Wellposedness of second order backward SDEs

By H. M. Soner, Nizar Touzi, Jianfeng Zhang. 2010

330 citations recorded by the source corpus.

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 Cheridito et al. (Commun. Pure Appl. Math. 60(7):1081–1110, 2007). In particular, we provide a fully nonlinear extension of the Feynman–Kac formula. Unlike (Cheridito et al. in Commun. Pure Appl. Math. 60(7):1081–1110, 2007), the alternative formulation of this paper insists that the equation must hold under a non-dominated 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 (Soner et al. in Dual Formulation of Second Order Target Problems. arXiv:1003.6050, 2009).

Subfields: Demography, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/1e9c438c48b7412817a8ce4747535fcc1979ec39

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Demography

In Finance

In Modeling and Simulation

Citations in the corpus

Measurability of semimartingale characteristics with respect to the probability law (2013, methodology): “his end, we ﬁrst apply Bichteler’s pathwise integration [5, Theorem 7.14], see also [15] for the same result in modern notation, to R Xi − dX j, for ﬁxed 1 ≤ i,j≤ d. This integration was also used in [19, 27, 28] in the context of continuous martingales; however, we have to elaborate on the construc”
Weak approximation of G-expectations (2011, unknown): “n. 1 Introduction The so-called G-expectation [12, 13, 14] is a nonlinear expectation advancing the notions of backward stochastic diﬀerential equations (BSDEs) [10] and g-expectations [11]; see also [2, 16] for a related theory of second order BSDEs. A G-expectation ξ→ EG(ξ) is a sublinear function”
ROBUST UTILITY MAXIMIZATION IN NONDOMINATED MODELS WITH 2BSDE: THE UNCERTAIN VOLATILITY MODEL (2015, methodology): “…definite matrices, i.e., (a − a) ∈ S>0d ) and we define the class PH := { P ∈ PS s.t. a ≤ ât ≤ a, dt × P − a.e. } , which is a particular case of definition 2.6 in Soner et al. (2012), the main differences here being that the two bounds on â are independent of the probability measures and…”
On viscosity solutions of path dependent PDEs (2011, background): “This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman–Kac formula to non-Markovian case.”
Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE (2012, background): “Moreover, G-expectation is closely related to second-order BSDE studied in [26], namely the process Yt = v(t,Bt ) satisfies a 2BSDE, which is formulated under a nondominated family of singular probability measures given by the law of X under P.”
Martingale optimal transport in the Skorokhod space (2014, background): “Following the structure introduced in [29,30], [6] considers a discrete time market in which a set of probability measures P is assumed.”
Quasi-sure Stochastic Analysis through Aggregation (2010, background): “We remark that Denis-Martini [5] and our paper [15] consider the class P W while Denis-Hu-Peng [6] and our paper [17] consider the class P S , up to some technical restriction of the diffusion coefficients.”
Martingale optimal transport and robust hedging in continuous time (2012, background): “As in the unconstrained model of [21, 22], the super-replication is defined not pathwise but rather probabilistically through quasi-sure inequalities.”
Dual Formulation of Second Order Target Problems (2010, methodology): “Indeed, we develop this theory in our accompanying work [21], including existence and uniqueness results with minimal assumptions.”
Martingale Representation Theorem for the G-Expectation (2010, background): “Research supported in part by NSF grant DMS 06-31366.”