By Aurélien Alfonsi
This booklet offers an summary of affine diffusions, from Ornstein-Uhlenbeck procedures to Wishart techniques and it considers a few similar diffusions akin to Wright-Fisher techniques. It makes a speciality of assorted simulation schemes for those tactics, specifically second-order schemes for the vulnerable errors. It additionally offers a few types, usually within the box of finance, the place those equipment are proper and offers a few numerical experiments.
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Additional info for Affine Diffusions and Related Processes: Simulation, Theory and Applications
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 . ˛1 ; : : : ; ˛d / 2 Nd , we denote by @˛ D @˛1 1 : : : @˛dd the operator that differentiates ˛i times with respect P to xi , and we set j˛j D dlD1 ˛l .
0 e ks=2 dM s ; t 0/ is a Brownian motion and conclude. 3. 9), check the (simpler) identity law e kt XQ x D Xtx on the marginal laws. 4 The Feller Condition For m 0, we introduce the stopping time m D infft 0; Xtx D mg with the 0 standard convention inf ; D C1. For m; m 0, we set m;m0 D min. m ; m0 /. The goal of this subsection is to study 0 and determine under which conditions the CIR never reaches zero. 1 that the process X x is nonnegative and we want to get necessary and sufficient conditions under which it is positive.
If, and only if 0 D infft 2a When 2 > 2a, we have 0 2 0; Xtx D 0g with inf ; D C1. s. 28) 0. 28) is well known as the Feller condition in the literature. 29, p. 0; C1/g. Here, we give a direct proof of this result for the CIR case. x/ D 0. The function s is increasing. 31) We have the following classical result. 5) and consider m; m such that 0 < m < x < m < 1: Then, we have P. Xtx^ m;m / is a bounded martingale and thus converges almost surely when t ! g. 15, p. 17 in ). x//2 < C1; 22 1 Real Valued Affine Diffusions and thus P.
Affine Diffusions and Related Processes: Simulation, Theory and Applications by Aurélien Alfonsi