In this paper we are interested to solve a class of quadratic BSDEs with jumps (QBSDEJs for short) of the following form: $\begin{array}{rl}{Y}_t& =\xi +{\int }_t^{T}H(Y_s,Z_s,U_s(\cdot \left)\right)\mathrm{d}s-{\int }_t^T{Z}_s\mathrm{d}{W}_s\\ & -{\int }_t^T{\int }_E{U}_s\left(e\right)\tilde{N}(\mathrm{d}s,\mathrm{d}e),\end{array}$Herein, the terminal data ξ will be assumed to be square integrable. Our study covers the following cases $H(y,z,u(\cdot \left)\right)=\left\{\begin{array}{l}f\left(y\right){\left|z\right|}^2+\left[u{]}_f\right(y)=:H_f(y,z,u(\cdot )\left)\right)\\ h\left(y,u(\cdot )\right)+cz+H_f(y,z,u(\cdot \left)\right))\\ a+b\left|y\right|+c\left|z\right|+\mathrm{d}{∥u(\cdot )∥}_{\nu ,1}+H_f(y,z,u(\cdot \left)\right)\\ cz+f\left(y\right){\left|z\right|}^2-{\int }_{E}u\left(e\right)\nu \left(\mathrm{d}e\right)\\ cz+f\left(y\right){\left|z\right|}^2\\ h\left(y,u(\cdot )\right)+cz+f\left(y\right){\left|z\right|}^2\\ H_0\left(r,X_r\right)+H_f(y,z,u(\cdot \left)\right)),\\ (X_r{)}_{r\ge 0}\text{is a Markov process}\end{array}\right.$where f is a measurable and integrable function, ${\left[u\right]}_f(\cdot )$ is a functional of the unknown processes $Y_{\cdot }$ and $U_{\cdot }(\cdot )$ to be defined later and h and $H_0$ enjoy some classical assumptions. The generators show quadratic growth in the Brownian component and non-linear functional form with respect to the jump term.

Existence or uniqueness of solutions as well as a comparison and strict comparison principles are established under no monotonicity condition in the third argument of the generator. Probabilistic representations of solutions to some classes of quadratic PIDE are given by means of solutions of these QBSDEJs. The main idea is to use a phase space transformation to transform our initial QBSDEJ to a standard BSDEJ without quadratic term.

在本文中，我们感兴趣的是求解一类具有以下形式的跳跃的二次 BSDE（简称 QBSDEJ）：$\begin{array}{rl}{是}_{吨}& =\xi +{\int }_{吨}^{吨}H({是}_s,Z_s,{ü}_s(\cdot \left)\right)\mathrm{d}s-{\int }_{吨}^{吨}{Z}_s\mathrm{d}{W}_s\\ & -{\int }_{吨}^{吨}{\int }_{乙}{ü}_s\left(e\right)\widetilde{ñ}(\mathrm{d}s,\mathrm{d}e),\end{array}$这里，将假设终端数据ξ是平方可积的。我们的研究涵盖以下案例$H(\mathrm{是的},z,你(\cdot \left)\right)=\left\{\begin{array}{l}F\left(\mathrm{是的}\right){\left|z\right|}^2+\left[你{]}_F\right(\mathrm{是的})=：H_F(\mathrm{是的},z,你(\cdot )\left)\right)\\ H\left(\mathrm{是的},你(\cdot )\right)+Cz+H_F(\mathrm{是的},z,你(\cdot \left)\right))\\ \mathrm{一种}+b\left|\mathrm{是的}\right|+C\left|z\right|+\mathrm{d}{∥你(\cdot )∥}_{\nu ,1}+H_F(\mathrm{是的},z,你(\cdot \left)\right)\\ Cz+F\left(\mathrm{是的}\right){\left|z\right|}^2-{\int }_{乙}你\left(e\right)\nu \left(\mathrm{d}e\right)\\ Cz+F\left(\mathrm{是的}\right){\left|z\right|}^2\\ H\left(\mathrm{是的},你(\cdot )\right)+Cz+F\left(\mathrm{是的}\right){\left|z\right|}^2\\ H_0\left(r,X_r\right)+H_F(\mathrm{是的},z,你(\cdot \left)\right)),\\ (X_r{)}_{r\ge 0}\text{是马尔可夫过程}\end{array}\right.$其中f是一个可测量且可积的函数，${\left[你\right]}_F(\cdot )$是未知过程的泛函${是}_{\cdot }$和${ü}_{\cdot }(\cdot )$稍后定义，h和$H_0$享受一些经典假设。生成器显示出布朗分量的二次增长和关于跳跃项的非线性函数形式。

在生成器的第三个参数中，在非单调性条件下建立了解的存在性或唯一性以及比较和严格比较原则。通过这些 QBSDEJ 的解，给出了某些类别的二次 PIDE 解的概率表示。主要思想是使用相空间变换将我们的初始 QBSDEJ 转换为没有二次项的标准 BSDEJ。