Waiting time analysis of foreign currency exchange rates: Beyond the renewal-reward theorem

Naoya Sazuka; Jun-ichi Inoue

Home/Archive/arXiv

arXiv · physics/0702003

Waiting time analysis of foreign currency exchange rates: Beyond the renewal-reward theorem

Naoya Sazuka·Jun-ichi Inoue

View original source ↗Read source document ↗Export metadata ↓

Abstract

We evaluate the average waiting time between observing the price of financial markets and the next price change, especially in an on-line foreign exchange trading service for individual customers via the internet. Basic technical idea of our present work is dependent on the so-called renewal-reward theorem. Assuming that stochastic processes of the market price changes could be regarded as a renewal process, we use the theorem to calculate the average waiting time of the process. In the conventional derivation of the theorem, it is apparently hard to evaluate the higher order moments of the waiting time. To overcome this type of difficulties, we attempt to derive the waiting time distribution Omega(s) directly for arbitrary time interval distribution (first passage time distribution) of the stochastic process P_{W}(tau) and observation time distribution P_{O}(t) of customers. Our analysis enables us to evaluate not only the first moment (the average waiting time) but also any order of the higher moments of the waiting time. Moreover, in our formalism, it is possible to model the observation of the price on the internet by the customers in terms of the observation time distribution P_{O}(t). We apply our analysis to the stochastic process of the on-line foreign exchange rate for individual customers from the Sony bank and compare the moments with the empirical data analysis.

Explore related subjects

physics.data-an physics.soc-ph q-fin.ST

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Naoya Sazuka, Jun-ichi Inoue. 2007-02-01. Waiting time analysis of foreign currency exchange rates: Beyond the renewal-reward theorem. https://arxiv.org/abs/physics/0702003

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

KEEP EXPLORING

Related papers

arXiv2026

How much data does a memory kernel need? Measurement requirements for fractional models of battery and corrosion degradation

How much data does a memory kernel need before a fit is meaningful? For a distributed-order model, the answer is about 3.1 decades of observation, more than standard battery-degradation tests provide. Battery capacity fade and corrosion-film growth depend on their history, so they are often modelled using general fractional calculus (GFC), where memory is described by an arbitrary Sonine kernel rather than a single power law. We tested this using two mechanistic models without fractional operators and de Levie's porous-electrode model, whose high-frequency order is exactly 0.5; our estimator recovered 0.500. A single aging curve can recover an effective order, alpha, to within 5% noise using a weak formulation, while separating the tempered parameters (alpha, lambda) requires several Arrhenius conditions, with errors improving from (0.25, 0.43) with one condition to (0.04, 0.07) with four. A distributed-order memory needs 3.1 decades of data to be resolved to 90%; below that, a two-order memory can collapse into one intermediate order that fits the data better than the true spectrum. For saturating passivation, however, a general kernel improves the fit substantially, raising R2 from 0.862 to 0.998. Across 20 cells from two public datasets, observation windows span only 1.23 to 2.24 decades, making the 3.1-decade result a practical design rule: it defines what must be measured before fitting a memory kernel, not only what cannot be fitted.

physics.data-an↗

arXiv2026

A Probability Model for Pentagonal Prism Dice Rolls

In the realm of probability theory and game design, the study of dice probabilities plays a crucial role. The featured dice are usually fair, so their probabilities can be predicted without rolling the dice. Pentagonal prismatic dice are not fair. Hence, the probabilities are more complicated. Our mathematical model is naturally derived from both the geometric properties of the dice and empirical evidence. Dice of varying sizes but with constant volume were designed. Using a 3D printer with 100 percent infill, the dice were printed and prepared. A machine that simulates dropping dice through a dice tower onto a hard surface was used to obtain about 41,000 results. From these data, a statistical model was constructed that gives probabilities of dice with varying height-to-radius ratios. The formula for the model is based on the centroid solid angle, but with two adjustment parameters due to varying energy and other requirements to transition between different resting aspects. Within the range of height-to-radius ratios tested, the data are within reasonable statistical error of the model.

physics.data-an↗

arXiv2026

Accelerating fits in high-energy physics and beyond

Data analysis in particle physics often involves optimizing a loss function that quantifies how well a theoretical model describes the observed data, in order to estimate the parameters of the model. This is referred to as fitting. As datasets grow larger and more expensive analysis methods are used, the time required to perform this parameter estimation can be a limiting factor. This work explores techniques for accelerating the fitting process, and measures their performance on a set of models representative of a variety of physics fits. We compare the speed and precision of variants of well-known optimization algorithms, namely Newton's method, BFGS, and the Migrad algorithm (commonly used in particle physics fits). We observe significant speedups when using some algorithms: most notably, some variants of BFGS are about ten times faster than Migrad, while achieving the same precision. For quasi-Newtonian algorithms like BFGS and Migrad, we find that the initial value of the Hessian matrix has a large impact on the runtime. We additionally discuss methods for accelerating the evaluation of the loss function and its derivatives, using algebraic simplifications, automatic differentiation, and modern hardware features. We explore techniques for achieving good precision with 32-bit operations, which can achieve significantly higher throughput on the GPU. Lastly we explore efficient parallelization strategies for the CPU and GPU. All of the techniques and algorithms in this paper are implemented in our new FastFit library, which was used for all of our benchmarks.

physics.data-an↗