Constructing Sublinear Expectations on Path Space Marcel Nutz∗ Ramon van Handel† January 21, 2013 Abstract We provide a general construction of time-consistent sublinear ex- pectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G- expectation, 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. Keywords Sublinear expectation; G-expectation; random G-expectation; Time- consistency; Optional sampling; Dynamic programming; Analytic set AMS 2000 Subject Classification 93E20; 60H30; 91B30; 28A05 1 Introduction d We study sublinear expectations on the space Ω= C0(R+, R ) of continuous arXiv:1205.2415v3 [math.PR] 11 Apr 2013 paths. Taking the dual point of view, we are interested in mappings P ξ 0(ξ)= sup E [ξ], 7→ E P ∈P where ξ is a random variable and is a set of probability measures, possibly P non-dominated. In fact, any sublinear expectation with certain continuity ∗Dept. of Mathematics, Columbia University, New York.
[email protected] The work of MN is partially supported by NSF grant DMS-1208985. †Sherrerd Hall rm. 227, Princeton University, Princeton.
[email protected] The work of RvH is partially supported by NSF grant DMS-1005575. 1 properties is of this form (cf. [10, Sect. 4]). Under appropriate assumptions on , we would like to construct a conditional expectation (ξ) at any P Eτ stopping time τ of the the filtration generated by the canonical process {Ft} B and establish the tower property ( (ξ)) = (ξ) for stopping times σ τ, (1.1) Eσ Eτ Eσ ≤ a property also known as time-consistency in this context.