2020/11/16 by Yuri Kifer, Kifer, Yuri
Economics, Econometrics and Finance · Mathematics · #34C29 #60F15 #60G40 #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2011.07907
openalex publication_date 2020/11/16 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/28
It is known that the slow motion Xε in the time-scaled multidimensional averaging setup \frac dXε(t)dt=\frac 1ε B(Xε(t), ξ(t/ε2))+b(Xε(t), ξ(t/\ve2)), t∈ [0,T] converges weakly as ε→ 0 to a diffusion process provided EB(x,ξ(s))≡ 0 where ξ is a sufficiently fast mixing stochastic process. In this paper we show that both Xε and a family of diffusions Ξε can be redefined on a common sufficiently rich probability space so that Esup0≤ t≤ T|Xε(t)-Ξε(t)|2M≤ C(M)ε^\del for some C(M),δ>0 and all M≥ 1, ε>0, where all Ξε, ε>0 have the same diffusion coefficients but underlying Brownian motions may change with ε. This is the first strong approximation result both in the above setup and at all when the limit is a nontrivial multidimensional diffusion. We obtain also a similar result for the corresponding discrete time averaging setup which was not considered before at all. As an application we consider Dynkin's games with path dependent payoffs involving a diffusion and obtain error estimates for computation of values of such games by means of such discrete time approximations which provides a more effective computational tool than the standard discretization of the diffusion itself.