Continuous Probability Measure Research Articles

EFFICIENT HEDGING OF EUROPEAN OPTIONS WITH ROBUST CONVEX LOSS FUNCTIONALS: A DUAL-REPRESENTATION FORMULA

Motivated by numerical representations of robust utility functionals, due to Maccheroni et al., we study the problem of partially hedging a European option H when a hedging strategy is selected through a robust convex loss functional L(·) involving a penalization term γ(·) and a class of absolutely continuous probability measures . We present three results. An optimization problem is defined in a space of stochastic integrals with value function EH(·). Extending the method of Föllmer and Leukerte, it is shown how to construct an optimal strategy. The optimization problem EH(·) as criterion to select a hedge, is of a “minimax” type. In the second, and main result of this paper, a dual-representation formula for this value is presented, which is of a “maxmax” type. This leads us to a dual optimization problem. In the third result of this paper, we apply some key arguments in the robust convex-duality theory developed by Schied to construct optimal solutions to the dual problem, if the loss functional L(·) has an associated convex risk measure ρL(·) which is continuous from below, and if the European option H is essentially bounded.

On the convergence of nonuniform ergodic means

for semiflows {Tt} with invariant measure μ was begun in [1], [2], where it was established that, for absolutely continuous probability measures ν on the semiaxis and bounded functions f , the quantities Ftf(x) as t → +∞ have, for μ-almost all x, the same limit as in the case of the classical uniform averaging in the Birkhoff–Khinchin theorem. The study of the averaging Ftf(x) was continued in [3] (see also [4, Chap. 10]), where it was found that, for unbounded functions f , it may happen that the nonuniform averaging does not converge. In that paper, additional conditions ensuring such a convergence were obtained. Similar questions for stochastic equations were considered in [5]. The case of averaging with an operator semigroup was studied in [6]. In the present paper, we study the convergence of the means Ftf(x) in Lp(μ), as well as some applications of the results obtained. In particular, we are referring to an analog of the ergodic Wiener–Wintner theorem. In contrast to [3], [5], [6], a weak mixing condition is imposed on the semigroup in a number of assertions in present paper. Suppose that (X,A , μ) is a probability space. Let us recall that the semigroup {Tt}t≥0 of transformations of the spaceX preserving the measure μ is said to be weakly mixing (see [7, p. 28]) if, for any of two functions f, g ∈ L2(μ), the following relation holds:

On nearly radial marginals of high-dimensional probability measures

Suppose that μ is an absolutely continuous probability measure on ℝ^n , for large n . Then μ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if n ≥ (C/ε)^{Cd} , then there exist d -dimensional marginals of μ that are ε -far from being spherically-symmetric, in an appropriate sense. Here C > 0 is a universal constant.

Robust efficient hedging for American options: The existence of worst case probability measures

Abstract Our problem here starts when the seller of an American option H has an initial capital c and aims to control shortfall risk by optimally trading in the market. We consider the case where the initial capital c is smaller than the superhedging cost of H. Thus, it is meaningful to consider efficient strategies of partial hedging. We investigate in the American case the approach of Föllmer and Leukert of efficiently hedging European options. At the same time, we consider situations where the model R is ambiguous. This implies that the seller´s preference is going to be represented by a robust loss functional, involving a whole class of absolutely continuous probability measures Q. We apply a construction due to Föllmer and Kramkov on the existence of Fatou-convergent sequences of convex combinations of supermartingales, to show that optimal strategies minimizing such a functional exist. We then show that the robust efficient hedging problem can be reduced into a non-robust problem of efficient hedging with respect to a worst case probability measure Q *∈Q, if Q satisfies a compactness condition and H is essentially bounded.

Uneven Splitting of Ham Sandwiches

Let m_1,...,m_n be continuous probability measures on R^n and a_1,...,a_n in [0,1]. When does there exist an oriented hyperplane H such that the positive half-space H^+ has m_i(H^+)=a_i for all i in [n]? It is well known that such a hyperplane does not exist in general. The famous ham sandwich theorem states that if a_i=1/2 for all i, then such a hyperplane always exists. In this paper we give sufficient criteria for the existence of H for general a_i in [0,1]. Let f_1,...,f_n:S^{n-1}->R^n denote auxiliary functions with the property that for all i the unique hyperplane H_i with normal v that contains the point f_i(v) has m_i(H_i^+)=a_i. Our main result is that if Im(f_1),...,Im(f_n) are bounded and can be separated by hyperplanes, then there exists a hyperplane H with m_i(H^+)=a_i for all i. This gives rise to several corollaries, for instance if the supports of m_1,...,m_n are bounded and can be separated by hyperplanes, then H exists for any choice of a_1,...,a_n in [0,1]. We also obtain results that can be applied if the supports of m_1,...,m_n overlap.

