1 Introduction
Backward Stochastic Differential Equations (BSDEs) were introduced (in the non-linear case) by Pardoux and Peng [21]. Precisely, given a data $(\xi ,f)$ of a square integrable random variable ξ and a progressively measurable function f, a solution to BSDE associated with data $(\xi ,f)$ is a pair of $\mathcal{F}_{t}$-adapted processes $(Y,Z)$ satisfying
These equations have attracted great interest due to their connections with mathematical finance [9, 10], stochastic control and stochastic games [3, 17] and partial differential equations [20, 22].
(1)
\[ Y_{t}=\xi +{\int _{t}^{T}}f(s,Y_{s},Z_{s})ds-{\int _{t}^{T}}Z_{s}dB_{s},\hspace{1em}0\le t\le T.\]In their seminal paper [21], Pardoux and Peng generalized such equations to the Lipschitz condition and proved existence and uniqueness results in a Brownian framework. Moreover, many efforts have been made to relax the Lipschitz condition on the coefficient. In this context, Bender and Kohlmann [2] considered the so-called stochastic Lipschitz condition introduced by El Karoui and Huang [8].
Further, El Karoui et al. [11] have introduced the notion of reflected BSDEs (RBSDEs in short), which is a BSDE but the solution is forced to stay above a lower barrier. In detail, a solution to such equations is a triple of processes $(Y,Z,K)$ satisfying
where L, the so-called barrier, is a given stochastic process. The role of the continuous increasing process K is to push the state process upward with the minimal energy, in order to keep it above L; in this sense, it satisfies ${\int _{0}^{T}}(Y_{t}-L_{t})dK_{t}=0$. The authors have proved that equation (2) has a unique solution under square integrability of the terminal condition ξ and the barrier L, and the Lipschitz property of the coefficient f.
(2)
\[ Y_{t}=\xi +{\int _{t}^{T}}f(s,Y_{s},Z_{s})ds+K_{T}-K_{t}-{\int _{t}^{T}}Z_{s}dB_{s},\hspace{1em}Y_{t}\ge L_{t}\hspace{2.5pt}0\le t\le T,\]RBSDEs have been proven to be powerful tools in mathematical finance [10], mixed game problems [6], providing a probabilistic formula for the viscosity solution to an obstacle problem for a class of parabolic partial differential equations [11].
Later, Cvitanic and Karatzas [6] studied doubly reflected BSDEs (DRBSDEs in short). A solution to such an equation related to a generator f, a terminal condition ξ and two barriers L and U is a quadruple of $(Y,Z,{K}^{+},{K}^{-})$ which satisfies
In this case, a solution Y has to remain between the lower barrier L and upper barrier U. This is achieved by the cumulative action of two continuous, increasing reflecting processes ${K}^{\pm }$. The authors proved the existence and uniqueness of the solution when $f(t,\omega ,y,z)$ is Lipschitz on $(y,z)$ uniformly in $(t,\omega )$. At the same time, one of the barriers L or U is regular or they satisfy the so-called Mokobodski condition, which turns out into the existence of a difference of a non-negative supermartingales between L and U. In addition, many efforts have been made to relax the conditions on f, L and U [1, 15, 16, 18, 19, 27, 29] or to deal with other issues [5, 12–14, 24].
(3)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \hspace{-28.45274pt}\left\{\begin{array}{l}Y_{t}=\xi +{\displaystyle \int _{t}^{T}}f(s,Y_{s},Z_{s})ds+\big({K_{T}^{+}}-{K_{t}^{+}}\big)-\big({K_{T}^{-}}-{K_{t}^{-}}\big)-{\displaystyle \int _{t}^{T}}Z_{s}dB_{s}\hspace{1em}\\{} L_{t}\le Y_{t}\le U_{t},\hspace{2.5pt}\forall t\le T\hspace{2.5pt}\text{and}\hspace{2.5pt}{\displaystyle \int _{0}^{T}}(Y_{t}-L_{t})d{K_{t}^{+}}={\displaystyle \int _{0}^{T}}(U_{t}-Y_{t})d{K_{t}^{-}}=0.\hspace{1em}\end{array}\right.\end{array}\]Let us have a look at the pricing problem of an American game option driven by Black–Scholes market model which is given by the following system of stochastic differential equations
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \left\{\begin{array}{l@{\hskip10.0pt}l}d{S_{t}^{0}}=r(t){S_{t}^{0}}dt,& {S_{0}^{0}}>0;\\{} dS_{t}=S_{t}\big(\big(r(t)+\theta (t)\sigma (t)\big)dt+\sigma (t)dB_{t}\big),& S_{0}>0,\end{array}\right.\end{array}\]
where $r(t)$ is the interest rate process, $\theta (t)$ is the risk premium process, $\sigma (t)$ is the volatility process of the market. The fair price of the American game option is defined by
\[ Y_{t}=\underset{\tau \in \mathrm{\Im }_{[0,T]}}{\inf }\underset{\nu \in \mathrm{\Im }_{[0,T]}}{\sup }\mathbb{E}\big[{e}^{-r(t)\sigma (t)\wedge \theta (t)}J(\tau ,\nu )|\mathcal{F}_{t}\big],\]
where $\mathrm{\Im }_{[0,T]}$ is the collection of all stopping times τ with values between 0 and T, and J is a Payoff given by
\[ J(\tau ,\nu )=U_{\nu }\mathbb{1}_{\{\nu <\tau \}}+L_{\tau }\mathbb{1}_{\{\tau \le \nu \}}+\xi \mathbb{1}_{\{\nu \wedge \tau =T\}}.\]
Here $r(t)$, $\sigma (t)$ and $\theta (t)$ are stochastic, moreover they are not bounded in general. So the existence results of Cvitanic and Karatzas [6], Li and Shi [19] with completely separated barriers cannot be applied.Motivated by the above works, the purpose of the present paper is to consider a class of DRBSDEs driven by a Brownian motion with stochastic Lipschitz coefficient. We try to get the existence and uniqueness of solutions to those DRBSDEs by means of the penalization method and the fixed point theorem. Furthermore, the comparison theorem for the solutions to DRBSDEs will be established.
The paper is organized as follows: in Section 2, we give some notations and assumptions needed in this paper. In Section 3, we establish the a priori estimates of solutions to DRBSDEs. In Section 4, we prove the existence and uniqueness of solutions to DRBSDEs via penalization method when one barrier is regular, in the first subsection, then we study the case when the barriers are completely separated, in the second subsection. In Section 5, we give the comparison theorem for the solutions to DRBSDEs. Finally, an Appendix is devoted to the special case of RBSDEs with lower barrier when the generator only depends on y; furthermore, the corresponding comparison theorem will be established under the stochastic Lipschitz coefficient.
2 Notations
Let $(\varOmega ,\mathcal{F},(\mathcal{F}_{t})_{t\le T},\mathbb{P})$ be a filtered probability space. Let $(B_{t})_{t\le T}$ be a d-dimensional Brownian motion. We assume that $(\mathcal{F}_{t})_{t\le T}$ is the standard filtration generated by the Brownian motion $(B_{t})_{t\le T}$.
We will denote by $|.|$ the Euclidian norm on ${\mathbb{R}}^{d}$.
Let’s introduce some spaces:
Let $\beta >0$ and $(a_{t})_{t\le T}$ be a non-negative $\mathcal{F}_{t}$-adapted process. We define the increasing continuous process $A(t)={\int _{0}^{t}}{a}^{2}(s)ds$, for all $t\le T$, and introduce the following spaces:
We consider the following conditions:
The coefficient $f:\varOmega \times [0,T]\times \mathbb{R}\times {\mathbb{R}}^{d}\longrightarrow \mathbb{R}$ satisfies
The two reflecting barriers L and U are two $\mathcal{F}_{t}$-adapted and continuous real-valued processes which satisfy
$(H2)$
$\forall t\in [0,T]$ $\forall (y,z,{y^{\prime }},{z^{\prime }})\in \mathbb{R}\times {\mathbb{R}}^{d}\times \mathbb{R}\times {\mathbb{R}}^{d}$, there are two non-negative $\mathcal{F}_{t}$-adapted processes μ and γ such that
$(H3)$
There exists $\epsilon >0$ such that ${a}^{2}(t):=\mu (t)+{\gamma }^{2}(t)\ge \epsilon $.
Definition 1.
Let $\beta >0$ and a be a non-negative $\mathcal{F}_{t}$-adapted process. A solution to DRBSDE is a quadruple $(Y,Z,{K}^{+},{K}^{-})$ satisfying (3) such that
3 A priori estimate
Lemma 1.
Let $\beta >0$ be large enough and assume $(H1)-(H6)$ hold. Let $(Y,Z,{K}^{+},{K}^{-})\in ({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}\times {\mathcal{S}}^{2}$ be a solution to DRBSDE with data $(\xi ,f,L,U)$. Then there exists a constant $C_{\beta }$ depending only on β such that
(4)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}|Y_{t}{|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}\big({a}^{2}(t)|Y_{t}{|}^{2}+|Z_{t}{|}^{2}\big)dt+{\big|{K_{T}^{+}}\big|}^{2}+{\big|{K_{T}^{-}}\big|}^{2}\Bigg]\\{} & \displaystyle \hspace{1em}\le C_{\beta }\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}\frac{|f(t,0,0){|}^{2}}{{a}^{2}(t)}dt\\{} & \displaystyle \hspace{2em}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}\big({\big|{L_{t}^{+}}\big|}^{2}+{\big|{U_{t}^{-}}\big|}^{2}\big)\Bigg].\end{array}\]Proof.
