Reflected Forward-Backward Stochastic Differential Equations Driven by G-Brownian Motion with Continuous Monotone Coefficients

Abstract

This paper is devoted to investigating the existence of solution to a class of reflected forward-backward stochastic differential equations driven by G-Brownian motion (G-RFBSDEs). We construct a solution to the equations by monotone convergence argument when the generator of backward equation and the drift of forward equation satisfy the monotonicity and uniformly continuous condition.

Appendix

Proof of (3.19) and (3.20)

Applying Itô’s formula to $|Y_{t}^{1,k}|^{2}$, we have

$$\begin{aligned} |Y_{0}^{1,k}|^{2}+\displaystyle \int _{0}^{T}|Z_{s}^{1,k}|^{2}d\langle B\rangle _{s}&=|\xi |^{2}+2\displaystyle \int _{0}^{T}Y_{s}^{1,k}f_{k}^{1}(s,w,Y_{s}^{1,k},Z_{s}^{1,k})ds\\&\quad -2\displaystyle \int _{0}^{T}Y_{s}^{1,k} Z_{s}^{1,k}dB_{s}+2\displaystyle \int _{0}^{T}Y_{s}^{1,k}dA_{s}^{1,k}. \end{aligned}$$

By Lemma 3.1, (H4) and f is Lipschitz continuous in z, we have

$$\begin{aligned} |f_{k}^{1}(s,w,y,z)|\le & {} |f(s,X_{s}^{0}(w),0,z)|+M(1+|y|)\nonumber \\\le & {} |f(s,0,0,0)|+M(2+|y|+|z|). \end{aligned}$$

(4.1)

Then, for $2\le \alpha <\beta$, by Lemma 2.2, Lemma 2.1 and Hölder’s inequality, we have

