \leq t \leq T} {| {\bar{Y}}_{t}^{n} |}^{2}) \leq 𝔼 (sup_{0 \leq t \leq T} {| {\bar{Y}}_{t}^{0} |}^{2}) + 𝔼 (sup_{0 \leq t \leq T} {| Y_{t}^{0} |}^{2}) < \infty,$ \matrix{{\sup}\limits_{n}\mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert \bar{Y}_{t}^{n}\right\vert^{2}\right) \leq \mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert \bar{Y}_{t}^{0}\right\vert^{2}\right) +\mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert Y_{t}^{0}\right\vert^{2}\right) <\infty,} then according to the Lebesgue's dominated convergence theorem, we deduce that ${({\bar{Y}}_{t}^{n})}_{n \geq 0}$ \left(\bar{Y}_{t}^{n}\right)_{n\geq 0} converges in 𝒮² (0, T, ℝ). We denote by $\bar{Y}$ \bar{Y} the limit of ${({\bar{Y}}_{t}^{n})}_{n \geq 0}$ \left(\bar{Y}_{t}^{n}\right)_{n\geq 0} .

On the other hand from Eq. (6), we deduce that $\begin{array}{l} {\bar{Y}}_{0}^{n + 1} & = {\bar{Y}}_{T}^{n + 1} + \int_{0}^{T} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ + \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} d {\bar{K}}_{s}^{n + 1} - \int_{t}^{T} {\bar{Z}}_{s}^{n + 1} {dW}_{s} . \end{array}$ \matrix{\bar{Y}_{0}^{n+1} =&\bar{Y}_{T}^{n+1}+\int_{0}^{T}E^{^{\prime}}\left(f\left(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n}\right) +\pi \left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right) \right) ds \\\hfill&+\int_{t}^{T}E^{^{\prime}}\left(g\left(s,\bar{Y}_{s}^{n+1},\left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\bar{Z}_{s}^{n+1}\right) \right) d\overleftarrow{B}_{s}+\int_{t}^{T}d\bar{K}_{s}^{n+1}-\int_{t}^{T}\bar{Z}_{s}^{n+1}dW_{s}.\hfill}

Applying Itô's formula, we obtain $\begin{array}{l} 𝔼 {| {\bar{Y}}_{0}^{n + 1} |}^{2} + 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq & 𝔼 {| {\bar{Y}}_{T}^{n + 1} |}^{2} + 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} d {\bar{K}}_{s}^{n + 1} + 𝔼 \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) | |}^{2} ds \\ + 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds . \end{array}$ \matrix{\mathbb{E}\left\vert \bar{Y}_{0}^{n+1}\right\vert^{2}+\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds &\leq &\mathbb{E}\left\vert \bar{Y}_{T}^{n+1}\right\vert^{2}+2\mathbb{E}\int_{0}^{T}\bar{Y}_{s}^{n+1}d\bar{K}_{s}^{n+1}+\mathbb{E}\int_{0}^{T}\left\vert \left\vert E^{^{\prime}}\left(g(s,\bar{Y}_{s}^{n+1},\left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\bar{Z}_{s}^{n+1})\right) \right\vert \right\vert^{2}ds \\&&+2\mathbb{E}\int_{0}^{T}\bar{Y}_{s}^{n+1}E^{^{\prime}}\left(f(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n})+\pi\left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right) \right) ds.}