Applying Itô’s formula and Young’s inequality, combined with the stochastic Lipschitz assumption $(H2)$ we can write
Taking expectation on both sides above, we get
and by the Burkholder–Davis–Gundy’s inequality we obtain
To conclude, we now give an estimate of ${{K_{T}^{+}}}^{2}$ and ${{K_{T}^{-}}}^{2}$. From the equation
The desired result is obtained by estimates (6), (8) and (9). □
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {e}^{\beta A(t)}|Y_{t}{|}^{2}+{\int _{t}^{T}}\beta {e}^{\beta A(s)}{a}^{2}(s)|Y_{s}{|}^{2}ds+{\int _{t}^{T}}{e}^{\beta A(s)}|Z_{s}{|}^{2}ds\\{} & \displaystyle \hspace{1em}\le {e}^{\beta A(T)}|\xi {|}^{2}+\frac{\beta }{2}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|Y_{s}{|}^{2}ds+\frac{2}{\beta }{\int _{t}^{T}}{e}^{\beta A(s)}\frac{|f(s,Y_{s},Z_{s}){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+2{\int _{t}^{T}}{e}^{\beta A(s)}Y_{s}d{K_{s}^{+}}-2{\int _{t}^{T}}{e}^{\beta A(s)}Y_{s}d{K_{s}^{-}}-2{\int _{t}^{T}}{e}^{\beta A(s)}Y_{s}Z_{s}dB_{s}\\{} & \displaystyle \hspace{1em}\le {e}^{\beta A(T)}|\xi {|}^{2}+\frac{\beta }{2}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|Y_{s}{|}^{2}ds+\frac{6}{\beta }{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|Y_{s}{|}^{2}ds\\{} & \displaystyle \hspace{2em}+\frac{6}{\beta }{\int _{t}^{T}}{e}^{\beta A(s)}|Z_{s}{|}^{2}ds+\frac{6}{\beta }{\int _{t}^{T}}{e}^{\beta A(s)}\frac{|f(s,0,0){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+2{\int _{t}^{T}}{e}^{\beta A(s)}Y_{s}d{K_{s}^{+}}-2{\int _{t}^{T}}{e}^{\beta A(s)}Y_{s}d{K_{s}^{-}}-2{\int _{t}^{T}}{e}^{\beta A(s)}Y_{s}Z_{s}dB_{s}.\end{array}\]
Using the fact that $d{K_{s}^{+}}=\mathbb{1}_{\{Y_{s}=L_{s}\}}d{K_{s}^{+}}$ and $d{K_{s}^{-}}=\mathbb{1}_{\{Y_{s}=U_{s}\}}d{K_{s}^{-}}$, we have
(5)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {e}^{\beta A(t)}|Y_{t}{|}^{2}+\bigg(\frac{\beta }{2}-\frac{6}{\beta }\bigg){\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|Y_{s}{|}^{2}ds+\bigg(1-\frac{6}{\beta }\bigg){\int _{t}^{T}}{e}^{\beta A(s)}|Z_{s}{|}^{2}ds\\{} & \displaystyle \hspace{1em}\le {e}^{\beta A(T)}|\xi {|}^{2}+\frac{6}{\beta }{\int _{t}^{T}}{e}^{\beta A(s)}\frac{|f(s,0,0){|}^{2}}{{a}^{2}(s)}ds+2{\int _{t}^{T}}{e}^{\beta A(s)}L_{s}d{K_{s}^{+}}\\{} & \displaystyle \hspace{2em}-2{\int _{t}^{T}}{e}^{\beta A(s)}U_{s}d{K_{s}^{-}}-2{\int _{t}^{T}}{e}^{\beta A(s)}Y_{s}Z_{s}dB_{s}.\end{array}\](6)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\Bigg[{\int _{0}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|Y_{s}{|}^{2}ds+{\int _{0}^{T}}{e}^{\beta A(s)}|Z_{s}{|}^{2}ds\Bigg]\\{} & \displaystyle \hspace{1em}\le c_{\beta }\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(s)}\frac{|f(s,0,0){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{L_{t}^{+}}\big|}^{2}+{\big|{K_{T}^{+}}\big|}^{2}+\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{U_{t}^{-}}\big|}^{2}+{\big|{K_{T}^{-}}\big|}^{2}\Bigg]\end{array}\](7)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}|Y_{t}{|}^{2}\\{} & \displaystyle \hspace{1em}\le \mathcal{C}_{\beta }\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(s)}\frac{|f(s,0,0){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+2{\int _{t}^{T}}{e}^{\beta A(s)}L_{s}d{K_{s}^{+}}-2{\int _{t}^{T}}{e}^{\beta A(s)}L_{s}d{K_{s}^{-}}\Bigg]\end{array}\](8)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \hspace{1em}\le \mathcal{C}_{\beta }\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(s)}\frac{|f(s,0,0){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}\big({\big|{L_{t}^{+}}\big|}^{2}+{\big|{U_{t}^{-}}\big|}^{2}\big)+{\big|{K_{T}^{+}}\big|}^{2}+{\big|{K_{T}^{-}}\big|}^{2}\Bigg].\end{array}\]
\[ {K_{T}^{+}}-{K_{T}^{-}}=Y_{0}-\xi -{\int _{0}^{T}}f(s,Y_{s},Z_{s})ds+{\int _{0}^{T}}Z_{s}dB_{s}\]
and the stochastic Lipschitz property $(H2)$, we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\big[{\big|{K_{T}^{+}}-{K_{T}^{-}}\big|}^{2}\big]\\{} & \displaystyle \hspace{1em}\le 4\mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}|Y_{t}{|}^{2}+|\xi {|}^{2}+\bigg(1+\frac{3}{\beta }\bigg){\int _{0}^{T}}{e}^{\beta A(s)}|Z_{s}{|}^{2}ds\\{} & \displaystyle \hspace{2em}+\frac{3}{\beta }{\int _{0}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|Y_{s}{|}^{2}ds+\frac{3}{\beta }{\int _{0}^{T}}{e}^{\beta A(s)}\frac{|f(s,0,0){|}^{2}}{{a}^{2}(s)}ds\Bigg].\end{array}\]
Combining this with (7), we derive that
(9)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{E}{\big|{K_{T}^{+}}\big|}^{2}+\mathbb{E}{\big|{K_{T}^{-}}\big|}^{2}& \displaystyle \le \mathfrak{C}_{\beta }\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(s)}\frac{|f(s,0,0){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{1em}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}\big({\big|{L_{t}^{+}}\big|}^{2}+{\big|{U_{t}^{-}}\big|}^{2}\big)\Bigg]+\frac{1}{2}\mathbb{E}{\big|{K_{T}^{+}}\big|}^{2}+\frac{1}{2}\mathbb{E}{\big|{K_{T}^{-}}\big|}^{2}.\end{array}\]4 Existence and uniqueness of solution
4.1 The obstacle U is regular
In this part, we apply the penalization method and the fixed point theorem to give the existence of the solution to the DRBSDE (3). We first consider the special case when the generator does not depend on $(y,z)$:
Theorem 1.
Assume that $\frac{g}{a}\in {\mathcal{H}}^{2}(\beta ,a)$ and $(H1)$–$(H6)$ hold. Then, the doubly reflected BSDE (3) with data $(\xi ,g,L,U)$ has a unique solution $(Y,Z,{K}^{+},{K}^{-})$ that belongs to $({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}\times {\mathcal{S}}^{2}$.
For all $n\in \mathbb{N}$, let $({Y}^{n},{Z}^{n},{K}^{n+})$ be the $\mathcal{F}_{t}$-adapted process with values in $({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}$ being a solution to the reflected BSDE with data $(\xi ,g(t)-n{(y-U_{t})}^{+},L)$. That is
We denote ${K_{t}^{n-}}:=n{\int _{0}^{t}}{({Y_{s}^{n}}-U_{s})}^{+}ds$ and ${g}^{n}(s,y):=g(s)-n{(y-U_{s})}^{+}$.
(10)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \hspace{-28.45274pt}\left\{\begin{array}{l}{Y_{t}^{n}}=\xi +{\displaystyle \int _{t}^{T}}g(s)ds-n{\displaystyle \int _{t}^{T}}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds+{K_{T}^{n+}}-{K_{t}^{n+}}-{\displaystyle \int _{t}^{T}}{Z_{s}^{n}}dB_{s}\\{} {Y_{t}^{n}}\ge L_{t},\hspace{2.5pt}\forall t\le T\hspace{2.5pt}\text{and}\hspace{2.5pt}{\displaystyle \int _{0}^{T}}\big({Y_{t}^{n}}-L_{t}\big)d{K_{t}^{n+}}=0.\end{array}\right.\end{array}\]We have divided the proof of Theorem 1 into sequence of lemmas.
Proof.
For all $n,m\ge 0$, let $({Y}^{n,m},{Z}^{n,m})$ be the solution to the following BSDE
\[ {Y_{t}^{n,m}}=\xi -{\int _{t}^{T}}\big\{g(s)+m{\big({Y_{s}^{n,m}}-L_{s}\big)}^{-}-n{\big({Y_{s}^{n,m}}-U_{s}\big)}^{+}\big\}ds-{\int _{t}^{T}}{Z_{s}^{n,m}}dB_{s}.\]
We denote ${\bar{Y}}^{n,m}={Y}^{n,m}-{U}^{m}$. Then we have
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {\bar{Y}_{t}^{n,m}}& \displaystyle =\xi -{U_{T}^{m}}+{\int _{t}^{T}}\big(g(s)+u_{m}(s)\big)ds-n{\int _{t}^{T}}{\big({\bar{Y}_{s}^{n,m}}-\big(U_{s}-{U_{s}^{m}}\big)\big)}^{+}ds\\{} & \displaystyle \hspace{1em}+m{\int _{t}^{T}}{\big({\bar{Y}_{s}^{n,m}}-\big(L_{s}-{U_{s}^{m}}\big)\big)}^{-}ds-{\int _{t}^{T}}\big({Z_{s}^{n,m}}-v_{n}(s)\big)dB_{s}.\end{array}\]
For $n\ge 0$, let $\mathcal{D}_{n}$ be the class of $\mathcal{F}_{t}$-progressively measurable process taking values in $[0,n]$. For $\nu \in \mathcal{D}_{n}$ and $\lambda \in \mathcal{D}_{m}$ we denote $R_{t}={e}^{-{\int _{0}^{t}}(\nu (s)+\lambda (s))ds}$. Applying Itô’s formula to $R_{t}{\bar{Y}_{t}^{n,m}}$ and using the same arguments as on page 2042 of [6], one can show that
\[ {\bar{Y}_{t}^{n,m}}\le \underset{\lambda \in \mathcal{D}_{m}}{ess sup}\underset{\nu \in \mathcal{D}_{n}}{ess inf}\mathbb{E}\Bigg[{\int _{t}^{T}}{e}^{-{\textstyle\int _{t}^{s}}(\nu (r)+\lambda (r))dr}\big|u_{m}(s)\big|ds|\mathcal{F}_{t}\Bigg].\]
From the assumption $(H6)(ii)$, we can write ${\bar{Y}_{t}^{n,m}}\vee 0\le \frac{C}{n}$. It follows that
□Lemma 3.
