arXiv subjects

Lane P. Hughston

Publications and source records attributed to Lane P. Hughston.

At least 19 recordsLinked to original sources

arXiv2026

Tonnetz Theory, Classical Harmony, and the Combinatorial Geometry of Abstract Musical Resources

In a previous submission, we established a fundamental relation between tone networks and configurations. It was shown that the Eulerian tonnetz can be represented by a $\{12_3\}$ of Daublebsky von Sterneck type D222. We also constructed a tonnetz for Tristan-genus chords (dominant sevenths and half-diminished sevenths) and showed that this tonnetz can be represented by a $\{12_3\}$ of type D228. In both constructions the associated Levi graphs play an important role. Here we look at the tonnetze associated with some other musical systems, thereby offering concrete examples of an abstract view of music as combinatorial geometry. First, we look at the tonal harmonies of the classical period. In the case of diatonic triads, we show the existence of a bipartite graph of type $\{7_3\}$ and girth four that represents the relations between the seven diatonic degrees and their pitch classes. In the case of diatonic seventh chords, we obtain a Fano configuration $\{7_3\}$, which gives a characterization of the voice-leading relations that hold between such chords. Next, we construct a tonnetz for pentatonic music based on the Desargues configuration $\{10_3\}$ and we construct a tonnetz for the 12-tone system based on the Cremona-Richmond configuration $\{15_3\}$. Both can be used as resources for compositions. Finally, we show that the relation between the chromatic pitch class set and the major triad set is represented by a D222. The minor triads are in one-to-one correspondence with the members of a certain class of hexacycles in the Levi graph of this configuration. In this way, the characteristic duality between major and minor triads in the tonnetz can be broken.

math.CO↗

arXiv2026

Quantum clock and Newtonian time

An extension of standard quantum mechanics is proposed in which the Newtonian time appearing as a parameter in the unitary evolution operator is replaced with the time shown by a `quantum clock'. Such a clock is defined by the following properties: (a) the time that the clock shows is nondecreasing, (b) the clock ticks at random Newtonian times with random tick sizes, and (c) on average the clock shows the Newtonian time. We show that the leading term in the evolution equation for the density matrix associated with any quantum clock gives the von Neumann equation. The leading correction to the von Neumann equation is given by the Lindblad equation generated by the Hamiltonian, but there are higher-order terms that generalize the von Neumann equation and the Lindblad terms. Modifications to the von Neumann equation are worked out in detail in a parametric family of models for which the tick sizes are gamma distributed. Lower bounds on the parameters of these quantum clock models are derived using the precision limit of an atomic clock. An anomalous term in the Ehrenfest theorem for a free particle is derived, which in principle can be used as a basis for testing such models.

quant-ph↗

arXiv2026

Configurations, Tessellations and Tone Networks

The tonnetz, which is commonly represented as a tessellation of the plane by a triangular network of tones, can also be represented as a bipartite graph of degree three with twelve vertices denoting major triads and twelve vertices denoting minor triads. We show that this Levi graph can be realized geometrically as a system of twelve points and twelve lines in $\mathbb R^2$ with the property that three points lie on each line and three lines pass through each point, in a configuration $\{12_3\}$ of Daublebsky von Sterneck type D222. This tonnetz configuration, alongside various generalizations thereof, can be used as a new basis for the composition and analysis of music.

math.CO↗

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↗

arXiv2024

Decoherence Implies Information Gain

It is shown that if the wave function of a quantum system undergoes an arbitrary random transformation such that the diagonal elements of the density matrix in the decoherence basis associated with a preferred observable remain constant, then (i) the off-diagonal elements of the density matrix become smaller in magnitude, and (ii) the state of the system gains information about the preferred observable from its environment in the sense that the uncertainty of the observable is reduced in the transformed state. These results do not depend on the details of how the system-environment interaction generates the random state transformation, and together imply that decoherence leads in general to information gain, not information loss.

quant-ph↗

arXiv2024

Brief Synopsis of the Scientific Career of T. R. Hurd

As an introduction to a Special Issue of International Journal of Theoretical and Applied Finance in Honour of the Memory of Thomas Robert Hurd we present a brief synopsis of Tom Hurd's scientific career and a bibliography of his scientific publications.