From assumption (H.7) and Yöung's inequality, we get (8) $\begin{array}{l} 2 𝔼 \int_{0}^{T} {\bar{Y}}_{t}^{n + 1} E^{'} [f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})] ds \\ \leq 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} [f_{s} (ω) + C (1 + | {\bar{Y}}_{s}^{n} | + | {({\bar{Y}}_{s}^{n})}^{'} | + | {\bar{Z}}_{s}^{n} |)] ds, \\ \leq 4 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + (4 C^{2} + 1) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds \\ + 16 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds, \\ = (24 C^{2} + 1) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds, \end{array}$ \matrix{\quad2\mathbb{E}\int_{0}^{T}\bar{Y}_{t}^{n+1}E^{^{\prime}}\left[ f(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n})\right]ds \notag \hfill\\\leq 2\mathbb{E}\int_{0}^{T}\bar{Y}_{s}^{n+1}E^{^{\prime}}\left[f_{s}\left(\omega \right) +C\left(1+\left\vert \bar{Y}_{s}^{n}\right\vert +\left\vert \left(\bar{Y}_{s}^{n}\right)^{^{\prime}}\right\vert+\left\vert \bar{Z}_{s}^{n}\right\vert \right) \right] ds, \notag \hfill\\\leq 4C^{2}\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\left(4C^{2}+1\right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}\right\vert^{2}ds \hfill\\\quad+16C^{2}\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\frac{1}{16}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds+\mathbb{E}\int_{0}^{T}\left\vert f_{s}\left(\omega \right) \right\vert^{2}ds, \notag \hfill\\=\left(24C^{2}+1\right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}\right\vert^{2}ds+\frac{1}{16}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds+\mathbb{E}\int_{0}^{T}\left\vert f_{s}\left(\omega \right) \right\vert^{2}ds, \notag \hfill} and from hypothesis(H.9) we get (9) $\begin{array}{l} 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ \leq 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (C (| δ {\bar{Y}}_{s}^{n + 1} | + | {(δ {\bar{Y}}_{s}^{n + 1})}^{'} | + | δ {\bar{Z}}_{s}^{n + 1} |)) ds, \\ \leq 4 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 4 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds + 8 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds \\ + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + 16 C^{2} 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds, \\ = (4 C + 28 C^{2}) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + \frac{1}{16} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds . \end{array}$ \matrix{\quad2\mathbb{E}\int_{0}^{T}\bar{Y}_{s}^{n+1}E^{^{\prime}}\left(\pi \left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right) \right) ds \notag \hfill\\\leq 2\mathbb{E}\int_{0}^{T}\bar{Y}_{s}^{n+1}E^{^{\prime}}\left(C\left(\left\vert \delta \bar{Y}_{s}^{n+1}\right\vert +\left\vert \left(\delta \bar{Y}_{s}^{n+1}\right)^{^{\prime}}\right\vert +\left\vert \delta \bar{Z}_{s}^{n+1}\right\vert \right) \right) ds, \notag \hfill\\\leq 4C\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+4C^{2}\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}\right\vert^{2}ds+8C^{2}\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds \hfill\\\quad+\frac{1}{8}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds+16C^{2}\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\frac{1}{16}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds, \notag \hfill\\=\left(4C+28C^{2}\right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}\right\vert^{2}ds+\frac{1}{8}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds+\frac{1}{16}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds. \notag \hfill}

Using the two inequalities (8) and (9), we obtain $\begin{array}{l} 2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ \leq (52 C^{2} + 4 C + 1) 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 2 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} |}^{2} ds \\ + \frac{1}{8} 𝔼 \int_{0}^{T} ({| {\bar{Z}}_{s}^{n + 1} |}^{2} + {| {\bar{Z}}_{s}^{n} |}^{2}) ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds . \end{array}$ \matrix{\hfill\quad2\mathbb{E}\int_{0}^{T}\bar{Y}_{s}^{n+1}E^{^{\prime}}\left(f(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n})+\pi\left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right) \right) ds \hfill\\\leq \left(52C^{2}+4C+1\right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+2\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}\right\vert^{2}ds \hfill\\+\frac{1}{8}\mathbb{E}\int_{0}^{T}\left(\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}+\left\vert \bar{Z}_{s}^{n}\right\vert^{2}\right)ds+\mathbb{E}\int_{0}^{T}\left\vert f_{s}\left(\omega \right) \right\vert^{2}ds.\hfill}

