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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}=\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}=\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 dWt and a backward stochastic integral dBt 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}=\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 Yt ≥ St a.s. for any t ∈ [0, T ]. The role of the nondecreasing continuous process (Kt)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}\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,
\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 {Wt, 0 ≤ t ≤ T} and {Bt, 0 ≤ t ≤ T} be two independent standard Brownian motion defined on (Ω, ℱ, P) with values in ℝd and ℝ, respectively.
Let
\mathcal{F}_{t}^{W}:=\sigma (W_{s};0\leq s\leq t)
, and
\mathcal{F}_{t,T}^{B}:=\sigma (B_{s}-B_{t};t\leq s\leq T),
, completed with P-null sets. We put,
\mathcal{F}_{t}:=\mathcal{F}_{t}^{W}\vee \mathcal{F}_{t,T}^{B}.
It should be noted that (ℱt) is not an increasing family of sub σ–fields, and hence it is not a filtration.
Let
\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
\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{\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
\theta \in L^{1}\left(\bar{\Omega},\mathcal{\bar{F}},\bar{P}\right)
, the variable θ (·, ω) : Ω → ℝ belongs to
L^{1}\left(\bar{\Omega},\mathcal{\bar{F}},\bar{P}\right)
, P (dω)−a.s,. We denote its expectation by
\acute{E}\left(\theta \left(\cdot,\omega \right) \right) =\int_{\Omega}\theta\left(\acute{\omega},\omega \right) P\left(d\acute{\omega}\right)
Let ℳ2 (0, T, ℝd) denote the set of d– dimensional, ℱt– progressively measurable processes {φt;t ∈ [0, T ]}, such that
\mathbb{E}\int_{0}^{T}\left\vert \varphi_{t}\right\vert^{2}dt<\infty
.
We denote by 𝒮2 (0, T, ℝd), the set of ℱt– adapted cádlág processes {φt; t ∈ [0, T]}, which satisfy 𝔼(sup0 ≤ t ≤ T|φt|2) < ∞.
𝒜2 set of continuous, increasing, ℱt-adapted process K: [0, T] × Ω → [0, +∞) with K0 = 0 and 𝔼(KT)2 < +∞.
𝕃2 set of ℱT- 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, ℝd) × ℳ2 (0, T, ℝd) × 𝒜2 and satisfies (2) such that:
\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) (St)t ≥ 0, is a continuous progressively measurable real valued process satisfying
\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], ST ≤ ξ, ℙ-almost surely.
(H.4)f : Ω × [0, T] × ℝ × ℝ × ℝd × ℝd → ℝ; g : Ω × [0, T] × ℝ × ℝ × ℝd × ℝd → ℝk be jointly measurable such that for any (y, y′, z, z′) ∈ ℝ × ℝ × ℝd × ℝd,
\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 \alpha \leq \frac{1}{2}
such that for every (ω, t) ∈ Ω × [0, T ] and (y, y′) ∈ ℝ2, (z, z′) ∈ ℝd × ℝd,
\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 \alpha \leq \frac{1}{2}
such that for every (ω, t) ∈ Ω × [0, T] and (y, y′) ∈ ℝ2, (z, z′) ∈ ℝd × ℝd,
\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, ℝd) × ℳ2 (0, T, ℝd) × 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 ft ∈ 𝕄2 (0, T, ℝd) such that
\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 × ℝd satisfying for y1 ≥ y2,
\left( y_{1}^{^{\prime}},y_{2}^{^{\prime}}\right) \in \left( \mathbb{R}\right)^{2}
, (z1, z2) ∈ (ℝd)2\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:
\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, ℝd) × ℳ2 (0, T, ℝd) × 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, ℝd). For a continuous function of finite variation
\tilde{K}
belong in 𝒜2we consider the processes\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:\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
The MF-RBDSDE (4) has a least one solution\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}
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
\left\vert \tilde{Y}_{t}^{-}\right\vert^{2}
, we have
\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\mathbb{E}\int_{t}^{T}\tilde{Y}_{s}^{-}d\tilde{K}_{s}\leq 0
, h(s) ≥ 0 and ξ ≥ 0, we get
\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
\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
\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
, ℙ – a.s., ∀t ∈ [0, T], which implies that
\tilde{Y}_{t}\geq 0
ℙ – a.s., ∀t ∈ [0, T].
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
\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
\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
\Lambda_{s}^{1}\geq 0
, because
\left(\bar{Y}_{t}^{0},\bar{Z}_{t}^{0}\right)
is the solution of Eq. (5), we get
\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}
. Now we want to prove
\bar{Y}_{t}^{n}\leq \bar{Y}_{t}^{n+1}
, for any n ≥ 0. We set
\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
\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
\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
\theta_{s}^{0}=\Lambda_{s}^{1}
, ∀n ≥ 0. According to it definition, one cas show that
\theta_{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,
\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
\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
\delta \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)\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
\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
, 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},\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 (ℱT, 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 \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
\left(\bar{Y}_{t}^{n}\right)_{n\geq 0}
is increasing and bounded in ℳ2 (0, T, ℝd). Since
\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
\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
\left(\bar{Y}_{t}^{n}\right)_{n\geq 0}
converges in 𝒮2 (0, T, ℝ). We denote by
\bar{Y}
the limit of
\left(\bar{Y}_{t}^{n}\right)_{n\geq 0}
.
On the other hand from Eq. (6), we deduce that
\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}
Using the two inequalities (8) and (9), we obtain
\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
\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\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
\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
\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 Cθ depending on α, ξ, C and θ, we derive
\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^{\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\alpha <1
, we obtain
\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
\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
\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
\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
\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}+\theta C_{2}<1
and notting
\mathbb{E}\int_{0}^{T}\left\vert \bar{Z}_{s}^{0}\right\vert^{2}ds<\infty
. Obtain
\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
\mathbb{E}\left\vert \bar{K}_{T}^{n+1}\right\vert^{2}<\infty.
Now we shall prove that
\left(\bar{Z}^{n},\bar{K}^{n}\right)
is a Cauchy sequence in ℳ2 (0, T, ℝd) × 𝒜2.
Applying Itô's formula to
\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
\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
\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}\left(d\bar{K}_{s}^{n}-d\bar{K}_{s}^{m}\right) \leq 0
, we obtain
\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
\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
\left(\bar{Y}^{n},\bar{Z}^{n},\bar{K}^{n}\right)
, we deduce that the
\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
\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
\left(\bar{Z}^{n}\right)_{n\geq 0}
is a Cauchy sequence in ℳ2 (0, T, ℝd). Then there exists Z ∈ ℳ2 (ℝd) such that
\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
\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\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}),}
\mathbb{\bar{P}}
– a.s., for all t ∈ [0, T] as n → ∞. Then follows by Lebesgue's dominated convergence theorem that
\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
\left(\tilde{Y}_{s},\tilde{Z}_{s},\Gamma_{s}^{n}\right)
converges in 𝒮2 (0, T, ℝ) × ℳ2 (0, T, ℝd) × ℳ2 (0, T, ℝ2) and
\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
\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 ℱt–mesurable process K wich value in ℝ such that
\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 {Kt; 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 Yt ≥ St, ∀t ∈ [0, T]. On the other hand, from the result of Saisho [8] (in 1987, p. 465), we have
\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},
\mathbb{\bar{P}}
– a.s., as n → ∞. Using the identite
\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}\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}^{\ast}
, for all n ≥ 0 and therefore, Y. ≤ Y* i.e., Y is the minimal solution.