72 results on '"Stochastic flow"'
Search Results
2. Probabilistic interpretation for Sobolev solutions of McKean–Vlasov partial differential equations
- Author
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Zhen Wu and Ruimin Xu
- Subjects
Statistics and Probability ,Stochastic flow ,Partial differential equation ,Generalization ,010102 general mathematics ,Probabilistic logic ,Term (logic) ,01 natural sciences ,Mathematics::Numerical Analysis ,Interpretation (model theory) ,Sobolev space ,010104 statistics & probability ,Stochastic differential equation ,Mathematics::Probability ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, we give a probabilistic interpretation of Sobolev solutions to parabolic semilinear McKean–Vlasov partial differential equations (PDEs for short) in terms of mean-field backward stochastic differential equations (BSDEs for short). This probabilistic interpretation can be viewed as a generalization of the Feynman–Kac formula. The method is based on the stochastic flow technique which is different from classical stochastic differential equations (SDEs for short) due to the influence of mean-field term in McKean–Vlasov SDEs.
- Published
- 2019
3. On the Completeness of Stochastic Flows Generated by Equations with Current Velocities
- Author
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Yu. E. Gliklikh and T. A. Shchichko
- Subjects
Statistics and Probability ,Stochastic flow ,Applied mathematics ,Statistics, Probability and Uncertainty ,Current (fluid) ,Completeness (statistics) ,Mathematics - Abstract
Sufficient conditions as well as necessary and sufficient ones are found for the completeness of the stochastic flow generated by equations with the so-called current velocities (Nelson's symmetric...
- Published
- 2019
4. Global sensitivity analysis for a stochastic flow problem
- Author
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Dmitriy Kolyukhin
- Subjects
Statistics and Probability ,Random field ,Darcy's law ,Stochastic flow ,020209 energy ,Applied Mathematics ,Sobol sequence ,02 engineering and technology ,010502 geochemistry & geophysics ,01 natural sciences ,Saturated porous medium ,Permeability (earth sciences) ,Global sensitivity analysis ,Log-normal distribution ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0105 earth and related environmental sciences ,Mathematics - Abstract
The paper is devoted to the modeling of a single-phase flow through saturated porous media. A statistical approach where permeability is considered as a lognormal random field is applied. The impact of permeability, random boundary conditions and wells pressure on the flow in a production well is studied. A numerical procedure to generate an ensemble of realizations of the numerical solution of the problem is developed. A global sensitivity analysis is performed using Sobol indices. The impact of different model parameters on the total model uncertainty is studied.
- Published
- 2018
5. Solving some Stochastic Partial Differential Equations driven by Lévy Noise using two SDEs. *
- Author
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Mrad Mohamed, Mrad, Mohamed, and Université Paris 13 (UP13)
- Subjects
Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,[QFIN]Quantitative Finance [q-fin] ,Euler scheme ,stochastic flow ,Garsia-Rodemich-Rumsey lemma ,SPDE driven by Lévy noise ,[QFIN] Quantitative Finance [q-fin] ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,strong approximation ,Modeling and Simulation ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,method of stochastic characteristics ,[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP] ,Utility-SPDE - Abstract
The method of characteristics is a powerful tool to solve some nonlinear second order stochastic PDEs like those satisfied by a consistent dynamic utilities, see [EM13, MM20]. In this situation the solution V (t, z) is theoretically of the formX t V (0,ξ t (z)) whereX and Y are solutions of a system of two SDEs,ξ is the inverse flow ofȲ and V (0, .) is the initial condition. Unfortunately this representation is not explicit except in simple cases whereX andȲ are solutions of linear equations. The objective of this work is to take advantage of this representation to establish a numerical scheme approximating the solution V using Euler approximations X N and ξ N of X and ξ. This allows us to avoid a complicated discretizations in time and space of the SPDE for which it seems really difficult to obtain error estimates. We place ourselves in the framework of SDEs driven by Lévy noise and we establish at first a strong convergence result, in L p-norms, of the compound approximation X N t (Y N t (z)) to the compound variable X t (Y t (z)), in terms of the approximations of X and Y which are solutions of two SDEs with jumps. We then apply this result to Utility-SPDEs of HJB type after inverting monotonic stochastic flows.
- Published
- 2021
6. Pathwise differentiability of reflected diffusions in convex polyhedral domains
- Author
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Kavita Ramanan and David Lipshutz
- Subjects
Statistics and Probability ,Derivative problem ,Directional derivative of the extended Skorokhod map ,93B35 ,01 natural sciences ,010104 statistics & probability ,60H07 ,Derivative process ,Primary: 60G17, 90C31, 93B35. Secondary: 60H07, 60H10, 65C30 ,Stochastic flow ,FOS: Mathematics ,65C30 ,0101 mathematics ,Mathematics ,010102 general mathematics ,Probability (math.PR) ,90C31 ,Pathwise differentiability ,60G17 ,Reflected diffusion ,60H10 ,Statistics, Probability and Uncertainty ,Sensitivity analysis ,Humanities ,Reflected Brownian motion ,Boundary jitter property ,Mathematics - Probability - Abstract
Reflected diffusions in convex polyhedral domains arise in a variety of applications, including interacting particle systems, queueing networks, biochemical reaction networks and mathematical finance. Under suitable conditions on the data, we establish pathwise differentiability of such a reflected diffusion with respect to its defining parameters --- namely, its initial condition, drift and diffusion coefficients, and (oblique) directions of reflection along the boundary of the domain. We characterize the right-continuous regularization of a pathwise derivative of the reflected diffusion as the pathwise unique solution to a constrained linear stochastic differential equation with jumps whose drift and diffusion coefficients, domain and directions of reflection depend on the state of the reflected diffusion. The proof of this result relies on properties of directional derivatives of the associated (extended) Skorokhod reflection map and their characterization in terms of a so-called derivative problem, and also involves establishing certain path properties of the reflected diffusion at nonsmooth parts of the boundary of the polyhedral domain, which may be of independent interest. As a corollary, we obtain a probabilistic representation for derivatives of expectations of functionals of reflected diffusions, which is useful for sensitivity analysis of reflected diffusions., Comment: 37 pages