Then, we get $\begin{array}{l} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq & 𝔼 {| ξ |}^{2} + 𝔼 \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) | |}^{2} ds \\ + C + 2 𝔼 \int_{0}^{T} 〈 {\bar{Y}}_{s}^{n + 1}, d {\bar{K}}_{s}^{n + 1} 〉 + \frac{1}{8} 𝔼 \int_{0}^{T} ({| {\bar{Z}}_{s}^{n + 1} |}^{2} + {| {\bar{Z}}_{s}^{n} |}^{2}) ds, \end{array}$ \matrix{\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds \leq &\mathbb{E}\left\vert \xi \right\vert^{2}+\mathbb{E}\int_{0}^{T}\left\vert\left\vert E^{^{\prime}}\left(g(s,\bar{Y}_{s}^{n+1},\left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\bar{Z}_{s}^{n+1})\right) \right\vert\right\vert^{2}ds \hfill\\&+C+2\mathbb{E}\int_{0}^{T}\langle \bar{Y}_{s}^{n+1},d\bar{K}_{s}^{n+1}\rangle +\frac{1}{8}\mathbb{E}\int_{0}^{T}\left(\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}+\left\vert \bar{Z}_{s}^{n}\right\vert^{2}\right)ds, \hfill} where $C = 2 𝔼 \int_{0}^{T} | {\bar{Y}}_{s}^{n} | ds + (52 C + 4 C + 1) \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 𝔼 \int_{0}^{T} {| f_{s} (ω) |}^{2} ds$ C=2\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}\right\vert ds+\left(52C+4C+1\right) \int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right \vert^{2}ds+\mathbb{E}\int_{0}^{T}\left\vert f_{s}\left(\omega \right) \right\vert^{2}ds .

Applying hypothesis (H. 11), we have $\begin{array}{l} 𝔼 \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) | |}^{2} ds \\ \leq 4 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + 2 α 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + 2 𝔼 \int_{0}^{T} {| | g (s, 0, 0, 0) | |}^{2} ds . \end{array}$ \matrix{\quad\hfill\mathbb{E}\int_{0}^{T}\left\vert \left\vert E^{^{\prime}}\left(g(s,\bar{Y}_{s}^{n+1},\left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\bar{Z}_{s}^{n+1})\right) \right\vert \right\vert^{2}ds \hfill\\\leq 4C\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+2\alpha \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds+2\mathbb{E}\int_{0}^{T}\left\vert \left\vert g(s,0,0,0)\right\vert \right\vert^{2}ds. \hfill}

Using Yöung's inequality, we obtain $2 𝔼 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} d {\bar{K}}_{s}^{n + 1} \leq 2 𝔼 \int_{0}^{T} S_{s} d {\bar{K}}_{s}^{n + 1} \leq \frac{1}{θ} 𝔼 (sup_{0 \leq t \leq T} {| S_{t} |}^{2}) + θ 𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} .$ \matrix{2\mathbb{E}\int_{0}^{T}\bar{Y}_{s}^{n+1}d\bar{K}_{s}^{n+1}\leq 2\mathbb{E}\int_{0}^{T}S_{s}d\bar{K}_{s}^{n+1}\leq \frac{1}{\theta}\mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert S_{t}\right\vert^{2}\right) +\theta \mathbb{E}\left\vert \bar{K}_{T}^{n+1}\right\vert^{2}.}

Therefore, there exists a constant C^θ depending on α, ξ, C and θ, we derive (10) $𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq C^{θ} + (\frac{1}{8} + 2 α) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + θ 𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2},$ \matrix{\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds\leqC^{\theta}+\left(\frac{1}{8}+2\alpha \right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds+\frac{1}{8}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds+\theta \mathbb{E}\left\vert \bar{K}_{T}^{n+1}\right\vert^{2},} where $C^{θ} = C + 𝔼 {| ξ |}^{2} + 4 C \int_{0}^{T} {| {\bar{Y}}_{s}^{n + 1} |}^{2} ds + \frac{1}{θ} 𝔼 ({sup}_{0 \leq t \leq T} {| S_{t} |}^{2}) + 2 𝔼 \int_{0}^{T} {| | g (s, 0, 0, 0) | |}^{2} ds$ C^{\theta}=C+\mathbb{E}\left\vert \xi \right\vert^{2}+4C\int_{0}^{T}\left\vert \bar{Y}_{s}^{n+1}\right\vert^{2}ds+\frac{1}{\theta}\mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert S_{t}\right\vert^{2}\right) +2\mathbb{E}\int_{0}^{T}\left\vert \left\vertg(s,0,0,0)\right\vert \right\vert^{2}ds .

