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Seminario del 2019
2019
13 novembre
Francesco Russo
nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA
Seminario di probabilità
The aim of this talk consists in introducing a formalism for the deterministic analysis associated
with backward stochastic differential equations driven by general càdlàg martingales, coupled with
a forward process. When the martingale is a standard Brownian motion, the natural deterministic analysis is
provided by the solution u of a semilinear PDE of parabolic type coupled with a function v which
is associated with the gradient ∇u, when u is of class C1 in space. When u is only a viscosity
solution of the PDE, the link associating v to u is not completely clear: sometimes in the literature
it is called the identification problem. We introduce in particular the notion of a decoupled mild
solution of a PDE, a IPDE, a path-dependent PDE or more generally a deterministic problem
associated with a BSDE. The idea is to introduce a suitable analysis to investigate the equivalent of the identification
problem first in a general Markovian setting with a class of examples. An interesting application
concerns the hedging problem under basis risk of a contingent claim g(XT,ST ), where S (resp.
X) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated
Föllmer-Schweizer decomposition. We revisit the case when the couple of price processes (X,S) is
a diffusion and we provide explicit expressions when (X,S) is an exponential of additive processes.
Extensions to non-Markovian (path-dependent) cases are discussed.