arXiv subjects

Levent A. Mengütürk

Publications and source records attributed to Levent A. Mengütürk.

4 recordsLinked to original sources

arXiv2026

Random-Bridges as Stochastic Transports for Generative Models

This paper motivates the use of random-bridges -- stochastic processes conditioned to take target distributions at fixed timepoints -- in the realm of generative modelling. Herein, random-bridges can act as stochastic transports between two probability distributions when appropriately initialized, and can display either Markovian or non-Markovian, and either continuous, discontinuous or hybrid patterns depending on the driving process. We show how one can start from general probabilistic statements and then branch out into specific representations for learning and simulation algorithms in terms of information processing. Our empirical results, built on Gaussian random bridges, produce high-quality samples in significantly fewer steps compared to traditional approaches, while achieving competitive Frechet inception distance scores. Our analysis provides evidence that the proposed framework is computationally cheap and suitable for high-speed generation tasks.

cs.LG↗

arXiv2026

Martingale Projections and Quantum Decoherence

We introduce so-called super/sub-martingale projections as a family of endomorphisms defined on unions of Polish spaces. Such projections allow us to identify martingales as collections of transformations that relate path-valued random variables to each other under conditional expectations. In this sense, super/sub-martingale projections are random functionals that (i) are boundedness preserving and (ii) satisfy a conditional expectation criterion similar to that of the classical martingale theory. As an application to the theory of open quantum systems, we prove (a) that any system-environment interaction that manifests a supermartingale projection on the density matrix gives rise to decoherence, and (b) that any system-environment interaction that manifests a submartingale projection gives rise an increase in Shannon-Wiener information. It follows (c) that martingale projections in an open quantum system give rise both to quantum decoherence and to information gain.

quant-ph↗

arXiv2021

Captive Jump Processes

We explicitly construct so-called captive jump processes. These are stochastic processes in continuous time, whose dynamics are confined by a time-inhomogeneous bounded domain. The drift and volatility of the captive processes depend on the domain's boundaries which in turn restricts the state space of the process. We show how an insurmountable shell may contain the captive jump process while the process may jump within the restricted domain. In a further development, we also show how - within a confined domain - inner time-dependent corridors can be introduced, which a captive jump process may leave only if the jumps reach far enough, while nonetheless being unable to ever penetrate the outer confining shell. Captive jump processes generalize the recently developed captive diffusion processes. In the case where a captive jump-diffusion is a continuous martingale or a pure-jump process, the uppermost confining boundary is non-decreasing, and the lowermost confining boundary is non-increasing. Captive jump processes could be considered to model phenomena such as electrons transitioning from one orbit (valence shell) to another, and quantum tunneling where stochastic wave-functions can "penetrate" boundaries (i.e., walls) of potential energy. We provide concrete, worked-out examples and simulations of the dynamics of captive jump processes in a geometry with circular boundaries, for demonstration.

math.PR↗

arXiv2020

Modulated Information Flows in Financial Markets

We model continuous-time information flows generated by a number of information sources that switch on and off at random times. By modulating a multi-dimensional Lévy random bridge over a random point field, our framework relates the discovery of relevant new information sources to jumps in conditional expectation martingales. In the canonical Brownian random bridge case, we show that the underlying measure-valued process follows jump-diffusion dynamics, where the jumps are governed by information switches. The dynamic representation gives rise to a set of stochastically-linked Brownian motions on random time intervals that capture evolving information states, as well as to a state-dependent stochastic volatility evolution with jumps. The nature of information flows usually exhibits complex behaviour, however, we maintain analytic tractability by introducing what we term the effective and complementary information processes, which dynamically incorporate active and inactive information, respectively. As an application, we price a financial vanilla option, which we prove is expressed by a weighted sum of option values based on the possible state configurations at expiry. This result may be viewed as an information-based analogue of Merton's option price, but where jump-diffusion arises endogenously. The proposed information flows also lend themselves to the quantification of asymmetric informational advantage among competitive agents, a feature we analyse by notions of information geometry.

math.PR↗