Quadratic BSDEs with jumps and related PIDEs.
Khelfallah, I. Madoui, M Eddahbi, N. . 2021
In this paper we are interested to solve a class of quadratic BSDEs with
jumps (QBSDEJs for short) of the following form:
\begin{equation*}
Y_{t}=\xi +\int_{t}^{T}H(Y_{s},Z_{s},U_{s}(\cdot
))ds-\int_{t}^{T}Z_{s}dW_{s}-\int_{t}^{T}\int_{E}U_{s}(e)\tilde{N}(ds,de),
\end{equation*}%
Herein, the terminal data $\xi $ will be assumed to be square integrable.
Our study covers the following cases
\begin{equation*}
H(y,z,u(\cdot ))=\left\{
\begin{array}{l}
f(y)\left\vert z\right\vert ^{2}+[u]_{f}(y)=:H_{f}(y,z,u(\cdot ))) \\
h\left( y,u(\cdot )\right) +cz+H_{f}(y,z,u(\cdot ))) \\
a+b\left\vert y\right\vert +c\left\vert z\right\vert +d\left\Vert u(\cdot
)\right\Vert _{\nu ,1}+H_{f}(y,z,u(\cdot )) \\
cz+f(y)\left\vert z\right\vert ^{2}-\int_{E}u(e)\nu (\mathrm{d}e) \\
cz+f(y)\left\vert z\right\vert ^{2} \\
h\left( y,u(\cdot )\right) +cz+f(y)\left\vert z\right\vert ^{2} \\
H_{0}\left( r,X_{r}\right) +H_{f}(y,z,u(\cdot )))\text{, }(X_{r})_{r\geq 0}%
\text{ is a Markov process}%
\end{array}%
\right.
\end{equation*}%
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.
We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages
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