Markov Processes and Applications: Algorithms, Networks, by Etienne Pardoux

By Etienne Pardoux

"This well-written publication presents a transparent and available remedy of the speculation of discrete and continuous-time Markov chains, with an emphasis in the direction of purposes. The mathematical remedy is distinct and rigorous with no superfluous information, and the consequences are instantly illustrated in illuminating examples. This ebook might be super necessary to anyone educating a path on Markov processes."
Jean-François Le Gall, Professor at Université de Paris-Orsay, France

Markov techniques is the category of stochastic techniques whose previous and destiny are conditionally autonomous, given their current nation. They represent very important versions in lots of utilized fields.

After an advent to the Monte Carlo procedure, this ebook describes discrete time Markov chains, the Poisson strategy and non-stop time Markov chains. It additionally offers a number of purposes together with Markov Chain Monte Carlo, Simulated Annealing, Hidden Markov types, Annotation and Alignment of Genomic sequences, regulate and Filtering, Phylogenetic tree reconstruction and Queuing networks. The final bankruptcy is an creation to stochastic calculus and mathematical finance.

Features include:

  • The Monte Carlo procedure, discrete time Markov chains, the Poisson strategy and non-stop time leap Markov processes.
  • An advent to diffusion tactics, mathematical finance and stochastic calculus.
  • Applications of Markov procedures to numerous fields, starting from mathematical biology, to monetary engineering and desktop science.
  • Numerous workouts and issues of suggestions to so much of them

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Additional resources for Markov Processes and Applications: Algorithms, Networks, Genome and Finance

Sample text

F is a collection of real numbers indexed by the d elements of the set E, in other words an element of Rd ), denoting g = −1/2 f , we have P˜ f 2 = (P −1/2 f )2x = P g 2 π ≤ g 2 π = f 2 . x∈E First, note that f is an eigenvector of P˜ if and only if g = −1/2 f is a right eigenvector of P , and g = 1/2 f is a left eigenvector of P associated with the same eigenvalue. We have that P˜ is a symmetric d × d matrix, whose norm is bounded by 1. Hence, from elementary results in linear algebra, P˜ admits the eigenvalues −1 ≤ λd ≤ λd−1 ≤ λ2 ≤ λ1 ≤ 1.

Show that, for all x ∈ T , hx = 1 + Pxy hy . y∈E Deduce the values of hx , x ∈ T . 9 Given 0 < p < 1, we consider an E = {1, 2, 3, 4}-valued Markov chain {Xn ; n ∈ N} with transition matrix P given by   p 1−p 0 0 0 0 p 1 − p . P = p 1 − p 0 0  0 0 p 1−p 1. Show that the chain {Xn } is irreducible and recurrent. 2. Compute its unique invariant probability π . 3. Show that the chain is aperiodic. Deduce that P n tends, as n → ∞, towards the matrix   π1 π2 π3 π4 π1 π2 π3 π4    π1 π2 π3 π4  .

11 in the symmetric case (p = 1/2). 11. Suppose for simplicity that X0 = x ∈ Z. For all a, b ∈ Z with a < x < b, let Ta,b = inf{n ≥ 0; Xn ∈]a, b[}, Ta = inf{n ≥ 0; Xn = a}, Tb = inf{n ≥ 0; Xn = b}. We note that n Xn∧Ta,b = x + Yk 1{Ta,b > k−1} . k=1 1. Show that the random variables Yk and 1{Ta,b > k−1} are independent. Deduce that EXn∧Ta,b = x. 2. Show that |Xn∧Ta,b | ≤ sup(|a|, |b|), Ta,b < ∞ almost surely, and EXTa,b = x. 3. Establish the identities P(XTa,b = a) = b−x , b−a P(XTa,b = b) = x−a .

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