Download Markov processes and applications: Algorithms, networks, by Etienne Pardoux PDF

By Etienne Pardoux

ISBN-10: 0470721863

ISBN-13: 9780470721865

ISBN-10: 0470772719

ISBN-13: 9780470772713

This well-written publication offers a transparent and available therapy of the speculation of discrete and continuous-time Markov chains, with an emphasis in the direction of functions. The mathematical remedy is specific and rigorous with no superfluous info, and the implications are instantly illustrated in illuminating examples. This publication should be super precious to anyone educating a direction on Markov tactics.

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Extra resources for Markov processes and applications: Algorithms, networks, genome and finance

Example text

Xm ∈ E, P (A ∩ {Xn+1 = x1 , . . , Xn+m = xm }|Xn = x) = P(A|Xn = x)Px (X1 = x1 , . . , Xm = xm ). MARKOV CHAINS 23 Proof It suffices to prove the result in the case where A = {X0 = y0 , X1 = y1 , . . , Xn = yn } (A is a finite or countable union of disjoint sets of that form, and the result in the general case will then follow from the σ -additivity of P). It suffices to consider the case yn = x, since otherwise both sides of the equality vanish. The left-hand side of the identity in the statement equals P(X0 = y0 , .

Xn )) ≥ E (f (X1 , . . , Xn )) E (g(X1 , . . , Xn )) . 3. Let h be a mapping from [0, 1]n into R, which is monotone in each of its arguments, and let U1 , . . , Un be independent U(0, 1) random variables. Show that cov (h(U1 , . . , Un ), h(1 − U1 , . . , 1 − Un )) ≤ 0, and show that the method of antithetic random variables reduces the variance in this case. 3) for the price of a put option. Deduce from the identity x = x + − (−x)+ the put–call parity relationship C − P = Eeσ Z − K, where the expectation Eeσ Z can be computed explicitly and equals exp(σ 2 /2).

In other words, for all A ∈ Fn and any m > 0, x1 , . . , xm ∈ E, P (A ∩ {Xn+1 = x1 , . . , Xn+m = xm }|Xn = x) = P(A|Xn = x)Px (X1 = x1 , . . , Xm = xm ). MARKOV CHAINS 23 Proof It suffices to prove the result in the case where A = {X0 = y0 , X1 = y1 , . . , Xn = yn } (A is a finite or countable union of disjoint sets of that form, and the result in the general case will then follow from the σ -additivity of P). It suffices to consider the case yn = x, since otherwise both sides of the equality vanish.

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Markov processes and applications: Algorithms, networks, genome and finance by Etienne Pardoux


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