Feynman’s Operational Calculi: Decomposing Disentanglings

Let X be a Banach space and suppose that A1,…,An are noncommuting (that is, not necessarily commuting) elements in ℒ(X), the space of bounded linear operators on X. Further, for each i∈{1,…,n}, let μi be a continuous probability measure on ℬ([0,1]), the Borel class of [0,1]. Each such n-tuple of operator-measure pairs (Ai,μi), i=1,…,n, determines an operational calculus or disentangling map\({\mathcal{T}}_{\mu_{1},\dots,\mu_{n}}\) from a commutative Banach algebra \({\mathbb{D}}(A_{1},\dots,A_{n})\) of analytic functions, called the disentangling algebra , into the noncommutative Banach algebra ℒ(X). The disentanglings are the central processes of Feynman’s operational calculi.

Points with large α-depth

We show that for every ϵ > 0 there exists an angle α = α ( ϵ ) between 0 and π, depending only on ϵ, with the following two properties: (1) For any continuous probability measure in the plane one can find two lines ℓ 1 and ℓ 2 , crossing at an angle of (at least) α, such that the measure of each of the two opposite quadrants of angle π − α , determined by ℓ 1 and ℓ 2 , is at least 1 2 − ϵ . (2) For any set P of n points in general position in the plane one can find two lines ℓ 1 and ℓ 2 , crossing at an angle of (at least) α and moreover at a point of P, such that in each of the two opposite quadrants of angle π − α , determined by ℓ 1 and ℓ 2 , there are at least ( 1 2 − ϵ ) n − 4 points of P.

A refined Jensen’s inequality in Hilbert spaces and empirical approximations

Let f : X → R be a convex mapping and X a Hilbert space. In this paper we prove the following refinement of Jensen’s inequality: E ( f | X ∈ A ) ≥ E ( f | X ∈ B ) for every A , B such that E ( X | X ∈ A ) = E ( X | X ∈ B ) and B ⊂ A . Expectations of Hilbert-space-valued random elements are defined by means of the Pettis integrals. Our result generalizes a result of [S. Karlin, A. Novikoff, Generalized convex inequalities, Pacific J. Math. 13 (1963) 1251–1279], who derived it for X = R . The inverse implication is also true if P is an absolutely continuous probability measure. A convexity criterion based on the Jensen-type inequalities follows and we study its asymptotic accuracy when the empirical distribution function based on an n -dimensional sample approximates the unknown distribution function. Some statistical applications are addressed, such as nonparametric estimation and testing for convex regression functions or other functionals.

Orthogonal equipartitions

Consider two absolutely continuous probability measures in the plane. A subdivision of the plane into k ⩾ 2 regions is equitable if every region has weight 1 / k in each measure. We show that, for any two probability measures in the plane and any integer k ⩾ 2 , there exists an equitable subdivision of the plane into k regions using at most k − 1 horizontal segments and at most k − 1 vertical segments. We also prove the existence of orthogonal equipartitions for point measures and present an efficient algorithm for computing an orthogonal equipartition.

Effectively closed sets of measures and randomness

We show that if a real x ∈ 2 ω is strongly Hausdorff H h -random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ -measure of the basic open cylinders shrinks according to h . The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π 1 0 -classes applied to closed sets of probability measures. We use the main result to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem.