- Published
- 2019
7. Hitting probabilities of a Brownian flow with Radial Drift
- Author
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Eyal Neuman, Carl Mueller, and Jong Jun Lee
- Subjects
Statistics and Probability ,37C10 ,Bessel process ,Statistics & Probability ,hitting ,Omega ,0101 Pure Mathematics ,Combinatorics ,Stochastic differential equation ,Stochastic flow ,60J45 ,FOS: Mathematics ,60H10 (Primary), 37H10, 60J45 (Secondary) ,Brownian motion ,60J60 ,Mathematics ,Image (category theory) ,Probability (math.PR) ,0104 Statistics ,Lipschitz continuity ,stochastic differential equations ,Flow (mathematics) ,Bounded function ,60H10 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
We consider a stochastic flow $\phi_t(x,\omega)$ in $\mathbb{R}^n$ with initial point $\phi_0(x,\omega)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|\phi_t(x)\|)}{\|\phi_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonempty interior, then the image of the set has positive probability of hitting the origin., Comment: 34 pages, 3 figures
- Published
- 2019
8. Numerical computation for backward doubly SDEs with random terminal time
- Author
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Wissal Sabbagh, Anis Matoussi, Laboratoire Manceau de Mathématiques (LMM), Le Mans Université (UM), Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), and Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Computation ,Euler scheme ,60H15, 60G46, 35H60 ,SPDEs ,Exit time ,and phrases: Backward Doubly Stochastic Differential Equation ,01 natural sciences ,Domain (mathematical analysis) ,Mathematics::Numerical Analysis ,010104 statistics & probability ,symbols.namesake ,Stochastic differential equation ,Mathematics::Probability ,Approximation error ,Stochastic flow ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Mathematics ,Dirichlet problem ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Sobolev space ,Monte carlo ,Dirichlet condition ,Dirichlet boundary condition ,method ,AMS 2000 subject classifications:Primary 60H15, 60G46 ,secondary 35H60 ,Euler's formula ,symbols ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Mathematics - Probability - Abstract
In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided., 38, Monte Carlo Methods and Applications (MCMA) 2016
- Published
- 2016
9. Regularity properties of the stochastic flow of a skew fractional Brownian motion
- Author
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Frank Proske, Oussama Amine, and David Baños
- Subjects
Statistics and Probability ,Fractional Brownian motion ,Stochastic flow ,Applied Mathematics ,Probability (math.PR) ,Mathematical analysis ,Skew ,Statistical and Nonlinear Physics ,Malliavin calculus ,Stochastic differential equation ,FOS: Mathematics ,Differentiable function ,Mathematics - Probability ,Mathematical Physics ,Mathematics - Abstract
In this paper we prove, for small Hurst parameters, the higher-order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis-based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.
- Published
- 2020
10. Time-Reversal of Coalescing Diffusive Flows and Weak Convergence of Localized Disturbance Flows
- Author
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James Bell
- Subjects
Statistics and Probability ,Brownian web ,Stochastic flow ,Disturbance (geology) ,Weak convergence ,Probability (math.PR) ,stochastic flow ,Mechanics ,Thermal diffusivity ,Physics::Fluid Dynamics ,Flow (mathematics) ,60F17 ,FOS: Mathematics ,dual flow ,coalescing flow ,Statistics, Probability and Uncertainty ,Diffusion (business) ,distrubance flow ,Arratia flow ,Brownian motion ,Mathematics - Probability ,time-reversed flow ,Mathematics - Abstract
We generalize the coalescing Brownian flow, aka the Brownian web, considered as a weak flow to allow varying drift and diffusivity in the constituent diffusion processes and call these flows coalescing diffusive flows. We then identify the time-reversal of each coalescing diffusive flow and provide two distinct proofs of this identification. One of which is direct and the other proceeds by generalizing the concept of a localized disturbance flow to allow varying size and shape of disturbances, we show these new flows converge weakly under appropriate conditions to a coalescing diffusive flow and identify their time-reversals., 57 pages
- Published
- 2018
11. On directional derivatives of Skorokhod maps in convex polyhedral domains
- Author
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David Lipshutz and Kavita Ramanan
- Subjects
Statistics and Probability ,Boundary (topology) ,93B35 ,reflected process ,stochastic flow ,Directional derivative ,oblique reflection ,01 natural sciences ,Domain (mathematical analysis) ,010104 statistics & probability ,Projection (mathematics) ,sensitivity analysis ,FOS: Mathematics ,Differentiable function ,0101 mathematics ,60G17, 90C31, 93B35 (Primary), 90B15 (Secondary) ,Mathematics ,010102 general mathematics ,Mathematical analysis ,Probability (math.PR) ,Regular polygon ,derivative problem ,90C31 ,directional derivative of the Skorokhod map ,Lipschitz continuity ,Extended Skorokhod problem ,90B15 ,boundary jitter property ,60G17 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Oblique reflection - Abstract
The study of both sensitivity analysis and differentiability of the stochastic flow of a reflected process in a convex polyhedral domain is challenging because the dynamics are discontinuous at the boundary of the domain and the boundary of the domain is not smooth. These difficulties can be addressed by studying directional derivatives of an associated extended Skorokhod map, which is a deterministic mapping that takes an unconstrained path to a suitably reflected version. In this work we develop an axiomatic framework for the analysis of directional derivatives of a large class of Lipschitz continuous extended Skorokhod maps in convex polyhedral domains with oblique directions of reflection. We establish existence of directional derivatives at a path whose reflected version satisfies a certain boundary jitter property, and also show that the right-continuous regularization of such a directional derivative can be characterized as the unique solution to a Skorokhod-type problem, where both the domain and directions of reflection vary (discontinuously) with time. A key ingredient in the proof is establishing certain contraction properties for a family of (oblique) derivative projection operators. As an application, we establish pathwise differentiability of reflected Brownian motion in the nonnegative quadrant with respect to the initial condition, drift vector, dispersion matrix and directions of reflection. The results of this paper are also used in subsequent work to establish pathwise differentiability of a much larger class of reflected diffusions in convex polyhedral domains., 58 pages, 3 figures