There exists a positive constant ${C^{\prime }_{\beta }}$ depending only on β such that for all $n\ge 0$
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}\big|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}{a}^{2}(t){\big|{Y_{t}^{n}}\big|}^{2}dt+{\int _{0}^{T}}{e}^{\beta A(t)}{\big|{Z_{t}^{n}}\big|}^{2}dt+{\big|{K_{T}^{n+}}\big|}^{2}\Bigg]\\{} & \displaystyle \hspace{1em}\le {C^{\prime }_{\beta }}\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}{\bigg|\frac{g(t)}{a(t)}\bigg|}^{2}dt\\{} & \displaystyle \hspace{2em}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}{\big|{U_{t}^{-}}\big|}^{2}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}{\big|{L_{t}^{+}}\big|}^{2}\Bigg].\end{array}\]
Proof.
Itô’s formula implies for $t\le T$:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \beta \mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}\big|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}\le \mathbb{E}{e}^{\beta A(T)}|\xi {|}^{2}+\frac{\beta }{2}\mathbb{E}{\int _{t}^{T}}{e}^{2\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}\big|}^{2}ds+\frac{2}{\beta }\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\frac{|g(s){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+2\mathbb{E}\Bigg[\underset{n\ge 0}{\sup }\underset{0\le t\le T}{\sup }n{\big({Y_{t}^{n}}-U_{t}\big)}^{+}{\int _{t}^{T}}{e}^{\beta A(s)}{U_{s}^{-}}ds\Bigg]+2\mathbb{E}\Bigg[{\int _{t}^{T}}{e}^{\beta A(s)}L_{s}d{K_{s}^{n+}}\Bigg].\end{array}\]
Here we used the fact that $-n{Y_{s}^{n}}{({Y_{s}^{n}}-U_{s})}^{+}\le n{U}^{-}{({Y_{s}^{n}}-U_{s})}^{+}$ and $d{K_{s}^{n+}}=\mathbb{1}_{\{{Y_{s}^{n}}=L_{s}\}}d{K_{s}^{n+}}$. We conclude, by the Burkholder–Davis–Gundy’s inequality, that
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}\big|}^{2}+\mathbb{E}{\int _{0}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}\big|}^{2}ds+\mathbb{E}{\int _{0}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}\le {c^{\prime }_{p}}\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(s)}\frac{|g(s){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}{\big|{U_{t}^{-}}\big|}^{2}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}{\big|{L_{t}^{+}}\big|}^{2}+{\big|{K_{T}^{n+}}\big|}^{2}\Bigg].\end{array}\]
In the same way as (9), we can prove that
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{E}{\big|{K_{T}^{n+}}\big|}^{2}& \displaystyle \le {\mathcal{C}^{\prime }_{p}}\mathbb{E}\Bigg[{e}^{\beta A(T)}|\xi {|}^{2}+{\int _{0}^{T}}{e}^{\beta A(s)}\frac{|g(s){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{1em}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}{\big|{U_{t}^{-}}\big|}^{2}+\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}{\big|{L_{t}^{+}}\big|}^{2}\Bigg].\end{array}\]
We obtain the desired result. □Lemma 4.
There exist two $\mathcal{F}_{t}$-adapted processes $(Y_{t})_{t\le T}$ and $({K_{t}^{+}})_{t\le T}$ such that ${Y}^{n}\searrow Y$, ${K}^{n+}\nearrow {K}^{+}$ and
Proof.
The comparison Theorem 5 (below) shows that ${Y_{t}^{0}}\ge {Y_{t}^{n}}\ge {Y_{t}^{n+1}}$ and ${K_{t}^{n+}}\le {K_{t}^{(n+1)+}}$ for all $t\le T$. Therefore, there exist processes Y and ${K}^{+}$ such that, as $n\to +\infty $, for all $t\le T$, ${Y_{t}^{n}}\searrow Y_{t}$ and ${K_{t}^{n+}}\nearrow {K_{t}^{+}}$. Since the process ${K}^{+}$ is continuous, it follows by Dini’s theorem that
□
Proof.
Since $Y_{t}\le {Y_{t}^{n}}\le {Y_{t}^{0}}$, we can replace $U_{t}$ by $U_{t}\vee {Y}^{0}$; that is, we may assume that $\mathbb{E}\sup _{0\le t\le T}{e}^{\beta A(t)}|U_{t}{|}^{2}<+\infty $.
Let $({\widetilde{Y}}^{n},{\widetilde{Z}}^{n},{\widetilde{K}}^{n})$ be the solution to the following Reflected BSDE associated with $(\xi ,g-n(y-U),L)$:
The comparison Theorem 5 shows that ${Y}^{n}\le {\widetilde{Y}}^{n}$ and $d{\widetilde{K}}^{n}\le d{K}^{n+}\le d{K}^{+}$. Let $\tau \le T$ be a stopping time. Then we can write
(11)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \hspace{-28.45274pt}\left\{\begin{array}{l}{\widetilde{Y}_{t}^{n}}=\xi +{\displaystyle \int _{t}^{T}}\big(g(s)-n\big({\widetilde{Y}_{s}^{n}}-U_{s}\big)\big)ds+{\widetilde{K}_{T}^{n}}-{\widetilde{K}_{t}^{n}}-{\displaystyle \int _{t}^{T}}{\widetilde{Z}_{s}^{n}}dB_{s}\\{} {\widetilde{Y}_{t}^{n}}\ge L_{t},\hspace{2.5pt}\forall t\le T\hspace{2.5pt}\text{and}\hspace{2.5pt}{\displaystyle \int _{0}^{T}}\big({\widetilde{Y}_{t}^{n}}-L_{t}\big)d{\widetilde{K}_{t}^{n}}=0.\end{array}\right.\end{array}\]
\[ {\widetilde{Y}_{\tau }^{n}}=\mathbb{E}\Bigg[{e}^{-n(T-\tau )}\xi +{\int _{\tau }^{T}}{e}^{-n(s-\tau )}\big(g(s)+nU_{s}\big)ds+{\int _{\tau }^{T}}{e}^{-n(s-\tau )}d{\widetilde{K}_{s}^{n}}|\mathcal{F}_{\tau }\Bigg].\]
Since $\mathbb{E}\sup _{0\le t\le T}{e}^{\beta A(t)}{U_{t}^{2}}<+\infty $, we obtain
\[ {e}^{-n(T-\tau )}\xi +n{\int _{\tau }^{T}}{e}^{-n(s-\tau )}U_{s}ds{\underset{n\to +\infty }{\overset{}{\to }}}\xi \mathbb{1}_{\tau =T}+U_{\tau }\mathbb{1}_{\tau <T}\hspace{1em}\mathbb{P}-\mathrm{a}.\mathrm{s}.\hspace{0.2778em}\mathrm{in}\hspace{0.2778em}{\mathcal{L}}^{2}\]
and the conditional expectation converges also in ${\mathcal{L}}^{2}$. Moreover,
\[ {\Bigg|{\int _{\tau }^{T}}{e}^{-n(s-\tau )}g(s)ds\Bigg|}^{2}\le {\int _{\tau }^{T}}{e}^{\beta A(s)}{\bigg|\frac{g(s)}{a(s)}\bigg|}^{2}ds{\int _{\tau }^{T}}{e}^{-2n(s-\tau )}{e}^{-\beta A(s)}{a}^{2}(s)ds.\]
Then
\[ {\int _{\tau }^{T}}{e}^{-n(s-\tau )}g(s)ds{\underset{n\to +\infty }{\overset{}{\to }}}0\hspace{1em}\mathbb{P}-\text{a.s. in}\hspace{2.5pt}{\mathcal{L}}^{2}.\]
In addition,
\[ 0\le {\int _{\tau }^{T}}{e}^{-n(s-\tau )}d{\widetilde{K}_{s}^{n}}\le {\int _{\tau }^{T}}{e}^{-n(s-\tau )}d{K_{s}^{+}}{\underset{n\to +\infty }{\overset{}{\to }}}0\hspace{2.5pt}\text{in}\hspace{2.5pt}{\mathcal{L}}^{1}.\]
Consequently,
\[ {\widetilde{Y}_{\tau }^{n}}{\underset{n\to +\infty }{\overset{}{\to }}}\xi \mathbb{1}_{\tau =T}+U_{\tau }\mathbb{1}_{\tau <T}\hspace{1em}\mathbb{P}\text{-a.s. in}\hspace{2.5pt}{\mathcal{L}}^{1}.\]
Therefore, $Y_{\tau }\le U_{\tau }$ $\mathbb{P}$-a.s. We deduce, from Theorem 86 page 220 in Dellacherie and Meyer [7], that $Y_{t}\le U_{t}$ for all $t\le T$ $\mathbb{P}$-a.s and then ${e}^{\beta A(t)}{({Y_{t}^{n}}-U_{t})}^{+}\searrow 0$ for all $t\le T$ $\mathbb{P}$-a.s. By Dini’s theorem, we have $\sup _{0\le t\le T}{e}^{\beta A(t)}{({Y_{t}^{n}}-U_{t})}^{+}\searrow 0$ $\mathbb{P}\text{-a.s.}$ and the result follows from the Lebesgue’s dominated convergence theorem. □Lemma 6.
There exist two processes $(Z_{t})_{t\le T}$ and $({K_{t}^{-}})_{t\le T}$ such that
Moreover,
Proof.
