The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model

Yuliya Mishura

Displaying similar documents to “The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model”

Moment Inequality for the Martingale Square Function

Adam Osękowski (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Consider the sequence ${(C ₙ)}_{n \geq 1}$ of positive numbers defined by C₁ = 1 and $C_{n + 1} = 1 + C ₙ ² / 4$ , n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

On the distance between ⟨X⟩ and $L^{\infty}$ in the space of continuous BMO-martingales

Litan Yan, Norihiko Kazamaki (2005)

Studia Mathematica

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Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, ${| | X | |}_{B M O} \equiv s u p_{T} | | E [| X_{\infty} - X_{T} | | ℱ_{T} {] | |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b (X) = b > 0 : s u p_{T} | | E [e x p (b ² (⟨ X ⟩_{\infty} - ⟨ X ⟩_{T})) | ℱ_{T}] {| |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q (X) ₜ = E [⟨ X ⟩_{\infty} | ℱ ₜ] - E [⟨ X ⟩_{\infty} | ℱ ₀]$ . We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{\infty}$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1 / 4 d ₁ (q (X), L^{\infty}) \leq b (X) \leq 4 / d ₁ (q (X), L^{\infty})$ hold for every continuous BMO-martingale X. ...

A correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”

Yongfeng Wu, Andrew Rosalsky, Andrei Volodin (2017)

Applications of Mathematics

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The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”.

Carthaginian enlargement of filtrations

Giorgia Callegaro, Monique Jeanblanc, Behnaz Zargari (2013)

ESAIM: Probability and Statistics

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This work is concerned with the theory of initial and progressive enlargements of a reference filtration $𝔽$ F with a random time. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an $𝔽$ F -martingale in the enlarged filtrations. Also, we address martingales’ characterization in the enlarged filtrations in terms of martingales in the reference filtration, as...

On the rate of convergence of difference schemes for the heat transfer equation on the solutions from $w_{2}^{s, s / 2}$

L. D. Ivanović, B. S. Jovanović, E. E. Süli (1984)

Matematički Vesnik

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On a relation between norms of the maximal function and the square function of a martingale

Masato Kikuchi (2013)

Colloquium Mathematicae

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Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For $f = {(f ₙ)}_{n \in ℤ ₊} \in ℳ$ , let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and ${| | M g | |}_{X} {\leq | | M f | |}_{X}$ , then ${| | S g | |}_{X} {\leq C | | S f | |}_{X}$ , where C is a constant independent of f and g.

Atomic decomposition of predictable martingale Hardy space with variable exponents

Zhiwei Hao (2015)

Czechoslovak Mathematical Journal

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This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let $(Ω, ℱ, ℙ)$ be a probability space and $p (\cdot) : Ω \to (0, \infty)$ be a $ℱ$ -measurable function such that $0 < {inf}_{x \in Ω} p (x) \leq {sup}_{x \in Ω} p (x) < \infty$ . It is proved that a predictable martingale Hardy space $𝒫_{p (\cdot)}$ has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy...

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.

A Note on the Burkholder-Rosenthal Inequality

Adam Osękowski (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate $∥ \sum_{k = 0}^{\infty} d f_{k} ∥_{p} \leq C_{p} ∥ (\sum_{k = 0}^{\infty} (| d f_{k} | ² | ℱ_{k - 1} {))}^{1 / 2} ∥_{p} + ∥ (\sum_{k = 0}^{\infty} | d f_{k} {|^{p})}^{1 / p} ∥_{p},$ with $C_{p} = O (p / l n p)$ as p → ∞.

Genealogies of regular exchangeable coalescents with applications to sampling

Vlada Limic (2012)

Annales de l'I.H.P. Probabilités et statistiques

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This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as $𝛯$ -coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained by the author in a previous work. Analogous results for the number of active mutation-free lineages and the combined lineage lengths are derived using the same martingale-based technique. They are given in terms of convergence...

A martingale approach to general Franklin systems

Anna Kamont, Paul F. X. Müller (2006)

Studia Mathematica

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We prove unconditionality of general Franklin systems in $L^{p} (X)$ , where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

On the completeness of $ℒ_{p}$ -spaces over a charge

Sreela Gangopadhyay (1990)

Colloquium Mathematicae

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Norm convergence of Fejér means of two-dimensional Walsh-Fourier series

Ushangi Goginava (2011)

Banach Center Publications

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The main aim of this paper is to prove that there exists a martingale $f \in H_{1 / 2}$ such that the Fejér means of the two-dimensional Walsh-Fourier series of f is not uniformly bounded in the space weak- $L_{1 / 2}$ .

Linearized plasticity is the evolutionary $Γ$ -limit of finite plasticity

Alexander Mielke, Ulisse Stefanelli (2013)

Journal of the European Mathematical Society

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We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via $Γ$ -convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

Noncommutative fractional integrals

Narcisse Randrianantoanina, Lian Wu (2015)

Studia Mathematica

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Let ℳ be a hyperfinite finite von Nemann algebra and ${(ℳ_{k})}_{k \geq 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${(ℳ_{k})}_{k \geq 1}$ . For a finite noncommutative martingale $x = {(x_{k})}_{1 \leq k \leq n} \subseteq L ₁ (ℳ)$ adapted to ${(ℳ_{k})}_{k \geq 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{α} x = \sum_{k = 1}^{n} ζ_{k}^{α} d x_{k}$ for an appropriate sequence ${(ζ_{k})}_{k \geq 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...

