An important Ito process in fiance is Ornstein-Uhlenbeck process (or Mean-reverting process). In fiance, It’s called as Hull-White model to describe a stochastic interest rate.
where is a 1-dim BM, and are positive processes. (1) has an explicit form of
In this below, we are going to consider an example of system of SDE with mean field effect.
Let be a given natural number, and -dimensional process satisfies
where Existence and uniqueness of the solution is well known with Lipschitz coefficients. We are going to study the asymptotic distribution of as goes to infinity. By summing up all equations, we observe that
for some standard Brownian motion . So we can write . We rewrite the equation as
Note that both in almost surely and in . Heuristically, one can replace by constant process , and have asymptotic process satisfying
This is indeed a solution of McKean-Vlasov equation of the following type, i.e. satisfies