- Published
- 2018
12. Reflected Brownian motion: selection, approximation and linearization
- Author
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Xue-Mei Li, Marc Arnaudon, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Warwick Mathematics Institute (WMI), and University of Warwick [Coventry]
- Subjects
60G ,Statistics and Probability ,Local time ,DOMAINS ,Statistics & Probability ,Boundary (topology) ,stochastic flow ,math.PR ,01 natural sciences ,010104 statistics & probability ,Stochastic differential equation ,PARTS FORMULAS ,Mathematics::Probability ,FOS: Mathematics ,THEOREM ,STOCHASTIC DIFFERENTIAL-EQUATIONS ,Boundary value problem ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Brownian motion ,Mathematics ,Science & Technology ,heat equation ,0104 Statistics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Derivative flow ,60H10, 60H30, 58J35 ,Weak derivative ,boundary ,SDES ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Heat equation for forms ,Reflected Brownian motion ,Diffusion process ,Physical Sciences ,MANIFOLDS ,Heat equation ,Statistics, Probability and Uncertainty ,INTEGRATION ,Mathematics - Probability ,HARMONIC-FUNCTIONS ,reflection - Abstract
We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownian flow as well as a corresponding stochastic damped transport process (Wt)(Wt), the limiting pair gives a probabilistic representation for solutions of the heat equations on differential 1-forms with the absolute boundary conditions. The transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary where it is driven by the boundary local time, and with its normal part erased at the end of the excursions to the boundary of the reflected Brownian motion. On the half line, this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution; on the half space it agrees with the construction of N. Ikeda and S. Watanabe [29] by Poisson point processes. The construction leads also to an approximation for the boundary local time, in the topology of uniform convergence but not in the semi-martingale topology, indicating the difficulty in proving convergence of solutions of a family of random ODE’s to the solution of a stochastic equation driven by the local time and with jumps. In addition, we obtain a differentiation formula for the heat semi-group with Neumann boundary condition and prove also that (Wt)(Wt) is the weak derivative of a family of reflected Brownian motions with respect to the initial point.
- Published
- 2017
13. Homeomorphic Property of the Stochastic Flow of a Natural Equation in Multi-dimensional Case
- Author
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Fatima Benziadi and Abdeldjabbar Kandouci
- Subjects
Statistics and Probability ,Pure mathematics ,Property (philosophy) ,Stochastic flow ,Multi dimensional ,Natural (archaeology) ,Mathematics - Published
- 2017
14. Quasi-invariance of the Stochastic Flow Associated to Itô’s SDE with Singular Time-Dependent Drift
- Author
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Dejun Luo
- Subjects
Statistics and Probability ,Pure mathematics ,symbols.namesake ,Stochastic flow ,Mathematics::Probability ,Lebesgue measure ,Flow (mathematics) ,Wiener process ,General Mathematics ,symbols ,Statistics, Probability and Uncertainty ,Drift coefficient ,Mathematics - Abstract
In this paper we consider the Ito SDE $$d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in {\mathbb R}^d,$$ where $W_t$ is a $d$-dimensional standard Wiener process and the drift coefficient $b:[0,T]\times{\mathbb R}^d\to{\mathbb R}^d$ belongs to $L^q(0,T;L^p({\mathbb R}^d))$ with $p\geq 2, q>2$ and $\frac dp +\frac 2q
- Published
- 2014
15. Hölder Flow and Differentiability for SDEs with Nonregular Drift
- Author
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Franco Flandoli, Ennio Fedrizzi, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Fedrizzi, E., and Flandoli, F.
- Subjects
Statistics and Probability ,Solution map ,Stochastic flow ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,16. Peace & justice ,01 natural sciences ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Strong solutions ,010104 statistics & probability ,Stochastic differential equation ,Flow (mathematics) ,Differentiable function ,0101 mathematics ,Statistics, Probability and Uncertainty ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
We prove the existence of a stochastic flow of Hölder homeomorphisms for solutions of SDEs with singular time dependent drift having only certain integrability properties. We also show that the solution map x → X x is differentiable in a weak sense.
- Published
- 2013
16. Quasi-invariant stochastic flows of SDEs with non-smooth drifts on compact manifolds
- Author
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Xicheng Zhang
- Subjects
Statistics and Probability ,DiPerna-Lions flow ,Riemannian manifold ,Applied Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Manifold ,Sobolev space ,Stochastic differential equation ,Flow (mathematics) ,Stochastic flow ,Hardy-Littlewood maximal function ,Modeling and Simulation ,Modelling and Simulation ,Almost everywhere ,Invariant measure ,Mathematics::Differential Geometry ,Invariant (mathematics) ,Sobolev drift ,Mathematics - Abstract
In this article we prove that stochastic differential equation (SDE) with Sobolev drift on a compact Riemannian manifold admits a unique ν -almost everywhere stochastic invertible flow, where ν is the Riemannian measure, which is quasi-invariant with respect to ν . In particular, we extend the well-known DiPerna-Lions flows of ODEs to SDEs on a Riemannian manifold.
- Published
- 2011
- Full Text
- View/download PDF
17. An Approximate Algorithm for the Robust Design in a Stochastic-Flow Network
- Author
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Yi-Kuei Lin and Shin-Guang Chen
- Subjects
Statistics and Probability ,Robust design ,Stochastic flow ,business.industry ,Product (mathematics) ,Node (networking) ,Approximation algorithm ,Context (language use) ,business ,Flow network ,Quality assurance ,Algorithm ,Mathematics - Abstract
This article proposes an approximate algorithm to solve the robust design problem in a stochastic-flow network. Conventionally, the robust design is an issue in quality engineering where the designed product functions well even in a versatile environment. However, in the context of a stochastic-flow network, it means the network functions well even in a node's failure situation. This can be solved by properly assigning capacity. Because the problem is known to be NP-hard, a relatively fast approximation algorithm would be beneficial. Some numerical examples are presented to illustrate the usefulness of the proposed approach.
