Showing papers in "Journal of the Optical Society of America in 2021"
TL;DR: In this article, the LQG homing problem is considered for Wiener processes with jumps that are exponentially distributed, and the objective is either to minimize (or maximize) the expected time spent by the controlled process in an interval [a, b], or to make the process leave this interval through a given endpoint.
Abstract: The LQG homing problem, in which a diffusion process is controlled until a certain event takes place, is considered for Wiener processes with jumps that are exponentially distributed. The objective is either to minimize (or maximize) the expected time spent by the controlled process in an interval [a, b], or to try to make the process leave this interval through a given endpoint. The integro-differential equation satisfied by the value function is transformed into a non-linear ordinary differential equation and is solved exactly in particular cases.
4 citations
TL;DR: The goal of this paper is to establish a connection between stochastically controlled-type processes, a concept reminiscent from rough paths theory, and the so-called weak Dirichlet processes.
Abstract: Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on deterministic techniques but there is still a probability in the background. The goal of this paper is to establish a connection between stochastically controlled-type processes, a concept reminiscent from rough paths theory, and the so-called weak Dirichlet processes. As a by-product, we present the connection between rough and Stratonovich integrals for cadlag weak Dirichlet processes integrands and continuous semimartingales integrators.
4 citations
TL;DR: In this article, the authors introduced a new centered mixed self-similar Gaussian process called the mixed generalized fractional Brownian motion, which could serve as a good model for a larger class of natural phenomena.
Abstract: To extend several known centered Gaussian processes, we introduce a new centered mixed self-similar Gaussian process called the mixed generalized fractional Brownian motion, which could serve as a good model for a larger class of natural phenomena. This process generalizes both the well-known mixed fractional Brownian motion introduced by Cheridito [7] and the generalized fractional Brownian motion introduced by Zili [29]. We study its main stochastic properties, its non-Markovian and non-stationarity characteristics and the conditions under which it is not a semimartingale. We prove the long-range dependence properties of this process.
3 citations
TL;DR: In a fuzzy hypothesis test the null hypothesis is rejected or accepted with a degree of rejection or acceptance using fuzzy significance level and fuzzy p-value, which is a fuzzy number constructed by a fuzzy estimator.
Abstract: In a fuzzy hypothesis test the null hypothesis is rejected or accepted with a degree of rejection or acceptance. A way to carry out such a test is to use fuzzy significance level and fuzzy p-value, which is a fuzzy number constructed by a fuzzy estimator. This approach is particularly useful in critical situations, where subtle comparisons between almost equal statistical quantities have to be made. In such cases the fuzzy hypotheses tests give better results than the crisp tests, since they give us the possibility of partial rejection or acceptance of H0 using a degree of rejection or acceptance obtained by ordering the fuzzy numbers of p-value and significance level.
3 citations
TL;DR: In this article, it was shown that the quantum decomposition of a classical random variable, or random field, is a very general phenomenon involving only an increasing filtration of Hilbert spaces and a family of Hermitean operators increasing by 1 the filter rate.
Abstract: We prove that the quantum decomposition of a classical random variable, or random field, is a very general phenomenon involving only an increasing filtration of Hilbert spaces and a family of Hermitean operators increasing by 1 the filtration. The creation, annihilation and preservation operators (CAP operators), defining the quantum decomposition of these Hermitean operators, satisfy commutation relations that generalize those of usual quantum mechanics. In fact there are two types of commutation relations (Type I and Type II). In Type I commutation relations the commutator is given by an operator–valued sesqui–linear form. The case when this operator–valued sesqui–linear form is scalar valued (multiples of the identity) characterizes the non–relativistic free Bose field and the associated commutation relations reduce to the Heisenberg ones. Type II commutation relations did not appear up to now because they are identically satisfied when the probability distribution of the random field is a product measure. In this sense they encode information on the self–interaction of the random field.
3 citations
TL;DR: In this article, the integration by parts formula for such jump processes is studied, and the strategy is based upon the calculus on Brownian motions via the Kolmogorov backward equations.
Abstract: Consider solutions to Marcus-type stochastic differential equations with jumps on the bundle of orthonormal frames O(M) over a Riemannian manifold M , and define the M -valued process by its canonical projection, which is parallel to the Eells-Elworthy-Malliavin construction of Brownian motions on M . In the present paper, the integration by parts formula for such jump processes is studied, and the strategy is based upon the calculus on Brownian motions via the Kolmogorov backward equations. The celebrated Bismut formula can be also obtained in our setting.