Chossing α such that $0 < \frac{1}{8} + 2 α < 1$ 0<\frac{1}{8}+2\alpha <1 , we obtain $𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq C^{θ} + \frac{1}{8} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + θ 𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} .$ \matrix{\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds\leq C^{\theta}+\frac{1}{8}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds+\theta \mathbb{E}\left\vert \bar{K}_{T}^{n+1}\right\vert^{2}.}

Moreover, since $\begin{array}{l} {\bar{K}}_{T}^{n + 1} = & {\bar{Y}}_{0}^{n + 1} - ξ - \int_{0}^{T} E^{'} (f (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) + π (s, δ {\bar{Y}}_{s}^{n + 1}, δ {({\bar{Y}}_{s}^{n + 1})}^{'}, δ {\bar{Z}}_{s}^{n + 1})) ds \\ - \int_{0}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n + 1}, {({\bar{Y}}_{s}^{n + 1})}^{'}, {\bar{Z}}_{s}^{n + 1})) d {\overset{\leftarrow}{B}}_{s} + \int_{t}^{T} {\bar{Z}}_{s}^{n + 1} {dW}_{s}, \end{array}$ \matrix{\bar{K}_{T}^{n+1} =&\bar{Y}_{0}^{n+1}-\xi -\int_{0}^{T}E^{^{\prime}}\left(f\left(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n}\right) +\pi \left(s,\delta \bar{Y}_{s}^{n+1},\delta \left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n+1}\right) \right) ds \hfill \\&-\int_{0}^{T}E^{^{\prime}}\left(g\left(s,\bar{Y}_{s}^{n+1},\left(\bar{Y}_{s}^{n+1}\right)^{^{\prime}},\bar{Z}_{s}^{n+1}\right) \right) d\overleftarrow{B}_{s}+\int_{t}^{T}\bar{Z}_{s}^{n+1}dW_{s}, \hfill} 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 $𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} \leq C_{1} + C_{2} (𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} + {| {\bar{Z}}_{s}^{n} |}^{2} ds) .$ \mathbb{E}\left\vert \bar{K}_{T}^{n+1}\right\vert^{2}\leq C_{1}+C_{2}\left(\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}+\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds\right)

Return to inequality (10), we get $𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds \leq C^{θ} + θ C_{1} + (\frac{1}{8} + θ C_{2}) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds + θ C_{2} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds,$ \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds\leq C^{\theta}+\theta C_{1}+\left(\frac{1}{8}+\theta C_{2}\right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds+\theta C_{2}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds, we chosing θ, such that θC₂ ≤ 1, we have $\begin{array}{l} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds & \leq C^{θ} + θ C_{1} + (\frac{1}{8} + θ C_{2}) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} |}^{2} ds \\ \leq (C^{θ} + θ C_{1}) \sum_{i = 0}^{i = n - 1} {(\frac{1}{8} + θ C_{2})}^{i} + {(\frac{1}{8} + θ C_{2})}^{n} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{0} |}^{2} ds . \end{array}$ \matrix{\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds & \leq C^{\theta}+\theta C_{1}+\left(\frac{1}{8}+\theta C_{2}\right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}\right\vert^{2}ds \hfill\\& \leq \left(C^{\theta}+\theta C_{1}\right) \sum_{i=0}^{i=n-1}\left(\frac{1}{8}+\theta C_{2}\right)^{i}+\left(\frac{1}{8}+\theta C_{2}\right)^{n}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{0}\right\vert^{2}ds.\hfill}

Now chossing θ such that $\frac{1}{8} + θ C_{2} < 1$ \frac{1}{8}+\theta C_{2}<1 and notting $𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{0} |}^{2} ds < \infty$ \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{0}\right\vert^{2}ds<\infty . Obtain $sup_{n \in ℕ} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n + 1} |}^{2} ds < \infty,$ \matrix{{\sup}\limits_{n\in \mathbb{N}}\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n+1}\right\vert^{2}ds<\infty,} consequently, we deduce $𝔼 {| {\bar{K}}_{T}^{n + 1} |}^{2} < \infty .$ \mathbb{E}\left\vert \bar{K}_{T}^{n+1}\right\vert^{2}<\infty. Now we shall prove that $({\bar{Z}}^{n}, {\bar{K}}^{n})$ \left(\bar{Z}^{n},\bar{K}^{n}\right) is a Cauchy sequence in ℳ² (0, T, ℝ^d) × 𝒜².

