Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0710.1606

Operator Methods, Abelian Processes and Dynamic Conditioning

Claudio Albanese
View original source ↗Read source document ↗Export metadata ↓

Abstract

A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field and leads to new numerical analysis techniques. 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 effectively 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.

Explore related subjects

math.PRmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Claudio Albanese. 2007-11-01. Operator Methods, Abelian Processes and Dynamic Conditioning. https://arxiv.org/abs/0710.1606

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Selection in the Zero-Noise Limit for High-Dimensional Diffusions with Measurable Drift

This paper investigates the zero-noise limit of high-dimensional small-noise diffusion processes governed by the stochastic differential equation (SDE) \begin{equation*} \mathrm{d}X_{t}^{\varepsilon }=b(X_{t}^{\varepsilon })\,\mathrm{d}t+\varepsilon \,\mathrm{d}W_{t},\quad X_{0}^{\varepsilon }=0,\quad \varepsilon >0, \end{equation*} where the drift coefficient $b$ is assumed to be measurable and bounded. Under this condition, the associated deterministic ordinary differential equation (ODE) $\dot{x}_{t}=b(x_{t})$ may admit multiple Filippov solutions, i.e., solutions in the sense of differential inclusions, due to the lack of Lipschitz continuity or uniqueness criteria. However, the introduction of non-degenerate additive noise restores well-posedness: the perturbed system admits a unique strong (pathwise) solution for each $\varepsilon >0$. Our analysis is based on the radial-spherical decomposition. We prove that, under certain assumptions, the radius dominates a one-dimensional comparison process and escapes to infinity at a linear rate, and the extrinsic angular martingale converges. Under an explicit separable gradient structure, expressed in terms of a radial potential $V(r,ω)=\langle b(rω),ω\rangle $, the angle converges, for every fixed noise, almost surely to the set of global angular maximizers of $V$, and its distance from that set tends to zero in probability as $\varepsilon\rightarrow0$. The limiting measure is compact, carried by immediate-departure Filippov paths, and, under a Lipschitz radial-graph condition, purely singular with Hausdorff dimension at most $d-1$. A continuous-mapping criterion for exit-time convergence is also established.

math.PR↗
arXiv2026

Asymptotic optimality of dynamic first-fit packing on the half-axis

We revisit a classical problem in dynamic storage allocation. Items arrive in a linear storage medium, modeled as a half-axis, at a Poisson rate $r$ and depart after an independent exponentially distributed unit mean service time. The arriving item sizes (lengths) are assumed to be independent and identically distributed (i.i.d.) from a common distribution $H$. A widely employed algorithm for allocating the items is the "first-fit" discipline, namely, each arriving item is placed in the left-most vacant interval large enough to accommodate it. In a seminal 1985 paper, Coffman, Kadota, and Shepp ([6]) proved that in the special case of unit length items (i.e. degenerate $H$), as $r$ tends towards infinity, the first-fit algorithm is asymptotically optimal in the following sense: the steady-state ratio of expected "empty space" (gaps between items) to expected occupied space tends towards $0$. In a sequel to [6], Coffman, Kadota, and Shepp ([5]) conjectured that the first-fit discipline is also asymptotically optimal for non-degenerate $H$. In this paper we provide the first proof of first-fit asymptotic optimality for non-degenerate distributions $H$ of item sizes. Our main result is for the case when $H$ is concentrated on countably many positive real sizes forming an increasing sequence that is either finite or goes to infinity, with the average item size being finite. We prove that under the first-fit discipline, as $r$ tends towards infinity, the steady-state packing configuration (scaled down by $r$) converges in distribution to the limiting packing configuration with smaller items on the left, larger items on the right, and with no gaps between. In particular, this proves asymptotic optimality of first-fit in the following sense: if $P$ is the expected occupied space, then in steady-state the empty space (scaled down by $r$) in $[0,P]$ vanishes.

math.PR↗
arXiv2026

Mind the jumps: well-posedness of semi-martingale 2BSDEs

We develop a well-posedness theory for second-order backward stochastic differential equations (2BSDEs) with jumps. This work covers two complementary notions of solution: an extrinsic and an intrinsic one. We also discuss the obstruction to aggregating jump integrands. The chosen framework allows for controlled diffusions with jumps, pure-jump processes, and discrete-time processes in a unified setting.

math.PR↗