For all $n\ge p\ge 0$ and $t\le T$, applying Itô’s formula and taking expectation yields that
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\Bigg[{e}^{\beta A(t)}{\big|{Y_{t}^{n}}\hspace{0.1667em}-\hspace{0.1667em}{Y_{t}^{p}}\big|}^{2}\hspace{0.1667em}+\hspace{0.1667em}\beta {\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}\hspace{0.1667em}-\hspace{0.1667em}{Y_{s}^{p}}\big|}^{2}ds\hspace{0.1667em}+{\int _{t}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}\hspace{0.1667em}-\hspace{0.1667em}{Z_{s}^{p}}\big|}^{2}ds\Bigg]\\{} & \displaystyle \hspace{1em}\le 2\mathbb{E}\Bigg[{\int _{t}^{T}}{e}^{\beta A(s)}{\big({Y_{s}^{p}}-U_{s}\big)}^{+}n{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds\Bigg]\\{} & \displaystyle \hspace{2em}+2\mathbb{E}\Bigg[{\int _{t}^{T}}{e}^{\beta A(s)}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}p{\big({Y_{s}^{p}}-U_{s}\big)}^{+}ds\Bigg]\\{} & \displaystyle \hspace{1em}\le \mathbb{E}{\Big[\underset{0\le t\le T}{\sup }{\big({e}^{\beta A(t)}{\big({Y_{t}^{p}}-U_{t}\big)}^{+}\big)}^{2}\Big]}^{\frac{1}{2}}\mathbb{E}{\Bigg[{\Bigg({\int _{t}^{T}}n{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds\Bigg)}^{2}\Bigg]}^{\frac{1}{2}}\\{} & \displaystyle \hspace{2em}+\mathbb{E}{\Big[\underset{0\le t\le T}{\sup }{\big({e}^{\beta A(t)}{\big({Y_{t}^{n}}-U_{t}\big)}^{+}\big)}^{2}\Big]}^{\frac{1}{2}}\mathbb{E}{\Bigg[{\Bigg({\int _{t}^{T}}p{\big({Y_{s}^{p}}-U_{s}\big)}^{+}ds\Bigg)}^{2}\Bigg]}^{\frac{1}{2}}\end{array}\]
since $({Y_{s}^{n}}-{Y_{s}^{p}})d({K_{s}^{n+}}-{K_{s}^{p+}})\le 0$. Therefore, using Lemmas 2 and 5, we obtain
\[ \mathbb{E}{\int _{0}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}-{Y_{s}^{p}}\big|}^{2}ds+\mathbb{E}{\int _{0}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}-{Z_{s}^{p}}\big|}^{2}ds{\underset{n,p\to +\infty }{\overset{}{\to }}}0.\]
It follows that $({Z}^{n})_{n\ge 0}$ is a Cauchy sequence in complete space ${\mathcal{H}}^{2}(\beta ,a)$. Then there exists an $\mathcal{F}_{t}$-progressively measurable process $(Z_{t})_{t\le T}$ such that the sequence $({Z}^{n})_{n\ge 0}$ tends toward Z in ${\mathcal{H}}^{2}(\beta ,a)$. On the other hand, by the Burkholder–Davis–Gundy’s inequality, one can derive that
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-{Y_{t}^{p}}\big|}^{2}\\{} & \displaystyle \hspace{1em}\le \mathbb{E}{\Big[\underset{0\le t\le T}{\sup }{\big({e}^{\beta A(t)}{\big({Y_{t}^{p}}-U_{t}\big)}^{+}\big)}^{2}\Big]}^{\frac{1}{2}}\mathbb{E}{\Bigg[{\Bigg({\int _{t}^{T}}n{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds\Bigg)}^{2}\Bigg]}^{\frac{1}{2}}\\{} & \displaystyle \hspace{2em}+\mathbb{E}{\Big[\underset{0\le t\le T}{\sup }{\big({e}^{\beta A(t)}{\big({Y_{t}^{n}}-U_{t}\big)}^{+}\big)}^{2}\Big]}^{\frac{1}{2}}\mathbb{E}{\Bigg[{\Bigg({\int _{t}^{T}}p{\big({Y_{s}^{p}}-U_{s}\big)}^{+}ds\Bigg)}^{2}\Bigg]}^{\frac{1}{2}}\\{} & \displaystyle \hspace{2em}+\frac{1}{2}\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-{Y_{t}^{p}}\big|}^{2}+2{c}^{2}\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}-{Z_{s}^{p}}\big|}^{2}ds\end{array}\]
where c is a universal non-negative constant. It follows that
\[ \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-{Y_{t}^{p}}\big|}^{2}{\underset{n,p\to +\infty }{\overset{}{\to }}}0\]
and then
\[ \mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-Y_{t}\big|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}{a}^{2}(t){\big|{Y_{t}^{n}}-Y_{t}\big|}^{2}dt\Bigg]{\underset{n\to +\infty }{\overset{}{\to }}}0.\]
Now, we set
One can show, at least for a subsequence (which we still index by n), that
The proof is completed. □Proof of Theorem 1.
Obviously, the process $(Y_{t},Z_{t},{K_{t}^{+}},{K_{t}^{-}})_{t\le T}$ satisfies, for all $t\le T$,
\[ Y_{t}=\xi +{\int _{t}^{T}}g(s)ds+\big({K_{T}^{+}}-{K_{t}^{+}}\big)-\big({K_{T}^{-}}-{K_{t}^{-}}\big)-{\int _{t}^{T}}Z_{s}dB_{s}.\]
Since ${Y_{t}^{n}}\ge L_{t}$ and from Lemma 5 we have $L_{t}\le Y_{t}\le U_{t}$.In the following, we want to show that
On the other hand, since the function $(Y_{t}(\omega )-L_{t}(\omega ))_{t\le T}$ is continuous, then there exists a sequence of non-negative step functions $({f}^{m}(\omega ))_{m\ge 0}$ which converges uniformly on $[0,T]$ to $Y_{t}(\omega )-L_{t}(\omega )$. That is
It follows that
Since ${\int _{0}^{T}}(U_{s}-{Y_{s}^{n}})n({Y_{s}^{n}}-U_{s})ds={\int _{0}^{T}}(U_{s}-{Y_{s}^{n}})d{K_{s}^{n-}}\le 0$ for each $n\ge 0$ $\mathbb{P}$-a.s. and for each $n,m\ge 0$, $n\ne m$,
Combining (13) and (14), we get ${\int _{0}^{T}}(U_{s}-Y_{s})d{K_{s}^{-}}\le 0\hspace{2.5pt}\mathbb{P}\text{-a.s.}$ Noting that $Y\le U$, we conclude that ${\int _{0}^{T}}(U_{s}-Y_{s})d{K_{s}^{-}}=0$. Consequently, $(Y_{t},Z_{t},{K_{t}^{+}},{K_{t}^{-}})$ is the solution to (3) associated to the data $(\xi ,g,L,U)$. □
\[ {\int _{0}^{T}}(Y_{t}-L_{t})d{K_{t}^{+}}={\int _{0}^{T}}(U_{t}-Y_{t})d{K_{t}^{-}}=0\hspace{1em}\mathbb{P}\text{-a.s.}\]
Note that
\[ {\int _{0}^{T}}(Y_{t}-L_{t})d{K_{t}^{+}}={\int _{0}^{T}}\big(Y_{t}-{Y_{t}^{n}}\big)d{K_{t}^{+}}+{\int _{0}^{T}}\big({Y_{t}^{n}}-L_{t}\big)\big(d{K_{t}^{+}}-d{K_{t}^{n+}}\big).\]
Let $\omega \in \varOmega $ be fixed. It follows from Lemma 4 that, for any $\varepsilon >0$, there exists $n(\omega )$ such that $\forall n\ge n(\omega )$, $Y_{t}(\omega )\le {Y_{t}^{n}}(\omega )+\varepsilon $. Hence
(12)
\[ {\int _{0}^{T}}\big(Y_{t}(\omega )-{Y_{t}^{n}}(\omega )\big)d{K_{t}^{+}}(\omega )\le \varepsilon {K_{T}^{+}}(\omega ).\]
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\int _{0}^{T}}\big(Y_{t}(\omega )-L_{t}(\omega )\big)d\big({K_{t}^{+}}(\omega )-{K_{t}^{n+}}(\omega )\big)\\{} & \displaystyle \hspace{1em}\le \varepsilon \big({K_{T}^{+}}(\omega )+{K_{T}^{n+}}(\omega )\big)+{\int _{0}^{T}}{f_{t}^{m}}(\omega )d\big({K_{t}^{+}}(\omega )-{K_{t}^{n+}}(\omega )\big).\end{array}\]
Further,
\[ \varepsilon \big({K_{T}^{+}}(\omega )+{K_{T}^{n+}}(\omega )\big){\underset{n\to +\infty }{\overset{}{\to }}}2\varepsilon {K_{T}^{+}}(\omega )\]
and, since $({f}^{m}(\omega ))_{m\ge 0}$ is a step function,
\[ {\int _{0}^{T}}{f_{t}^{m}}(\omega )d\big({K_{t}^{+}}(\omega )-{K_{t}^{n+}}(\omega )\big){\underset{m\to +\infty }{\overset{}{\to }}}0.\]
Therefore, we have
\[ \underset{n\to +\infty }{\limsup }{\int _{0}^{T}}\big({Y_{t}^{n}}-L_{t}\big)d\big({K_{t}^{+}}-{K_{t}^{n+}}\big)\le 2\varepsilon {K_{T}^{+}}(\omega ).\]
From (12) we deduce that
The arbitrariness of ε and $Y\ge L$, show that ${\int _{0}^{T}}(Y_{t}-L_{t})d{K_{t}^{+}}=0$. Further, by Lemma 4 and the result treated on p. 465 of Saisho [25] we can write
(13)
\[ {\int _{0}^{T}}\big(U_{s}-{Y_{s}^{n}}\big)n\big({Y_{s}^{n}}-U_{s}\big)ds{\underset{n\to +\infty }{\overset{}{\to }}}{\int _{0}^{T}}(U_{s}-Y_{s})d{K_{s}^{-}}.\]
\[ \mathbb{E}\Bigg[\Bigg|{\int _{0}^{T}}\big({Y_{s}^{n}}-{Y_{s}^{m}}\big)d{K_{s}^{m-}}\Bigg|\Bigg]\le \mathbb{E}\Big[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}\big|{Y_{t}^{n}}-{Y_{t}^{m}}\big|{K_{T}^{m-}}\Big]{\underset{n,m\to +\infty }{\overset{}{\to}}}0.\]
Then we have
(14)
\[ \underset{n\to +\infty }{\limsup }{\int _{0}^{T}}\big(U_{s}-{Y_{s}^{n}}\big)d{K_{t}^{n-}}\le 0\hspace{1em}\mathbb{P}\text{-a.s.}\]We can now state the main result:
Theorem 2.
Assume $(H1)$–$(H6)$ hold for a sufficient large β. Then DRBSDE (3) has a unique solution $(Y,Z,{K}^{+},{K}^{-})$ that belongs to $({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}\times {\mathcal{S}}^{2}$.
Proof.
