On Multivariate Bernstein Polynomials

Foupouagnigni, M.; Wouodjié, Merlin Mouafo

doi:10.3390/math8091397

Cited by 12 publications

(5 citation statements)

References 31 publications

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“…The second quantity in the bracket only includes the situation that no agent selects the lowest spread and thus converges to 0. The first quantity converges to P(x * ), by the property of multivariate Bernstein polynomials and the continuity of P(•); see a recent survey in Foupouagnigni and Mouafo Wouodjié (2020).…”

Section: A Proofs Of Resultsmentioning

confidence: 99%

Cooperation between Independent Market Makers

Han¹

2022

Preprint

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With the digitalization of the financial market, dealers are increasingly handling marketmaking activities by algorithms. Recent antitrust literature raises concerns on collusion caused by artificial intelligence. This paper studies the possibility of cooperation between market makers via independent Q-learning. Market making with inventory risk is formulated as a repeated general-sum game. Under a stag-hunt type payoff, we find that market makers can learn cooperative strategies without communication. In general, high spreads can have the largest probability even when the lowest spread is the unique Nash equilibrium. Moreover, introducing more agents into the game does not necessarily eliminate the presence of supra-competitive spreads.

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Section: A Proofs Of Resultsmentioning

confidence: 99%

Cooperation between Independent Market Makers

Han¹

2022

Preprint

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“…By Theorem 4.2, there exists a sequence {f α, ∆;n }, for n k ∈ N, of multivariate Bernstein zipper α-fractal functions that converges to f for any given non-negative function f ∈ C(I). By [22], B n is a positive linear operator and thus B n f (X) ≥ 0, for all X ∈ I, which implies the positivity of Φ n .…”

Section: Multivariate Bernstein Zipper Fractal Functionmentioning

confidence: 97%

“…To get the convergence of multivariate α-fractal function f α, ∆,b to f without altering the scaling function α, we take as base functions b multivariate Bernstein polynomials B n f (X) [19,22] of f . The n := (n 1 , ..., n m )-th Bernstein polynomial for f ∈ C(I) is given by…”

Section: Multivariate Bernstein Zipper Fractal Functionmentioning

confidence: 99%

“…These multivariate zipper fractal functions are more general than the existing classical and fractal approximants. Based on the existence of ZFIF, we construct a novel class of multivariate Bernstein zipper α-fractal functions using Bernstein polynomials B n1,...,nm f [19,22] as base functions in its IFS f α, n1n2...nm for a given f ∈ C m k=1 I k , where the I k are bounded and closed intervals in R. Multivariate Bernstein zipper α-fractal functions f α, n1n2...nm converge uniformly to f as n i → ∞ for all i, without altering the scaling functions. We prove that the multivariate Bernstein polynomial B n1,...,nm f is Lipschitz if so is f .…”

Section: Introductionmentioning

confidence: 99%

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Multivariate Zipper Fractal Functions

Kumar¹,

Chand²,

Massopust³

2022

Preprint

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A novel approach to zipper fractal interpolation theory for functions of several variables is proposed. We develop multivariate zipper fractal functions in a constructive manner. We then perturb a multivariate function to construct its zipper α-fractal varieties through free choices of base functions, scaling functions, and a binary matrix called signature.In particular, we propose a multivariate Bernstein zipper fractal function and study its approximation properties such as shape preserving aspects, non-negativity, and coordinate-wise monotonicity. In addition, we derive bounds for the graph of multivariate zipper fractal functions by imposing conditions on the scaling factors and the Hölder exponent of the associated germ function and base function. The Lipschitz continuity of multivariate Bernstein functions is also studied in order to obtain estimates for the box dimension of multivariate Bernstein zipper fractal functions.

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“…Furthermore, by suitably choosing the orders (N 1 , N 2 ) of lifting, such a function g can be used to approximate any continuous functions (a base , x base ) → f (a base , x base ) to arbitrary accuracy on any compact set C. (E.g., see the paper Foupouagnigni and Mouafo Wouodjié (2020) for results on the approximation (multivariate) functions with Bernstein polynomials). Note that besides polynomial type basis (EC.1) that yields the form (EC.2), we can also use other bases such as Fourier or Haar.…”

Section: Ec1 Nonlinear Relationship Representations and Extensions To...mentioning

confidence: 99%

Doubly High-Dimensional Contextual Bandits: An Interpretable Model for Joint Assortment-Pricing

Cai,

Chen,

Wainwright

et al. 2023

SSRN Journal

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On Multivariate Bernstein Polynomials

Cited by 12 publications

References 31 publications

Cooperation between Independent Market Makers

Cooperation between Independent Market Makers

Multivariate Zipper Fractal Functions

Doubly High-Dimensional Contextual Bandits: An Interpretable Model for Joint Assortment-Pricing

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