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A new formula for the ultimate ruin probability in the Cramér-Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples. [ABSTRACT FROM AUTHOR]

We describe an approach to the description of the time-development of the process of strong field ionization of atoms based on the calculation of the joint probability of occurrence of two events, event B being finding atom in the ionized state after the end of the laser pulse, event A being finding a particular value of a given physical observable at a moment of time inside the laser pulse duration. As an example of such an physical observable we consider lateral velocity component of the electron's velocity. Our approach allows us to study time-evolution of the lateral velocity distribution for the ionized electron during the interval of the laser pulse duration. We present results of such a study for the cases of target atomic systems with short range Yukawa and Coulomb interactions. [ABSTRACT FROM AUTHOR]

We describe an approach to the description of the time-development of the process of strong field ionization of atoms based on the calculation of the joint probability of occurrence of two events, event B being finding atom in the ionized state after the end of the laser pulse, event A being finding a particular value of a given physical observable at a moment of time inside the laser pulse duration. As an example of such an physical observable we consider lateral velocity component of the electron's velocity. Our approach allows us to study time-evolution of the lateral velocity distribution for the ionized electron during the interval of the laser pulse duration. We present results of such a study for the cases of target atomic systems with short range Yukawa and Coulomb interactions. [ABSTRACT FROM AUTHOR]

Our aim is to make a step toward clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In Schrödinger’s words, this is entanglement of knowledge which can be extracted via conditional measurements. In particular, quantum probabilities are interpreted as conditional ones (as, e.g., by Ballentine). We restrict considerations to perfect conditional correlations (PCC) induced by measurements (‘EPR entanglement’). Such entanglement is coupled to the pairs of observables with the projection type state update as the back action of measurement. In this way, we determine a special class of entangled states. One of our aims is to decouple the notion of entanglement from the compound systems. The rigid association of entanglement with the state of a few body systems stimulated its linking with quantum nonlocality (‘spooky action at a distance’). However, already by Schrödinger entanglement was presented as knotting of knowledge (about statistics) for one observable A with knowledge about another observable B. [ABSTRACT FROM AUTHOR]

The collapse of a quantum state can be understood as a mathematical way to construct a joint probability density even for operators that do not commute. We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. The familiar collapse picture, in which a quantum state collapses after each measurement as a way to construct a joint probability density for consecutive measurements, is equivalent to a no-collapse picture in which LĂĽders transformers applied to subsequent measurements construct a quantum-mechanicsâ€"free subsystem of quantum non-demolition operators, not as a dynamical process but as an alternative mathematical model for the same consecutive measurements. The no-collapse picture is particularly simpler when we apply signal analysis to millions or billions of consecutive measurements. [ABSTRACT FROM AUTHOR]

The sports betting industry has been growing at a phenomenal rate and has many similarities to the financial market in that a payout is made contingent on an outcome of an event. Despite this, there has been little to no mathematical focus on the potential ruin of bookmakers. In this paper, the expected profit of a bookmaker and probability of multiple soccer matches are observed via Dirac notations and Feynman's path calculations. Furthermore, we take the unforeseen circumstances into account by subjecting the betting process to more uncertainty. A perturbed betting process, set by modifying the conventional stochastic process, is handled to scale and manage this uncertainty. [ABSTRACT FROM AUTHOR]

Recent theories developing broad notions of context and its effects on inference are becoming increasingly important in fields as diverse as cognitive psychology, information science and quantum information theory and computing. Here we introduce a novel and general approach to the characterisation of contextuality using the techniques of Chu spaces and Channel Theory viewed as general theories of information flow. This involves introducing three essential components into the formulism: events, conditions and measurement systems. Incorporating these factors in relationship to conditional probabilities leads to information flows both in the setting of Chu spaces and Channel Theory. The latter provides a representation of semantic content using local logics from which conditionals can be derived. We employ these features to construct cone-cocone diagrams, commutativity of which enforces inferential coherence. With these we build a scale-free architecture incorporating a Bayesian-like hierarchical structure, in which there is an interpretation of active inference and Markov blankets. We compare this architecture with other theories of contextuality which we briefly review. We also show that this development of ideas conveniently accommodates negative probabilities, leading to the notion of signed information flow, and address how quantum contextuality can be interpreted within this model. Finally, we relate contextuality to the Frame Problem, another way of characterising a fundamental limitation on the observational and inferential capabilities of finite agents. [ABSTRACT FROM AUTHOR]

Uncertainty is an intrinsic part of life; most events, affairs, and questions are uncertain. A key problem in behavioral sciences is how the mind copes with uncertain information. Quantum probability theory offers a set of principles for inference, which align well with intuition about psychological processes in certain cases: cases when it appears that inference is contextual, the mental state changes as a result of previous judgments, or there is interference between different possibilities. We motivate the use of quantum theory in cognition and its key characteristics. For each of these characteristics, we review relevant quantum cognitive models and empirical support. The scope of quantum cognitive models encompasses fallacies in decision-making (such as the conjunction fallacy or the disjunction effect), question order effects, conceptual combination, evidence accumulation, perception, over-/underdistribution effects in memory, and more. Quantum models often formalize psychological ideas previously expressed in heuristic terms, allow unified explanations of previously disparate findings, and have led to several surprising, novel predictions. We also cast a critical eye on quantum models and consider some of their shortcomings and issues regarding their further development. [ABSTRACT FROM AUTHOR]

Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. Assuming that our usual inference procedure is working rationally for every set of logical propositions represented in terms of commuting projectors on a given Hilbert space, we extend the logical interpretation to quantum mechanics and derive the Born rule. Our result implies that, from epistemological viewpoints, we can regard quantum mechanics as a natural extension of classical probability theory. [ABSTRACT FROM AUTHOR]