Artykuły w czasopismach: „Measure-Valued stochastic processes”

1

Panpan, Ren, i Wang Fengyu. "Stochastic analysis for measure-valued processes". SCIENTIA SINICA Mathematica 50, nr 2 (3.01.2020): 231. http://dx.doi.org/10.1360/ssm-2019-0225.

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Dawson, Donald A., i Zenghu Li. "Stochastic equations, flows and measure-valued processes". Annals of Probability 40, nr 2 (marzec 2012): 813–57. http://dx.doi.org/10.1214/10-aop629.

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Dorogovtsev, Andrey A. "Stochastic flows with interaction and measure-valued processes". International Journal of Mathematics and Mathematical Sciences 2003, nr 63 (2003): 3963–77. http://dx.doi.org/10.1155/s0161171203301073.

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We consider the new class of the Markov measure-valued stochastic processes with constant mass. We give the construction of such processes with the family of the probabilities which describe the motion of single particles. We also consider examples related to stochastic flows with the interactions and the local times for such processes.

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Méléard, Sylvie, i Sylvie Roelly. "Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations". Mathematische Nachrichten 154, nr 1 (1991): 141–56. http://dx.doi.org/10.1002/mana.19911540112.

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Dorogovtsev, Andrey A. "Measure-valued Markov processes and stochastic flows on abstract spaces". Stochastics and Stochastic Reports 76, nr 5 (październik 2004): 395–407. http://dx.doi.org/10.1080/10451120422331292216.

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Mailler, Cécile, i Denis Villemonais. "Stochastic approximation on noncompact measure spaces and application to measure-valued Pólya processes". Annals of Applied Probability 30, nr 5 (październik 2020): 2393–438. http://dx.doi.org/10.1214/20-aap1561.

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HE, HUI. "FLEMING–VIOT PROCESSES IN AN ENVIRONMENT". Infinite Dimensional Analysis, Quantum Probability and Related Topics 13, nr 03 (wrzesień 2010): 489–509. http://dx.doi.org/10.1142/s0219025710004127.

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We consider a new type of lookdown processes where spatial motion of each individual is influenced by an individual noise and a common noise, which could be regarded as an environment. Then a class of probability measure-valued processes on real line ℝ is constructed. The sample path properties are investigated: the values of this new type process are either purely atomic measures or absolutely continuous measures according to the existence of individual noise. When the process is absolutely continuous with respect to Lebesgue measure, we derive a new stochastic partial differential equation for the density process. At last we show that such processes also arise from normalizing a class of measure-valued branching diffusions in a Brownian medium as the classical result that Dawson–Watanabe superprocesses, conditioned to have total mass one, are Fleming–Viot superprocesses.

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Yurachkivs’kyi, A. P. "Generalization of one problem of stochastic geometry and related measure-valued processes". Ukrainian Mathematical Journal 52, nr 4 (kwiecień 2000): 600–613. http://dx.doi.org/10.1007/bf02515399.

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Feldman, Raisa E., i Srikanth K. Iyer. "Non-Gaussian Density Processes Arising from Non-Poisson Systems of Independent Brownian Motions". Journal of Applied Probability 35, nr 1 (marzec 1998): 213–20. http://dx.doi.org/10.1239/jap/1032192564.

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The Brownian density process is a Gaussian distribution-valued process. It can be defined either as a limit of a functional over a Poisson system of independent Brownian particles or as a solution of a stochastic partial differential equation with respect to Gaussian martingale measure. We show that, with an appropriate change in the initial distribution of the infinite particle system, the limiting density process is non-Gaussian and it solves a stochastic partial differential equation where the initial measure and the driving measure are non-Gaussian, possibly having infinite second moment.

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Feldman, Raisa E., i Srikanth K. Iyer. "Non-Gaussian Density Processes Arising from Non-Poisson Systems of Independent Brownian Motions". Journal of Applied Probability 35, nr 01 (marzec 1998): 213–20. http://dx.doi.org/10.1017/s0021900200014807.

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The Brownian density process is a Gaussian distribution-valued process. It can be defined either as a limit of a functional over a Poisson system of independent Brownian particles or as a solution of a stochastic partial differential equation with respect to Gaussian martingale measure. We show that, with an appropriate change in the initial distribution of the infinite particle system, the limiting density process is non-Gaussian and it solves a stochastic partial differential equation where the initial measure and the driving measure are non-Gaussian, possibly having infinite second moment.

