Abstract
It is known since (Theory Probab. Appl. 11 (1966) 390–406) that the slow motion Xε in the time-scaled multidimensional averaging setup dXε(t) =1 B(Xε(t), ξ(t/ε2)) + b(Xε(t), ξ(t/ε2)), t ∈ [0, T ] dt ε 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 E sup0≤t≤T |Xε(t) − Ξε(t)|2M ≤ C(M)εδ 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. We obtain also a similar result for the corresponding discrete time averaging setup. 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.
Original language | English |
---|---|
Pages (from-to) | 103-147 |
Number of pages | 45 |
Journal | Annals of Applied Probability |
Volume | 34 |
Issue number | 1 A |
DOIs | |
State | Published - Feb 2024 |
Bibliographical note
Publisher Copyright:© 2024 Institute of Mathematical Statistics. All rights reserved.
Keywords
- Averaging
- diffusion approximation
- Dynkin game
- φ-mixing