Quasi-sure Stochastic Analysis through Aggregation

By Mete Soner, Nizar Touzi, Jianfeng Zhang. 2010

190 citations recorded by the source corpus.

This paper is on developing stochastic analysis simultaneously under a general family of probability measures that are not dominated by a single probability measure. The interest in this question originates from the probabilistic representations of fully nonlinear partial differential equations and applications to mathematical finance. The existing literature relies either on the capacity theory (Denis and Martini), or on the underlying nonlinear partial differential equation (Peng). In both approaches, the resulting theory requires certain smoothness, the so-called quasi-sure continuity, of the corresponding processes and random variables in terms of the underlying canonical process. In this paper, we investigate this question for a larger class of ``non-smooth" processes, but with a restricted family of non-dominated probability measures. For smooth processes, our approach leads to similar results as in previous literature, provided the restricted family satisfies an additional density property.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/0146116d9ce1e47024d5f9c1a525c69a42fbabb1

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Citations in the corpus

General Rough integration, Levy Rough paths and a Levy--Kintchine type formula (2012, unknown): “[44]3), but free of implicit semimartingale features.”
Pathwise construction of stochastic integrals (2011, background): “[12], is to impose a strong separability assumption on the set P, which allows to glue together directly the processes {(P ) ∫ H dX}P∈P .”
Nonlinear Lévy Processes and their Characteristics (2016, unknown): “Another concept related to nonlinear Lévy processes is the so-called second order backward stochastic differential equation with jumps as introduced in [11], partially extending the formulation of [24] to the case with jumps using an approach along the lines of [23, 25].”
Dual Formulation of Second Order Target Problems (2010, methodology): “11 in [23], P̂ is in fact a dominating measure of P(A), and thus of PA H .”
Wellposedness of second order backward SDEs (2010, background): “by W P), we recall from [15] that PS = { P ∈ PW : FW P P = F } , (2.”
On the Robust superhedging of measurable claims (2013, unknown): “The next progress was achieved by Soner, Touzi and Zhang [13] who introduced a restriction of the set of non-dominated measures so as to guarantee that the predictable representation property holds true under each measure.”
Constructing Sublinear Expectations on Path Space (2012, methodology): “variable Eτ(ξ) such that the property (1.2) holds; this is the aggregation problem. Severe restrictions are necessary to construct Eτ(ξ) directly by gluing together the right hand sides in (1.2); cf. [3, 23]. We shall use a diﬀerent starting point, which will lead both to a general construction of t”
Martingale Representation Theorem for the G-Expectation (2010, background): “Indeed, recently the authors developed such a generalization and a duality theory for 2BSDEs using probabilistic constructions similar to quasi-sure analysis [19, 20, 21].”
Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix (2016, unknown): “10 in [34] that there exists a PΘ quasi surely aggregated solution X, i.”
ROBUST UTILITY MAXIMIZATION IN NONDOMINATED MODELS WITH 2BSDE: THE UNCERTAIN VOLATILITY MODEL (2015, background): “However, this problem was solved in lemma 2.4 of Soner, Touzi, and Zhang (2011), which implies that for every P ∈ PH, we can always consider a version of our processes which is progressively measurable for the completion of F+ under P. 2.2.”