q-fin.MF↗

arXiv2024

Information-Based Trading

We consider a pair of traders in a market where the information available to the second trader is a strict subset of the information available to the first trader. The traders make prices based on the information available concerning a security that pays a random cash flow at a fixed time $T$ in the future. Market information is modelled in line with the scheme of Brody, Hughston & Macrina (2007, 2008) and Brody, Davis, Friedman & Hughston (2009). The risk-neutral distribution of the cash flow is known to the traders, who make prices with a fixed multiplicative bid-offer spread and report their prices to a game master who declares that a trade has been made when the bid price of one of the traders crosses the offer price of the other. We prove that the value of the first trader's position is strictly greater than that of the second. The results are analyzed by use of simulation studies and generalized to situations where (a) there is a hierarchy of traders, (b) there are multiple successive trades, and (c) there is inventory aversion.

q-fin.MF↗

arXiv2023

Valuation of a Financial Claim Contingent on the Outcome of a Quantum Measurement

We consider a rational agent who at time $0$ enters into a financial contract for which the payout is determined by a quantum measurement at some time $T>0$. The state of the quantum system is given in the Heisenberg representation by a known density matrix $\hat p$. How much will the agent be willing to pay at time $0$ to enter into such a contract? In the case of a finite dimensional Hilbert space, each such claim is represented by an observable $\hat X_T$ where the eigenvalues of $\hat X_T$ determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state $\hat q$ which is equivalent to the physical state $\hat p$ on null spaces such that the pricing function $Π_{0T}$ takes the form $Π_{0T}(\hat X_T) = P_{0T}\,{\rm tr} ( \hat q \hat X_T) $ for any claim $\hat X_T$, where $P_{0T}$ is the one-period discount factor. By "equivalent" we mean that $\hat p$ and $\hat q$ share the same null space: thus, for any $|ξ\rangle \in \mathcal H$ one has $\langle \bar ξ| \hat p | ξ\rangle = 0$ if and only if $\langle \bar ξ| \hat q | ξ\rangle = 0$. We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen-Specker theorem in such a setting and we look at the problem of forming portfolios of such contracts. Finally, we consider multi-period contracts.

quant-ph↗

arXiv2023

Mathematical Foundations of Complex Tonality

Equal temperament, in which semitones are tuned in the irrational ratio of $2^{1/12} : 1$, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of $2$, $3$, and $5$, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system. The tritone, so problematic in just temperament, given ambiguously by the ratios $\tfrac{45}{32}$, $\tfrac{64}{45}$, $\tfrac{36}{25}$, $\tfrac{25}{18}$, none satisfactory, is in our scheme represented by the complex ratio $1 + \rm{i} : 1$. The major and minor whole tones, given by intervals of $\tfrac{9}{8}$ and $\tfrac{10}{9}$, can each be factorized into products of complex semitones, giving us a major complex semitone $\tfrac{3}{4}(1 + \rm{i})$ and a minor complex semitone $\tfrac{1}{3}(3 + \rm{i})$. The perfect third, given by the interval $\tfrac{5}{4}$, factorizes into the product of a complex whole tone $\tfrac{1}{2}(1 + 2\rm{i})$ and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of low powers of Gaussian primes leads to the construction of a complete system of major and minor scales in all keys.

cs.SD↗

arXiv2023

Lévy Models for Collapse of the Wave Function

Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.

quant-ph↗

arXiv2021

On the Pricing of Storable Commodities

This paper introduces an information-based model for the pricing of storable commodities such as crude oil and natural gas. The model uses the concept of market information about future supply and demand as a basis for valuation. Physical ownership of a commodity is taken to provide a stream of convenience dividends equivalent to a continuous cash flow. The market filtration is assumed to be generated jointly by (i) current and past levels of the dividend rate, and (ii) partial information concerning the future of the dividend flow. The price of a commodity is the expectation under a suitable pricing measure of the totality of the discounted risk-adjusted future convenience dividend, conditional on the information provided by the market filtration. In the situation where the dividend rate is modelled by an Ornstein-Uhlenbeck process, the prices of options on commodities can be derived in closed form. The approach that we present can be applied to other assets that yield potentially negative effective cash flows, such as real estate, factories, refineries, mines, and power generating plants.