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Bahlali, Seïd, Brahim Mezerdi i Boualem Djehiche. "Approximation and optimality necessary conditions in relaxed stochastic control problems". Journal of Applied Mathematics and Stochastic Analysis 2006 (1.06.2006): 1–23. http://dx.doi.org/10.1155/jamsa/2006/72762.

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We consider a control problem where the state variable is a solution of a stochastic differential equation (SDE) in which the control enters both the drift and the diffusion coefficient. We study the relaxed problem for which admissible controls are measure-valued processes and the state variable is governed by an SDE driven by an orthogonal martingale measure. Under some mild conditions on the coefficients and pathwise uniqueness, we prove that every diffusion process associated to a relaxed control is a strong limit of a sequence of diffusion processes associated to strict controls. As a consequence, we show that the strict and the relaxed control problems have the same value function and that an optimal relaxed control exists. Moreover we derive a maximum principle of the Pontriagin type, extending the well-known Peng stochastic maximum principle to the class of measure-valued controls.

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Ben Gherbal, H., A. Redjil i O. Kebiri. "THE RELAXED MAXIMUM PRINCIPLE FOR G-STOCHASTIC CONTROL SYSTEMS WITH CONTROLLED JUMPS". Advances in Mathematics: Scientific Journal 11, nr 12 (24.12.2022): 1313–43. http://dx.doi.org/10.37418/amsj.11.12.11.

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This paper is concerned with optimal control of systems driven by G-stochastic differential equations (G-SDEs), with controlled jump term. We study the relaxed problem, in which admissible controls are measure-valued processes and the state variable is governed by an G-SDE\ driven by a counting measure valued process called relaxed Poisson measure such that the compensator is a product measure. Under some conditions on the coefficients, using the G-chattering lemma, we show that the strict and the relaxed control problems have the same value function. Additionally, we derive a maximum principle for this control problem.

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13

Chen, X. "Condition for intersection occupation measure to be absolutely continuous". Ukrains’kyi Matematychnyi Zhurnal 72, nr 9 (22.09.2020): 1304–12. http://dx.doi.org/10.37863/umzh.v72i9.6278.

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UDC 519.21 Given the i.i.d. -valued stochastic processes with the stationary increments, a minimal condition is provided for the occupation measure to be absolutely continuous with respect to the Lebesgue measure on An isometry identity related to the resulting density (known as intersection local time) is also established.

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Dette, Holger, i Bettina Reuther. "Some Comments on Quasi-Birth-and-Death Processes and Matrix Measures". Journal of Probability and Statistics 2010 (2010): 1–23. http://dx.doi.org/10.1155/2010/730543.

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We explore the relation between matrix measures and quasi-birth-and-death processes. We derive an integral representation of the transition function in terms of a matrix-valued spectral measure and corresponding orthogonal matrix polynomials. We characterize several stochastic properties of quasi-birth-and-death processes by means of this matrixmeasure and illustrate the theoretical results by several examples.

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15

Schmidt, Klaus D. "The Andersen-Jessen theorem revisited". Mathematical Proceedings of the Cambridge Philosophical Society 102, nr 2 (wrzesień 1987): 351–61. http://dx.doi.org/10.1017/s0305004100067360.

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AbstractFor stochastic processes which are induced by a signed measure, the Andersen-Jessen theorem asserts almost sure convergence and yields the identification of the limit. This result has been extended to real and vector-valued stochastic processes which are induced by a finitely additive set function or a set function process. In the present paper, we study the structure of such induced stochastic processes in order to locate the Andersen-Jessen theorem and its extensions in the family of convergence theorems for martingales and their generalizations. As an application of these results, we also show that the Andersen-Jessen theorem and its extensions can be deduced from the convergence theorems for conditional expectations and positive supermartingales.

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Evans, Steven N., i Edwin A. Perkins. "Measure-Valued Branching Diffusions with Singular Interactions". Canadian Journal of Mathematics 46, nr 1 (1.02.1994): 120–68. http://dx.doi.org/10.4153/cjm-1994-004-6.

