About the # Function

1976 ◽  
Vol 19 (4) ◽  
pp. 455-460
Author(s):  
G. T. Klincsek
Keyword(s):  
Stochastic Process ◽  
Stopping Time ◽  
Decreasing Rearrangement ◽  
Image Position

AbstractThe use of decreasing rearrangement formulas, and particularly that of the weak N inequality, is illustrated by deriving from Eτ |f-f(τ -)|≤Eτu (where ft is some stochastic process and τ arbitrary stopping time) the estimate ||f||≤Const||u|| in the class of structureless norms with finite dual Hardy bound.The basic estimate is

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

On two integral equations of queueing theory

1967 ◽  
Vol 4 (2) ◽  
pp. 343-355 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Stochastic Process ◽  
Integral Equations ◽  
Positive Constant ◽  
Mathematical Description ◽  
Queueing Theory ◽  
Image Position

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process {vn, n = 1,2, …}, recursively defined by K is a positive constant, τ1, τ2, …; Σ1, Σ2, …; are independent, non-negative variables, with τ1, τ2,…, identically distributed, similarly, the variables Σ1, Σ2, …, are identically distributed.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (04) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t>0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

The definition of a multi-dimensional generalization of shot noise

1971 ◽  
Vol 8 (01) ◽  
pp. 128-135 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Stochastic Process ◽  
Probability Distribution ◽  
Absolute Convergence ◽  
List Type ◽  
Type Number ◽  
Homogeneous Poisson Process ◽  
Cauchy Principal ◽  
Principal Value ◽  
Definition Of ◽  
Image Position

The paper studies the formally defined stochastic process where {tj } is a homogeneous Poisson process in Euclidean n-space En and the a.e. finite Em -valued function f(·) satisfies |f(t)| = g(t) (all |t | = t), g(t) ↓ 0 for all sufficiently large t → ∞, and with either m = 1, or m = n and f(t)/g(t) =t/t. The convergence of the sum at (*) is shown to depend on (i) (ii) (iii) . Specifically, finiteness of (i) for sufficiently large X implies absolute convergence of (*) almost surely (a.s.); finiteness of (ii) and (iii) implies a.s. convergence of the Cauchy principal value of (*) with the limit of this principal value having a probability distribution independent of t when the limit in (iii) is zero; the finiteness of (ii) alone suffices for the existence of this limiting principal value at t = 0.

Differences, Derivatives, and Decreasing Rearrangements

1967 ◽  
Vol 19 ◽  
pp. 1153-1178 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Finite Sequence ◽  
Real Numbers ◽  
Decreasing Rearrangement ◽  
Image Position ◽  
Decreasing Rearrangements ◽  
The Given

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.

Research on Stock Analysis Based on Stochastic Process

2012 ◽  
Vol 433-440 ◽  
pp. 5967-5974
Author(s):  
Pei Ze Li
Keyword(s):  
Stochastic Process ◽  
Stock Prices ◽  
Stock Price ◽  
Statistical Physics ◽  
Stopping Time ◽  
Mathematical Finance ◽  
Market Condition ◽  
Price Process ◽  
Physics Model ◽  
Statistical Physics Model

In stock market, the stock prices directly reflects market condition, therefore, the research on stock price process is one of the research contents of mathematical finance. In this paper by using the election model of statistical physics model to study the stock price fluctuation . This paper first applying stochastic process theory to establish election model, then the election model and stopping time theory are applied to establish stock profit process, we get the stock price process.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (4) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτE (max0≤t≤τXt − c τ), where X = (Xt)t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | Xt = g* (max0≤t≤sXs)} where s ↦ g*(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g*(s) ∼ ((Δ − 1) / K Δ)1 / Δs1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (supt>0Xt) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

Rate of convergence for computing expectations of stopping functionals of an α-mixing process

1998 ◽  
Vol 30 (2) ◽  
pp. 425-448
Author(s):  
Mohamed Ben Alaya ◽  
Gilles Pagès
Keyword(s):  
Monte Carlo ◽  
Probability Space ◽  
Random Number ◽  
Shift Operator ◽  
Probability Distributions ◽  
Random Number Generator ◽  
Stopping Time ◽  
Mixing Process ◽  
Canonical Process ◽  
Image Position

The shift method consists in computing the expectation of an integrable functional F defined on the probability space ((ℝd)N, B(ℝd)⊗N, μ⊗N) (μ is a probability measure on ℝd) using Birkhoff's Pointwise Ergodic Theorem, i.e. as n → ∞, where θ denotes the canonical shift operator. When F lies in L2(FT, μ⊗N) for some integrable enough stopping time T, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributions P on ((ℝd)N, B(ℝd)⊗N), namely when the canonical process (Xn)n∊N is P-stationary, α-mixing and fulfils Ibragimov's assumption for some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distribution Pν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.

The behavior of the ratio of a small-noise Markov chain to its deterministic approximation

1985 ◽  
Vol 17 (4) ◽  
pp. 731-747
Author(s):  
Norman Kaplan ◽  
Thomas Darden
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Discrete Time ◽  
Small Noise ◽  
Fisher Model ◽  
Image Position

For each N≧1, let {XN(t, x), t≧0} be a discrete-time stochastic process with XN(0) = x. Let FN(y) = E(XN(t + 1) | XN(t) = y), and define YN(t, x) = FN(YN(t – 1, x)), t≧1 and YN(0, x) = x. Assume that in a neighborhood of the origin FN(y) = mNy(l + O(y)) where mN> 1, and define for δ> 0 and x> 0, υN(δ, x) = inf{t:xmtN>δ}. Conditions are given under which, for θ> 0 and ε> 0, there exist constants δ > 0 and L <∞, depending on εand 0, such that This result together with a result of Kurtz (1970), (1971) shows that, under appropriate conditions, the time needed for the stochastic process {XN(t, 1/N), t≧0} to escape a δ -neighborhood of the origin is of order log Νδ /log mN. To illustrate the results the Wright-Fisher model with selection is considered.

Stochastic processes relating to particles distributed in a continuous infinity of states

1950 ◽  
Vol 46 (4) ◽  
pp. 595-602 ◽  
Author(s):  
Alladi Ramakrishnan
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Stochastic Processes ◽  
Stochastic Variable ◽  
Stochastic Problems ◽  
Image Position ◽  
Number Of Particles

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and this distribution varies with another parameter t (which may be continuous or discrete; if t represents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If the E-space were discrete and the states represented by E1, E2, …, then it would be possible to define a functionrepresenting the probability that there are ν1 particles in E1, ν2 particles in E2, …, at t. The variation of π with t is governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variableswith the help of the π function which yields also the correlation between the stochastic variables νi.

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