A NUMERICAL QUADRATURE APPROACH TO OPTION VALUATION IN WATER MARKETS

The standard Black-Scholes approach to option valuation becomes cumbersome and may fail to yield a solution when applied to non-standard options such as those emerging in water markets. An alternative tool, numerical quadrature, avoids some restrictive assumptions of the Black-Scholes framework and can more easily price options with complex structures.Research Methods/ Statistical Methods,

Steering the distribution of agents in mean-field and cooperative games

The purpose of this work is to pose and solve the problem to guide a collection of weakly interacting dynamical systems (agents, particles, etc.) to a specified terminal distribution. The framework is that of mean-field and of cooperative games. A terminal cost is used to accomplish the task; we establish that the map between terminal costs and terminal probability distributions is onto. Our approach relies on and extends the theory of optimal mass transport and its generalizations.Comment: 20 pages, 8 figure

Optimal transport over a linear dynamical system

We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator ($\dot x(t) = u(t)$) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schroedinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases

On the Relation Between Optimal Transport and Schr\uf6dinger Bridges: A Stochastic Control Viewpoint

We take a new look at the relation between the optimal transport problem and the Schr\uf6dinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou\u2013Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schr\uf6dinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a Fisher information functional. We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior. This can be seen as the zero-noise limit of Schr\uf6dinger bridges when the prior is any Markovian evolution.We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided

Entropic and displacement interpolation: a computational approach using the Hilbert metric

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -- it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for interpolation of distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of recent developments in physics, probability theory, image processing, time-series analysis, and several other fields. In spite of extensive work and theoretical developments, the computation of OMT for large scale problems has remained a challenging task. An alternative framework for interpolating distributions, rooted in statistical mechanics and large deviations, is that of Schroedinger bridges (entropic interpolation). This may be seen as a stochastic regularization of OMT and can be cast as the stochastic control problem of steering the probability density of the state-vector of a dynamical system between two marginals. In this approach, however, the actual computation of flows had hardly received any attention. In recent work on Schroedinger bridges for Markov chains and quantum evolutions, we noted that the solution can be efficiently obtained from the fixed-point of a map which is contractive in the Hilbert metric. Thus, the purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which i) leads to a new proof of a classical result on Schroedinger bridges and ii) provides an efficient computational scheme for both, Schroedinger bridges and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.Comment: 20 pages, 7 figure

Steering state statistics with output feedback

Consider a linear stochastic system whose initial state is a random vector with a specified Gaussian distribution. Such a distribution may represent a collection of particles abiding by the specified system dynamics. In recent publications, we have shown that, provided the system is controllable, it is always possible to steer the state covariance to any specified terminal Gaussian distribution using state feedback. The purpose of the present work is to show that, in the case where only partial state observation is available, a necessary and sufficient condition for being able to steer the system to a specified terminal Gaussian distribution for the state vector is that the terminal state covariance be greater (in the positive-definite sense) than the error covariance of a corresponding Kalman filter.Comment: 10 pages, 2 figure

Efficient robust routing for single commodity network flows

We study single commodity network flows with suitable robustness and efficiency specs. An original use of a maximum entropy problem for distributions on the paths of the graph turns this problem into a steering problem for Markov chains with prescribed initial and final marginals. From a computational standpoint, viewing scheduling this way is especially attractive in light of the existence of an iterative algorithm to compute the solution. The present paper builds on [13] by introducing an index of efficiency of a transportation plan and points, accordingly, to efficient-robust transport policies. In developing the theory, we establish two new invariance properties of the solution (called bridge) \u2013 an iterated bridge invariance property and the invariance of the most probable paths. These properties, which were tangentially mentioned in our previous work, are fully developed here. We also show that the distribution on paths of the optimal transport policy, which depends on a \u201ctemperature\u201d parameter, tends to the solution of the \u201cmost economical\u201d but possibly less robust optimal mass transport problem as the temperature goes to zero. The relevance of all of these properties for transport over networks is illustrated in an example

Water production rates and activity of interstellar comet 2I/Borisov

We observed the interstellar comet 2I/Borisov using the Neil Gehrels-Swift Observatory's Ultraviolet/Optical Telescope. We obtained images of the OH gas and dust surrounding the nucleus at six epochs spaced before and after perihelion (-2.56 AU to 2.54 AU). Water production rates increased steadily before perihelion from $(7.0\pm1.5)\times10^{26}$ molecules s$^{-1}$ on Nov. 1, 2019 to $(10.7\pm1.2)\times10^{26}$ molecules s$^{-1}$ on Dec. 1. This rate of increase in water production rate is quicker than that of most dynamically new comets and at the slower end of the wide range of Jupiter-family comets. After perihelion, the water production rate decreased to $(4.9\pm0.9)\times10^{26}$ molecules s$^{-1}$ on Dec. 21, which is much more rapidly than that of all previously observed comets. Our sublimation model constrains the minimum radius of the nucleus to 0.37 km, and indicates an active fraction of at least 55% of the surface. $A(0)f\rho$ calculations show a variation between 57.5 and 105.6 cm with a slight trend peaking before the perihelion, lower than previous and concurrent published values. The observations confirm that 2I/Borisov is carbon-chain depleted and enriched in NH$_2$ relative to water.Comment: 12 pages, 3 figures, 2 tables, submitted to ApJ

Low-profile and wearable energy harvester based on plucked piezoelectric cantilevers

The Pizzicato Energy Harvester (EH) introduced the technique of frequency up-conversion to piezoelectric EHs wearable on the lateral side of the knee-joint. The operation principle is to pluck the piezoelectric bimorphs with plectra so that they produce electrical energy during the ensuing mechanical vibrations. The device presented in this work is, in some ways, an evolution of the earlier Pizzicato: it is a significantly more compact and lighter device; the central hub holds 16 piezoelectric bimorphs shaped as trapezoids, which permits a sleek design and potentially increased energy output for the same bimorph area. Plectra were formed by Photochemical Machining of a 100-μm-thick steel sheet. To avoid the risk of short-circuiting, the plectra were electrically passivated by sputtering a 100 nm layer of ZrO2. Bench tests with the steel plectra showed a very large energy generation. Polyimide plectra were also manufactured with a cutting plotter from a 125μm-thick film. Besides bench tests, a volunteer wore the device while walking on flat ground or climbing stairs, with a measured energy output of approximately 0.8 mJ per step. Whereas most of the tests were performed by the traditional method of discharging the rectified output from the EH onto a resistive load, tests were performed also with a circuit offering a stabilised 3.3 V supply. The circuit produced a stable 0.1 mA supply during running gait with kapton plectra