Simulation of Stochastic Processes 4.1 Stochastic processes

Universitext, 1995

Journal of Statistical Computation and Simulation, 2013

Between the first undergraduate course in probability and the first graduate course that uses measure theory, there are a number of courses that teach Stochastic Processes to students with many different interests and with varying degrees of mathematical sophistication. To allow readers (and instructors) to choose their own level of detail, many of the proofs begin with a nonrigorous answer to the question "Why is this true?" followed by a Proof that fills in the missing details. As it is possible to drive a car without knowing about the working of the internal combustion engine, it is also possible to apply the theory of Markov chains without knowing the details of the proofs. It is my personal philosophy that probability theory was developed to solve problems, so most of our effort will be spent on analyzing examples. Readers who want to master the subject will have to do more than a few of the twenty dozen carefully chosen exercises.

UNITEXT for Physics

For simplicity again, in this chapter we will only consider processes with just one component. We already remarked in the Sect. 6.3 that a white noise is a singular process whose main properties can be traced back to the non differentiability of some processes. As a first example we have shown indeed that the Poisson impulse process (6.63) and its associated compensated version (6.65) are white noises entailed by the formal derivation respectively of a simple Poisson process N (t) and of its compensated variant N (t). In the same vein we have shown then in the Example 6.22 that also the formal derivative of the Wiener process W (t)-not differentiable according to the Proposition 6.18-meets the conditions (6.69) to be a white noise, and in the Appendix H we also hinted that the role of the fluctuating force B(t) in the Langevin equation (6.78) for the Brownian motion is actually played by such a white noiseẆ (t). We can now give a mathematically more cogent justification for this identification in the framework of the Markovian diffusions. The Langevin equation (6.78) is a particular case of the more general equatioṅ X (t) = a(X (t), t) + b(X (t), t) Z (t) (8.1) where a(x, t) and b(x, t) are given functions and Z (t) is a process with E [Z (t)] = 0 and uncorrelated with X (t). From a formal integration of (8.1) we find X (t) = X (t 0) + t t 0 a(X (s), s) ds + t t 0 b(X (s), s)Z (s) ds so that, being X (t) assembled as a combination of Z (s) values with t 0 < s < t, to secure the non correlation of X (t) and Z (t) we should intuitively require also the non correlation of Z (s) and Z (t) for every pair s = t. Since moreover Z (t) is presumed

Teaching problems on Statistics of Stochastic Processes are given, with theoretical grounds.

1998

functions (cf. Révész (1990), pp. 33–34). Definition 1.1. The function ψ belongs to the upper-upper class of {V (t); t ≥ 0} (ψ(t) ∈ UUC(V (t))) if for almost all ω ∈ Ω there exists a t0 = t0(ω) &gt; 0 such that V (t) &lt; ψ(t) for all t &gt; t0. Definition 1.2. The function ψ belongs to the upper-lower class of {V (t); t ≥ 0} (ψ(t) ∈ ULC(V (t))) if for almost all ω ∈ Ω there exists a sequence of positive numbers 0 &lt; t1 = t1(ω) &lt; t2 = t2(ω) &lt; · · · with tn → ∞ such that V (ti) ≥ ψ(ti), i = 1, 2, . . . . Definition 1.3. The function ψ belongs to the lower-upper class of {V (t); t ≥ 0} (ψ(t) ∈ LUC(V (t))) if for almost all ω ∈ Ω there exists a sequence of positive numbers 0 &lt; t1 = t1(ω) &lt; t2 = t2(ω) &lt; · · · with tn → ∞ such that V (ti) ≤ ψ(ti), i = 1, 2, . . . . Definition 1.4. The function ψ belongs to the lower-lower class of {V (t); t ≥ 0} (ψ(t) ∈ LLC(V (t))) if for almost all ω ∈ Ω there exists a t0 = t0(ω) &gt; 0 such that V (t) &gt; ψ(t) for all t &gt; t0. Assum...

International Encyclopedia of Statistical Science, 2011

Abril JC () Asymptotic expansions for time series problems with applications to moving average models. PhD Thesis, The

Random Operators and Stochastic Equations, 2003

Stochastic Processes and their Applications, 1998

Research Triangle Park, VC 27709-2211 y ol '9. 6 mA4s.DJ I SUPP NTART NOT95 The view, opinions and/or f Indings contained In this report are those of the author(s) and should not be construed as an offici1al Deprm:t of the Army poition, plicy or decision uMless so byd e We invustiate when sampling a stochastic process X = VW :) t 2: 0) at the time of an independent point procin, 0, eads to the same empirical distribution as the time average limiting distribution of X. Two case awe cmnsred. The first is when X is an asymptoticaly stationary ergodic process and %0 satisfis a mixin and coupling candition Ln this case, the entire limiting distributions in function swce are shown to be the sain The second am is when X Is only assumed to have a contaat fiite time averag and* is assumed a poaltive recurrent renewal proces with a spreed-out cycie length dist ribhation. In this non-stgodic cane, the averages are shown to be the same when some futhber conditions are pieced on X and 0. ILn WAMMM OF PAGIS 10 Time average, event average hidependent sampfing, asymptoically argodic. IL m COrN OFMIATO UPON

2011

We analyze here different types of fractional differential equations, under the assumption that their fractional order ν ∈ (0, 1] is random with probability density n(ν). We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process N(t), t > 0. We prove that, for a particular (discrete) choice of n(ν), it leads to a process with random time, defined as N( T ν 1, ν 2 (t)), t > 0. The distribution of the random time argument T ν 1, ν 2 (t) can be expressed, for any fixed t, in terms of convolutions of stable-laws. The new process N( T ν 1, ν 2 ) is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of N( T ν 1, ν 2 ), as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see ). In view of the previous results it is natural to consider diffusion-type fractional equations of distributed order. We present here an approach to their solutions in terms of composition of the Brownian motion B(t), t > 0 with the random time T ν 1, ν 2 . We thus provide an alternative to the constructions presented in Mainardi and Pagnini and in Chechkin et al. [6], at least in the double-order case.