Given $(\phi ,\psi )\in {\mathfrak{B}}^{2}$, consider the following DRBSDE :
From $(H2)$ and $(H3)$, we have
(15)
\[ \left\{\begin{array}{l}Y_{t}=\xi \hspace{0.1667em}+{\displaystyle \int _{t}^{T}}f(s,\phi _{s},\psi _{s})ds\hspace{0.1667em}+\hspace{0.1667em}({K_{T}^{+}}-{K_{t}^{+}})-({K_{T}^{-}}-{K_{t}^{-}})\hspace{0.1667em}-{\displaystyle \int _{t}^{T}}Z_{s}dB_{s}\hspace{1em}t\hspace{0.1667em}\le \hspace{0.1667em}T\\{} L_{t}\le Y_{t}\le U_{t},\hspace{2.5pt}\forall t\le T\hspace{2.5pt}\text{and}\hspace{2.5pt}{\displaystyle \int _{0}^{T}}(Y_{t}-L_{t})d{K_{t}^{+}}={\displaystyle \int _{0}^{T}}(U_{t}-Y_{t})d{K_{t}^{-}}=0.\end{array}\right.\]
\[ {\big|f(t,\phi _{t},\psi _{t})\big|}^{2}\le 3\big(a{(t)}^{4}|\phi _{t}{|}^{2}+a{(t)}^{2}|\psi _{t}{|}^{2}+{\big|f(t,0,0)\big|}^{2}\big).\]
It follows from $(H4)$ that $\frac{f}{a}\in {\mathcal{H}}^{2}(\beta ,a)$ and then (15) has a unique solution $(Y,Z,{K}^{+},{K}^{-})$.We define a mapping
\[\begin{array}{c@{\hskip10.0pt}c@{\hskip10.0pt}c@{\hskip10.0pt}c@{\hskip10.0pt}c}\varphi & :& {\mathfrak{B}}^{2}& \longrightarrow & {\mathfrak{B}}^{2}\\{} & & (\phi ,\psi )& \longmapsto & (Y,Z)\end{array}\]
Let $\varphi (\phi ,\psi )=(Y,Z)$ and $\varphi ({\phi ^{\prime }},{\psi ^{\prime }})=({Y^{\prime }},{Z^{\prime }})$ where $(Y,Z,{K}^{+},{K}^{-})$ (resp. $({Y^{\prime }},{Z^{\prime }},{K}^{{+^{\prime }}},{K}^{{-^{\prime }}})$) is the unique solution to the DRBSDE associated with data $(\xi ,f(.,\phi ,\psi ),L,U)$ (resp. $(\xi ,f(.,{\phi ^{\prime }},{\psi ^{\prime }}),L,U)$). Denote $\Delta \varGamma =\varGamma -{\varGamma ^{\prime }}$ for $\varGamma =Y,Z,{K}^{+},{K}^{-},\phi ,\psi $ and $\Delta f_{t}=f(t,{\phi ^{\prime }}_{t},{\psi ^{\prime }}_{t})-f(t,\phi _{t},\psi _{t})$. Applying Itô’s formula to ${e}^{\beta A(t)}|\Delta Y_{t}{|}^{2}$ and taking expectation we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}{e}^{\beta A(t)}|\Delta Y_{t}{|}^{2}+\beta \mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|\Delta Y_{s}{|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}|\Delta Z_{s}{|}^{2}ds\\{} & \displaystyle \hspace{1em}\le 2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\Delta Y_{s}\Delta f_{s}ds\\{} & \displaystyle \hspace{1em}\le \alpha \beta \mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s)|\Delta Y_{s}{|}^{2}ds+\frac{2}{\alpha \beta }\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\big({a}^{2}(s)|\Delta \phi _{s}{|}^{2}+|\Delta \psi _{s}{|}^{2}\big)ds.\end{array}\]
We have used the fact that $\Delta Y_{s}d(\Delta {K_{s}^{+}}-\Delta {K_{s}^{-}})\le 0$. Choosing $\alpha \beta =4$ and $\beta >5$, we can write
\[ {\big\| \varphi (\phi ,\psi )\big\| _{\beta }^{2}}\le \frac{1}{2}{\big\| (\phi ,\psi )\big\| _{\beta }^{2}}.\]
It follows that φ is a strict contraction mapping on ${\mathfrak{B}}^{2}$ and then φ has a unique fixed point which is the solution to the DRBSDE (3). □Remark 1.
If we consider $U=+\infty $, we obtain the BSDE with one continuous reflecting barrier L, then we proved the existence and uniqueness of the solution to RBSDE (2) by means of a penalization method. Before this work, Wen Lü [26] showed the existence and uniqueness result for this class of equations via the Snell envelope notion.
4.2 Completely separated barriers
In this section we will prove the existence of solution to (3) when the barriers are completely separated, i.e., $L_{t}<U_{t}$, $\forall t\le T$. Then
We will show the existence by the general penalization method. We first consider the special case when the generator does not depend on $(y,z)$:
Let $({Y}^{n},{Z}^{n})\in ({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)$ be solution to the following BSDE
We denote ${K_{t}^{n+}}:=n{\int _{0}^{t}}{({Y_{s}^{n}}-L_{s})}^{-}ds$, ${K_{t}^{n-}}:=n{\int _{0}^{t}}{({Y_{s}^{n}}-U_{s})}^{+}ds$, ${K_{t}^{n}}={K_{t}^{n+}}-{K_{t}^{n-}}$ and ${f}^{n}(s,y)=f(s)-n{(y-U_{s})}^{+}+n{(y-L_{s})}^{-}$.
$(\mathcal{H}7)$
there exists a continuous semimartingale
with $h\in {\mathcal{H}}^{2}(0,a)$ and ${V}^{\pm }\in {\mathcal{S}}^{2}$ (${V_{0}^{\pm }}=0$) are two nondecreasing continuous processes, such that
(17)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {Y_{t}^{n}}& \displaystyle =\xi +{\int _{t}^{T}}f(s)ds-n{\int _{t}^{T}}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds+n{\int _{t}^{T}}{\big({Y_{s}^{n}}-L_{s}\big)}^{-}ds\\{} & \displaystyle \hspace{1em}-{\int _{t}^{T}}{Z_{s}^{n}}dB_{s}.\end{array}\]Now let us derive the uniform a priori estimates of $({Y}^{n},{Z}^{n},{K}^{n+},{K}^{n-})$.
Lemma 7.
There exists a positive constant κ independent of n such that, $\forall n\ge 0$,
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}\big|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}{a}^{2}(t){\big|{Y_{t}^{n}}\big|}^{2}dt\\{} & \displaystyle \hspace{1em}+{\int _{0}^{T}}{e}^{\beta A(t)}{\big|{Z_{t}^{n}}\big|}^{2}dt+{\big|{K_{T}^{n+}}\big|}^{2}+{\big|{K_{T}^{n-}}\big|}^{2}\Bigg]\le \kappa .\end{array}\]
Proof.
Consider the RBSDE with data $(\xi ,f,L)$. That is,
From Appendix A there exists a unique triplet of processes $(\overline{Y},\overline{Z},\overline{K})\in ({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}$ being the solution to RBSDE (18). We consider the penalization equation associated with the RBSDE (18), for $n\in \mathbb{N}$,
(18)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \left\{\begin{array}{l}\overline{Y}_{t}=\xi +{\displaystyle \int _{t}^{T}}f(s)ds+\overline{K}_{T}-\overline{K}_{t}-{\displaystyle \int _{t}^{T}}\overline{Z}_{s}dB_{s}\\{} \overline{Y}_{t}\ge L_{t},\hspace{2.5pt}\forall t\le T\hspace{2.5pt}\text{and}\hspace{2.5pt}{\displaystyle \int _{0}^{T}}(\overline{Y}_{t}-L_{t})d\overline{K}_{t}=0.\end{array}\right.\end{array}\]
\[ {\overline{Y}_{t}^{n}}=\xi +{\int _{t}^{T}}f(s)ds+n{\int _{t}^{T}}{\big({\overline{Y}_{s}^{n}}-L_{s}\big)}^{-}ds-{\int _{t}^{T}}{\overline{Z}_{s}^{n}}dB_{s}.\]
The Remark 2 implies that ${\overline{Y}_{t}^{0}}\le {\overline{Y}_{t}^{n}}\le {\overline{Y}}^{n+1}$ and ${Y_{t}^{n}}\le {\overline{Y}_{t}^{n}}$ for all $t\le T$. Therefore, as $n\longrightarrow +\infty $ for all $t\le T$, ${\overline{Y}_{t}^{n}}\nearrow \overline{Y}_{t}$. Hence ${Y_{t}^{n}}\le \overline{Y}_{t}$.Similarly, we consider the RBSDE with data $(\xi ,f,U)$. There exists a unique triplet of processes $(\underline{Y},\underline{Z},\underline{K})\in ({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}$, which satisfies
By the penalization equation associated with the RBSDE (19)
On the other hand, using Itô’s formula and taking expectation implies for $t\le T$:
Now we need to estimate $\mathbb{E}{[{\int _{t}^{T}}n{({Y_{s}^{n}}-U_{s})}^{+}ds]}^{2}+\mathbb{E}{[{\int _{t}^{T}}n{({Y_{s}^{n}}-L_{s})}^{-}ds]}^{2}$. For this, let us consider the following stopping times
On the other hand, using the assumption $(\mathcal{H}7)$, we get
In the same way, we obtain
Combining (23), (24) with (21), we obtain the desired result. □
(19)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \left\{\begin{array}{l}\underline{Y}_{t}=\xi +{\displaystyle \int _{t}^{T}}f(s)ds-(\underline{K}_{T}-\underline{K}_{t})-{\displaystyle \int _{t}^{T}}\underline{Z}_{s}dB_{s}\\{} \underline{Y}_{t}\le U_{t},\hspace{2.5pt}\forall t\le T\hspace{2.5pt}\text{and}\hspace{2.5pt}{\displaystyle \int _{0}^{T}}(U_{t}-\underline{Y}_{t})d\underline{K}_{t}=0.\end{array}\right.\end{array}\]
\[ {\underline{Y}_{t}^{n}}=\xi +{\int _{t}^{T}}f(s)ds-n{\int _{t}^{T}}{\big({\underline{Y}_{s}^{n}}-U_{s}\big)}^{+}ds-{\int _{t}^{T}}{\underline{Z}_{s}^{n}}dB_{s}\]
and the Remark 2, we deduce that ${Y_{t}^{n}}\ge \underline{Y}_{t}$ for all $t\le T$. Then we can write
(20)
\[ \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}\big|}^{2}\le \max \Big\{\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}|\overline{Y}_{t}{|}^{2},\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}|\underline{Y}_{t}{|}^{2}\Big\}\le \kappa .\]
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \beta \mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}\big|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}\le \mathbb{E}{e}^{\beta A(T)}|\xi {|}^{2}+2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{Y_{s}^{n}}f(s)ds\\{} & \displaystyle \hspace{2em}-2n\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{Y_{s}^{n}}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds+2n\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{Y_{s}^{n}}{\big({Y_{s}^{n}}-L_{s}\big)}^{-}ds\\{} & \displaystyle \hspace{1em}\le \mathbb{E}{e}^{\beta A(T)}|\xi {|}^{2}+\frac{\beta }{2}\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}\big|}^{2}ds+\frac{2}{\beta }\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\frac{|f(s){|}^{2}}{{a}^{2}(s)}ds\\{} & \displaystyle \hspace{2em}+2n\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{U_{s}^{-}}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds+2n\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{L_{s}^{+}}{\big({Y_{s}^{n}}-L_{s}\big)}^{-}ds.\end{array}\]