- Published
- 2010
18. Reliability evaluation of a revised stochastic flow network with uncertain minimum time
- Author
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Yi-Kuei Lin
- Subjects
Statistics and Probability ,Arc (geometry) ,Mathematical optimization ,Stochastic flow ,Minimum time ,Boundary (topology) ,Condensed Matter Physics ,Reliability (statistics) ,Lead time ,Mathematics - Abstract
This paper constructs a revised stochastic flow network to model a realistic computer network in which each arc has a lead time and a stochastic capacity. The minimum time to send a specified amount of data through the network is thus uncertain. Hence, this paper mainly proposes an approach for evaluating the system reliability that d units of data can be transmitted through k minimal paths simultaneously within the time threshold T . The idea of lower boundary points for ( d , T ), the minimal system states satisfying the demand d within the time threshold T , is proposed firstly. All system states meeting the time and demand requirements can be represented as the union of subsets generated from all lower boundary points for ( d , T ), and thus the system reliability is computed quickly.
- Published
- 2010
19. Central Limit Theorems for a Super-Diffusion over a Stochastic Flow
- Author
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Mei Zhang
- Subjects
Statistics and Probability ,Stochastic flow ,General Mathematics ,Gaussian ,Mathematical analysis ,Degenerate energy levels ,Motion (geometry) ,symbols.namesake ,Mathematics::Probability ,symbols ,Limit (mathematics) ,Statistics, Probability and Uncertainty ,Diffusion (business) ,Mathematics ,Central limit theorem ,Superprocess - Abstract
Central limit theorems of the occupation time of a superprocess over a stochastic flow are proved. For the critical and higher dimensions d≥4, the limits are Gaussian variables. For d=3, the limit is conditional Gaussian. When the stochastic flow disappears, the results degenerate to those for the ordinary super-Brownian motion.
- Published
- 2009
20. Dispersion of volume under the action of isotropic Brownian flows
- Author
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Michael Scheutzow, Georgi Dimitroff, and Publica
- Subjects
Statistics and Probability ,60F05, 60G15, 60G60, 62H30 ,Stochastic differential equation ,Stochastic process ,Applied Mathematics ,Probability (math.PR) ,Mathematical analysis ,Isotropy ,Asymptotic distribution ,Vague convergence ,Mathematics::Probability ,Stochastic flow ,Modeling and Simulation ,Modelling and Simulation ,Isotropic Brownian flow ,FOS: Mathematics ,Asymptotic normality ,Almost surely ,Martingale (probability theory) ,Random variable ,Mathematics - Probability ,Brownian motion ,Mathematics - Abstract
We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and -- under slightly stronger assumptions -- asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive., Comment: To appear in "Stochastic processes and their applications"
- Published
- 2009
- Full Text
- View/download PDF
21. Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps
- Author
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Huijie Qiao and Xicheng Zhang
- Subjects
Statistics and Probability ,Stochastic flow ,Stochastic process ,Applied Mathematics ,Mathematical analysis ,Non-Lipschitz ,Lipschitz continuity ,Continuity property ,Stochastic partial differential equation ,Stochastic differential equation ,Modeling and Simulation ,Modelling and Simulation ,Martingale (probability theory) ,SDEs with jumps ,Homeomorphism flow ,Exponential martingale ,Mathematics - Abstract
In this paper we study the continuity property as well as the homeomorphism property for the solutions of multidimensional stochastic differential equations with jumps and non-Lipschitz coefficients with respect to the initial values.
- Published
- 2008
- Full Text
- View/download PDF
22. KUNITA-TYPE STOCHASTIC FLOWS OF HOMEOMORPHISMS IN EUCLIDEAN SPACE
- Author
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Zongxia Liang
- Subjects
Statistics and Probability ,Discrete mathematics ,Stochastic differential equation ,Semimartingale ,Stochastic flow ,Euclidean space ,Applied Mathematics ,Statistical and Nonlinear Physics ,Almost surely ,Type (model theory) ,Mathematical Physics ,Mathematics - Abstract
In this paper we prove Kunita-type stochastic differential equation (SDE) [Formula: see text], t > s, driven by a spatial continuous semimartingale F (x, t) =(F1 (x, t), …, Fm (x, t)), x ∈ ℜm, with local characteristic (a, b), in the sense of Kunita (Chap. 3 of Ref. 10), can produce a stochastic flow of homeomorphisms of ℜm into itself almost surely under the (a, b) satisfies non-Lipschitz conditions. This result bases on recent works12 by the author.
- Published
- 2007
23. Stochastic flow approach to Dupire’s formula
- Author
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Benjamin Jourdain
- Subjects
Statistics and Probability ,Stochastic flow ,Mathematical finance ,Duality (mathematics) ,Probabilistic logic ,Exponential function ,Adjoint equation ,Local volatility ,Probabilistic proof ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mathematical economics ,Finance ,Mathematics - Abstract
The equivalent probabilistic formulation of Dupire’s PDE is the put-call duality equality. In local volatility models including exponential Levy jumps, we give a direct probabilistic proof for this result based on stochastic flow arguments. This approach also enables us to check the equivalent probabilistic formulation of various generalizations of Dupire’s PDE recently obtained by Pironneau [C. R. Acad. Sci. Paris Ser. I 344(2) 127–133 (2007)] by the adjoint equation technique in the case of complex options.
- Published
- 2007
24. On The Invariant Measure of a Positive Recurrent Diffusion in $${\mathbb{R}}$$
- Author
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Michele L. Baldini
- Subjects
Statistics and Probability ,Discrete mathematics ,Combinatorics ,Distribution (mathematics) ,Stochastic flow ,Mathematics::Probability ,General Mathematics ,Invariant measure ,Diffeomorphism ,Statistics, Probability and Uncertainty ,Diffusion (business) ,Measure (mathematics) ,Mathematics - Abstract
Given a one-dimensional positive recurrent diffusion governed by the Stratonovich SDE $${X_t=x+\int_0^t\sigma(X_s)\bullet\hbox{d}b(s)+\int_0^t m(X_s)\hbox{d}s}$$ , we show that the associated stochastic flow of diffeomorphisms focuses as fast as $${exp (-2t\int_{\mathbb{R}}\frac{m^2}{\sigma^2} d\Pi)}$$ , where $${d\Pi}$$ is the finite stationary measure. Moreover, if the drift is reversed and the diffeomorphism is inverted, then the path function so produced tends, independently of its starting point, to a single (random) point whose distribution is $${d\Pi}$$ . Applications to stationary solutions of X t , asymptotic behavior of solutions of SPDEs and random attractors are offered.
