48,401 research outputs found
OPERATOR METHODS, ABELIAN PROCESSES AND DYNAMIC CONDITIONING
A mathematical framework for Continuous Time Finance based on operator algebraic
methods oers 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 underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates 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 Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can eectively use block-factorization algorithms. Finally, we discuss
the method of dynamic conditioning that allows one to dynamically correlate over possibly
even hundreds of processes in a numerically noiseless framework while preserving marginal
distributions
Reconstruction of potential energy profiles from multiple rupture time distributions
We explore the mathematical and numerical aspects of reconstructing a
potential energy profile of a molecular bond from its rupture time
distribution. While reliable reconstruction of gross attributes, such as the
height and the width of an energy barrier, can be easily extracted from a
single first passage time (FPT) distribution, the reconstruction of finer
structure is ill-conditioned. More careful analysis shows the existence of
optimal bond potential amplitudes (represented by an effective Peclet number)
and initial bond configurations that yield the most efficient numerical
reconstruction of simple potentials. Furthermore, we show that reconstruction
of more complex potentials containing multiple minima can be achieved by
simultaneously using two or more measured FPT distributions, obtained under
different physical conditions. For example, by changing the effective potential
energy surface by known amounts, additional measured FPT distributions improve
the reconstruction. We demonstrate the possibility of reconstructing potentials
with multiple minima, motivate heuristic rules-of-thumb for optimizing the
reconstruction, and discuss further applications and extensions.Comment: 20 pages, 9 figure
Affinity and Fluctuations in a Mesoscopic Noria
We exhibit the invariance of cycle affinities in finite state Markov
processes under various natural probabilistic constructions, for instance under
conditioning and under a new combinatorial construction that we call ``drag and
drop''. We show that cycle affinities have a natural probabilistic meaning
related to first passage non-equilibrium fluctuation relations that we
establish.Comment: 30 pages, 1 figur
- …