Asymptotic pricing in large financial markets

The problem of hedging and pricing sequences of contingent claims in large financial markets is studied. Connection between asymptotic arbitrage and behavior of the α-quantile price is shown. The large Black–Scholes model is carefully examined.      

Optimal martingale measures for defaultable assets

We model a defaultable asset as solution to a stochastic differential equation driven by both a Brownian motion and the counting process martingale associated to the one-jump process. We discuss in this framework the minimal entropy martingale measure as well as the linear Esscher and the minimal martingale measure. In particular we deal with some rather delicate verification issues.     

On martingale tail sums for the path length in random trees

For a martingale (X_n) converging almost surely to a random variable X, the sequence (X_n– X) is called martingale tail sum. Recently, Neininger (Random Structures Algorithms 46 (2015), 346–361) proved a central limit theorem for the martingale tail sum of Régnier's martingale for the path length in random binary search trees. Grübel and Kabluchko (in press) gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane‐oriented recursive trees. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 493–508, 2017     

A theorem on martingale selection for relatively open convex set-valued random sequences

For set-valued random sequences (G _n) _n=0 ^N with relatively open convex values G _n(ω), we prove a new test for the existence of a sequence (x _n) _n=0 ^N of selectors adapted to the filtration and admitting an equivalent martingale measure. The statement is formulated in terms of the supports of regular upper conditional distributions of G _n. This is a strengthening of the main result proved in our previous paper [1], where the openness of the set G _n(ω) was assumed and a possible weakening of this condition was discussed.     

Minimal-Variance Hedging in Large Financial Markets: Random Fields Approach

We study a large financial market where the discounted asset prices are modeled by martingale random fields. This approach allows the treatment of both the cases of a market with a countable amount of assets and of a market with a continuum amount. We discuss conditions for these markets to be complete and we study the minimal variance hedging problem both in the case of full and partial information. An explicit representation of the minimal variance hedging portfolio is suggested. Techniques of stochastic differentiation are applied to achieve the main results. Examples of large market models with a countable number of assets are considered according to the literature and an example of market model with a continuum of assets is taken from the bond market.     

Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces

For a blockwise martingale difference sequence of random elements {Vn , n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form limn→∞∑ n i=1 Vi /gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.     

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

On the negative value of information in informationally inefficient markets: Calculations for large number of traders

In informationally inefficient markets, classical decision theory assumes the value of information to be positive. Recent developments, however, contradict this paradigm. Schredelseker [Schredelseker, K., 2001. Is the usefulness approach useful? Some reflections on the utility of public information. In: McLeay, S., Riccaboni, A. (Eds.), Contemporary Issues in Accounting Regulation, Kluwer Academic Publishers, Boston, pp. 135–153] proposed a simulation model wherein a single security is traded among non-cooperating and asymetrically informed traders. One of the main results was the fact that badly informed traders could expect higher returns than traders with more information. But Schredelseker was able to give exact results for a small number of traders only. The aim of this paper is to give reliable results for a sufficiently large number of traders for both the expected gain and the probability of gain larger than zero. We are using combinatorial methods in order to get exact results for badly informed traders and simulation techniques for results of traders with higher level of information. The exact results are used (error between exact results and simulation results for the first traders) to determine the number of samples which have to be drawn with the simulation algorithm.     

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable,dependent claim occurrences

This paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite‐time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete‐time compound binomial risk model, for which claim occurrences are independent and identically distributed.     

Stability criteria in terms of two measures for functional differential equation with variable‐time impulses

This paper focuses on the stability in terms of two measures for functional differential equation with variable‐time impulses. Being different from most of existing literatures, the impulses of functional differential equation are assumed to be closely associated to the current state. We propose a new comparison principle for the considered systems and establish several stability criteria in terms of two measures. Also, the theoretical results are applied in a class of delayed neural network systems with variable‐tine impulses, and numerical simulations are introduced to illustrate the effectiveness of our results. Copyright © 2014 John Wiley & Sons, Ltd.     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .     

Central limit theorem and weak law of large numbers with rates for martingales in Banach spaces

This paper is concerned with large-$\text{O}$ error estimates concerning convergence in distribution as well as norm convergence for Banach space-valued martingale difference sequences. Indeed, two general limit theorems equipped with rates of convergence for such difference sequences are established. Applications of these lead to the central limit theorem and the weak law of large numbers with rates for Banach space-valued martingales.