By Aurélien Alfonsi
This ebook supplies an summary of affine diffusions, from Ornstein-Uhlenbeck methods to Wishart approaches and it considers a few similar diffusions resembling Wright-Fisher strategies. It makes a speciality of various simulation schemes for those tactics, specially second-order schemes for the susceptible errors. It additionally provides a few types, as a rule within the box of finance, the place those tools are proper and gives a few numerical experiments.
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Extra info for Affine Diffusions and Related Processes: Simulation, Theory and Applications
Xtx ; t 2 Œ0; T /. Wt ; t 2 Œ0; T /, and we can define it intrinsically. XOt ; t 2 Œ0; T / matters. Getting accurate schemes for the weak error is then sufficient for this use. 2 Strong Approximations The main way to construct strong discretization schemes is to use iterated stochastic Taylor expansion. To fix the ideas, let us assume for a while that the dimension d D 1 and that the coefficients b and are smooth. 5) can be seen as the approximation where Xsx is replaced by Xtxi . s ti /dW s ti ti We can see this scheme as the addition of some corrective terms to the Euler scheme.
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. m;m < C1/ D 1. x/ C p Since the map x 7! m/P. m/P. 32). 15. We consider an increasing (resp. x; C1/N (resp. 0; x/N ) such that mn !
However, it still remains many open questions. In particular, is the rate of order 2=3 " optimal or could we prove an order of convergence closer to 1? Also, is it possible to extend this result to functions F that are less regular than Lipschitz? 3 Beyond the Euler Scheme: Strong and Weak High Order Approximations The Euler-Maruyama discretization scheme is simple and easy to implement in practice. However, one may wish to use other discretization schemes to accelerate the convergence. To do so, two different approaches exist in the literature.
Affine Diffusions and Related Processes: Simulation, Theory and Applications by Aurélien Alfonsi