Hence
(21)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{\beta }{2}\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}\big|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}\le \mathbb{E}{e}^{\beta A(T)}|\xi {|}^{2}+\frac{2}{\beta }\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\bigg|\frac{f(s)}{a(s)}\bigg|}^{2}ds+\frac{1}{\alpha }\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{2\beta A(t)}\big({\big|{L_{t}^{+}}\big|}^{2}+{\big|{U_{t}^{-}}\big|}^{2}\big)\\{} & \displaystyle \hspace{2em}+\alpha \mathbb{E}{\Bigg[{\int _{t}^{T}}n{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds\Bigg]}^{2}+\alpha \mathbb{E}{\Bigg[{\int _{t}^{T}}n{\big({Y_{s}^{n}}-L_{s}\big)}^{-}ds\Bigg]}^{2}.\end{array}\]
\[ \left\{\begin{array}{l@{\hskip10.0pt}l}\tau _{0}=0,\\{} \tau _{2l+1}=\inf \big\{t>\tau _{2l}\hspace{0.2778em}\hspace{0.2778em}|\hspace{0.2778em}\hspace{0.2778em}{Y_{t}^{n}}\le L_{t}\big\}\wedge T,& l\ge 0\\{} \tau _{2l+2}=\inf \big\{t>\tau _{2l+1}\hspace{0.2778em}\hspace{0.2778em}|\hspace{0.2778em}\hspace{0.2778em}{Y_{t}^{n}}\ge U_{t}\big\}\wedge T,& l\ge 0.\end{array}\right.\]
Since Y, L and U are continuous processes and $L<U$, $\tau _{l}<\tau _{l+1}$ on the set $\{\tau _{l+1}<T\}$. In addition the sequence $(\tau _{l})_{l\ge 0}$ is of stationary type (i.e. $\forall \omega \in \varOmega $, there exists $l_{0}(\omega )$ such that $\tau _{l_{0}}(\omega )=T$). Indeed, let us set $G=\{\omega \in \varOmega ,\tau _{l}(\omega )<T,\hspace{2.5pt}l\ge 0\}$, and we will show that $\mathbb{P}(G)=0$. We assume that $\mathbb{P}(G)>0$, therefore for $\omega \in G$, we have $Y_{\tau _{2l+1}}\le L_{\tau _{2l+1}}$ and $Y_{\tau _{2l}}\ge U_{\tau _{2l}}$. Since $(\tau _{l})_{l\ge 0}$ is nondecreasing sequence then $\tau _{l}\nearrow \tau $, hence $U_{\tau }\le Y_{\tau }\le L_{\tau }$ which is contradiction since $L<U$. We deduce that $\mathbb{P}(G)=0$. Obviously ${Y}^{n}\ge L$ on the interval $[\tau _{2l},\tau _{2l+1}]$, then the BSDE (17) becomes
(22)
\[ {Y_{\tau _{2l}}^{n}}={Y_{\tau _{2l+1}}^{n}}+{\int _{\tau _{2l}}^{\tau _{2l+1}}}f(s)ds-n{\int _{\tau _{2l}}^{\tau _{2l+1}}}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds-{\int _{\tau _{2l}}^{\tau _{2l+1}}}{Z_{s}^{n}}dB_{s}.\]
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {Y_{\tau _{2l}}^{n}}& \displaystyle \ge H_{\tau _{2l}}\hspace{2.5pt}\text{on}\hspace{2.5pt}\{\tau _{2l}<T\}\hspace{1em}\text{and}\hspace{1em}{Y_{\tau _{2l}}^{n}}=H_{\tau _{2l}}=\xi \hspace{2.5pt}\text{on}\hspace{2.5pt}\{\tau _{2l}=T\},\\{} \displaystyle {Y_{\tau _{2l+1}}^{n}}& \displaystyle \le H_{\tau _{2l+1}}\hspace{5pt}\text{on}\hspace{2.5pt}\{\tau _{2l+1}<T\}\hspace{1em}\text{and}\hspace{1em}{Y_{\tau _{2l+1}}^{n}}=H_{\tau _{2l+1}}=\xi \hspace{2.5pt}\text{on}\hspace{2.5pt}\{\tau _{2l+1}=T\}.\end{array}\]
From (22) and the definition of process H we obtain
\[\begin{array}{r@{\hskip0pt}l}\displaystyle n{\int _{\tau _{2l}}^{\tau _{2l+1}}}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds& \displaystyle \le H_{\tau _{2l+1}}-H_{\tau _{2l}}+{\int _{\tau _{2l}}^{\tau _{2l+1}}}f(s)ds-{\int _{\tau _{2l}}^{\tau _{2l+1}}}{Z_{s}^{n}}dB_{s}\\{} & \displaystyle \le {\int _{\tau _{2l}}^{\tau _{2l+1}}}\big(h_{s}-{Z_{s}^{n}}\big)dB_{s}+{\int _{\tau _{2l}}^{\tau _{2l+1}}}\big|f(s)\big|ds+{V_{\tau _{2l+1}}^{-}}-{V_{\tau _{2l}}^{-}}.\end{array}\]
By summing in l, using the fact that ${Y}^{n}\le U$ on the interval $[\tau _{2l+1},\tau _{2l+2}]$, we can write for $t\le T$
(23)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{E}{\Bigg[n{\int _{t}^{T}}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}ds\Bigg]}^{2}& \displaystyle \le 4\Bigg(\mathbb{E}{\int _{t}^{T}}|h_{s}{|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A_{s}}{\big|{Z_{s}^{n}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}+\frac{T}{\beta }\mathbb{E}{\int _{t}^{T}}{e}^{\beta A_{s}}\frac{|f(s){|}^{2}}{{a}^{2}(s)}ds+\mathbb{E}{\big|{V_{T}^{-}}\big|}^{2}\Bigg).\end{array}\](24)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{E}{\Bigg[n{\int _{t}^{T}}{\big({Y_{s}^{n}}-L_{s}\big)}^{-}ds\Bigg]}^{2}& \displaystyle \le 4\Bigg(\mathbb{E}{\int _{t}^{T}}|h_{s}{|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A_{s}}{\big|{Z_{s}^{n}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}+\frac{T}{\beta }\mathbb{E}{\int _{t}^{T}}{e}^{\beta A_{s}}\frac{|f(s){|}^{2}}{{a}^{2}(s)}ds+\mathbb{E}{\big|{V_{T}^{+}}\big|}^{2}\Bigg).\end{array}\]Proof.
Consider the following BSDE for each $n\in \mathbb{N}$
It is easily seen that
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {\widehat{Y}_{t}^{n}}& \displaystyle =\xi +{\int _{t}^{T}}f(s)ds+n{\int _{t}^{T}}\big(L_{s}-{\widehat{Y}_{s}^{n}}\big)ds-{\int _{t}^{T}}{\widehat{Z}_{s}^{n}}dB_{s}\\{} & \displaystyle =\xi +{\int _{t}^{T}}f(s)ds+n{\int _{t}^{T}}{\big({\widehat{Y}_{s}^{n}}-L_{s}\big)}^{-}ds-n{\int _{t}^{T}}{\big(L_{s}-{\widehat{Y}_{s}^{n}}\big)}^{-}ds\hspace{0.1667em}-{\int _{t}^{T}}{\widehat{Z}_{s}^{n}}dB_{s}.\end{array}\]
By the Remark 2, we have ${Y_{t}^{n}}\ge {\widehat{Y}_{t}^{n}}$ for all $t\le T$. Let ν be a stopping time such that $\nu \le T$. Then
(25)
\[ {\widehat{Y}_{\nu }^{n}}=\mathbb{E}\Bigg[{e}^{-n(T-\nu )}\xi +{\int _{\nu }^{T}}{e}^{-n(s-\nu )}f(s)ds+n{\int _{\nu }^{T}}{e}^{-n(s-\nu )}L_{s}ds|\mathcal{F}_{\nu }\Bigg].\]
\[ {e}^{-n(T-\nu )}\xi +n{\int _{\nu }^{T}}{e}^{-n(s-\nu )}L_{s}ds{\underset{n\to +\infty }{\overset{}{\to }}}\xi \mathbb{1}_{\nu =T}+L_{\nu }\mathbb{1}_{\nu <T}\hspace{2em}\mathbb{P}\text{-a.s. in}\hspace{2.5pt}{\mathcal{L}}^{2}.\]
Moreover, the conditional expectation converges also in ${\mathcal{L}}^{2}$. In addition, by the Hölder inequality, we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\Bigg|{\int _{\nu }^{T}}{e}^{-n(s-\nu )}f(s)ds\Bigg|}^{2}\\{} & \displaystyle \hspace{1em}\le \Bigg({\int _{\nu }^{T}}{e}^{\beta A(s)}{\bigg|\frac{f(s)}{a(s)}\bigg|}^{2}ds\Bigg)\Bigg({\int _{\nu }^{T}}{e}^{-2n(s-\nu )-\beta A(s)}{a}^{2}(s)ds\Bigg){\underset{n\to +\infty }{\overset{}{\to }}}0.\end{array}\]
Thus ${\int _{\nu }^{T}}{e}^{-n(s-\nu )}f(s)ds{\underset{n\to +\infty }{\overset{}{\to }}}0$ $\mathbb{P}\text{-a.s. in}\hspace{5pt}{\mathcal{L}}^{2}$.Now, we denote
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\widehat{y}_{t}^{n}}:={e}^{-n(T-t)}\xi +{\int _{t}^{T}}{e}^{-n(s-t)}\big(f(s)+nL_{s}\big)ds,\\{} & \displaystyle {\tilde{y}_{t}^{n}}:={e}^{-n(T-t)}L_{T}+{\int _{t}^{T}}{e}^{-n(s-t)}\big(f(s)+nL_{s}\big)ds\end{array}\]
and
By the fact that L is uniformly continuous on $[0,T]$, it can be shown that the sequence $({X_{t}^{n}})_{n\ge 1}$ uniformly converges in t, and the same for $({X_{t}^{n-}})_{n\ge 1}$. Lebesgue’s dominated convergence theorem implies that
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{n\to +\infty }{\lim }\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{\big({\widehat{y}_{t}^{n}}-L_{t}\big)}^{-}\big|}^{2}=\underset{n\to +\infty }{\lim }\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{\big({\tilde{y}_{t}^{n}}-L_{t}\big)}^{-}\big|}^{2}\\{} & \displaystyle \hspace{1em}\le 2\underset{n\to +\infty }{\lim }\mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{X_{t}^{n-}}\big|}^{2}+\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\Bigg|{\int _{t}^{T}}{e}^{-n(s-t)}f(s)ds\Bigg|}^{2}\Bigg]=0.\end{array}\]
So, from (25), Jensen’s inequality and Doob’s maximal quadratic inequality (see Theorem 20, p. 11 in [23]), we have
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{\big({\widehat{Y}_{t}^{n}}-L_{t}\big)}^{-}\big|}^{2}& \displaystyle \le \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|\mathbb{E}\big[{\big({\widehat{y}_{t}^{n}}-L_{t}\big)}^{-}|\mathcal{F}_{t}\big]\big|}^{2}\\{} & \displaystyle \le 4\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{\big({\widehat{y}_{t}^{n}}-L_{t}\big)}^{-}\big|}^{2}{\underset{n\to +\infty }{\overset{}{\to }}}0.\end{array}\]
From the fact that ${Y_{t}^{n}}\ge {\widehat{Y}_{t}^{n}}$ for all $t\le T$ we deduce that
\[ \underset{n\to +\infty }{\lim }\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{\big({Y_{t}^{n}}-L_{t}\big)}^{-}\big|}^{2}=0.\]
Similarly to proof of the Lemma 5, we can obtain
□Lemma 9.