Applying Itô's formula to ${| δ {\tilde{Y}}_{s}^{n, m} |}^{2} = {| {\tilde{Y}}_{s}^{n} - {\tilde{Y}}_{s}^{m} |}^{2}$ \left\vert \delta \tilde{Y}_{s}^{n,m}\right\vert^{2}=\left\vert \tilde{Y}_{s}^{n}-\tilde{Y}_{s}^{m}\right\vert^{2} , we have $\begin{array}{l} 𝔼 {| {\bar{Y}}_{t}^{n} - {\bar{Y}}_{t}^{m} |}^{2} + 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds = & 2 𝔼 \int_{0}^{T} ({\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m}) E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) ds + 2 \int_{0}^{T} {\bar{Y}}_{s}^{n + 1} (d {\bar{K}}_{s}^{n} - d {\bar{K}}_{s}^{m}) \\ + \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) - g (s, {\bar{Y}}_{s}^{m}, {({\bar{Y}}_{s}^{m})}^{'}, {\bar{Z}}_{s}^{m})) | |}^{2} ds . \end{array}$ \matrix{\mathbb{E}\left\vert \bar{Y}_{t}^{n}-\bar{Y}_{t}^{m}\right\vert^{2}+\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}-\bar{Z}_{s}^{m}\right\vert^{2}ds &=2\mathbb{E}\int_{0}^{T}\left(\bar{Y}_{s}^{n}-\bar{Y}_{s}^{m}\right)E^{^{\prime}}\left(\Gamma_{s}^{n}-\Gamma_{s}^{m}\right) ds+2\int_{0}^{T}\bar{Y}_{s}^{n+1}\left(d\bar{K}_{s}^{n}-d\bar{K}_{s}^{m}\right)\hfill \\ &+\int_{0}^{T}\left\vert \left\vert E^{^{\prime}}\left(g\left(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n}\right)-g\left(s,\bar{Y}_{s}^{m},\left(\bar{Y}_{s}^{m}\right)^{^{\prime}},\bar{Z}_{s}^{m}\right) \right) \right\vert \right\vert^{2}ds. \hfill} where $Γ_{s}^{n} = f (s, {\bar{Y}}_{s}^{n - 1}, {({\bar{Y}}_{s}^{n - 1})}^{'}, {\bar{Z}}_{s}^{n - 1}) + π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n})$ \Gamma_{s}^{n}=f(s,\bar{Y}_{s}^{n-1},\left(\bar{Y}_{s}^{n-1}\right)^{^{\prime}},\bar{Z}_{s}^{n-1})+\pi \left(s,\delta \bar{Y}_{s}^{n},\delta\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n}\right) . Since $\int_{0}^{T} {\bar{Y}}_{s}^{n + 1} (d {\bar{K}}_{s}^{n} - d {\bar{K}}_{s}^{m}) \leq 0$ \int_{0}^{T}\bar{Y}_{s}^{n+1}\left(d\bar{K}_{s}^{n}-d\bar{K}_{s}^{m}\right) \leq 0 , we obtain $\begin{array}{l} 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds & \leq 2 𝔼 \int_{0}^{T} ({\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m}) E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) ds \\ + \int_{0}^{T} {| | E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) - g (s, {\bar{Y}}_{s}^{m}, {({\bar{Y}}_{s}^{m})}^{'}, {\bar{Z}}_{s}^{m})) | |}^{2} ds . \end{array}$ \matrix{\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}-\bar{Z}_{s}^{m}\right\vert^{2}ds &\leq 2\mathbb{E}\int_{0}^{T}\left(\bar{Y}_{s}^{n}-\bar{Y}_{s}^{m}\right) E^{^{\prime}}\left(\Gamma_{s}^{n}-\Gamma_{s}^{m}\right)ds \hfill \\ &+\int_{0}^{T}\left\vert \left\vert E^{^{\prime}}\left(g\left(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n}\right)-g\left(s,\bar{Y}_{s}^{m},\left(\bar{Y}_{s}^{m}\right)^{^{\prime}},\bar{Z}_{s}^{m}\right) \right) \right\vert \right\vert^{2}ds. \hfill}