- Published
- 2006
25. A stochastic flow arising in the study of local times
- Author
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Jon Warren
- Subjects
Statistics and Probability ,Geometric Brownian motion ,Stochastic flow ,Mathematical finance ,Probability (math.PR) ,Mathematical analysis ,Topology ,Mathematics::Probability ,Probability theory ,60J55 ,60J60 ,Local time ,FOS: Mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Analysis ,Brownian motion ,Mathematics - Abstract
A stochastic flow of homeomorphisms of the real line previously studied by Bass and Burdzy is shown to arise in describing a Brownian motion conditional on knowing its local times on hitting a fixed level. This makes it possible to connect Ray-Knight type results for the flow with the classical Ray-Knight theorems for Brownian motion., Comment: 11 pages
- Published
- 2005
26. Sticky flows on the circle and their noises
- Author
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Yves Le Jan and Olivier Raimond
- Subjects
Statistics and Probability ,Stochastic flow ,Mathematical finance ,Mathematical analysis ,Condensed Matter::Soft Condensed Matter ,Nonlinear Sciences::Chaotic Dynamics ,Physics::Fluid Dynamics ,Noise ,Mathematics::Probability ,Flow (mathematics) ,Diffusion flow ,Probability theory ,Calculus ,Statistics, Probability and Uncertainty ,Analysis ,Brownian motion ,Mathematics - Abstract
This paper gives a construction of sticky flows on the circle. Sticky flows give examples of stochastic flows of kernels that interpolates between Arratia’s coalescing flow and the deterministic diffusion flow. They are associated with systems of sticky independent Brownian particles on the circle, for some fixed parameter of stickyness. It is proved that the noise generated by Brownian sticky flows is black. A new proof of the fact that the noise of Arratia’s coalescing flow is black is given.
- Published
- 2004
27. Stochastic flows associated to coalescent processes
- Author
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Jean Bertoin and Jean-Françcois Le Gall
- Subjects
Statistics and Probability ,Fleming–Viot process ,Stochastic flow ,Stochastic process ,Mathematical finance ,010102 general mathematics ,01 natural sciences ,Coalescent theory ,Combinatorics ,010104 statistics & probability ,Mathematics::Probability ,Quantitative Biology::Populations and Evolution ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Coalescent process ,Analysis ,Mathematics - Abstract
We study a class of stochastic flows connected to the coalescent processes that have been studied recently by Mohle, Pitman, Sagitov and Schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size. We define a bridge to be a right-continuous process (B(r),r[0,1]) with nondecreasing paths and exchangeable increments, such that B(0)=0 and B(1)=1. We show that flows of bridges are in one-to-one correspondence with the so-called exchangeable coalescents. This yields an infinite-dimensional version of the classical Kingman representation for exchangeable partitions of ℕ. We then propose a Poissonian construction of a general class of flows of bridges and identify the associated coalescents. We also discuss an important auxiliary measure-valued process, which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized Fleming-Viot process.
- Published
- 2003
28. Limit theorems for continuous-time branching flows
- Author
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Rugang Ma and Hui He
- Subjects
Statistics and Probability ,Discrete mathematics ,continuous time ,60J80 ,superprocess ,General Mathematics ,Mathematical analysis ,State (functional analysis) ,nonlocal branching ,branching process ,Branching (linguistics) ,Scaling limit ,Mathematics Subject Classification ,Flow (mathematics) ,Stochastic flow ,60J68 ,60G57 ,High Energy Physics::Experiment ,Limit (mathematics) ,Statistics, Probability and Uncertainty ,discrete state ,Branching process ,Superprocess ,Mathematics - Abstract
We construct a flow of continuous time and discrete state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses over the positive half line studied in Li (2012). Mathematics Subject Classification (2010): Primary 60J68, 60J80; secondary 60G57
- Published
- 2014
29. On differentiability of stochastic flow for а multidimensional SDE with discontinuous drift
- Author
-
Andrey Pilipenko and Olga Aryasova
- Subjects
Statistics and Probability ,Stochastic flow ,Probability (math.PR) ,Derivative ,Diffusion matrix ,Continuous additive functional ,Identity (music) ,Bounded variation ,FOS: Mathematics ,60J65 ,Applied mathematics ,60H10 ,Differentiable function ,Differentiability with respect to initial data ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Vector-valued function ,Mathematics - Probability ,Mathematics - Abstract
We consider a $d$-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data., 15 pages
- Published
- 2014
30. Lyapunov exponents of Poisson shot-noise velocity fields
- Author
-
Erhan Çinlar and Mine Caglar
- Subjects
Lyapunov function ,Statistics and Probability ,Poisson shot-noise ,Stochastic process ,Applied Mathematics ,Mathematical analysis ,Shot noise ,Lyapunov exponent ,Poisson distribution ,Turbulence ,symbols.namesake ,Compact space ,Stationary ,Position (vector) ,Stochastic flow ,Modeling and Simulation ,Modelling and Simulation ,symbols ,Homogeneous ,Vector field ,Mathematics - Abstract
We consider the Lyapunov exponents of flows generated by a class of Markovian velocity fields. The existence of the exponents is obtained for flows on a compact set, but with the most general form of the velocity field. As a particular class, we study the homogeneous and incompressible flows. In this case, the exponents are nonrandom, free of the initial position of the particle path, and their sum is zero. We numerically compute the top Lyapunov exponent on R 2 for a range of parameters to conjecture that it is strictly positive.
- Published
- 2001
- Full Text
- View/download PDF
31. Ray–Knight theorems related to a stochastic flow
- Author
-
Yueyun Hu and Jon Warren
- Subjects
Ray–Knight theorems ,Statistics and Probability ,Pure mathematics ,Exponential distribution ,Stochastic flow ,Stochastic process ,Flow ,Jacobi processes ,Applied Mathematics ,Mathematical analysis ,Tanaka's formula ,Mathematics::Probability ,Modeling and Simulation ,Modelling and Simulation ,Martingale (probability theory) ,Bifurcation ,Brownian motion ,Mathematics - Abstract
We study a stochastic flow of C 1 -homeomorphisms of R . At certain stopping times, the spatial derivative of the flow is a diffusion in the space variable and its generator is given. This answers several questions posed in a previous study by Bass and Burdzy (1999, Ann. Probab. 27, 50–108) .