For each $n\ge p\ge 0$, we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-{Y_{t}^{p}}\big|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}{a}^{2}(t){\big|{Y_{t}^{n}}-{Y_{t}^{p}}\big|}^{2}dt\\{} & \displaystyle \hspace{1em}+{\int _{0}^{T}}{e}^{\beta A(t)}{\big|{Z_{t}^{n}}-{Z_{t}^{p}}\big|}^{2}dt+\underset{0\le t\le T}{\sup }{\big|{K_{t}^{n}}-{K_{t}^{p}}\big|}^{2}\Bigg]{\underset{n,p\to +\infty }{\overset{}{\to }}}0.\end{array}\]
Proof.
Itô’s formula implies that
On the other hand, by the Burkholder–Davis–Gundy’s inequality, we get
From the equation
we can conclude that
The proof is completed. □
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-{Y_{t}^{p}}\big|}^{2}+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\big(\beta {a}^{2}(s){\big|{Y_{s}^{n}}-{Y_{s}^{p}}\big|}^{2}+{\big|{Z_{s}^{n}}-{Z_{s}^{p}}\big|}^{2}\big)ds\\{} & \displaystyle \hspace{1em}\le 2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\big({Y_{s}^{n}}-{Y_{s}^{p}}\big)\big(d{K_{s}^{n+}}-d{K_{s}^{p+}}\big)\\{} & \displaystyle \hspace{2em}-2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\big({Y_{s}^{n}}-{Y_{s}^{p}}\big)\big(d{K_{s}^{n-}}-d{K_{s}^{p-}}\big)\\{} & \displaystyle \hspace{1em}\le 2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big({Y_{s}^{n}}-L_{s}\big)}^{-}d{K_{s}^{p+}}+2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big({Y_{s}^{p}}-L_{s}\big)}^{-}d{K_{s}^{n+}}\\{} & \displaystyle \hspace{2em}+2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big({Y_{s}^{n}}-U_{s}\big)}^{+}d{K_{s}^{p-}}+2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big({Y_{s}^{p}}-U_{s}\big)}^{+}d{K_{s}^{n-}}.\end{array}\]
Hence
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \beta \mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}-{Y_{s}^{p}}\big|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}-{Z_{s}^{p}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}\le 2\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big({Y_{t}^{n}}-L_{t}\big)}^{-}{K_{T}^{p+}}+2\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big({Y_{t}^{p}}-L_{t}\big)}^{-}{K_{T}^{n+}}\\{} & \displaystyle \hspace{2em}+2\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big({Y_{t}^{n}}-U_{t}\big)}^{+}{K_{T}^{p-}}+2\mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big({Y_{t}^{p}}-U_{t}\big)}^{+}{K_{T}^{n-}}.\end{array}\]
Lemma 8 implies that
(26)
\[ \mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{a}^{2}(s){\big|{Y_{s}^{n}}-{Y_{s}^{p}}\big|}^{2}ds+\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}{\big|{Z_{s}^{n}}-{Z_{s}^{p}}\big|}^{2})ds{\underset{n,p\to +\infty }{\overset{}{\to }}}0.\](27)
\[ \mathbb{E}\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-{Y_{t}^{p}}\big|}^{2}{\underset{n,p\to +\infty }{\overset{}{\to }}}0.\]The main result of this section is the following:
Theorem 3.
Assume that $L<U$. Then the DRBSDE (3) has a unique solution $(Y,Z,{K}^{+},{K}^{-})$ that belongs to $({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}\times {\mathcal{S}}^{2}$.
Proof.
From Lemma 9, we obtain that there exists an adapted process $(Y,Z,K)\in ({\mathcal{S}}^{2}(\beta ,a)\cap {\mathcal{S}}^{2,a}(\beta ,a))\times {\mathcal{H}}^{2}(\beta ,a)\times {\mathcal{S}}^{2}$ such that
Then, passing to the limit as $n\to +\infty $ in the equation
(30)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\Bigg[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}{\big|{Y_{t}^{n}}-Y_{t}\big|}^{2}+{\int _{0}^{T}}{e}^{\beta A(t)}{a}^{2}(t){\big|{Y_{t}^{n}}-Y_{t}\big|}^{2}dt\\{} & \displaystyle \hspace{1em}+{\int _{0}^{T}}{e}^{\beta A(t)}{\big|{Z_{t}^{n}}-Z_{t}\big|}^{2}dt+\underset{0\le t\le T}{\sup }{\big|{K_{t}^{n}}-K_{t}\big|}^{2}\Bigg]{\underset{n\to +\infty }{\overset{}{\to }}}0.\end{array}\]
\[ {Y_{t}^{n}}=\xi +{\int _{t}^{T}}f(s)ds+{K_{T}^{n}}-{K_{t}^{n}}-{\int _{t}^{T}}{Z_{s}^{n}}dB_{s},\]
we obtain
Let $\tau \le T$ be a stopping time, by Lemma 7 we obtain that the sequences ${K_{\tau }^{n\pm }}$ are bounded in ${\mathcal{L}}^{2}$, consequently, there exist $\mathcal{F}_{\tau }$-measurable random variables ${K_{\tau }^{\pm }}$ in ${\mathcal{L}}^{2}$, such that there exist the subsequences of ${K_{\tau }^{n\pm }}$ weakly converging in ${K_{\tau }^{\pm }}$.Now we set $\mathcal{K}_{\tau }={K_{\tau }^{+}}-{K_{\tau }^{-}}$. By [28] (Mazu’s Lemma, p. 120), there exists, for every $n\in \mathbb{N}$, an integer $N\ge n$ and a convex combination ${\sum _{j=n}^{N}}{\zeta _{j}^{\tau ,n}}({K_{\tau }^{\pm }})_{j}$ with ${\zeta _{j}^{\tau ,n}}\ge 0$ and ${\sum _{j=n}^{N}}{\zeta _{j}^{\tau ,n}}=1$ such that
Denoting ${\mathcal{K}_{\tau }^{n}}={\mathcal{K}_{\tau }^{n+}}-{\mathcal{K}_{\tau }^{n-}}$, it follows that
Thanks to (30), we have $\| {K_{\tau }^{n}}-K_{\tau }\| _{{\mathcal{L}}^{2}}<\varepsilon $ for all $\varepsilon >0$. Therefore
Combining (32) and (33), we obtain $\mathcal{K}_{\tau }=K_{\tau }$ a.s. Therefore, from Theorem 86, p. 220 in [7] we have $\mathcal{K}_{t}=K_{t}$ for all $t\le T$. On the other hand, (31) implies that, for $\tau =T$, there exists a subsequence of ${\mathcal{K}_{T}^{n+}}:={\sum _{j=n}^{N}}{\zeta _{j}^{T,n}}({K_{T}^{+}})_{j}$ (resp. ${\mathcal{K}_{T}^{n-}}:={\sum _{j=n}^{N}}{\zeta _{j}^{T,n}}({K_{T}^{-}})_{j}$) converging a.s. to ${K_{T}^{+}}$ (resp. ${K_{T}^{-}}$). Then for $\mathbb{P}$-a.s. $\omega \in \varOmega $, the sequence ${\mathcal{K}_{T}^{n+}}(\omega )$ (resp. ${\mathcal{K}_{T}^{n-}}(\omega )$) is bounded. Using Theorem 4.3.3, p. 88 in [4], there exists a subsequence of ${\mathcal{K}_{t}^{n+}}(\omega )$ (resp. ${\mathcal{K}_{t}^{n-}}(\omega )$) tending to ${K_{t}^{+}}(\omega )$ (resp. ${K_{t}^{-}}(\omega )$), weakly.