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AbstractThe usual super-Brownian motion is a measure-valued process that arises as a high density limit of a system of branching Brownian particles in which the branching mechanism is critical. In this work we consider analogous processes that model the evolution of a system of two such populations in which there is inter-species competition or predation.We first consider a competition model in which inter-species collisions may result in casualties on both sides. Using a Girsanov approach, we obtain existence and uniqueness of the appropriate martingale problem in one dimension. In two and three dimensions we establish existence only. However, we do show that, in three dimensions, any solution will not be absolutely continuous with respect to the law of two independent super-Brownian motions. Although the supports of two independent super-Brownian motions collide in dimensions four and five, we show that there is no solution to the martingale problem in these cases.We next study a prédation model in which collisions only affect the "prey" species. Here we can show both existence and uniqueness in one, two and three dimensions. Again, there is no solution in four and five dimensions. As a tool for proving uniqueness, we obtain a representation of martingales for a super-process as stochastic integrals with respect to the related orthogonal martingale measure.We also obtain existence and uniqueness for a related single population model in one dimension in which particles are killed at a rate proportional to the local density. This model appears as a limit of a rescaled contact process as the range of interaction goes to infinity.

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17

Shiga, Tokuzo. "A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes". Journal of Mathematics of Kyoto University 30, nr 2 (1990): 245–79. http://dx.doi.org/10.1215/kjm/1250520071.

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Fekete, D., J. Fontbona i A. E. Kyprianou. "Skeletal stochastic differential equations for continuous-state branching processes". Journal of Applied Probability 56, nr 4 (grudzień 2019): 1122–50. http://dx.doi.org/10.1017/jpr.2019.67.

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AbstractIt is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

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LYTVYNOV, EUGENE. "ORTHOGONAL DECOMPOSITIONS FOR LÉVY PROCESSES WITH AN APPLICATION TO THE GAMMA, PASCAL, AND MEIXNER PROCESSES". Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, nr 01 (marzec 2003): 73–102. http://dx.doi.org/10.1142/s0219025703001031.

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It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists of representing any square-integrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of this paper are to develop the three approaches in the case of a general (ℝ-valued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general case.

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SORKIN, RAFAEL D. "Toward a fundamental theorem of quantal measure theory". Mathematical Structures in Computer Science 22, nr 5 (6.09.2012): 816–52. http://dx.doi.org/10.1017/s0960129511000545.

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In this paper we address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets and a particle moving on the finite lattice ℤn. In both cases, the dynamics can be coded into a vector-valued measure μ on Ω, the space of all histories. Initially, μ is just defined on special subsets of Ω called cylinder events, and we would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally μ is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events that in a suitable sense converges to the event whose measure we are seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.

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Ninouh, Abdelhakim, Boulakhras Gherbal i Nassima Berrouis. "Existence of optimal controls for systems of controlled forward-backward doubly SDEs". Random Operators and Stochastic Equations 28, nr 2 (1.06.2020): 93–112. http://dx.doi.org/10.1515/rose-2020-2031.

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AbstractWe wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod {\mathbb{D}} equipped with the S-topology of Jakubowski. Moreover, when the Roxin-type convexity condition is fulfilled, we prove that the optimal relaxed control is in fact strict. Secondly, we prove the existence of a strong optimal controls for a linear forward-backward doubly SDEs. Furthermore, we establish necessary as well as sufficient optimality conditions for a control problem of this kind of systems. This is the first theorem of existence of optimal controls that covers the forward-backward doubly systems.

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Li, Yiting, i Xin Sun. "On fluctuations for random band Toeplitz matrices". Random Matrices: Theory and Applications 04, nr 03 (lipiec 2015): 1550012. http://dx.doi.org/10.1142/s2010326315500124.

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In this paper, we study two one-parameter families of random band Toeplitz matrices: [Formula: see text] where (1) a0 = 0, {a1, a2, …} in An(t) are independent random variables and a-i = ai; (2) a0(t) = 0, {a1(t), a2(t), …} in Bn(t) are independent copies of the standard Brownian motion at time t and a-i(t) = ai(t). As t varies, the empirical measures μ(An(t)) and μ(Bn(t)) are measure valued stochastic processes. The purpose of this paper is to study the fluctuations of μ(An(t)) and μ(Bn(t)) as n goes to ∞. Given a monomial f(x) = xp with p ≥ 2, the corresponding rescaled fluctuations of μ(An(t)) and μ(Bn(t)) are [Formula: see text] respectively. We will prove that the above equations converge to centered Gaussian families {Zp(t)} and {Wp(t)} respectively. The covariance structure 𝔼[Zp(t1)Zq(t2)] and 𝔼[Wp(t1)Wq(t2)] are obtained for all p, q ≥ 2, t1, t2 ≥ 0, and are both homogeneous polynomials of t1 and t2 for fixed p, q. In particular, Z2(t) is the Brownian motion and Z3(t) is the same as W2(t) up to a constant. The main method of this paper is the moment method.

