By Aurélien Alfonsi
This ebook provides an summary of affine diffusions, from Ornstein-Uhlenbeck tactics to Wishart strategies and it considers a few comparable diffusions similar to Wright-Fisher procedures. It makes a speciality of varied simulation schemes for those approaches, specially second-order schemes for the susceptible blunders. It additionally provides a few types, 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
Xti ; 0 Ä i Ä n/. Of course, there are many different possible criteria to quantify these distances. We will mainly use the two following ones. XOti ; 0 Ä i Ä n/ for the process X is said to have a strong error of order > 0 if Ä 9C > 0; 8n 2 N ; E max kXOti 0Äi Än It has a weak error of order support, X ti k Ä C : n > 0 if for any C 1 function f W Rd ! XT /j Ä C : n Since a C 1 function with compact support is Lipschitz, we observe that the weak order of convergence is equal or higher than the strong order of convergence.
Xt ; t 2 Œ0; T /2 < 1, the Central p Limit Theorem gives a confidence interval whose size is proportional to 1= K. Thus, the Monte-Carlo method motivates the need to simulate the process X . Of course, it is not possible in practice to generate full continuous paths. At best, we can only generate the process for a finite number of times. For a time horizon T > 0 and n 2 N , we consider then the regular time grid ti D iTn , 0 Ä i Ä n. To implement in practice the Monte-Carlo method, we have to make two further approximations.
Of course, it is not possible in practice to generate full continuous paths. At best, we can only generate the process for a finite number of times. For a time horizon T > 0 and n 2 N , we consider then the regular time grid ti D iTn , 0 Ä i Ä n. To implement in practice the Monte-Carlo method, we have to make two further approximations. Xti ; 0 Ä i Ä n/. ti /; 0 Ä i Ä n/. Xti ; 0 Ä i Ä n/: K kD1 Let XOti D XOt1i for 0 Ä i Ä n. T // and eliminate the approximation error of F by FO . Xtki ; 0 Ä i Ä n/ of the process, the second term of the error disappears.