(31)
\[ {\mathcal{K}_{\tau }^{n\pm }}:={\sum \limits_{j=n}^{N}}{\zeta _{j}^{\tau ,n}}\big({K_{\tau }^{\pm }}\big)_{j}{\underset{n\to +\infty }{\overset{}{\to }}}{K_{\tau }^{\pm }}.\](32)
\[ \mathbb{E}{\big|{\mathcal{K}_{\tau }^{n}}-\mathcal{K}_{\tau }\big|}^{2}{\underset{n\to +\infty }{\overset{}{\to }}}0.\]
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \big\| {\mathcal{K}_{\tau }^{n}}-K_{\tau }\big\| _{{\mathcal{L}}^{2}}& \displaystyle =\Bigg\| {\sum \limits_{j=n}^{N}}{\zeta _{j}^{\tau ,n}}\big(\big({K_{\tau }^{\pm }}\big)_{j}-K_{\tau }\big)\Bigg\| _{{\mathcal{L}}^{2}}\\{} & \displaystyle \le {\sum \limits_{j=n}^{N}}{\zeta _{j}^{\tau ,n}}\big\| \big({K_{\tau }^{\pm }}\big)_{j}-K_{\tau }\big\| _{{\mathcal{L}}^{2}}<\varepsilon .\end{array}\]
Hence
(33)
\[ \mathbb{E}{\big|{\mathcal{K}_{\tau }^{n}}-K_{\tau }\big|}^{2}{\underset{n\to +\infty }{\overset{}{\to }}}0.\]On the other hand, by the definition of stopping times $(\tau _{l})_{l\ge 0}$, we have
Since ${\int _{0}^{T}}({Y_{t}^{n}}-L_{t})d{K_{t}^{n+}}\le 0$, $\forall n\ge 0$ a.s., and $\forall n,m\ge 0$, $n\ne m$,
Combining (34) and (35), we get ${\int _{0}^{T}}(Y_{t}-L_{t})d{K_{t}^{+}}\le 0\hspace{2.5pt}\mathbb{P}\text{-a.s.}$ Noting that $Y\ge L$, we conclude that ${\int _{0}^{T}}(Y_{t}-L_{t})d{K_{t}^{+}}=0$. By a similar consideration, we can prove ${\int _{0}^{T}}(U_{t}-Y_{t})d{K_{t}^{-}}=0$.
\[ \left\{\begin{array}{l@{\hskip10.0pt}l}{Y_{t}^{n}}>L_{t},& on[\tau _{2l},\tau _{2l+1}[;\\{} {Y_{t}^{n}}<U_{t},& on[\tau _{2l+1},\tau _{2l+2}[.\end{array}\right.\]
Then
\[ L_{t}\mathbb{1}_{[\tau _{2i},\tau _{2i+1}]}(t)\le {Y_{t}^{n}}\le U_{t}\mathbb{1}_{[\tau _{2i+1},\tau _{2i+2}]}(t).\]
By summing in i, $i=0,\dots ,l$ and passing to limit in n, we obtain $L_{t}\le Y_{t}\le U_{t}$. Now, we would have to show the Skorokhod’s conditions. Indeed, since ${\mathcal{K}_{t}^{n+}}(\omega )$ tends to ${K_{t}^{+}}(\omega )$, using the result treated in p. 465 of [25] we can write
(34)
\[ {\int _{0}^{T}}\big({Y_{t}^{n}}(\omega )-L_{t}(\omega )\big)d{\mathcal{K}_{t}^{n+}}(\omega ){\underset{n\to +\infty }{\overset{}{\to }}}{\int _{0}^{T}}\big(Y_{t}(\omega )-L_{t}(\omega )\big)d{K_{t}^{+}}(\omega ).\]
\[ \mathbb{E}\Bigg[\Bigg|{\int _{0}^{T}}\big({Y_{t}^{n}}-{Y_{t}^{m}}\big)d{K_{t}^{m+}}\Bigg|\Bigg]\le \mathbb{E}\Big[\underset{0\le t\le T}{\sup }{e}^{\beta A(t)}\big|{Y_{t}^{n}}-{Y_{t}^{m}}\big|{K_{T}^{m+}}\Big]{\underset{n,m\to +\infty }{\overset{}{\to }}}0,\]
then by
\[ {\int _{0}^{T}}\big({Y_{t}^{n}}-L_{t}\big)d{K_{t}^{m+}}={\int _{0}^{T}}\big({Y_{t}^{n}}-{Y_{t}^{m}}\big)d{K_{t}^{m+}}+{\int _{0}^{T}}\big({Y_{t}^{m}}-L_{t}\big)d{K_{t}^{m+}}\]
we have
(35)
\[ \underset{n\to +\infty }{\limsup }{\int _{0}^{T}}\big({Y_{t}^{n}}-L_{t}\big)d{\mathcal{K}_{t}^{n+}}\le 0\hspace{1em}\mathbb{P}\text{-a.s.}\]Finally, using the fixed point theorem we construct a strict contraction mapping φ on ${\mathfrak{B}}^{2}$ and conclude that $(Y_{t},Z_{t},{K_{t}^{+}},{K_{t}^{-}})$ is the unique solution to DRBSDE (3) associated with data $(\xi ,f,L,U)$. □
5 Comparison theorem
In this section we prove a comparison theorem for the DRBSDE under the stochastic Lipschitz assumptions on generators.
Theorem 4.
Proof.
Let $\bar{\mathrm{\Re }}={\mathrm{\Re }}^{1}-{\mathrm{\Re }}^{2}$ for $\mathrm{\Re }=Y,Z,{K}^{+},{K}^{+},\xi $ and
Applying the Meyer–Itô formula (Theorem 66, p. 210 in [23]), there exists a continuous nondecreasing process $(\mathcal{A}_{t})_{t\le T}$ such that
-
• $\zeta _{t}=\mathbb{1}_{\{\bar{Y}_{t}\ne 0\}}\displaystyle\frac{{f}^{1}(t,{Y_{t}^{1}},{Z_{t}^{1}})-{f}^{1}(t,{Y_{t}^{2}},{Z_{t}^{1}})}{\bar{Y}_{t}}$;
-
• $\eta _{t}=\mathbb{1}_{\{\bar{Z}_{t}\ne 0\}}\displaystyle\frac{{f}^{1}(t,{Y_{t}^{2}},{Z_{t}^{1}})-{f}^{1}(t,{Y_{t}^{2}},{Z_{t}^{2}})}{\bar{Z}_{t}}$;
-
• $\delta _{t}={f}^{1}(t,{Y_{t}^{2}},{Z_{t}^{2}})-{f}^{2}(t,{Y_{t}^{2}},{Z_{t}^{2}})$.
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {\big|{\bar{Y}_{t}^{+}}\big|}^{2}& \displaystyle =2{\int _{t}^{T}}{\bar{Y}_{s}^{+}}(\zeta _{s}\bar{Y}_{s}+\eta _{s}\bar{Z}_{s}+\delta _{s})ds-2{\int _{t}^{T}}{\bar{Y}_{s}^{+}}\bar{Z}_{s}dB_{s}\\{} & \displaystyle \hspace{1em}+2{\int _{t}^{T}}{\bar{Y}_{s}^{+}}d{\bar{K}_{s}^{+}}-2{\int _{t}^{T}}{\bar{Y}_{s}^{+}}d{\bar{K}_{s}^{-}}-(\mathcal{A}_{T}-\mathcal{A}_{t}).\end{array}\]
Suppose in addition that
\[ \mathbb{E}{\int _{0}^{T}}\mu _{t}dt<+\infty \hspace{1em}\hspace{2.5pt}\text{and}\hspace{2.5pt}\hspace{1em}\mathbb{E}{\int _{0}^{T}}|\gamma _{t}{|}^{2}dt<+\infty .\]
Let $\{\varGamma _{t,s},0\le t\le s\le T\}$ be the process defined as
\[ \varGamma _{t,s}=\exp \Bigg\{{\int _{t}^{s}}\bigg(\zeta _{u}-\frac{1}{2}|\eta _{u}{|}^{2}\bigg)du+{\int _{t}^{s}}\eta _{u}dB_{u}\Bigg\}>0\]
being a solution to the linear stochastic differential equation
\[ \varGamma _{t,s}=1+{\int _{t}^{s}}\zeta _{u}\varGamma _{t,u}du+{\int _{t}^{s}}\eta _{u}\varGamma _{t,u}dB_{u}.\]
Applying the integration by parts and taking expectation yield
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\big[{e}^{\beta A(t)}{\big|{\bar{Y}_{t}^{+}}\big|}^{2}\big]+\beta \mathbb{E}{\int _{0}^{T}}{e}^{\beta A(s)}\varGamma _{t,s}{a}^{2}(s){\big|{\bar{Y}_{s}^{+}}\big|}^{2}ds\\{} & \displaystyle \hspace{1em}\le \mathbb{E}\Bigg[{\int _{t}^{T}}{e}^{\beta A(s)}\varGamma _{t,s}\zeta _{s}{\big|{\bar{Y}_{s}^{+}}\big|}^{2}ds\Bigg]+2\mathbb{E}\Bigg[{\int _{t}^{T}}{e}^{\beta A(s)}\varGamma _{t,s}\delta _{s}{\bar{Y}_{s}^{+}}ds\Bigg]\\{} & \displaystyle \hspace{2em}+2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\varGamma _{t,s}{\bar{Y}_{s}^{+}}d{K_{s}^{+}}-2\mathbb{E}{\int _{t}^{T}}{e}^{\beta A(s)}\varGamma _{t,s}{\bar{Y}_{s}^{+}}d{K_{s}^{-}}.\end{array}\]
Remark that
\[ {\bar{Y}_{s}^{+}}d{\bar{K}_{s}^{+}}=\big({L_{s}^{1}}-{Y_{s}^{2}}\big)\mathbb{1}_{{Y_{s}^{1}}>{Y_{s}^{2}}}d{K_{s}^{1+}}-\big({Y_{s}^{1}}-{L_{s}^{2}}\big)\mathbb{1}_{{Y_{s}^{1}}>{Y_{s}^{2}}}d{K_{s}^{2+}}\le 0\]
and
\[ {\bar{Y}_{s}^{+}}d{\bar{K}_{s}^{-}}=\big({Y_{s}^{1}}-{U_{s}^{2}}\big)\mathbb{1}_{{Y_{s}^{1}}>{Y_{s}^{2}}}d{K_{s}^{2-}}-\big({U_{s}^{1}}-{Y_{s}^{2}}\big)\mathbb{1}_{{Y_{s}^{1}}>{Y_{s}^{2}}}d{K_{s}^{1-}}\le 0.\]
Since $\delta _{s}\le 0$ and $|\zeta _{s}|\le {a}^{2}(s)$, one can derive that
It follows that ${\bar{Y}_{t}^{+}}=0$, i.e ${Y_{t}^{1}}\le {Y_{t}^{2}}$ for all $t\le T$ a.s. □