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Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. (English) Zbl 1312.60052
The authors study the stability and strong convergence for general quadratic BSDEs and quadratic semimartingales. A quadratic BSDE has the property that the coefficient $$g$$ satisfies the quadratic structure equation. Its solution is a quadratic Itō semimartingale where the predictable process which has finite variation also satisfies this structure equation. The authors characterize quadratic semimartingales via an exponential transform where the generator of the BSDE is transformed in a driver with linear-quadratic growth and identify this transformation as quadratic submartingales which are defined by the multiplicative and additive decomposition. Under mild assumptions on the integrabilibility of the exponential terminal value, they derive a general stability result with strong convergence in martingale parts, from $$\mathbb{H}^1$$ to BMO-martingales, and convergence in total variation of the finite variation parts. The stability is characterized by an almost sure-convergence regarding to the strong convergence of the martingale parts. When the quadratic semimartingale is bounded, the results are new and direct and within a BSDE-framework in $$\mathbb{H}^1$$. For BSDE-like quadratic semimartingales, the authors prove that also the limit process is a BSDE-like semimartingale. Existence results are also established in a more general framework using inf-convolution. In contrast to the approach of M. Kobylanski [Ann. Probab. 28, No. 2, 558–602 (2000; Zbl 1044.60045)], the authors do not assume bounded solutions. These results can be extended to jump processes.

##### MSC:
 60G48 Generalizations of martingales 60G07 General theory of stochastic processes 60G44 Martingales with continuous parameter 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H99 Stochastic analysis 91B26 Auctions, bargaining, bidding and selling, and other market models 91B16 Utility theory
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