Sample paths of demimartingales

1984 ◽  
pp. 365-373 ◽  
Author(s):  
Thomas E. Wood
Keyword(s):  
Sample Paths

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

scholarly journals SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1074
Author(s):  
Vincent Wagner ◽  
Nicole Erika Radde
Keyword(s):  
Reaction System ◽  
Stochastic Simulation Algorithm ◽  
Test Bed ◽  
Biochemical Reaction Networks ◽  
Full System ◽  
Catalytic Systems ◽  
Reaction Systems ◽  
Sample Paths ◽  
Alternative Description ◽  
Linear Differential

The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution over all configurations. The Stochastic Simulation Algorithm (SSA) is a method to simulate sample paths from this stochastic process. Both approaches are only applicable for small systems, characterized by few reactions and small numbers of molecules. For larger systems, the CME is computationally intractable due to a large number of possible configurations, and the SSA suffers from large reaction propensities. In our study, we focus on catalytic reaction systems, in which substrates are converted by catalytic molecules. We present an alternative description of these systems, called SiCaSMA, in which the full system is subdivided into smaller subsystems with one catalyst molecule each. These single catalyst subsystems can be analyzed individually, and their solutions are concatenated to give the solution of the full system. We show the validity of our approach by applying it to two test-bed reaction systems, a reversible switch of a molecule and methyltransferase-mediated DNA methylation.

A multi-level splitting algorithm based on differential evolution

2018 ◽  
Vol 09 (02) ◽  
pp. 1850021
Author(s):  
Liping Wang ◽  
Wenhui Fan
Keyword(s):  
Differential Evolution ◽  
Rare Event ◽  
Sampling Strategy ◽  
Splitting Algorithm ◽  
Algorithm Efficiency ◽  
Event Simulation ◽  
Sample Paths ◽  
Level Splitting ◽  
Key Factor ◽  
Multi Level

Multi-level splitting algorithm is a famous rare event simulation (RES) method which reaches rare set through splitting samples during simulation. Since choosing sample paths is a key factor of the method, this paper embeds differential evolution in multi-level splitting mechanism to improve the sampling strategy and precision, so as to improve the algorithm efficiency. Examples of rare event probability estimation demonstrate that the new proposed algorithm performs well in convergence rate and precision for a set of benchmark cases.

Diffusion Processes and Their Sample Paths.

1996 ◽  
Vol 91 (436) ◽  
pp. 1754
Author(s):  
PALE ◽  
Kiyosi Ito ◽  
Henry P. McKean
Keyword(s):  
Diffusion Processes ◽  
Sample Paths

Genetic Drift in an Infinite Population: The Pseudohitchhiking Model

Genetics ◽  
2000 ◽  
Vol 155 (2) ◽  
pp. 909-919 ◽  
Author(s):  
John H Gillespie
Keyword(s):  
Genetic Drift ◽  
Stochastic Dynamics ◽  
Diffusion Theory ◽  
Stochastic Effects ◽  
Stochastic Force ◽  
Infinite Population ◽  
Sample Paths ◽  
Neutral Locus ◽  
Linked Selection ◽  
Number Of Branches

Abstract Selected substitutions at one locus can induce stochastic dynamics that resemble genetic drift at a closely linked neutral locus. The pseudohitchhiking model is a one-locus model that approximates these effects and can be used to describe the major consequences of linked selection. As the changes in neutral allele frequencies when hitchhiking are rapid, diffusion theory is not appropriate for studying neutral dynamics. A stationary distribution and some results on substitution processes are presented that use the theory of continuous-time Markov processes with discontinuous sample paths. The coalescent of the pseudohitchhiking model is shown to have a random number of branches at each node, which leads to a frequency spectrum that is different from that of the equilibrium neutral model. If genetic draft, the name given to these induced stochastic effects, is a more important stochastic force than genetic drift, then a number of paradoxes that have plagued population genetics disappear.

scholarly journals p-variation of Ornstein–Uhlenbeck type processes

2007 ◽  
Vol 47 ◽  
Author(s):  
Martynas Manstavičius
Keyword(s):  
Total Variation ◽  
Lévy Process ◽  
Levy Process ◽  
Sample Paths ◽  
Variation Index

The p-variation of sample paths of Ornstein–Uhlenbeck type processes is investigated. It is shown that the p-variation index of such a process is the same as the p-variation index of the driving Lévy process, provided this process is of unbounded total variation.

scholarly journals Lattice Paths, Sampling Without Replacement, and Limiting Distributions

10.37236/156 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer ◽  
H. Prodinger
Keyword(s):  
Generating Functions ◽  
Probability Distributions ◽  
Random Variable ◽  
Phase Changes ◽  
Lattice Paths ◽  
Sampling Without Replacement ◽  
Sample Paths ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Weighted Lattice Paths

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.

PATH PROPERTIES OF THE LINEAR MULTIFRACTIONAL STABLE MOTION

Fractals ◽  
2005 ◽  
Vol 13 (02) ◽  
pp. 157-178 ◽  
Author(s):  
STILIAN STOEV ◽  
MURAD S. TAQQU
Keyword(s):  
Heavy Tails ◽  
Hölder Regularity ◽  
Self Similarity ◽  
Similarity Parameter ◽  
Stable Motion ◽  
Multifractional Brownian Motion ◽  
Sample Paths ◽  
Hölder Exponents ◽  
Fractional Stable Motion ◽  
Holder Regularity

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t). We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Hölder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Hölder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Hölder exponents. The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Hölder regularity of their paths.

scholarly journals A Simulation Approach to Statistical Estimation of Multiperiod Optimal Portfolios

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Hiroshi Shiraishi
Keyword(s):  
Portfolio Choice ◽  
Value Function ◽  
Sample Path ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Time Dependency ◽  
Portfolio Rebalancing ◽  
Sample Paths ◽  
Optimal Investment Strategy ◽  
The Value Function

This paper discusses a simulation-based method for solving discrete-time multiperiod portfolio choice problems under AR(1) process. The method is applicable even if the distributions of return processes are unknown. We first generate simulation sample paths of the random returns by using AR bootstrap. Then, for each sample path and each investment time, we obtain an optimal portfolio estimator, which optimizes a constant relative risk aversion (CRRA) utility function. When an investor considers an optimal investment strategy with portfolio rebalancing, it is convenient to introduce a value function. The most important difference between single-period portfolio choice problems and multiperiod ones is that the value function is time dependent. Our method takes care of the time dependency by using bootstrapped sample paths. Numerical studies are provided to examine the validity of our method. The result shows the necessity to take care of the time dependency of the value function.

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