Dual formulation of second order target problems

This paper provides a new formulation of second order stochastic target problems introduced in [SIAM J. Control Optim. 48 (2009) 2344-2365] by modifying the reference probability so as to allow for different 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 differential equations. In particular, in the Markov case, the dual problem is known to be connected to a fully nonlinear, parabolic partial differential equation and this connection can be viewed as a stochastic representation for all nonlinear, scalar, second order, parabolic equations with a convex Hessian dependence.Comment: Published in at http://dx.doi.org/10.1214/12-AAP844 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 by Cheridito et.al. In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike the earlier papers, 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 our accompanying paperComment: 36 page

Large liquidity expansion of super-hedging costs

We consider a financial market with liquidity cost as in \c{C}etin, Jarrow and Protter [2004], where the supply function $S^{\epsilon}(s,\nu)$ depends on a parameter $\epsilon\geq 0$ with $S^0(s,\nu)=s$ corresponding to the perfect liquid situation. Using the PDE characterization of \c{C}etin, Soner and Touzi [2010] of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of $\epsilon$. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option

Large liquidity expansion of super-hedging costs.

We consider a financial market with liquidity cost as in Cetin, Jarrow and Protter [3] where the supply function S"(s; ) depends on a parameter " 0 with S0(s; ) = s corresponding to the perfect liquid situation. Using the PDE characterization of Cetin, Soner and Touzi [6] of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of ". In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option.Super-replication; liquidity; viscosity solutions; asymptotic expansions;

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 Cheridito etal. (Commun. Pure Appl. Math. 60(7):1081-1110, 2007). In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike (Cheridito etal. 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 etal. in Dual Formulation of Second Order Target Problems. arXiv:1003.6050, 2009

A stochastic representation for the level set equations

Abstract A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data