Much has been written about theory and practice in the law, and the tension between practitioners and theorists. Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases. These arguments are not revisited. Instead the Essay explores the working and interaction of theory and practice, practitioners and theorists. The Essay starts with a story about solving a legal issue using our intellectual tools - theory, practice, and their progenies: experience and "gut." Next the Essay elaborates on the nature of theory, practice, experience and "gut." The third part of the Essay discusses theories that are helpful to practitioners and those that are less helpful. The Essay concludes that practitioners theorize, and theorists practice. They use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ. Theory, practice, experience and "gut" help us think, remember, decide and create. They complement each other like the two sides of the same coin: distinct but inseparable.

Of Theory and Practice

In this paper, we present a new strategy for pricing average value options, i.e. options whose payoff depends on the average price of the underlying asset over a fixed period leading up to the maturity date. Such options are of particular interest and importance for thinly-traded assets (e.g. crude oil), since price manipulation is inhibited, and both the investor and issuer enjoy a welcome degree of protection from the vagaries of the market. These options are often implicit in a bond contract, although they also appear in a straightforward form. Our results suggest that the price of an average-value option will always be lower than that of a standard European option. Our pricing strategy involves Monte Carlo simulation with variance reduction elements and offers an enhanced pricing method to both arbitragers and hedgers, as well as to the issuers of such bonds.

A Pricing Method for Options Based on Average Asset Values

Many applications, such as marine navigation, land vehicles location, etc., require real time precise positioning under medium or long baseline conditions. In this contribution, we develop a model of real-time kinematic decimeter-level positioning with BeiDou Navigation Satellite System (BDS) triple-frequency signals over medium distances. The ambiguities of two extra-wide-lane (EWL) combinations are fixed first, and then a wide lane (WL) combination is reformed based on the two EWL combinations for positioning. Theoretical analysis and empirical analysis is given of the ambiguity fixing rate and the positioning accuracy of the presented method. The results indicate that the ambiguity fixing rate can be up to more than 98% when using BDS medium baseline observations, which is much higher than that of dual-frequency Hatch-Melbourne-Wubbena (HMW) method. As for positioning accuracy, decimeter level accuracy can be achieved with this method, which is comparable to that of carrier-smoothed code differential positioning method. Signal interruption simulation experiment indicates that the proposed method can realize fast high-precision positioning whereas the carrier-smoothed code differential positioning method needs several hundreds of seconds for obtaining high precision results. We can conclude that a relatively high accuracy and high fixing rate can be achieved for triple-frequency WL method with single-epoch observations, displaying significant advantage comparing to traditional carrier-smoothed code differential positioning method.

https://www.mdpi.com/1424-8220/16/1/1/pdf

Instantaneous Real-Time Kinematic Decimeter-Level Positioning with BeiDou Triple-Frequency Signals over Medium Baselines

The analysis of time series: an introduction (4th edition), by C. Chatfield. Pp 241. £13·50. 1989. ISBN 0-412-31820-2 (Chapman and Hall)

PPP-RTK is integer ambiguity resolution-enabled precise point positioning. In this contribution, we present the principles of PPP-RTK, together with a review of different mechanizations that have been proposed in the literature. By application of $$\mathcal {S}$$
 -system theory, the estimable parameters of the different methods are identified and compared. Their interpretation is essential for gaining a proper insight into PPP-RTK in general, and into the role of the PPP-RTK corrections in particular. We show that PPP-RTK is a relative technique for which the ‘single-receiver user’ integer ambiguities are in fact double-differenced ambiguities. We determine the transformational links between the different methods and their PPP-RTK corrections, thereby showing how different PPP-RTK methods can be mixed between network and users. We also present and discuss four different estimators of the PPP-RTK corrections. It is shown how they apply to the different PPP-RTK models, as well as why some of the proposed estimation methods cannot be accepted as PPP-RTK proper. We determine analytical expressions for the variance matrices of the ambiguity-fixed and ambiguity-float PPP-RTK corrections. This gives important insight into their precision, as well as allows us to discuss which parts of the PPP-RTK correction variance matrix are essential for the user and which are not.

/pdf/review-and-principles-of-ppp-rtk-methods-2qrskwdxim.pdf

Review and principles of PPP-RTK methods

The concept of integer ambiguity resolution-enabled Precise Point Positioning (PPP-RTK) relies on appropriate network information for the parameters that are common between the single-receiver user that applies and the network that provides this information. Most of the current methods for PPP-RTK are based on forming the ionosphere-free combination using dual-frequency Global Navigation Satellite System (GNSS) observations. These methods are therefore restrictive in the light of the development of new multi-frequency GNSS constellations, as well as from the point of view that the PPP-RTK user requires ionospheric corrections to obtain integer ambiguity resolution results based on short observation time spans. The method for PPP-RTK that is presented in this article does not have above limitations as it is based on the undifferenced, uncombined GNSS observation equations, thereby keeping all parameters in the model. Working with the undifferenced observation equations implies that the models are rank-deficient; not all parameters are unbiasedly estimable, but only combinations of them. By application of $$\mathcal {S}$$
 -system theory the model is made of full rank by constraining a minimum set of parameters, or S-basis. The choice of this S-basis determines the estimability and the interpretation of the parameters that are transmitted to the PPP-RTK users. As this choice is not unique, one has to be very careful when comparing network solutions in different $$\mathcal {S}$$
 -systems; in that case the S-transformation, which is provided by the $$\mathcal {S}$$
 -system method, should be used to make the comparison. Knowing the estimability and interpretation of the parameters estimated by the network is shown to be crucial for a correct interpretation of the estimable PPP-RTK user parameters, among others the essential ambiguity parameters, which have the integer property which is clearly following from the interpretation of satellite phase biases from the network. The flexibility of the $$\mathcal {S}$$
 -system method is furthermore demonstrated by the fact that all models in this article are derived in multi-epoch mode, allowing to incorporate dynamic model constraints on all or subsets of parameters.

