By Aurélien Alfonsi
This ebook supplies an summary of affine diffusions, from Ornstein-Uhlenbeck techniques to Wishart procedures and it considers a few similar diffusions comparable to Wright-Fisher techniques. It specializes in diversified simulation schemes for those procedures, specifically second-order schemes for the susceptible blunders. It additionally provides a few versions, often within the box of finance, the place those tools are proper and gives a few numerical experiments.
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Extra resources for Affine Diffusions and Related Processes: Simulation, Theory and Applications
First, we observe that the order of the weak convergence is one. It is then strictly better than the order of the strong convergence which is equal to one half. 1=n/. n 1 /. XTx / at a rate of order 2. 2n/). This approach has been studied in detail by Kebaier  to determine the optimal number of samples to generate for each time step. A generalization of this approach called the Multilevel Monte-Carlo algorithm has been proposed by Giles . It involves more than two different time steps.
E t 1/ 2 . T t / 2 k C . T kC u cQ 2u e t ct;T 2 / u 2 ct;T Â kC t/ 2u ! e t 1/ ; for the second equality. 12, we get that ct;T rt follows under PT a chi-square distribution with 2kÂ2 degrees of freedom and noncentrality dt;T r0 . xI ; d / the cumulative distribution function of a chi-square distribution with > 0 degrees of freedom and noncentrality d 0. r0 ; T1 / 2 Â cT0 ;T1 r ? r0 ; T0 / 2 T0 /I 2kÂ Â cT0 ;T0 r ? T1 2 Ã ; dT0 ;T1 r0 T0 /I 2kÂ 2 Ã ; dT0 ;T0 r0 : Of course, a similar formula holds for puts on zero coupon bond and we thus get explicit formulas for any floorlet or caplet within the CIR model.
Dz/; i for any bounded measurable function f W Rd ! R. XO ti ; 0 Ä i Ä n/ is thus entirely determined by its initial value and its transition probabilities. dz/ or even the random variable XO tx . x/> . We now present a framework to analyze the weak error that has been proposed in Alfonsi  and is convenient for affine diffusions. 1 The Weak Error Analysis We first introduce some notations. 1) takes its values, is a subset of Rd . For 1 Ä i Ä d , @i is the partial differential operator with respect to the i -th coordinate xi .