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HOYLE, EDWARD, ANDREA MACRINA i LEVENT ALI MENGÜTÜRK. "MODULATED INFORMATION FLOWS IN FINANCIAL MARKETS". International Journal of Theoretical and Applied Finance 23, nr 04 (czerwiec 2020): 2050026. http://dx.doi.org/10.1142/s0219024920500260.

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We model continuous-time information flows generated by a number of information sources that switch on and off at random times. By modulating a multi-dimensional Lévy random bridge over a random point field, our framework relates the discovery of relevant new information sources to jumps in conditional expectation martingales. In the canonical Brownian random bridge case, we show that the underlying measure-valued process follows jump-diffusion dynamics, where the jumps are governed by information switches. The dynamic representation gives rise to a set of stochastically-linked Brownian motions on random time intervals that capture evolving information states, as well as to a state-dependent stochastic volatility evolution with jumps. The nature of information flows usually exhibits complex behavior, however, we maintain analytic tractability by introducing what we term the effective and complementary information processes, which dynamically incorporate active and inactive information, respectively. As an application, we price a financial vanilla option, which we prove is expressed by a weighted sum of option values based on the possible state configurations at expiry. This result may be viewed as an information-based analogue of Merton’s option price, but where jump-diffusion arises endogenously. The proposed information flows also lend themselves to the quantification of asymmetric informational advantage among competitive agents, a feature we analyze by notions of information geometry.

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"Asymptotics for a class of measure-valued stochastic processes". Stochastic Processes and their Applications 21, nr 1 (grudzień 1985): 2–3. http://dx.doi.org/10.1016/0304-4149(85)90229-7.

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"Measure valued diffusion processes associated with stochastic processes of Fleming-Viot type". Stochastic Processes and their Applications 21, nr 1 (grudzień 1985): 26. http://dx.doi.org/10.1016/0304-4149(85)90273-x.

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YURACHKIVSKY, A. "Asymptotic study of measure-valued processes related to stochastic geometry". Random Operators and Stochastic Equations 9, nr 2 (2001). http://dx.doi.org/10.1515/rose.2001.9.2.121.

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Louvet, Apolline. "Stochastic measure-valued models for populations expanding in a continuum". ESAIM: Probability and Statistics, 13.12.2022. http://dx.doi.org/10.1051/ps/2022020.

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We model spatially expanding populations by means of two spatial Λ-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the ∞-parent SLFV. In order to do so, we fill empty areas with type 0 "ghost" individuals with a strong selective disadvantage against "real" type 1 individuals, quantified by a parameter k. The reproduction of ghost individuals is interpreted as local extinction events due to stochasticity in reproduction. When k → +∞, the limiting process, corresponding to the ∞-parent SLFV, is reminiscent of stochastic growth models from percolation theory, but is associated to tools making it possible to investigate the genetic diversity in a population sample. In this article, we provide a rigorous construction of the ∞-parent SLFV, and show that it corresponds to the limit of the k-parent SLFV when k → +∞. In order to do so, we introduce an alternative construction of the k-parent SLFV which allows us to couple SLFVs with different selection strengths and is of interest in its own right. We exhibit three different characterizations of the ∞-parent SLFV, which are valid in different settings and link together population genetics models and stochastic growth models.

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28

Wang, Zhiyuan, Qianli Zhou i Yong Deng. "Belief entropy rate: a method to measure the uncertainty of interval-valued stochastic processes". Applied Intelligence, 5.01.2023. http://dx.doi.org/10.1007/s10489-022-04407-1.

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Büke, Burak, i Wenyi Qin. "Many-Server Queues with Random Service Rates: A Unified Framework Based on Measure-Valued Processes". Mathematics of Operations Research, 12.07.2022. http://dx.doi.org/10.1287/moor.2022.1280.