q-fin.PR↗

arXiv2021

Quantum Measurement of Space-Time Events

The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.

quant-ph↗

arXiv2021

Lévy-Ito Models in Finance

We present an overview of the broad class of financial models in which the prices of assets are Lévy-Ito processes driven by an $n$-dimensional Brownian motion and an independent Poisson random measure. The Poisson random measure is associated with an $n$-dimensional Lévy process. Each model consists of a pricing kernel, a money market account, and one or more risky assets. We show how the excess rate of return above the interest rate can be calculated for risky assets in such models, thus showing the relationship between risk and return when asset prices have jumps. The framework is applied to a variety of asset classes, allowing one to construct new models as well as interesting generalizations of familiar models.

q-fin.MF↗

arXiv2020

Pricing with Variance Gamma Information

In the information-based pricing framework of Brody, Hughston and Macrina, the market filtration $\{ \mathcal F_t\}_{t\geq 0}$ is generated by an information process $\{ ξ_t\}_{t\geq0}$ defined in such a way that at some fixed time $T$ an $\mathcal F_T$-measurable random variable $X_T$ is "revealed". A cash flow $H_T$ is taken to depend on the market factor $X_T$, and one considers the valuation of a financial asset that delivers $H_T$ at $T$. The value $S_t$ of the asset at any time $t\in[0,T)$ is the discounted conditional expectation of $H_T$ with respect to $\mathcal F_t$, where the expectation is under the risk neutral measure and the interest rate is constant. Then $S_{T^-} = H_T$, and $S_t = 0$ for $t\geq T$. In the general situation one has a countable number of cash flows, and each cash flow can depend on a vector of market factors, each associated with an information process. In the present work, we construct a new class of models for the market filtration based on the variance-gamma process. The information process is obtained by subordinating a particular type of Brownian random bridge with a gamma process. The filtration is taken to be generated by the information process together with the gamma bridge associated with the gamma subordinator. We show that the resulting extended information process has the Markov property and hence can be used to price a variety of different financial assets, several examples of which are discussed in detail.

q-fin.MF↗

arXiv2020

Optimal Hedging in Incomplete Markets

We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a Lévy-Ito market, where assets are driven jointly by an $n$-dimensional Brownian motion and an independent Poisson random measure on an $n$-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of an expected least squared-error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.

q-fin.MF↗

arXiv2019

Theory of Cryptocurrency Interest Rates

A term structure model in which the short rate is zero is developed as a candidate for a theory of cryptocurrency interest rates. The price processes of crypto discount bonds are worked out, along with expressions for the instantaneous forward rates and the prices of interest-rate derivatives. The model admits functional degrees of freedom that can be calibrated to the initial yield curve and other market data. Our analysis suggests that strict local martingales can be used for modelling the pricing kernels associated with virtual currencies based on distributed ledger technologies.

q-fin.MF↗

arXiv2016

Quantum State Reduction

We propose an energy-driven stochastic master equation for the density matrix as a dynamical model for quantum state reduction. In contrast, most previous studies of state reduction have considered stochastic extensions of the Schrödinger equation, and have introduced the density matrix as the expectation of the random pure projection operator associated with the evolving state vector. After working out properties of the reduction process we construct a general solution to the energy-driven stochastic master equation. The solution is obtained by the use of nonlinear filtering theory and takes the form of a completely positive stochastic map.

quant-ph↗

arXiv2016

Lévy-Vasicek Models and the Long-Bond Return Process

The classical derivation of the well-known Vasicek model for interest rates is reformulated in terms of the associated pricing kernel. An advantage of the pricing kernel method is that it allows one to generalize the construction to the Lévy-Vasicek case, avoiding issues of market incompleteness. In the Lévy-Vasicek model the short rate is taken in the real-world measure to be a mean-reverting process with a general one-dimensional Lévy driver admitting exponential moments. Expressions are obtained for the Lévy-Vasicek bond prices and interest rates, along with a formula for the return on a unit investment in the long bond, defined by $L_t = \lim_{T \rightarrow \infty} P_{tT} / P_{0T}$, where $P_{tT}$ is the price at time $t$ of a $T$-maturity discount bond. We show that the pricing kernel of a Lévy-Vasicek model is uniformly integrable if and only if the long rate of interest is strictly positive.

q-fin.MF↗