$$\begin{aligned}&\underline{\sigma }^{\alpha }\hat{{\mathbb {E}}}\left[ \left( \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] \le \hat{{\mathbb {E}}}\left[ \left( \int _{0}^{T}|Z_{s}^{1,k}|^{2}d\langle B\rangle _{s}\right) ^{\frac{\alpha }{2}}\right] \nonumber \\&\quad \le 2^{\frac{3\alpha }{2}-2}\{\hat{{\mathbb {E}}}[|\xi |^{\alpha }] +\hat{{\mathbb {E}}}\left[ |\int _{0}^{T}Y_{s}^{1,k}f_{k}^{1}(s,w,Y_{s}^{1,k},Z_{s}^{1,k})ds|^{\frac{\alpha }{2}}\right] \nonumber \\&\quad +\hat{{\mathbb {E}}}\left[ |\int _{0}^{T}Y_{s}^{1,k}Z_{s}^{1,k}dB_{s}|^{\frac{\alpha }{2}}\right] +\hat{{\mathbb {E}}}\left[ |\int _{0}^{T}Y_{s}^{1,k}dA_{s}^{1,k}|^{\frac{\alpha }{2}}\right] \}\nonumber \\&\quad \le 2^{\frac{3\alpha }{2}-2}\{\hat{{\mathbb {E}}}[|\xi |^{\alpha }] +\hat{{\mathbb {E}}}[\sup _{0\le s\le T}|Y_{s}^{1,k}|^{\frac{\alpha }{2}}\left( \int _{0}^{T}|f(s,0,0,0)| +M(2+|Y_{s}^{1,k}|+|Z_{s}^{1,k}|)ds\right) ^{\frac{\alpha }{2}}]\nonumber \\&\quad +C_{\frac{\alpha }{2}}\overline{\sigma }^{\frac{\alpha }{2}}\hat{{\mathbb {E}}}\left[ \left( \int _{0}^{T}|Y_{s}^{1,k}|^{2}|Z_{s}^{1,k} |^{2}ds\right) ^{\frac{\alpha }{4}}\right] +\hat{{\mathbb {E}}}[\sup _{0\le s\le T}|Y_{s}^{1,k}|^{\frac{\alpha }{2}} |A_{T}^{1,k}|^{\frac{\alpha }{2}}]\}\nonumber \\&\quad \le 2^{\frac{3\alpha }{2}-2}\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }] +2^{\frac{5\alpha }{2}-4}(\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }])^{\frac{1}{2}}\nonumber \\&\quad \cdot \{\hat{{\mathbb {E}}}\left[ \left( \int _{0}^{T}|f(s,0,0,0)|ds\right) ^{\alpha }\right] +(2MT)^{\alpha }+(MT)^{\alpha } \hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }]\nonumber \\&\quad +M^{\alpha }T^{\frac{\alpha }{2}}\hat{{\mathbb {E}}}\left[ \left( \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] \} ^{\frac{1}{2}}+2^{\frac{3\alpha }{2}-2}C_{\frac{\alpha }{2}}\overline{\sigma }^{\frac{\alpha }{2}} (\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }])^{\frac{1}{2}}\left( \hat{{\mathbb {E}}} \left[ \left( \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] \right) ^{\frac{1}{2}}\nonumber \\&\quad +2^{\frac{3\alpha }{2}-2}(\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }])^{\frac{1}{2}} (\hat{{\mathbb {E}}}[|A_{T}^{1,k}|^{\alpha }])^{\frac{1}{2}} \nonumber \\&\quad \le 2^{\frac{3\alpha }{2}-2}\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }] +2^{\frac{5\alpha }{2}-4}T^{\frac{\alpha -1}{2}}(\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k} |^{\alpha }])^{\frac{1}{2}}\cdot \left( \hat{{\mathbb {E}}}\left[ \int _{0}^{T}|f(s,0,0,0)|^{\alpha }ds\right] \right) ^{\frac{1}{2}}\nonumber \\&\quad +(2MT)^{\frac{\alpha }{2}}2^{\frac{5\alpha }{2}-4}(\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }]) ^{\frac{1}{2}}+(MT)^{\frac{\alpha }{2}}2^{\frac{5\alpha }{2}-4}\hat{{\mathbb {E}}}[\sup _{0\le s \le T} |Y_{s}^{1,k}|^{\alpha }]\nonumber \\&\quad +\left( 2^{\frac{3\alpha }{2}-2}C_{\frac{\alpha }{2}}\overline{\sigma }^{\frac{\alpha }{2}}+2^{\frac{5\alpha }{2} -4}M^{\frac{\alpha }{2}}T^{\frac{\alpha }{4}}\right) \left( \hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }]\right) ^{\frac{1}{2}}\left( \hat{{\mathbb {E}}}\left[ \left( \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] \right) ^{\frac{1}{2}}\nonumber \\&\quad +2^{\frac{3\alpha }{2}-2}(\hat{{\mathbb {E}}}[\sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }])^{\frac{1}{2}} (\hat{{\mathbb {E}}}[|A_{T}^{1,k}|^{\alpha }])^{\frac{1}{2}}. \end{aligned}$$

(4.2)

On the other hand,

$$\begin{aligned} A_{T}^{1,k}= Y_{0}^{1,k}-\xi -\displaystyle \int _{0}^{T}f^{1}_{k}(s,w,Y_{s}^{1,k},Z_{s}^{1,k})ds+\displaystyle \int _{0}^{T}Z_{s}^{1,k}dB_{s}. \end{aligned}$$

By Lemma 2.1, (4.1) and by simple calculation, we get

$$\begin{aligned}&\hat{{\mathbb {E}}}[|A_{T}^{1,k}|^{\alpha }] \le 4^{\alpha -1}\{2\hat{{\mathbb {E}}}\left[ \sup _{0\le t \le T}|Y_{t}^{1,k}|^{\alpha }\right] \nonumber \\&\quad +\hat{{\mathbb {E}}}\left[ |\displaystyle \int _{0}^{T}f^{1}_{k}(s,w,Y_{s}^{1,k},Z_{s}^{1,k})ds|^{\alpha }\right] +\hat{{\mathbb {E}}}[|\displaystyle \int _{0}^{T}Z_{s}^{1,k}dB_{s}|^{\alpha }\}\nonumber \\&\quad \le 2^{2\alpha -1}\hat{{\mathbb {E}}}\left[ \sup _{0\le t \le T}|Y_{t}^{1,k}|^{\alpha }\right] +4^{2\alpha -2}T^{\alpha -1}\hat{{\mathbb {E}}}\left[ \displaystyle \int _{0}^{T}|f(s,0,0,0)|^{\alpha }ds\right] +4^{2\alpha -2}(2MT)^{\alpha }\nonumber \\&\quad +4^{2\alpha -2}(MT)^{\alpha }\hat{{\mathbb {E}}}\left[ \sup _{0\le t\le T}|Y_{t}^{1,k}|^{\alpha }\right] +(4^{2\alpha -2}M^{\alpha }T^{\frac{\alpha }{2}}+4^{\alpha -1}C_{\alpha }\bar{\sigma }^{\alpha }) \hat{{\mathbb {E}}}\left[ \left( \displaystyle \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] . \end{aligned}$$

