29 research outputs found
On the convergence of monotone schemes for path-dependent PDE
We propose a reformulation of the convergence theorem of monotone numerical
schemes introduced by Zhang and Zhuo for viscosity solutions of path-dependent
PDEs, which extends the seminal work of Barles and Souganidis on the viscosity
solution of PDE. We prove the convergence theorem under conditions similar to
those of the classical theorem in the work of Barles and Souganidis. These
conditions are satisfied, to the best of our knowledge, by all classical
monotone numerical schemes in the context of stochastic control theory. In
particular, the paper provides a unified approach to prove the convergence of
numerical schemes for non-Markovian stochastic control problems, second order
BSDEs, stochastic differential games etc.Comment: 28 page
Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
We study a class of stochastic target games where one player tries to find a
strategy such that the state process almost-surely reaches a given target, no
matter which action is chosen by the opponent. Our main result is a geometric
dynamic programming principle which allows us to characterize the value
function as the viscosity solution of a non-linear partial differential
equation. Because abstract mea-surable selection arguments cannot be used in
this context, the main obstacle is the construction of measurable
almost-optimal strategies. We propose a novel approach where smooth
supersolutions are used to define almost-optimal strategies of Markovian type,
similarly as in ver-ification arguments for classical solutions of
Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed
by an exten-sion of Krylov's method of shaken coefficients. We apply our
results to a problem of option pricing under model uncertainty with different
interest rates for borrowing and lending.Comment: To appear in MO
Measurability of Semimartingale Characteristics with Respect to the Probability Law
Given a c\`adl\`ag process on a filtered measurable space, we construct a
version of its semimartingale characteristics which is measurable with respect
to the underlying probability law. More precisely, let be
the set of all probability measures under which is a semimartingale. We
construct processes which are jointly measurable in time,
space, and the probability law , and are versions of the semimartingale
characteristics of under for each . This result
gives a general and unifying answer to measurability questions that arise in
the context of quasi-sure analysis and stochastic control under the weak
formulation.Comment: 37 pages; forthcoming in 'Stochastic Processes and their
Applications