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We consider many-server queueing systems with heterogeneous exponential servers, for which the service rate of each server is a random variable drawn from a given distribution. We develop a framework for analyzing the heavy-traffic diffusion limits of these queues using measure-valued stochastic processes. We introduce the measure-valued fairness process, which denotes the proportion of cumulative idleness experienced by servers whose rates fall in a Borel subset of the support of the service rates. It can be shown that these processes do not converge in the usual Skorokhod-J1 topology. Hence, we introduce a new notion of convergence based on shifted versions of these processes. We also introduce some useful martingales to identify limiting fairness processes under different routing policies. To demonstrate the power of our framework, we show how it can be used to prove diffusion limits for parallel server systems with within-pool heterogeneity.

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Mezerdi, Meriem, i Brahim Mezerdi. "On the maximum principle for relaxed control problems of nonlinear stochastic systems". Advances in Continuous and Discrete Models 2024, nr 1 (20.03.2024). http://dx.doi.org/10.1186/s13662-024-03803-w.

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AbstractWe consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng’s maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable.

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Calvia, Alessandro, i Giorgio Ferrari. "Nonlinear Filtering of Partially Observed Systems Arising in Singular Stochastic Optimal Control". Applied Mathematics & Optimization 85, nr 2 (kwiecień 2022). http://dx.doi.org/10.1007/s00245-022-09822-x.

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AbstractThis paper deals with a nonlinear filtering problem in which a multi-dimensional signal process is additively affected by a process $$\nu $$ ν whose components have paths of bounded variation. The presence of the process $$\nu $$ ν prevents from directly applying classical results and novel estimates need to be derived. By making use of the so-called reference probability measure approach, we derive the Zakai equation satisfied by the unnormalized filtering process, and then we deduce the corresponding Kushner–Stratonovich equation. Under the condition that the jump times of the process $$\nu $$ ν do not accumulate over the considered time horizon, we show that the unnormalized filtering process is the unique solution to the Zakai equation, in the class of measure-valued processes having a square-integrable density. Our analysis paves the way to the study of stochastic control problems where a decision maker can exert singular controls in order to adjust the dynamics of an unobservable Itô-process.

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Endo, Taiki, Makoto Katori i Noriyoshi Sakuma. "Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases". Symmetry, Integrability and Geometry: Methods and Applications, 2.07.2022. http://dx.doi.org/10.3842/sigma.2022.049.

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We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the R-transforms and the S-transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the R-transforms and the S-transforms are simpler than those for the Cauchy transforms and useful for explicit calculations including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.

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"Transition functions, paths and path integrals". Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 434, nr 1890 (8.07.1991): 41–63. http://dx.doi.org/10.1098/rspa.1991.0079.

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The main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory. Section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which Kolmogorov achieved via PDE theory (the theory of partial differential equations) and his criterion for path continuity. Section 2 describes how Itô calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process X by solving an SDE (stochastic differential equation) driven by brownian motion. (Of course, SDEs have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems.) Though §2 begins heuristically, the mathematics is then tightened up. This paper is, after all, a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation. Once Kolmogorov’s precise language is available, §2 then takes a quick sight-seeing trip through some of the great developments by Doob, Itô and their successors. (In an age in which so many do simulations of Itô equations, I have explained precisely in the briefest possible fashion what the exact theory is. It is easy and usable.) You will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems, and of what the Malliavin calculus is about. You will, however, be advised on guide-books on these and other areas (including physicists’ favourites: large deviations, measure-valued diffusions, etc.), so that you can later explore at your leisure. Confession : the paper consists very largely of selected tracks (remixed!) from the album (Rogers & Williams 1987 Diffusions, Markov processes and martingales ; Chichester: Wiley). Tributes to Kolmogorov’s work in probability and statistics have appeared in (every book ever written on probability and in) Ann. Probability 17 (1989), 815-964, Ann. Statist. 18 (1990), 987-1031, Bull. Lond. math. Soc. . 22 (1990), 31-100, Teor. Veroyatnost Primenen 34 (1989), no. 1, Usp. mat. Nauk 43 (1988), no. 6. The official biography by Shiryaev will be a wonderful volume. For me, this paper is a further expression of my thanks to Kolmogorov and (as he would have wished) to Lévy, Doob and Itô too.

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Artykuły w czasopismach na temat „Measure-Valued stochastic processes”

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