- Published
- 2000
- Full Text
- View/download PDF
32. Existence of global stochastic flow and attractors for Navier–Stokes equations
- Author
-
Marek Capiński and Nigel J. Cutland
- Subjects
Statistics and Probability ,Partial differential equation ,Stochastic flow ,Mathematical finance ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Torus ,Multiplicative noise ,Physics::Fluid Dynamics ,Attractor ,Statistics, Probability and Uncertainty ,Navier–Stokes equations ,Analysis ,Mathematics - Abstract
For 2-D stochastic Navier-Stokes equations on the torus with multiplicative noise we construct a perfect cocycle and show the existence of global random compact attractors. The equations considered do not admit a pathwise method of solution.
- Published
- 1999
33. Drift estimation for Brownian flows
- Author
-
L. Piterbarg
- Subjects
Statistics and Probability ,Mathematical optimization ,Mean squared error ,Stochastic process ,Applied Mathematics ,Estimator ,Eulerian path ,Lagrangian data ,Diffusion ,Physics::Fluid Dynamics ,symbols.namesake ,Diffusion process ,Modelling and Simulation ,Stochastic flow ,Modeling and Simulation ,symbols ,Applied mathematics ,Diffusion (business) ,Constant (mathematics) ,Estimation ,Brownian motion ,Maximum likelihood ,Mathematics - Abstract
The problem of estimating the drift of a stochastic flow given Lagrangian observations is an estimation problem for a multidimensional diffusion with a degenerate diffusion matrix. The maximum-likelihood estimator of the constant drift is considered. A long-time asymptotic of its mean-square error (MSE) is computed. It is shown that the time-space average of the observed Lagrangian velocities has the same asymptotic. These estimators are compared to the least-squares estimator based on Eulerian data. In the most important, for applications, two-dimensional case the Lagrangian estimator is typically preferable for incompressible flows, while for flows close to potential the Eulerian estimator is better.
- Published
- 1998
34. Ito formula forC 1-functions of semimartingales
- Author
-
Pierre Vallois and Francesco Russo
- Subjects
Statistics and Probability ,Stochastic flow ,Semimartingale ,Mathematics::Probability ,Mathematical finance ,Mathematical analysis ,Hölder condition ,Statistics, Probability and Uncertainty ,Itō's lemma ,Analysis ,Stochastic integral ,Homeomorphism ,Mathematics - Abstract
We establish an Ito formula forC 1 functions of processes whose time reversal are semimartingales and forC 1 functions whose first derivatives are Holder continuous of any parameter and the process comes out from a stochastic flow of homeomorphism.
- Published
- 1996
35. Generalized self-intersection local time for a superprocess over a stochastic flow
- Author
-
Aaron Heuser
- Subjects
Statistics and Probability ,self-intersection ,60J80 ,Stochastic flow ,Probability (math.PR) ,Spatial motion ,Multiplicity (mathematics) ,stochastic flow ,Constructive ,Article ,local time ,Mathematics::Probability ,Local time ,60J68 ,60H15 ,FOS: Mathematics ,Superprocess ,60G57 ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Mathematics - Abstract
This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows., Comment: Published in at http://dx.doi.org/10.1214/11-AOP653 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2012
36. Localization of solutions to stochastic porous media equations: finite speed of propagation
- Author
-
Viorel Barbu and Michael Roeckner
- Subjects
Statistics and Probability ,Discrete mathematics ,Stochastic flow ,energy method ,Interval (mathematics) ,stochastic flow ,Omega ,Wiener process ,symbols.namesake ,porous media equation ,60H15 ,35R60 ,Energy method ,symbols ,Statistics, Probability and Uncertainty ,Porous medium ,Mathematics ,Mathematical physics - Abstract
It is proved that the solutions to the slow diffusion stochastic porous media equation $dX-{\Delta}( |X|^{m-1}X )dt=\sigma(X)dW_t,$ $ 1< m\le 5,$ in $\mathcal{O}\subset\mathbb{R}^d,\ d=1,2,3,$ have the property of finite speed of propagation of disturbances for $\mathbb{P}\text{-a.s.}$ ${\omega}\in{\Omega}$ on a sufficiently small time interval $(0,t({\omega}))$.
- Published
- 2012
37. Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property
- Author
-
Cyril Labbé, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Property (philosophy) ,Genealogy ,Branching (linguistics) ,60G09 ,Continuous-state branching process ,60J25 ,Stochastic flow ,FOS: Mathematics ,Subordinator ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Branching process ,Entire population ,60J80 ,60J80, 60G09, 60J25 ,Probability (math.PR) ,Representation (systemics) ,Measure-valued process ,State (functional analysis) ,16. Peace & justice ,Eve ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Flow (mathematics) ,Lookdown process ,High Energy Physics::Experiment ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Partition - Abstract
We encode the genealogy of a continuous-state branching process associated with a branching mechanism $\Psi$ - or $\Psi$-CSBP in short - using a stochastic flow of partitions. This encoding holds for all branching mechanisms and appears as a very tractable object to deal with asymptotic behaviours and convergences. In particular we study the so-called Eve property - the existence of an ancestor from which the entire population descends asymptotically - and give a necessary and sufficient condition on the $\Psi$-CSBP for this property to hold. Finally, we show that the flow of partitions unifies the lookdown representation and the flow of subordinators when the Eve property holds., Comment: 43 pages
- Published
- 2012
- Full Text
- View/download PDF
38. On properties of a flow generated by an SDE with discontinuous drift
- Author
-
Andrey Pilipenko and Olga Aryasova
- Subjects
Statistics and Probability ,Discrete mathematics ,60J65, 60H10 ,Stochastic flow ,Probability (math.PR) ,Mathematical analysis ,differentiability with respect to initial data ,Function (mathematics) ,stochastic flow ,local times ,Physics::Fluid Dynamics ,Flow (mathematics) ,Mathematics::Probability ,Bounded variation ,FOS: Mathematics ,60J65 ,Differentiable function ,60H10 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Mathematics - Abstract
We consider a stochastic flow on $\mathds{R}$ generated by an SDE with its drift being a function of bounded variation. We show that the flow is differentiable with respect to the initial conditions. Asymptotic properties of the flow are studied., 19 pages
- Published
- 2012
39. A quasi likelihood for integral data on birth and death on flows
- Author
-
Michael J. Phelan
- Subjects
Statistics and Probability ,Continuous-time stochastic process ,Mathematical optimization ,Interacting particle system ,Applied Mathematics ,Integral data ,Quasi likelihood ,Markov process ,Poisson process ,Tracking (particle physics) ,Birth–death process ,Point process ,Physics::Fluid Dynamics ,symbols.namesake ,Flow (mathematics) ,Modeling and Simulation ,Stochastic flow ,Modelling and Simulation ,Compound Poisson process ,symbols ,Applied mathematics ,Mathematics - Abstract
Birth and Death on a flow refers to a particle system on a stochastic flow. Particles are born in a point process and move on the flow subject to position-dependent killing. They die eventually and leave the flow. The particle process is a measure-valued, Markov process tracking these motions. Its law depends on the distribution of births, the coefficients of the flow, and the rate of killing. We parametrize the system and derive a quasi-likelihood for chronicles of integral data on the particle process.