Convergence in the dual of certain $K M_{p}$ -spaces

Larry Kitchens, Charles Swartz (1974)

Colloquium Mathematicae

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On risk reserve under distribution constraints

Mariusz Michta (2000)

Discussiones Mathematicae Probability and Statistics

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The purpose of this work is a study of the following insurance reserve model: $R (t) = η + \int_{0}^{t} p (s, R (s)) d s + \int_{0}^{t} σ (s, R (s)) d W_{s} - Z (t)$ , t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: $i n f_{0 \leq t \leq T} P R (t) \geq c \geq γ$ is considered.

Sharp moment inequalities for differentially subordinated martingales

Adam Osękowski (2010)

Studia Mathematica

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We determine the optimal constants $C_{p, q}$ in the moment inequalities ${| | g | |}_{p} \leq C_{p, q} {| | f | |}_{q}$ , 1 ≤ p< q< ∞, where f = (fₙ), g = (gₙ) are two martingales, adapted to the same filtration, satisfying |dgₙ| ≤ |dfₙ|, n = 0,1,2,..., with probability 1. Furthermore, we establish related sharp estimates ||g||₁ ≤ supₙΦ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.

A law of the iterated logarithm for general lacunary series

Charles N. Moore, Xiaojing Zhang (2012)

Colloquium Mathematicae

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We prove a law of the iterated logarithm for sums of the form $\sum_{k = 1}^{N} a_{k} f (n_{k} x)$ where the $n_{k}$ satisfy a Hadamard gap condition. Here we assume that f is a Dini continuous function on ℝⁿ which has the property that for every cube Q of sidelength 1 with corners in the lattice ℤⁿ, f vanishes on ∂Q and has mean value zero on Q.

On the Rademacher maximal function

Mikko Kemppainen (2011)

Studia Mathematica

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This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^{p}$ -boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^{p}$ -boundedness is shown not to depend on the underlying measure space or the filtration. Martingale...

Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

Adam Osękowski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be a conditionally symmetric martingale and let S(f) denote its square function. (i) For p,q > 0, we determine the best constants $C_{p, q}$ such that $s u p_{n} {(| f ₙ |}^{p} {) / (1 + S ₙ ² (f))}^{q} \leq C_{p, q}$ . Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2,... and q > 0, we determine the best constants $C_{N, q}^{'}$ such that $s u p_{n} (f ₙ^{2 N - 1}) {(1 + S ₙ ² (f))}^{q} \leq C_{N, q}^{'}$ . These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if...

A note on necessary and sufficient conditions for convergence of the finite element method

Kučera, Václav

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In this short note, we present several ideas and observations concerning finite element convergence and the role of the maximum angle condition. Based on previous work, we formulate a hypothesis concerning a necessary condition for $O (h)$ convergence and show a simple relation to classical problems in measure theory and differential geometry which could lead to new insights in the area.

Hedging in complete markets driven by normal martingales

Youssef El-Khatib, Nicolas Privault (2003)

Applicationes Mathematicae

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This paper aims at a unified treatment of hedging in market models driven by martingales with deterministic bracket $⟨ M, M ⟩_{t}$ , including Brownian motion and the Poisson process as particular cases. Replicating hedging strategies for European, Asian and Lookback options are explicitly computed using either the Clark-Ocone formula or an extension of the delta hedging method, depending on which is most appropriate.

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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$L_{\infty}$ -Convergence of Finite Element Approximation

J. A. Nitsche (1975)

Publications mathématiques et informatique de Rennes

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On the convergence of sequences of iterates of random-valued vector functions

Rafał Kapica (2007)

Annales Polonici Mathematici

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Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates $f ⁿ : X \times Ω^{ℕ} \to X$ defined by f¹(x,ω ) = f(x,ω₁), $f^{n + 1} (x, ω) = f (f ⁿ (x, ω), ω_{n + 1})$ .

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $π = \sum_{n = 0}^{\infty} ((5 n + 3) n! (2 n)!) / (2^{n - 1} (3 n + 2)!)$ with convergence as $13 . 5^{- n}$ , in much the same way as the Euler transformation gives $π = \sum_{n = 0}^{\infty} (2^{n + 1} n! n!) / (2 n + 1)!$ with convergence as $2^{- n}$ . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

On the rate of convergence of the Bézier-type operators

Grażyna Anioł (2006)

Bollettino dell'Unione Matematica Italiana

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For bounded functions $f$ on an interval $I$ , in particular, for functions of bounded p-th power variation on $I$ there is estimated the rate of pointwise convergence of the Bezier-type modification of the discrete Feller operators. In the main theorem the Chanturiya modulus of variation is used.

Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Péter Kórus, Ferenc Móricz (2010)

Studia Mathematica

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We investigate the convergence behavior of the family of double sine integrals of the form $\int_{0}^{\infty} \int_{0}^{\infty} f (x, y) s i n u x s i n v y d x d y$ , where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals $\int_{a ₁}^{b ₁} \int_{a ₂}^{b ₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and $b_{j} > a_{j} \geq 0$ , j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial...

A lower bound in the law of the iterated logarithm for general lacunary series

Charles N. Moore, Xiaojing Zhang (2014)

Studia Mathematica

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We prove a lower bound in a law of the iterated logarithm for sums of the form $\sum_{k = 1}^{N} a_{k} f (n_{k} x + c_{k})$ where f satisfies certain conditions and the $n_{k}$ satisfy the Hadamard gap condition $n_{k + 1} / n_{k} \geq q > 1$ .