Approximations to permutation and exchangeable processes

We obtain weighted approximations by a Brownian bridge to permutation and exchangeable processes and to appropriately defined inverse processes. Our results provide as special cases useful weighted approximations to the uniform empirical and quantile processes and to generalized bootstrapped versions of these processes. A number of other applications are discussed. Our approach is based on the Skorokhod embedding for martingales.     

On weighted inequalities for martingale transform operators

For 1 < p < ∞, the almost surely finiteness of is a necessary and sufficient condition in order to have almost surely convergence of the sequences {E(f|?_n)} with f ∈ L^p(v dP). This condition is also equivalent to have weighted inequalities from L^p(v dP) into L^p(u dP) for some weight u for Doob's maximal function, square function and generalized Burkholder martingale transforms. Similarly, E(u|?₁) < ∞ turns out to be necessary and sufficient for the above weighted inequalities to hold for some v.     

Time-space polynomial martingales cenerated by a discrete-time martingale

We investigate, for a given martingaleM={M _n: n0}, the conditions for the existence of polynomialsP(·,·) of two variables, time and space, and of arbitrary degree in the latter, such that{P(n, M _n)} is a martingale for the natural filtration ofM. Denoting by the vector space of all such polynomials, we ask, in particular, when such a sequence can be chosen so as to span . A complete necessary and sufficient condition is obtained in the case whenM has independent increments. For generalM, we obtain a necessary condition which entails, under mild additional hypotheses, thatM is necessarily Markovian. Considering a slightly more general class of polynomials than we obtain necessary and sufficient conditions in the case of general martingales also. It is moreover observed that in most of the cases, the set determines the law of the martingale in a certain sense.The research of this author was supported by the National Board of Higher Mathematics, Bombay, India.     

Bounds for tail probabilities of martingales using skewness and kurtosis

Let M _n = X ₁ + ⋯ + X _n be a sum of independent random variables such that X _k ⩽ 1, and EX _k ² = σ _k ² for all k. Hoeffding [15, Theorem 3] proved that

Doob's Decomposition Theorem for Fuzzy (Super) Submartingales

Abstract

The notions of fuzzy random variables and fuzzy (super) submartingales are introduced. In this paper we provide the necessary and sufficient conditions of Doob's decomposition for fuzzy (super) submartingales. Finally, we discuss the decomposition of fuzzy (super) submartingales on R, and an example is given which explains that not every fuzzy (super) submartingale has Doob's decomposition.     

Convex bounds on multiplicative processes, with applications to pricing in incomplete markets

Extremal distributions have been extensively used in the actuarial literature in order to derive bounds on functionals of the underlying risks, such as stop-loss premiums or ruin probabilities, for instance. In this paper, the idea is extended to a dynamic setting. Specifically, convex bounds on multiplicative processes are derived. Despite their relative simplicity, the extremal processes are shown to produce reasonably accurate bounds on option prices in the classical trinomial model for incomplete markets.     

Stochastic integration for operator valued processes on hilbert spaces and on nuclear spaces

The representation of a nuclear space valued square integrable martingale by means of another nuclear space valued square integrable martingale is given in terms of stochastic inegrals of operator valued processes. The construction of the stochastic integral goes through that of operator valued processes on Hilbert spaces. A new approach is given for the Hilbertian case, so that only the integration of Hilbert-Schmidt operator valued processes is needed to represent square integrable martingales     

The Method of Upper and Lower Solutions of Stochastic Differential Equations and Applications

Abstract

The purpose of this article is to consider a stochastic integral equation driven by semimartingale with discontinuous and increasing drift part. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered topological space. Finally we present some applications in finance.