- Published
- 1994
- Full Text
- View/download PDF
40. Stochastic flows acting on Schwartz distributions
- Author
-
H. Kunita
- Subjects
Statistics and Probability ,Stochastic partial differential equation ,Stochastic differential equation ,Geometric Brownian motion ,Continuous-time stochastic process ,Stochastic flow ,General Mathematics ,Local time ,Mathematical analysis ,Stochastic calculus ,Statistics, Probability and Uncertainty ,Differential (mathematics) ,Mathematics - Abstract
We define the compositionT°ϕ s,t of a Schwartz distributionT with a stochastic flow ϕ s,t generated by a stochastic differential equation. Then we establish a generalized Ito's formula for the composite processesT(t)°ϕ s,t andT(t)°ϕ −1 , which describe a differential rule with respect to timet. The formula is then applied to two problems. One is the regularity of semigroups induced by the stochastic flow. The other is the existence and the continuity of the local time with respect to the spatial parameter, of a one dimensional stochastic flow.
- Published
- 1994
41. Lack of strong completeness for stochastic flows
- Author
-
Michael Scheutzow and Xue-Mei Li
- Subjects
Statistics and Probability ,Pure mathematics ,37C10 ,Statistics & Probability ,homogenization ,weak completeness ,stochastic differential equation ,math.PR ,Null set ,Stochastic differential equation ,SYSTEMS ,Stochastic flow ,Completeness (order theory) ,FOS: Mathematics ,Initial value problem ,Almost surely ,QA ,Mathematics ,Science & Technology ,0104 Statistics ,Probability (math.PR) ,35B27 ,DIFFERENTIAL-EQUATIONS ,Lipschitz continuity ,EXISTENCE ,Flow (mathematics) ,Bounded function ,Physical Sciences ,MANIFOLDS ,strong completeness ,60H10 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If -- in addition -- the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition $x$, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently $x$, then the maximal flow is called {\em strongly complete}. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a 2-dimensional SDE with coefficients which are even bounded (and smooth) and which is {\em not} strongly complete thus answering the question in the negative.
- Published
- 2011
42. Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration
- Author
-
Clément Foucart, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,media_common.quotation_subject ,Immigration ,Population ,stochastic flow ,Lambda ,genealogy for a population with immigration ,92D25 ,01 natural sciences ,Coalescent theory ,Combinatorics ,010104 statistics & probability ,Mathematics::Probability ,60G09 ,60G09, 60J27 ,60J25 ,Calculus ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,0101 mathematics ,education ,Exchangeable partition ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,media_common ,Superprocess ,coalescent theory ,Exchangeable partitions ,education.field_of_study ,Stochastic flow ,Applied Mathematics ,010102 general mathematics ,Probability (math.PR) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Stochastic flows ,Population model ,Merge (version control) ,Mathematics - Probability ,coming down from infinity - Abstract
Coalescents with multiple collisions (also called Lambda-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Imagine an infinite population with immigration labelled at each generation by N:={1,2,...}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focussing on simple distinguished coalescents, i.e such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0,1] denoted by M=(\Lambda_{0},\Lambda_{1}). We call them M-coalescents. We show by martingale arguments that the condition of coming down from infinity for the M-coalescent coincides with that obtained by Schweinsberg for the \Lambda-coalescent. In the same vein as Bertoin and Le Gall, M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures \Lambda_{0} and \Lambda_{1} specify respectively the reproduction and the immigration. The coming down from infinity of the M-coalescent will be interpreted as the initial types extinction: after a certain time, all individuals are immigrant children., Comment: 30 pages
- Published
- 2011
43. Weak convergence of the localized disturbance flow to the coalescing Brownian flow
- Author
-
James Norris and Amanda Turner
- Subjects
Statistics and Probability ,Coalescence (physics) ,Brownian web ,Laplace transform ,Weak convergence ,Mathematical analysis ,Probability (math.PR) ,Separable space ,Mathematics::Probability ,60F17 ,Stochastic flow ,coalescing Brownian motions ,Jump ,FOS: Mathematics ,Statistics, Probability and Uncertainty ,Arratia flow ,Martingale (probability theory) ,Mathematics - Probability ,Brownian motion ,Mathematics - Abstract
We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence., Comment: Published at http://dx.doi.org/10.1214/13-AOP845 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:0810.0211
- Published
- 2011
- Full Text
- View/download PDF
44. An asymptotically exact decomposition of coupled Brownian systems
- Author
-
Kerry W. Fendick
- Subjects
Statistics and Probability ,Stochastic flow ,General Mathematics ,Decomposition (computer science) ,Geometry ,Queueing system ,Brownian excursion ,Statistical physics ,Statistics, Probability and Uncertainty ,Brownian motion ,Mathematics - Abstract
Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.