On the estimability of parameters in undifferenced, uncombined GNSS network and PPP-RTK user models by means of S-system theory

During the last decade the technique of CORS-based Network RTK has been proven capable of providing cm-level positioning accuracy for rovers receiving GNSS correction data from the CORS network. The technique relies on successful integer carrier-phase ambiguity resolution at both network and user level: at the level of the CORS network, ambiguity resolution is a prerequisite in order to determine very precise atmospheric corrections for (mobile) users, while these users need to resolve their integer ambiguities (with respect to a certain CORS network master reference station) to obtain precise cm-level positioning accuracy. In case of Network RTK a user thus needs corrections from the network, plus the GNSS data of one of the CORS stations. In practice, there exists a variety of implementations of the Network RTK concept, of which VRS, FKP and MAC are best known [1, 2, 3, 4]. In this contribution we discuss a closely related concept, PPP-RTK, and show its performance on two CORS networks.

PPP-RTK: Results of CORS Network-Based PPP with Integer Ambiguity Resolution

In this contribution, a novel un-differenced (UD) (PPP-RTK) concept, i.e. a synthesis of Precise Point Positioning and Network-based Real-Time Kinematic concept, is introduced. In the first step of our PPP-RTK approach, the UD GNSS observations from a regional reference network are processed based upon re-parameterised observation equations, corrections for satellite clocks, phase biases and (interpolated) atmospheric delays are calculated and provided to users. In the second step, these network-based corrections are used at the user site to restore the integer nature of his UD phase ambiguities, which makes rapid and high accuracy user positioning possible. The proposed PPP-RTK approach was tested using two GPS CORS networks with inter-station distances ranging from 60 to 100 km. The first test network is the northern China CORS network and the second is the Australian Perth CORS network. In the test of the first network, a dual-frequency PPP-RTK user receiver was used, while in the test of the second network, a low-cost, single-frequency PPP-RTK user receiver was used. The performance of fast ambiguity resolution and the high accuracy positioning of the PPP-RTK results are demonstrated.

/pdf/a-novel-un-differenced-ppp-rtk-concept-21wf380zku.pdf

A novel un-differenced PPP-RTK concept

A joint-processing model for multi-GNSS (GPS, GLONASS, BDS and GALILEO) precise point positioning (PPP) is proposed, in which raw code and phase observations are used. In the proposed model, inter-system biases (ISBs) and GLONASS code inter-frequency biases (IFBs) are carefully considered, among which GLONASS code IFBs are modeled as a linear function of frequency numbers. To get the full rank function model, the unknowns are re-parameterized and the estimable slant ionospheric delays and ISBs/IFBs are derived and estimated simultaneously. One month of data in April, 2015 from 32 stations of the International GNSS Service (IGS) Multi-GNSS Experiment (MGEX) tracking network have been used to validate the proposed model. Preliminary results show that RMS values of the positioning errors (with respect to external double-difference solutions) for static/kinematic solutions (four systems) are 6.2 mm/2.1 cm (north), 6.0 mm/2.2 cm (east) and 9.3 mm/4.9 cm (up). One-day stabilities of the estimated ISBs described by STD values are 0.36 and 0.38 ns, for GLONASS and BDS, respectively. Significant ISB jumps are identified between adjacent days for all stations, which are caused by the different satellite clock datums in different days and for different systems. Unlike ISBs, the estimated GLONASS code IFBs are quite stable for all stations, with an average STD of 0.04 ns over a month. Single-difference experiment of short baseline shows that PPP ionospheric delays are more precise than traditional leveling ionospheric delays.

Multi-GNSS precise point positioning (MGPPP) using raw observations

A synthesis of two prevailing global navigation satellite system positioning technologies, namely the precise point positioning and the network-based real-time kinematic, results in the emergence of the PPP-RTK, enabling single-receiver users to achieve high positioning accuracy with reasonable timeliness through integer ambiguity resolution. The realization of PPP-RTK needs to accomplish two sequential tasks. The first task is to determine a class of corrections including, among others, the satellite phase biases (SPBs) at the network level. With these corrections, the second task, then, is to solve for the ambiguity-fixed, absolute position at the user level. In this contribution, we revisit three variants (geometry-free, geometry-fixed and geometry-plus-satellite-clock-fixed) of the undifferenced and uncombined PPP-RTK network model and then point out their implications for practical use. We also carry out a case study using multi-day, dual-frequency global positioning system data from the crustal movement observation network of China stations, aiming to figure out what are the most appropriate linear combinations of the SPBs to be transmitted to the users from the viewpoint of decorrelation, and to assess the static and kinematic positioning performance.

Baocheng Zhang

Papers

On the estimability of parameters in undifferenced, uncombined GNSS network and PPP-RTK user models by means of S-system theory

PPP-RTK: Results of CORS Network-Based PPP with Integer Ambiguity Resolution

A novel un-differenced PPP-RTK concept

Multi-GNSS precise point positioning (MGPPP) using raw observations

PPP-RTK based on undifferenced and uncombined observations: theoretical and practical aspects