Dual Formulation of Second Order Target Problems

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

158 citations recorded by the source corpus.

This paper provides a new formulation of second order stochastic target problems introduced in [19] by modifying the reference probability so as to allow for difierent scales. This new ingredient enables us to prove a dual formulation of the target problem as the supremum of the solutions of standard backward stochastic difierential equations. In particular, in the Markov case, the dual problem is known to be connected to a fully nonlinear, parabolic partial difierential equation and this connection can be viewed as a stochastic representation for all nonlinear, scalar, second order, parabolic equations with a convex Hessian dependence.

Subfields: Demography, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/2780e6889a1940948138c2acb8647a136352aecc

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Citations in the corpus

Measurability of semimartingale characteristics with respect to the probability law (2013, methodology): “This integration was also used in [19, 27, 28] in the context of continuous martingales; however, we have to elaborate on the construction to find a Borel-measurable version.”
Wellposedness of second order backward SDEs (2010, methodology): “1 Connection with the second order stochastic target problem [17] Let (Y, Z) be a solution of 2BSDE (3.”
ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES (2014, background): “[5, 16, 20, 30, 36, 38, 39, 41, 44, 45], among others, or when all paths are considered possible for the stock and options are also traded; see, e.”
Martingale Representation Theorem for the G-Expectation (2010, methodology): “A martingale representation theorem for this theory is proved by using the techniques and the results established in [18] for the second order stochastic target problems and the second order backward stochastic differential equations.”
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].”
Optimal stopping under adverse nonlinear expectation and related games (2012, background): “More precisely, following [33], we assume that P is a subset of PS , the set of all local martingale laws of the form”
UTILITY MAXIMIZATION UNDER MODEL UNCERTAINTY IN DISCRETE TIME (2013, background): “The passage from the one-period markets to the original market is achieved via measurable selections and in particular the theory of lower semianalytic functions; it draws from [6, 24, 25, 33].”
Quasi-sure Stochastic Analysis through Aggregation (2010, background): “es the same result as the quasi-sure analysis. These type of results, of course, require continuity of all the maps in an appropriate sense. The details of this approach are investigated in our paper [16], see also Remark 7.5 in the context of 4 the application to the hedging problem under uncertain”
On the Robust superhedging of measurable claims (2013, unknown): “Proof (ii) has already been proved in [15].”
Arbitrage and duality in nondominated discrete-time models (2013, background): “In the setting where no options are traded and P is supposed to consist of martingale measures in the rst place, there is a substantial literature on superhedging under volatility uncertainty, where S is a (continuous-time) process with continuous trajectories; see, among others, [8, 27, 40, 41, 42,”
Stochastic control for a class of nonlinear kernels and applications (2015, background): “In the case of a fully nonlinear PPDE, such a result has been recently achieved by Ren, Touzi and Zhang [82].”
Superreplication under Volatility Uncertainty for Measurable Claims (2012, methodology): “The set PS was introduced in [19] and its elements have several nice properties; in particular, they have the predictable representation property which plays an important role in the proof of the duality result below (see also Section 4).”
Random G-expectations. (2010, methodology): “2) is strongly inspired by the formulation of stochastic target problems in Soner, Touzi and Zhang [17].”
A complete representation theorem for G-martingales (2012, background): “[23,24] for the closely related theory of second-order backward stochastic differential equations (SDEs).”
Option pricing with linear market impact and nonlinear Black–Scholes equations (2018, unknown): “Indeed observe that the elliptic operator F(γ ) in (2) is convex, and then applying the reasoning of [35] based on the Legendre transform representation of F , one can show formally that a solution to (2) will also be solution of the following variational problem:”
Constructing Sublinear Expectations on Path Space (2012, methodology): “Taking inspiration from [23], we then define Eτ (ξ)(ω) := sup P∈P(τ,ω) E [ξ], ω ∈ Ω”
Optimal Stopping under Nonlinear Expectation (2012, unknown): “Then, as in [14], we define for all ω ∈ Ω a probability measure P on F τ(ω) T by: P(E) := PωFτ  {ω⊗τ(ω) ω ′ : ω ∈ E}  , ∀E ∈ F τ(ω) T , (A.”
Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012, background): “Since σ > 0, as discussed in [31] X t,ω,k and Bt induce the same P0-augmented filtration, and thus there exists k̃ ∈ Kt such that k̃(X t,ω,k · ) = k, P0-a.”