 
Summary: OPERATOR METHODS, ABELIAN PROCESSES AND DYNAMIC
CONDITIONING
CLAUDIO ALBANESE
Abstract. A mathematical framework for Continuous Time Finance based on operator al
gebraic methods offers a new direct and entirely constructive perspective on the field. It also
leads to new numerical analysis techniques which can take advantage of the emerging massively
parallel GPU architectures which are uniquely suited to execute large matrix manipulations.
This is partly a review paper as it covers and expands on the mathematical framework un
derlying a series of more applied articles. In addition, this article also presents a few key new
theorems that make the treatment selfcontained. Stochastic processes with continuous time
and continuous space variables are defined constructively by establishing new convergence es
timates for Markov chains on simplicial sequences. We emphasize high precision computability
by numerical linear algebra methods as opposed to the ability of arriving to analytically closed
form expressions in terms of special functions. Path dependent processes adapted to a given
Markov filtration are associated to an operator algebra. If this algebra is commutative, the
corresponding process is named Abelian, a concept which provides a far reaching extension of
the notion of stochastic integral. We recover the classic CameronDysonFeynmanGirsanov
ItoKacMartin theorem as a particular case of a broadly general blockdiagonalization algo
rithm. This technique has many applications ranging from the problem of pricing cliquets
to targetredemptionnotes and volatility derivatives. NonAbelian processes are also relevant