(4.3)

Combining (4.2) and (4.3), then there exists a constant $C'$ independent of k, such that

$$\begin{aligned}&\underline{\sigma }^{\alpha }\hat{{\mathbb {E}}}\left[ \left( \displaystyle \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] \nonumber \\&\quad \le C'\left\{ \hat{{\mathbb {E}}}\left[ \sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }\right] +\left( \hat{{\mathbb {E}}}\left[ \sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }\right] \right) ^{\frac{1}{2}}+\hat{{\mathbb {E}}}\left[ \displaystyle \int _{0}^{T}|f(s,0,0,0)|^{\alpha }ds\right] \right\} \nonumber \\&\quad +C''\left( \hat{{\mathbb {E}}}\left[ \sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }\right] \right) ^{\frac{1}{2}}(\hat{{\mathbb {E}}} \left[ \left( \displaystyle \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds)^{\frac{\alpha }{2}}\right] \right) ^{\frac{1}{2}} \nonumber \\&\quad \le C'\{\hat{{\mathbb {E}}}[\sup _{0\le t \le T}|Y_{t}^{1,k}|^{\alpha }]+\left( \hat{{\mathbb {E}}}\left[ \sup _{0\le t \le T}|Y_{t}^{1,k}|^{\alpha }\right] \right) ^{\frac{1}{2}}+\hat{{\mathbb {E}}}\left[ \displaystyle \int _{0}^{T}|f(s,0,0,0)|^{\alpha }ds\right] \}\nonumber \\&\quad +\frac{(C'')^{2}}{4\varepsilon }\hat{{\mathbb {E}}}\left[ \sup _{0\le t \le T}|Y_{t}^{1,k}|^{\alpha }\right] +\varepsilon \hat{{\mathbb {E}}}\left[ \left( \displaystyle \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] \end{aligned}$$

(4.4)

where $C''=2^{\frac{3\alpha }{2}-2}C_{\frac{\alpha }{2}}\bar{\sigma }^{\frac{\alpha }{2}}+2^{\frac{5\alpha }{2}-4} (2^{\alpha }+1)M^{\frac{\alpha }{2}}T^{\frac{\alpha }{4}}+2^{\frac{5\alpha }{2}-3}C_{\alpha }^{\frac{1}{2}}\bar{\sigma }^{\frac{\alpha }{2}}.$ Here, we have used Young’s inequality with some $\varepsilon >0$.

Choosing $\varepsilon <\underline{\sigma }^{\alpha }$, then there exists a constant $C'''$ independent of k, such that

$$\begin{aligned}&\hat{{\mathbb {E}}}\left[ \left( \displaystyle \int _{0}^{T}|Z_{s}^{1,k}|^{2}ds\right) ^{\frac{\alpha }{2}}\right] \\&\quad \le C'''\left\{ \hat{{\mathbb {E}}}\left[ \sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }\right] +\left( \hat{{\mathbb {E}}}\left[ \sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }\right] \right) ^{\frac{1}{2}} +\hat{{\mathbb {E}}}\left[ \displaystyle \int _{0}^{T}|f(s,0,0,0)|^{\alpha }ds\right] \right\} . \end{aligned}$$

Moreover, by (4.3), we also have

$$\begin{aligned} \hat{{\mathbb {E}}}[|A_{T}^{1,k}|^{\alpha }]&\le C''''\left\{ \hat{{\mathbb {E}}}\left[ \sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }\right] +\left( \hat{{\mathbb {E}}}\left[ \sup _{0\le s \le T}|Y_{s}^{1,k}|^{\alpha }\right] \right) ^{\frac{1}{2}}\right. \\&\quad \left. +\hat{{\mathbb {E}}}\left[ \displaystyle \int _{0}^{T}|f(s,0,0,0)|^{\alpha }ds\right] \right\} , \end{aligned}$$

where $C''''$ is a constant independent of k. $\square$