| Abstrakt: |
In this paper we are interested to solve a class of quadratic BSDEs with jumps (QBSDEJs for short) of the following form: Y t = ξ + ∫ t T H (Y s , Z s , U s (⋅)) d s − ∫ t T Z s d W s − ∫ t T ∫ E U s (e) N ~ (d s , d e) , Herein, the terminal data ξ will be assumed to be square integrable. Our study covers the following cases H (y , z , u (⋅)) = f (y) z 2 + [ u ] f (y) =: H f (y , z , u (⋅))) h y , u (⋅) + c z + H f (y , z , u (⋅))) a + b y + c z + d u (⋅) ν , 1 + H f (y , z , u (⋅)) c z + f (y) z 2 − ∫ E u (e) ν (d e) c z + f (y) z 2 h y , u (⋅) + c z + f (y) z 2 H 0 r , X r + H f (y , z , u (⋅))) , (X r ) r ≥ 0 is a Markov process where f is a measurable and integrable function, u f (⋅) is a functional of the unknown processes Y ⋅ and U ⋅ (⋅) 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. [ABSTRACT FROM AUTHOR] |
