This thesis comprises two independent parts.
In the first part, we develop a pathwise calculus for functionals of integer-valued
measures and extend the framework of Functional Itô Calculus to functionals of
integer-valued random measures by constructing a ’stochastic derivative’ operator
with respect to such integer-valued random measures. This allows us to obtain
weak martingale representation formulae holding beyond the class of Poisson random
measures, and allowing for random and time-dependent compensators. We study
the behaviour of this operator and compare it with other previous approaches in
the literature, providing in passing a review of the various Malliavin approaches for
jump processes. Finally, some examples of computations are provided.
The second part is oriented towards nonparametric statistics, with a financial
application as our main goal: we aim at recovering a surface of FX call options
on a pegged currency such as the Hong Kong dollar against the U.S. dollar, based
on a small number of noisy measurements (the market bid-ask quotes). Inspiring
ourselves from the Compressed Sensing literature, we develop a methodology that
aims at recovering an arbitrage-free call surface. We first apply this methodology, based on tensor polynomial decomposition of the surface, to a sparse set of
call-option prices on the S&P500, recovering the call option prices within desired
tolerance, as well as a smooth local-volatility surface. On a pegged currency such
as the HKD/USD, it appears that tensor polynomials may not be an adequate way
to model the smiles across maturities. Modifying the methodology in favour of
structure-preserving functions, we apply the new methodology to our HKD/USD
dataset, recovering the smiles, and the corresponding state-price density.