117 research outputs found
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 depends on
a parameter with 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 .
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
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