The Tail Behavior Of ARCH Model

In this thesis,We study the tail of the stationary distribution of an ARCH(l) model ,the ARCH(1) model is:where α₀> 0 and α₁ > 0. In the article of Engle(1982) ,He defined the ARCH(l) model and supposed (ε_n)_n∈N are i.i.d normal random variables , but in our thesis , we define a stochastic equation:and consider the (ε_n)_n∈N are i.i.d symmetric random variables.As to above model,we introduce the required assumptions on the (ε_n)_n∈N and distinguish between the so-called general conditions and the technical conditions.the following,we proved that (X_n)_n∈N has a unique stationary distribution and the stationary distribution is continuous and symmetric.In the end,We consider the stationary distribution has a Pareto-like tail and the tail index depends on A and the distribution of the (ε_n)_n∈N,the following is its elementary resultsIn (X_n)_n∈N , Let F(x) := P(X > x), x > 0, is the right tail of the stationary diatribution function.Thenwhereand κ is given as the unique positive solution to...