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A modification of a classical Bachelier model by letting a stock price absorb at zero is revisited. Alternative proofs are given to derive option pricing formulas under the modified Bachelier model and numerical comparison with the Black–Scholes formula is provided. Quantile hedging methodology is developed for both classical and modified Bachelier models and application to pricing the pure endowment with fixed guarantee life insurance contracts is demonstrated, both theoretically and by means of a numerical example.

Two optimal stopping problems for geometric random walks with the observer’s power payoff function, on the finite and infinite horizons, are solved. For these problems, an explicit form of the cut value and also optimal stopping rules are established. It is proved that the optimal stopping rules are nonrandomized thresholds and describe the corresponding free boundary. An explicit form of the free boundary is presented.

What are the optimal times for an Internet of Things (IoT) device to act as a blockchain miner? The aim is to minimize the energy consumed by low-power IoT devices that log their data into a secure (tamper-proof) distributed ledger. We formulate the energy-efficient blockchain mining for IoT devices as a multiple-stopping time partially observed Markov decision process (POMDP) to maximize the probability of adding a block in the blockchain; we also present a model to optimize the number of stops (mining instants). In general, POMDPs are computationally intractable to solve, but we show mathematically using submodularity that the optimal mining policy has a useful structure: 1) it is monotone in belief space, and 2) it exhibits a threshold structure, which divides the belief space into two connected sets. Exploiting the structural results, we formulate a computationally-efficient linear mining policy for the blockchain-enabled IoT device. We present a policy gradient technique to optimize the parameters of the linear mining policy. Finally, we use synthetic and real Bitcoin datasets to study the performance of our proposed mining policy. We demonstrate the energy efficiency achieved by the optimal linear mining policy in contrast to other heuristic strategies.

This paper attempts to study the optimal stopping time for semi-Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the value functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value function and develop an effective iterative algorithm for computing it. Moreover, we show that the optimal and {\epsilon}-optimal stopping time can be characterized by the hitting time of the special sets. Finally, to illustrate the validity of our results, an example of a maintenance system is presented in the end., Comment: 24 pages, 0 figure

It is proved that the law of a possibly killed Levy process X , seen up to and including (resp. up to strictly before) a stopping time, determines already the law of X (resp. up to a compound Poisson component and killing).

Background The Elliot wave principle commonly characterizes the impulsive and corrective wave trends for both financial market trends and electrocardiograms. The impulsive wave trends of electrocardiograms can annotate several wave components of heart-beats including pathological heartbeat waveforms. The stopping time inquires which ordinal element satisfies the assumed mathematical condition within a numerical set. The proposed work constitutes several algorithmic states in reinforcement learning from the stopping time decision, which determines the impulsive wave trends. Each proposed algorithmic state is applicable to any relevant algorithmic state in reinforcement learning with fully numerical explanations. Because commercial electrocardiographs still misinterpret myocardial infarctions from extraordinary electrocardiograms, a novel algorithm needs to be developed to evaluate myocardial infarctions. Moreover, differential diagnosis for right ventricle infarction is required to contraindicate a medication such as nitroglycerin. Methods The proposed work implements the stopping time theory to impulsive wave trend distribution. The searching process of the stopping time theory is equivalent to the actions toward algorithmic states in reinforcement learning. The state value from each algorithmic state represents the numerically deterministic annotated results from the impulsive wave trend distribution. The shape of the impulsive waveform is evaluated from the interoperable algorithmic states via least-first-power approximation and approximate entropy. The annotated electrocardiograms from the impulsive wave trend distribution utilize a structure of neural networks to approximate the isoelectric baseline amplitude value of the electrocardiograms, and detect the conditions of myocardial infarction. The annotated results from the impulsive wave trend distribution consist of another reinforcement learning environment for the evaluation of impulsive waveform direction. Results The accuracy to discern myocardial infarction was found to be 99.2754% for the data from the comma-separated value format files, and 99.3579% for those containing representative beats. The clinical dataset included 276 electrocardiograms from the comma-separated value files and 623 representative beats. Conclusions Our study aims to support clinical interpretation on 12-channel electrocardiograms. The proposed work is suitable for a differential diagnosis under infarction in the right ventricle to avoid contraindicated medication during emergency. An impulsive waveform that is affected by myocardial infarction or the electrical direction of electrocardiography is represented as an inverse waveform.

The article deals with an arbitrary elastic 3D-system (body) that performs a controlled finite rotation with respect to some fixed axis and small nonstationary oscillations. The system oscillations occur due to external load (power control) or inertial load of the rotational transportation of the carrying body (kinematic control). The linear equations of oscillations are used in normal coordinates, in which motion is represented by eigenmodes of vibrations for system that is free in the rotation angle (including system rotation as a solid body in the case of power control) and for system fixed in rotation angle in the case of kinematic control. It is assumed that the (power or inertial)load acting on the system is proportional to some controlling finite time function from a certain class. The purpose of this article is to solve the problem of system rotation for a certain time from one rest position to another at a given finite angle using the given control function and to eliminate the elastic oscillations on the several lowest eigenmodes at the stopping time. The relations between the time of the system rotation under the action of a given control function and the eigenmodes frequencies for oscillations being eliminated are obtained on the basis of the exact solutions of the equations in normal coordinates. These relations satisfy the zero initial and final conditions. They are “tuned” by minimizing the positive definite quadratic form written for them by varying the system parameters to fulfill these relations simultaneously for several eigenfrequencies. As an example, the calculations for a model of a symmetrical spacecraft with two identical elastic solar cell panels consisting of four planar non-deformable sections connected by elastic hinges are carried out for comparison and analysis of the results accuracy. The finite rolling motion of the system with the damping at the stopping time of rotation for several (from one to three) lowest eigenmodes of antisymmetric vibrations is considered. The comparisons of the initial equations of motion for the system in generalized coordinates using several simple control functions and the found parameters of the “tuned” system with numerical solutions are accomplished.

Evaluation measures are more or less explicitly based on user models which abstract how users interact with a ranked result list and how they accumulate utility from it. However, traditional measures typically come with a hard-coded user model which can be, at best, parametrized. Moreover, they take a deterministic approach which leads to assign a precise score to a system run.In this paper, we take a different angle and, by relying on Markov chains and random walks, we propose a new family of evaluation measures which are able to accommodate for different and flexible user models, allow for simulating the interaction of different users, and turn the score into a random variable which more richly describes the performance of a system. We also show how the proposed framework allows for instantiating and better explaining some state-of-the-art measures, like AP, RBP, DCG, and ERR.