scholarly journals Strong approximation of continuous local martingales by simple random walks

2004 ◽  
Vol 41 (1) ◽  
pp. 101-126
Author(s):  
B. Székely ◽  
T. Szabados
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Rates Of Convergence ◽  
Quadratic Variation ◽  
Local Martingale ◽  
Motion Time ◽  
Time Changes ◽  
Necessary And Sufficient ◽  
Nested Sequence ◽  
The Given

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

scholarly journals Brownian representations of cylindrical continuous local martingales

2018 ◽  
Vol 21 (02) ◽  
pp. 1850013
Author(s):  
Ivan Yaroslavtsev
Keyword(s):  
Brownian Motion ◽  
Stochastic Integral ◽  
Sufficient Conditions ◽  
Closed Operator ◽  
Time Change ◽  
Necessary And Sufficient Conditions ◽  
Local Martingale ◽  
Necessary And Sufficient

In this paper, we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular, we consider the class of cylindrical martingales with closed operator-generated covariations. We also prove that for every cylindrical continuous local martingale [Formula: see text] there exists a time change [Formula: see text] such that [Formula: see text] is Brownian representable.

scholarly journals Quadratic variation, models, applications and lessons

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dilip B. Madan ◽  
King Wang
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Physical Process ◽  
Continuous Time ◽  
Fourier Transforms ◽  
Quadratic Variation ◽  
Market Prices ◽  
Variance Swap ◽  
Time Changes ◽  
Quadratic Variations

<p style="text-indent:20px;">Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.</p>

scholarly journals On the First Passage time for Brownian Motion Subordinated by a Lévy Process

2009 ◽  
Vol 46 (1) ◽  
pp. 181-198 ◽  
Author(s):  
T. R. Hurd ◽  
A. Kuznetsov
Keyword(s):  
Brownian Motion ◽  
Approximation Scheme ◽  
First Passage Time ◽  
Gaussian Model ◽  
Passage Time ◽  
Financial Mathematics ◽  
First Passage ◽  
Financial Modeling ◽  
Motion Time ◽  
Normal Inverse Gaussian

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{&#x0394;}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

scholarly journals Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

2021 ◽  
Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Processes ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Bounded Geometry ◽  
And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

scholarly journals Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ryuya Namba
Keyword(s):  
Random Walks ◽  
Quadratic Forms ◽  
Volume Growth ◽  
Moderate Deviation ◽  
Point Of View ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Geometric Point ◽  
Rate Functions ◽  
The Given

AbstractModerate deviation principles (MDPs) for random walks on covering graphs with groups of polynomial volume growth are discussed in a geometric point of view. They deal with any intermediate spatial scalings between those of laws of large numbers and those of central limit theorems. The corresponding rate functions are given by quadratic forms determined by the Albanese metric associated with the given random walks. We apply MDPs to establish laws of the iterated logarithm on the covering graphs by characterizing the set of all limit points of the normalized random walks.

A General Analysis of Sequential Social Learning

2021 ◽  
Author(s):  
Itai Arieli ◽  
Manuel Mueller-Frank
Keyword(s):  
Social Learning ◽  
Utility Function ◽  
Posterior Probability ◽  
General Analysis ◽  
Sufficient Condition ◽  
Important Class ◽  
General Utility ◽  
Learning Game ◽  
Necessary And Sufficient ◽  
The Given

This paper analyzes a sequential social learning game with a general utility function, state, and action space. We show that asymptotic learning holds for every utility function if and only if signals are totally unbounded, that is, the support of the private posterior probability of every event contains both zero and one. For the case of finitely many actions, we provide a sufficient condition for asymptotic learning depending on the given utility function. Finally, we establish that for the important class of simple utility functions with finitely many actions and states, pairwise unbounded signals, which generally are a strictly weaker notion than unbounded signals, are necessary and sufficient for asymptotic learning.

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