By the Hölder inequality and hypothesis (H.11), we deduce that $\begin{matrix} (1 - α) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds \leq 2 𝔼 {(\int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds)}^{\frac{1}{2}} 𝔼 {(\int_{0}^{T} {| E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) |}^{2} ds)}^{\frac{1}{2}} \\ + 2 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds . \end{matrix}$ \matrix{\left(1-\alpha \right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}-\bar{Z}_{s}^{m}\right\vert^{2}ds &\leq 2\mathbb{E}\left(\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}-\bar{Y}_{s}^{m}\right\vert^{2}ds\right)^{\frac{1}{2}}\mathbb{E}\left(\int_{0}^{T}\left\vert E^{^{\prime}}\left(\Gamma_{s}^{n}-\Gamma_{s}^{m}\right) \right\vert^{2}ds\right)^{\frac{1}{2}} \hfill \\&+2C\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}-\bar{Y}_{s}^{m}\right\vert^{2}ds.\hfill}

The boundedness of the sequence $({\bar{Y}}^{n}, {\bar{Z}}^{n}, {\bar{K}}^{n})$ \left(\bar{Y}^{n},\bar{Z}^{n},\bar{K}^{n}\right) , we deduce that the $Λ = {sup}_{n \in ℕ} [𝔼 \int_{0}^{T} E^{'} {| Γ_{s}^{n} |}^{2} ds] < \infty$ \Lambda ={\sup}_{n\in \mathbb{N}}\left[ \mathbb{E}\int_{0}^{T}E^{^{\prime}}\left\vert \Gamma_{s}^{n}\right\vert^{2}ds\right] <\infty , this yields that $(1 - α) 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m} |}^{2} ds \leq 4 Λ 𝔼 {(\int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds)}^{\frac{1}{2}} + 2 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} - {\bar{Y}}_{s}^{m} |}^{2} ds,$ \matrix{\left(1-\alpha \right) \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}-\bar{Z}_{s}^{m}\right\vert^{2}ds\leq 4\Lambda \mathbb{E}\left(\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}-\bar{Y}_{s}^{m}\right\vert^{2}ds\right)^{\frac{1}{2}}+2C\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}-\bar{Y}_{s}^{m}\right\vert^{2}ds,} which yields that ${({\bar{Z}}^{n})}_{n \geq 0}$ \left(\bar{Z}^{n}\right)_{n\geq 0} is a Cauchy sequence in ℳ² (0, T, ℝ^d). Then there exists Z ∈ ℳ² (ℝ^d) such that $𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - Z_{s} |}^{2} ds \to 0 as n \to \infty .$ \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}-Z_{s}\right\vert^{2}ds\rightarrow 0\text{}as\text{}n\rightarrow \infty.

On the other hand, by Burkhölder-Davis-Gundy inequality, we get ${\begin{array}{l} 𝔼 sup_{0 \leq t \leq T} {| \int_{t}^{T} {\bar{Z}}_{s}^{n} {dW}_{s} - \int_{t}^{T} Z_{s} {dW}_{s} |}^{2} \leq 𝔼 \int_{t}^{T} {| {\bar{Z}}_{s}^{n} - Z_{s} |}^{2} ds \to 0, as n \to \infty, \\ 𝔼 sup_{0 \leq t \leq T} {| \int_{t}^{T} E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n})) - E^{'} (g (s, Y_{s}, {(Y_{s})}^{'}, Z_{s})) |}^{2} \\ \leq 2 C 𝔼 \int_{0}^{T} {| {\bar{Y}}_{s}^{n} - Y_{s} |}^{2} ds + α 𝔼 \int_{0}^{T} {| {\bar{Z}}_{s}^{n} - Z_{s} |}^{2} ds \to 0, as n \to \infty . \end{array}$ \left\{\matrix{\mathbb{E}{\sup}_{0\leq t\leq T}\left\vert \int_{t}^{T}\bar{Z}_{s}^{n}dW_{s}-\int_{t}^{T}Z_{s}dW_{s}\right\vert^{2}\leq \mathbb{E}\int_{t}^{T}\left\vert \bar{Z}_{s}^{n}-Z_{s}\right\vert^{2}ds\rightarrow 0,\text{}as\text{}n\rightarrow \infty, \\ \mathbb{E}{\sup}_{0\leq t\leq T}\left\vert \int_{t}^{T}E^{^{\prime}}\left(g(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n})\right) -E^{^{\prime}}\left(g(s,Y_{s},\left(Y_{s}\right)^{^{\prime}},Z_{s})\right) \right\vert^{2}\hfill \\ \leq 2C\mathbb{E}\int_{0}^{T}\left\vert \bar{Y}_{s}^{n}-Y_{s}\right\vert^{2}ds+\alpha \mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{n}-Z_{s}\right\vert^{2}ds\rightarrow 0,\text{}as\text{}n\rightarrow \infty. \hfill}\right.

