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105
Wellposedness of second order backward SDEs
"... We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested in [4]. In particular, we provide a fully nonlin ..."
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Cited by 50 (11 self)
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We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested in [4]. In particular, we provide a fully nonlinear extension of the FeynmanKac formula. Unlike [4], the alternative formulation of this paper insists that the equation must hold under a nondominated family of mutually singular probability measures. The key argument is a stochastic representation, suggested by the optimal control interpretation, and analyzed in the accompanying paper [17]. Key words: Backward SDEs, nondominated family of mutually singular measures, viscosity solutions for second order PDEs.
On Representation Theorem of GExpectations and Paths of GBrownian Motion
 Acta Mathematicae Applicatae Sinica, English Series
, 2009
"... We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ: θ ∈ Θ} to represent an important sublinear expectation — Gexpectation E[·]. We also give a concrete approximation of a bounded continuous function X(ω) by an increasing sequence of cy ..."
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Cited by 36 (14 self)
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We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ: θ ∈ Θ} to represent an important sublinear expectation — Gexpectation E[·]. We also give a concrete approximation of a bounded continuous function X(ω) by an increasing sequence of cylinder functions Lip(Ω) in order to prove that Cb(Ω) belongs to the E[  · ]completion of the Lip(Ω).
A New Central Limit Theorem under Sublinear Expectations
 in arXiv:math.PR/0803.2656vl 18
, 2008
"... We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of Gdistributions is introduced which generalizes our Gnormaldistribution in the sense that meanuncertainty can be also described. W present our new resul ..."
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Cited by 34 (8 self)
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We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of Gdistributions is introduced which generalizes our Gnormaldistribution in the sense that meanuncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of meanuncertainty. 1
Stopping times and related Itos calculus with GBrownian motion
 Stoch. Proc. and their App
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Superhedging and dynamic risk measures under volatility uncertainty
 SIAM J. CONTROL OPTIM
, 2012
"... We consider dynamic sublinear expectations (i.e., timeconsistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The ..."
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Cited by 27 (12 self)
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We consider dynamic sublinear expectations (i.e., timeconsistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.
Constructing Sublinear Expectations on Path Space
, 2013
"... We provide a general construction of timeconsistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional Gexpectation of a Borelmeasurable (rather than quasicontinuous) random variable, a generalization of the random Gexpectation, and an optional ..."
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Cited by 26 (12 self)
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We provide a general construction of timeconsistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional Gexpectation of a Borelmeasurable (rather than quasicontinuous) random variable, a generalization of the random Gexpectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.
A quasisure approach to the control of nonMarkovian stochastic differential equations
, 2012
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