1,721,020 research outputs found
Recurrence of transformations with absolutely continuous invariant measures
No full textLet X = [0, 1]. If an ergodic transformation T : X-->X preserves an absolutely continuous probability measure rho(x) dx with rho(x) > 0, then it is shown that for almost every x is an element of X, [GRAPHICS] Define the kth first return time R-k(x) = min sless than or equal to1:\T(s)x-x\ less than or equal to 1/2(k)} and the kth recurrence error by epsilon(k) (x) = \T(R k (x))x-x\. Then it is shown that [GRAPHICS] (C) 2002 Elsevier Science Inc. All rights reserved
Spectral types of skewed Bernoulli shift
No full textFor the transformation T : x bar right arrow kx (mod 1) for k greater than or equal to 2, it is proved that a real-valued function f(x) of modulus 1 is not a multiplicative coboundary if the discontinuities 0 < x(1) < ... < x(n) less than or equal to 1 of f(x) are k-adic points and x(1) greater than or equal to 1/k. It is also proved that the weakly mixing skew product transformations arising from Bernoulli shifts have Lebesgue spectrum
Limit properties of continuous self-exciting processes
No full textWe introduce a self-exciting continuous process based on Brownian motion, and derive its limit properties. We find conditions when the limit behaviors of the given process and its associated Hawkes process agree. The Kolmogorov-Smirnov test was applied to check the statistical similarity of the two processes.
Recurrence speed of multiples of an irrational number
No full textLet 0 < <theta> < 1 be irrational and T(<theta>)x = x + theta mod 1 on (0, 1). Consider the partition Q(n) = {((i-1)/2(n), i/2(n)) : 1 less than or equal to i less than or equal to 2(n)} and let Q(n)(x) denote the interval in Q(n) containing x. Define two versions of the first return time: J(n)(x) = min{j greater than or equal to 1 : parallel tox - T(theta)(j)x parallel to = parallel toj . theta parallel to < 1/2(n)} where <parallel>t parallel to = min(n is an element ofZ) vertical bart -n vertical bar, and K-n(x) = min{j greater than or equal to 1 : T-theta(j) x is an element of Q(n)(x)}. We show that log J(n)/n --> 1 and log K-n(x)/n --> 1 a.e. as n --> infinity for a.e. theta
Pricing of American lookback spread options
No full textWe find the closed form formula for the price of the perpetual American lookback spread option, whose payoff is the difference of the running maximum and minimum prices of a single asset. We solve an optimal stopping problem related to both maximum and minimum. We show that the spread option is equivalent to some fixed strike options on some domains, find the exact form of the optimal stopping region, and obtain the solution of the resulting partial differential equations. The value function is not differentiable. However, we prove the verification theorem due to the monotonicity of the maximum and minimum processes.
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