Therefore, from the properieties of f and π $Γ_{s}^{n} = f (s, {\bar{Y}}_{s}^{n - 1}, {({\bar{Y}}_{s}^{n - 1})}^{'}, {\bar{Z}}_{s}^{n - 1}) + π (s, δ {\bar{Y}}_{s}^{n}, δ {({\bar{Y}}_{s}^{n})}^{'}, δ {\bar{Z}}_{s}^{n}) \to f (s, Y_{s}, {(Y_{s})}^{'}, Z_{s}),$ \matrix{\Gamma_{s}^{n}=f(s,\bar{Y}_{s}^{n-1},\left(\bar{Y}_{s}^{n-1}\right)^{^{\prime}},\bar{Z}_{s}^{n-1})+\pi \left(s,\delta \bar{Y}_{s}^{n},\delta\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\delta \bar{Z}_{s}^{n}\right)\rightarrow f(s,Y_{s},\left(Y_{s}\right)^{^{\prime}},Z_{s}),}

$\bar{ℙ}$ \mathbb{\bar{P}} – a.s., for all t ∈ [0, T] as n → ∞. Then follows by Lebesgue's dominated convergence theorem that $𝔼 \int_{0}^{T} {| E^{'} (Γ_{s}^{n} - f (s, Y_{s}, {(Y_{s})}^{'}, Z_{s})) |}^{2} ds \to 0, n \to \infty$ \mathbb{E}\int_{0}^{T}\left\vert E^{^{\prime}}\left(\Gamma_{s}^{n}-f(s,Y_{s},\left(Y_{s}\right)^{^{\prime}},Z_{s})\right)\right\vert^{2}ds\rightarrow 0,\text{}n\rightarrow \infty

