10,262 research outputs found
Some integral inequalities for functions with (n−1)st derivatives of bounded variation
AbstractIn this paper, we generalize Cerone’s results, and a unified treatment of error estimates for a general inequality satisfying f(n−1) being of bounded variation is presented. We derive the estimates for the remainder terms of the mid-point, trapezoid, and Simpson formulas. All constants of the errors are sharp. Applications in numerical integration are also given
The filtering equations revisited
The problem of nonlinear filtering has engendered a surprising number of
mathematical techniques for its treatment. A notable example is the
change-of--probability-measure method originally introduced by Kallianpur and
Striebel to derive the filtering equations and the Bayes-like formula that
bears their names. More recent work, however, has generally preferred other
methods. In this paper, we reconsider the change-of-measure approach to the
derivation of the filtering equations and show that many of the technical
conditions present in previous work can be relaxed. The filtering equations are
established for general Markov signal processes that can be described by a
martingale-problem formulation. Two specific applications are treated
Almost sure optimal hedging strategy
In this work, we study the optimal discretization error of stochastic
integrals, in the context of the hedging error in a multidimensional It\^{o}
model when the discrete rebalancing dates are stopping times. We investigate
the convergence, in an almost sure sense, of the renormalized quadratic
variation of the hedging error, for which we exhibit an asymptotic lower bound
for a large class of stopping time strategies. Moreover, we make explicit a
strategy which asymptotically attains this lower bound a.s. Remarkably, the
results hold under great generality on the payoff and the model. Our analysis
relies on new results enabling us to control a.s. processes, stochastic
integrals and related increments.Comment: Published in at http://dx.doi.org/10.1214/13-AAP959 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Lattice point problems and distribution of values of quadratic forms
For d-dimensional irrational ellipsoids E with d >= 9 we show that the number
of lattice points in rE is approximated by the volume of rE, as r tends to
infinity, up to an error of order o(r^{d-2}). The estimate refines an earlier
authors' bound of order O(r^{d-2}) which holds for arbitrary ellipsoids, and is
optimal for rational ellipsoids. As an application we prove a conjecture of
Davenport and Lewis that the gaps between successive values, say s<n(s), s,n(s)
in Q[Z^d], of a positive definite irrational quadratic form Q[x], x in R^d, are
shrinking, i.e., that n(s) - s -> 0 as s -> \infty, for d >= 9. For comparison
note that sup_s (n(s)-s) 0, for rational Q[x] and
d>= 5. As a corollary we derive Oppenheim's conjecture for indefinite
irrational quadratic forms, i.e., the set Q[Z^d] is dense in R, for d >= 9,
which was proved for d >= 3 by G. Margulis in 1986 using other methods.
Finally, we provide explicit bounds for errors in terms of certain
characteristics of trigonometric sums.Comment: 51 pages, published versio
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