Splitting multidimensional necklaces

The well-known “splitting necklace theorem” of Alon [N. Alon, Splitting necklaces, Adv. Math. 63 (1987) 247–253] says that each necklace with k ⋅ a i beads of color i = 1 , … , n , can be fairly divided between k thieves by at most n ( k − 1 ) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [ 0 , 1 ] where beads of given color are interpreted as measurable sets A i ⊂ [ 0 , 1 ] (or more generally as continuous measures μ i ). We demonstrate that Alon's result is a special case of a multidimensional consensus division theorem about n continuous probability measures μ 1 , … , μ n on a d-cube [ 0 , 1 ] d . The dissection is performed by m 1 + ⋯ + m d = n ( k − 1 ) hyperplanes parallel to the sides of [ 0 , 1 ] d dividing the cube into m 1 ⋅ ⋯ ⋅ m d elementary cuboids (parallelepipeds) where the integers m i are prescribed in advance.

On classification of singular measures and fractal properties of quasi-self-affine measures in R 2

A multidimensional classification of singularly continuous (w.r.t. the Lebesgue measure) probability measures in R2 is introduced and a theorem on canonical representation of such measures is proven. A class of random elements on the unit square which is defined by a system of partitions generated by the Q*-representation of real numbers is introduced and studied in details. Conditions for the discreteness, absolute continuity resp. singular continuity (w.r.t. Lebesgue measure) of the corresponding probability measures are found. Metric, topological and fractal properties of the spectra of the corresponding probability distributions are investigated. A class of transformations preserving the Hausdorff-Besicovitch dimension of any subset of the unit square (DP-transformations) is also studied.

On the multimodality of random probability measures

Nonparametric methods for density estimation are examined here. Within a Bayesian setting the construction of an absolutely continuous random probability measure is often required for nonparametric statistical analysis. To achieve this we propose a partial convexification procedure of a process, such as the Dirichlet, resulting in a multimodal distribution function with a finite expected number of modes. In agreement with convexity theory results, it is shown that the derived random probability measure admits a density with respect to Lebesgue measure.

Multifractal analysis of singularly continuous probability measures

Multifractal analysis of singularly continuous probability measures

Multifractal Analysis of Singularly Continuous Probability Measures

We analyze correlations between different approaches to the definition of the Hausdorff dimension of singular probability measures on the basis of fractal analysis of essential supports of these measures. We introduce characteristic multifractal measures of the first and higher orders. Using these measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures.

Continuity of an optimal transport in Monge problem

Given two absolutely continuous probability measures f ± in R 2 , we consider the classical Monge transport problem, with the Euclidean distance as cost function. We prove the existence of a continuous optimal transport, under the assumptions that (the densities of) f ± are continuous and strictly positive in the interior part of their supports, and that such supports are convex, compact, and disjoint. We show through several examples that our statement is nearly optimal. Moreover, under the same hypotheses, we also obtain the continuity of the transport density.

Fractal properties of singular probability distributions with independent [formula omitted]-digits

The problem of the determination of the Hausdorff dimension of sets via the special class of coverings generated by special partitions, called Q * -partitions, of the unit interval is discussed. Fractal properties of singular continuous probability measures with independent “ Q * -symbols” are studied in details. Explicit formulae for the determination of the Hausdorff dimension of such measures as well as for the determination of the Hausdorff dimension of the topological supports of the corresponding measures are given.

Density estimation by orthogonal series in an infinite dimensional space: Application to processes of diffusion type I

In this paper, we will be concerned with the estimation of the probability density of a diffusion process with respect to the Wiener measure. Since a diffusion process can be viewed as a random variable taking its values in an infinite dimensional space, this problem can be viewed as a special case of estimating the probability density of a random variable with values in an infinite dimensional space. However, obtaining the absolute continuity of probability measures in infinite dimension is difficult. A density estimator based on a Fourier–Hermite orthogonal series is investigated, which is a generalization of the classical density estimator of a real valued random variable. We establish that our estimator converges in integrated and simple mean square, under suitable conditions. Rates of convergence are also given.

Decay of correlations in one-dimensional dynamics

We consider multimodal C 3 interval maps f satisfying a summability condition on the derivatives D n along the critical orbits which implies the existence of an absolutely continuous f-invariant probability measure μ. If f is non-renormalizable, μ is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence ( D n ) as n→∞. We also give sufficient conditions for μ to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.

On a criterion of Riemannian distance for singularity and absolute continuity of probability measures

Parametric statistical models, with suitable regularity conditions, have a natural Riemannian manifold structure given by the Fisher information metric. This paper is concerned with the Riemannian distance for geometric conditions on singularity or absolute continuity of probability measures which depend on the parameters.