2 citations
2 citations
TL;DR: In this paper, a type decomposition formula for call option prices for the Barndorff-Nielsen and Shephard model with infinite active jumps is provided, which is the first result on the Al\`os type decompositions formula for models with continuous active jumps.
Abstract: The objective is to provide an Al\`os type decomposition formula of call option prices for the Barndorff-Nielsen and Shephard model: an Ornstein-Uhlenbeck type stochastic volatility model driven by a subordinator without drift. Al\`os (2012) introduced a decomposition expression for the Heston model by using Ito's formula. In this paper, we extend it to the Barndorff-Nielsen and Shephard model. As far as we know, this is the first result on the Al\`os type decomposition formula for models with infinite active jumps.
2 citations
TL;DR: In this paper, the authors extend Varadhan's construction of the Edwards model for polymers to fractional Brownian loops and fractional brownian starbursts, and show that the Edwards density under a renormalizaion is an integrable function for the case Hd ≤ 1.
Abstract: We extend Varadhan’s construction of the Edwards model for polymers to fractional Brownian loops and fractional Brownian starbursts. We show that, as in the fBm case, the Edwards density under a renormalizaion is an integrable function for the case Hd ≤ 1.
2 citations
TL;DR: In this paper, the authors show how one can construct non-Gaussian Martingale processes, which have marginal distributions that tend to the Cauchy distribution in the large volatility limit.
Abstract: The application of the Cauchy distribution has often been discussed as a potential model of the financial markets. In particular the way in which single extreme, or "Black Swan'', events can impact long term historical moments, is often cited. In this article we show how one can construct Martingale processes, which have marginal distributions that tend to the Cauchy distribution in the large volatility limit. This provides financial justification to the approaches investigated by other authors, and highlights an example of how quantum probability can be used to construct non-Gaussian Martingales. We go on to illustrate links with hyperbolic diffusion, and discuss the insight this provides.
1 citations
TL;DR: This article derives semimartingale dynamics for a semi-Markov chain and gives them in a new vector form which explicitly exhibits the times at which jump-events occur and the probabilities of state transitions.
Abstract: In this article we investigate estimation for a partially observed semi-Markov chain, or a Hidden semi-Markov Model (HsMM). We derive semimartingale dynamics for a semi-Markov chain and give them in a new vector form which explicitly exhibits the times at which jump-events occur and the probabilities of state transitions. However, the most important result is the new vector lattice state-space representation for a general finite-state, discrete-time semi-Markov chain. On this space the semi-Markov chain and its occupation times are a Markov process with dynamics described by finite matrices. These representations are new. Finite dimensional recursive filters are derived for a HsMM.
TL;DR: In this paper, the authors studied the limit behavior of the Wishart matrix and proved that it converges in law to a diagonal random matrix whose diagonal elements are random variables in the second Wiener chaos.
Abstract: We analyze the limit behavior of the Wishart matrix Wn,d = Xn,dX n,d constructed from an n × d random matrix Xn,d whose entries are given by the increments of the Hermite process. These entries are correlated on the same row, independent from one row to another and their probability distribution is different on different rows. We prove that the Wishart matrix converges in law, as d → ∞, to a diagonal random matrix whose diagonal elements are random variables in the second Wiener chaos. We also estimate the Wasserstein distance associated to this convergence.
TL;DR: In this article, an anticipating stochastic integral was constructed by linearly decomposing a class of non-measurable random variables and applied to the derivation of the Itô formula.
Abstract: In this paper, we construct an anticipating stochastic integral by linearly decompose a class of non Ft-measurable random variables. The result is applied to the derivation of the Itô formula.
TL;DR: In this paper, the authors estimate the rate of convergence in the central limit theorem for a sequence of Pareto variables Xk with shape parameter r. If r ≤ 4, E(|X1|) = ∞ and the Berry-Esseen theorem cannot be applied.
Abstract: We estimate the rate of convergence in the central limit theorem for a sequence of iid Pareto variables Xk with shape parameter r. If r ≤ 4, E(|X1|) = ∞ and the Berry-Esseen theorem cannot be applied. In these cases the rate of convergence is very slow and can be expressed as a function of r.