- Published
- 1993
45. Stochastic equations, flows and measure-valued processes
- Author
-
Zenghu Li and Donald A. Dawson
- Subjects
Statistics and Probability ,superprocess ,Generalization ,stochastic flow ,92D25 ,Poisson distribution ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,symbols.namesake ,60G09 ,Mathematics::Probability ,60J25 ,strong solution ,FOS: Mathematics ,60J68 ,Applied mathematics ,Limit (mathematics) ,Uniqueness ,0101 mathematics ,Mathematics ,Superprocess ,generalized Fleming–Viot process ,Probability (math.PR) ,Stochastic equation ,010102 general mathematics ,coalescent ,16. Peace & justice ,Scaling limit ,Poincaré conjecture ,symbols ,Statistics, Probability and Uncertainty ,continuous-state branching process ,Mathematics - Probability ,immigration - Abstract
We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming--Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincar\'{e} Probab. Statist. 41 (2005) 307--333]. Two scaling limit theorems for the generalized Fleming--Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147--181]., Comment: Published in at http://dx.doi.org/10.1214/10-AOP629 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2010
46. Capacity Expansion in Stochastic Flow Networks
- Author
-
George S. Fishman and Christos Alexopoulos
- Subjects
Statistics and Probability ,Stochastic flow ,Computer science ,Applied mathematics ,Management Science and Operations Research ,Statistics, Probability and Uncertainty ,Industrial and Manufacturing Engineering - Abstract
Sensitivity analysis represents an important aspect of network flow design problems. For example, what is the incremental increase in system flow of increasing the diameters of specified pipes in a water distribution network? Although methods exist for solving this problem in the deterministic case, no comparable methodology has been available when the network's arc capacities are subject to random variation. This paper provides this methodology by describing a Monte Carlo sampling plan that allows one to conduct a sensitivity analysis for a variable upper bound on the flow capacity of a specified arc. The proposed plan has two notable features. It permits estimation of the probabilities of a feasible flow for many values of the upper bound on the arc capacity from a single data set generated by the Monte Carlo method at a single value of this upper bound. Also, the resulting estimators have considerably smaller variancesthan crude Monte Carlo sampling would produce in the same setting. The success of the technique follows from the use of lower and upper bounds on each probability of interest where the bounds are generated from an established method of decomposing the capacity state space.
- Published
- 1992
47. Some properties of superprocesses under a stochastic flow
- Author
-
Carl Mueller, Jei Xiong, and Kijung Lee
- Subjects
Random environment ,Statistics and Probability ,60J80 ,Stochastic flow ,Lebesgue measure ,Probability (math.PR) ,Stochastic partial differential equation ,Mathematics - Analysis of PDEs ,Singularity ,Mathematics::Probability ,60G57, 60H15, Secondary 60J80 ,FOS: Mathematics ,60H15 ,Superprocess ,60G57 ,Applied mathematics ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Snake representation ,Mathematics - Probability ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
For a superprocess under a stochastic flow, we prove that it has a density with respect to the Lebesgue measure for d=1 and is singular for d>1. For d=1, a stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov's L_p-theory for linear SPDE. A snake representation for this superprocess is established. As applications of this representation, we prove the compact support property for general d and singularity of the process when d>1.
- Published
- 2009
48. Attractors and Expansion for Brownian Flows
- Author
-
Michael Scheutzow, Georgi Dimitroff, and Publica
- Subjects
60G60 ,Statistics and Probability ,chaining ,60G90 ,Mathematical analysis ,Probability (math.PR) ,stochastic differential equation ,37H10 ,Upper and lower bounds ,attractor ,Stochastic differential equation ,Stochastic flow ,Bounded function ,Chaining ,Attractor ,FOS: Mathematics ,60H10 ,Ball (mathematics) ,Statistics, Probability and Uncertainty ,Volatility (finance) ,Mathematics - Probability ,Brownian motion ,Mathematics - Abstract
We show that a stochastic flow which is generated by a stochastic differential equation on $\mathbb{R}^d$ with bounded volatility has a random attractor provided that the drift component in the direction towards the origin is larger than a certain strictly positive constant $\beta$ outside a large ball. Using a similar approach, we provide a lower bound for the linear growth rate of the inner radius of the image of a large ball under a stochastic flow in case the drift component in the direction away from the origin is larger than a certain strictly positive constant $\beta$ outside a large ball. To prove the main result we use chaining techniques in order to control the growth of the diameter of subsets of the state space under the flow.
- Published
- 2009
- Full Text
- View/download PDF
49. Martingale representation and hedging policies
- Author
-
Robert J. Elliott, P. Ekkehard Kopp, and David B. Colwell
- Subjects
Statistics and Probability ,Mathematical optimization ,Function space ,Mathematics::Optimization and Control ,stochastic flow ,01 natural sciences ,010104 statistics & probability ,Mathematics::Probability ,Computer Science::Computational Engineering, Finance, and Science ,Modelling and Simulation ,0502 economics and business ,0101 mathematics ,Martingale representation theorem ,Mathematics ,martingale representation ,050208 finance ,Girsanov's theorem ,Markov chain ,Applied Mathematics ,diffusion ,05 social sciences ,hedging portfolio ,Doob's martingale inequality ,Modeling and Simulation ,Local martingale ,Martingale difference sequence ,Martingale (probability theory) ,Mathematical economics ,Martingale pricing - Abstract
The integrand, when a martingale under an equivalent measure is represented as a stochastic integral, is determined by elementary methods in the Markov situation. Applications to hedging portfolios in finance are described.
- Published
- 1991
50. Stochastic Flow-Shop Scheduling with Lateness-Related Performance Measures
- Author
-
Chung Yee Lee and Chen-Sin Lin
- Subjects
Statistics and Probability ,Mathematical optimization ,Stochastic flow ,Due date ,Computer science ,Scheduling (production processes) ,Stochastic optimization ,Flow shop scheduling ,Management Science and Operations Research ,Statistics, Probability and Uncertainty ,Industrial and Manufacturing Engineering - Abstract
In this paper we consider stochastic flow-shop scheduling with reference to certain lateness-related performance measures. We show that for various assumptions on the distribution of job-processing times of a flow shop, certain scheduling policies following the stochastic analogy of the Earliest Due Date (EDD) rule yield optimal results.
- Published
- 1991
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