225 research outputs found
A Benamou-Brenier formulation of martingale optimal transport
We introduce a Benamou-Brenier formulation for the continuous-time martingale
optimal transport problem as a weak length relaxation of its discrete-time
counterpart. By the correspondence between classical martingale problems and
Fokker-Planck equations, we obtain an equivalent PDE formulation for which
basic properties such as existence, duality and geodesic equations can be
analytically studied, yielding corresponding results for the stochastic
formulation. In the one dimensional case, sufficient conditions for finiteness
of the cost are also given and a link between geodesics and porous medium
equations is partially investigated
Backward SDE Representation for Stochastic Control Problems with Non Dominated Controlled Intensity
We are interested in stochastic control problems coming from mathematical
finance and, in particular, related to model uncertainty, where the uncertainty
affects both volatility and intensity. This kind of stochastic control problems
is associated to a fully nonlinear integro-partial differential equation, which
has the peculiarity that the measure characterizing the
jump part is not fixed but depends on a parameter which lives in a compact
set of some Euclidean space . We do not assume that the family
is dominated. Moreover, the diffusive part can be
degenerate. Our aim is to give a BSDE representation, known as nonlinear
Feynman-Kac formula, for the value function associated to these control
problems. For this reason, we introduce a class of backward stochastic
differential equations with jumps and partially constrained diffusive part. We
look for the minimal solution to this family of BSDEs, for which we prove
uniqueness and existence by means of a penalization argument. We then show that
the minimal solution to our BSDE provides the unique viscosity solution to our
fully nonlinear integro-partial differential equation.Comment: arXiv admin note: text overlap with arXiv:1212.2000 by other author
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