scholarly journals Space-distribution PDEs for path independent additive functionals of McKean–Vlasov SDEs

2020 ◽  
Vol 23 (03) ◽  
pp. 2050018
Author(s):  
Panpan Ren ◽  
Feng-Yu Wang
Keyword(s):  
Product Space ◽  
Space Variable ◽  
Space Distribution ◽  
Nonlinear Pdes ◽  
Probability Measures ◽  
Girsanov Transformation ◽  
Additive Functionals ◽  
Drift Term ◽  
Path Independence ◽  
Girsanov Transformations

Let [Formula: see text] be the space of probability measures on [Formula: see text] with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space [Formula: see text] equipped with the usual derivative in space variable and Lions’ derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path independence of Girsanov transformations is identified with nonlinear PDEs on [Formula: see text] whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.

scholarly journals Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps

2021 ◽  
Vol 24 (01) ◽  
pp. 2150006
Author(s):  
Huijie Qiao ◽  
Jiang-Lun Wu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Recent Work ◽  
Probability Measures ◽  
Brownian Motions ◽  
Additive Functionals ◽  
Path Independence ◽  
Independent Property

In this paper, the path independent property of additive functionals of McKean–Vlasov stochastic differential equations with jumps is characterized by nonlinear partial integro-differential equations involving [Formula: see text]-derivatives with respect to probability measures introduced by Lions. Our result extends the recent work16 by Ren and Wang where their concerned McKean–Vlasov stochastic differential equations are driven by Brownian motions.

Mackey Continuity of Characteristic Functionals

2002 ◽  
Vol 9 (1) ◽  
pp. 83-112
Author(s):  
S. Kwapień ◽  
V. Tarieladze
Keyword(s):  
Product Space ◽  
Locally Convex Space ◽  
Inner Product ◽  
Nuclear Space ◽  
Probability Measures ◽  
Linear Kernel ◽  
Characteristic Functional ◽  
Initial Space ◽  
Radon Probability Measure ◽  
Locally Convex

Abstract Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigmacompact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey-continuous characteristic functional.

scholarly journals The Girsanov Linearization Method for Stochastically Driven Nonlinear Oscillators

2006 ◽  
Vol 74 (5) ◽  
pp. 885-897 ◽  
Author(s):  
Nilanjan Saha ◽  
D. Roy
Keyword(s):  
Numerical Integration ◽  
Nonlinear Oscillators ◽  
Linearization Method ◽  
Statistical Moments ◽  
Probability Measures ◽  
Numerical Accuracy ◽  
Girsanov Transformation ◽  
Nonlinear Part ◽  
Integration Schemes ◽  
Nikodym Derivative

For most practical purposes, the focus is often on obtaining statistical moments of the response of stochastically driven oscillators than on the determination of pathwise response histories. In the absence of analytical solutions of most nonlinear and higher-dimensional systems, Monte Carlo simulations with the aid of direct numerical integration remain the only viable route to estimate the statistical moments. Unfortunately, unlike the case of deterministic oscillators, available numerical integration schemes for stochastically driven oscillators have significantly poorer numerical accuracy. These schemes are generally derived through stochastic Taylor expansions and the limited accuracy results from difficulties in evaluating the multiple stochastic integrals. As a numerically superior and semi-analytic alternative, a weak linearization technique based on Girsanov transformation of probability measures is proposed for nonlinear oscillators driven by additive white-noise processes. The nonlinear part of the drift vector is appropriately decomposed and replaced, resulting in an exactly solvable linear system. The error in replacing the nonlinear terms is then corrected through the Radon-Nikodym derivative following a Girsanov transformation of probability measures. Since the Radon-Nikodym derivative is expressible in terms of a stochastic exponential of the linearized solution and computable with high accuracy, one can potentially achieve a remarkably high numerical accuracy. Although the Girsanov linearization method is applicable to a large class of oscillators, including those with nondifferentiable vector fields, the method is presently illustrated through applications to a few single- and multi-degree-of-freedom oscillators with polynomial nonlinearity.

A Comparison of Methods for Constructing Probability Measures on Infinite Product Spaces

1987 ◽  
Vol 30 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Charles W. Lamb
Keyword(s):  
Probability Measure ◽  
Conditional Probability ◽  
Product Space ◽  
Infinite Product ◽  
Probability Measures ◽  
Classical Theorem ◽  
Comparison Of Methods ◽  
Product Spaces ◽  
Probability Functions ◽  
Finite Dimensional

AbstractThe construction, from a consistent family of finite dimensional probability measures, of a probability measure on a product space when the marginal measures are perfect is shown to follow from a classical theorem due to Ionescu Tulcea and known results on the existence of regular conditional probability functions.

DIRICHLET PROBLEM WITH STOCHASTIC COEFFICIENTS

2005 ◽  
Vol 05 (04) ◽  
pp. 555-568 ◽  
Author(s):  
INGO BULLA
Keyword(s):  
Sobolev Space ◽  
Existence And Uniqueness ◽  
Product Space ◽  
Space Variable ◽  
Dirichlet Condition ◽  
Elliptic Partial Differential Equations ◽  
Random Coefficients ◽  
Stochastic Variable ◽  
Random Data ◽  
Tensor Product Space

We consider general second-order linear elliptic partial differential equations having random coefficients and random data and fulfilling the homogeneous Dirichlet condition. We prove the existence and uniqueness of the weak solution in a certain tensor product space which is suitably completed to make it a Hilbert space. The factors of this space are a Sobolev space of functions depending on the space variable and a general Sobolev space of functions depending on the stochastic variable.

scholarly journals GENERALIZED BESSEL AND FRAME MEASURES

2020 ◽  
Vol 35 (1) ◽  
pp. 217
Author(s):  
Fariba Zeinal Zadeh Farhadi ◽  
Mohammad Sadegh Asgari ◽  
Mohammad Reza Mardanbeigi ◽  
Mahdi Azhini
Keyword(s):  
Product Space ◽  
Borel Measure ◽  
Finite Measure ◽  
Inner Product Space ◽  
Inner Product ◽  
Probability Measures ◽  
Bessel Sequence ◽  
Finite Borel Measure ◽  
Conjugate Exponents ◽  
Finite Measures

Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we have introduced $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $ q $-Bessel sequence and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We have constructed a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We have showed that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using the convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We have presented a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.

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