Since $({\tilde{Y}}_{s}, {\tilde{Z}}_{s}, Γ_{s}^{n})$ \left(\tilde{Y}_{s},\tilde{Z}_{s},\Gamma_{s}^{n}\right) converges in 𝒮² (0, T, ℝ) × ℳ² (0, T, ℝ^d) × ℳ² (0, T, ℝ²) and $\begin{array}{l} 𝔼 (sup_{0 \leq t \leq T} {| {\bar{K}}_{t}^{n} - {\bar{K}}_{t}^{m} |}^{2}) \leq & 𝔼 {| {\bar{Y}}_{0}^{n} - {\bar{Y}}_{0}^{m} |}^{2} + 𝔼 sup_{0 \leq t \leq T} {| {\bar{Y}}_{t}^{n} - {\bar{Y}}_{t}^{m} |}^{2} + 𝔼 \int_{0}^{T} {| E^{'} (Γ_{s}^{n} - Γ_{s}^{m}) |}^{2} ds \\ + 𝔼 sup_{0 \leq t \leq T} {| \int_{0}^{t} E^{'} (g (s, {\bar{Y}}_{s}^{n}, {({\bar{Y}}_{s}^{n})}^{'}, {\bar{Z}}_{s}^{n}) - g (s, {\bar{Y}}_{s}^{m}, {({\bar{Y}}_{s}^{m})}^{'}, {\bar{Z}}_{s}^{m})) d \overset{\leftarrow}{B_{s}} |}^{2} \\ + 𝔼 sup_{0 \leq t \leq T} {| \int_{0}^{t} ({\bar{Z}}_{s}^{n} - {\bar{Z}}_{s}^{m}) {dW}_{s} |}^{2} \end{array}$ \matrix{\mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert \bar{K}_{t}^{n}-\bar{K}_{t}^{m}\right\vert^{2}\right) \leq &\mathbb{E}\left\vert \bar{Y}_{0}^{n}-\bar{Y}_{0}^{m}\right\vert^{2}+\mathbb{E}{\sup}\limits_{0\leq t\leq T}\left\vert \bar{Y}_{t}^{n}-\bar{Y}_{t}^{m}\right\vert^{2}+\mathbb{E}\int_{0}^{T}\left\vert E^{^{\prime}}\left(\Gamma_{s}^{n}-\Gamma_{s}^{m}\right) \right\vert^{2}ds \hfill\\&+\mathbb{E}{\sup}\limits_{0\leq t\leq T}\left\vert \int_{0}^{t}E^{^{\prime}}\left(g(s,\bar{Y}_{s}^{n},\left(\bar{Y}_{s}^{n}\right)^{^{\prime}},\bar{Z}_{s}^{n})-g(s,\bar{Y}_{s}^{m},\left(\bar{Y}_{s}^{m}\right)^{^{\prime}},\bar{Z}_{s}^{m})\right) d\overleftarrow{B_{s}}\right\vert^{2}\hfill \\&+\mathbb{E}{\sup}\limits_{0\leq t\leq T}\left\vert \int_{0}^{t}\left(\bar{Z}_{s}^{n}-\bar{Z}_{s}^{m}\right) dW_{s}\right\vert^{2}\hfill} for any n ≥ 0, we deduce from Bukhölder-Davis-Gundy inequality that $𝔼 (sup_{0 \leq t \leq T} {| {\bar{K}}_{t}^{n} - {\bar{K}}_{t}^{m} |}^{2}) \to 0,$ \mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert \bar{K}_{t}^{n}-\bar{K}_{t}^{m}\right\vert^{2}\right) \rightarrow 0, as n → ∞. Consequently, there exists a ℱ_t–mesurable process K wich value in ℝ such that $𝔼 (sup_{0 \leq t \leq T} {| {\bar{K}}_{t}^{n} - K_{t} |}^{2}) \to 0,$ \mathbb{E}\left({\sup}\limits_{0\leq t\leq T}\left\vert \bar{K}_{t}^{n}-K_{t}\right\vert^{2}\right) \rightarrow 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, ${\bar{Y}}_{t}^{n} \geq S_{t}$ \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 $\int_{0}^{T} ({\bar{Y}}_{s}^{n} - S_{s}) d {\bar{K}}_{s}^{n} \to \int_{0}^{T} (Y_{s} - S_{s}) {dK}_{s},$ \int_{0}^{T}\left(\bar{Y}_{s}^{n}-S_{s}\right) d\bar{K}_{s}^{n}\rightarrow\int_{0}^{T}\left(Y_{s}-S_{s}\right) dK_{s},

$\bar{ℙ}$ \mathbb{\bar{P}} – a.s., as n → ∞. Using the identite $\int_{0}^{T} ({\bar{Y}}_{s}^{n} - S_{s}) d {\bar{K}}_{s}^{n} = 0$ \int_{0}^{T}\left(\bar{Y}_{s}^{n}-S_{s}\right) d\bar{K}_{s}^{n}=0 , for all n ≥ 0 we conclude that $\int_{0}^{T} (Y_{s} - S_{s}) {dK}_{s} \geq 0$ \int_{0}^{T}\left(Y_{s}-S_{s}\right) 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 ${\bar{Y}}_{\cdot}^{n} \leq Y_{\cdot}^{*}$ \bar{Y}_{\cdot}^{n}\leq Y_{\cdot}^{\ast} , for all n ≥ 0 and therefore, Y. ≤ Y* i.e., Y is the minimal solution.

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
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Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

Published Online: Oct 28, 2020

Page range: 205 - 216

Received: May 14, 2019

Accepted: Mar 30, 2020

DOI: https://doi.org/10.2478/amns.2020.2.00038

KeywordsReflected Backward doubly stochastic differential equations, Mean-field, Minimal solution, Comparison theorem, Discontinuous generator

© 2020 Mostapha Abdelouahab Saouli, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Reflected Backward doubly stochastic differential equations, Mean-field, Minimal solution, Comparison theorem, Discontinuous generator