2024/01/10 by Yuri Kifer, Kifer, Yuri
Economics, Econometrics and Finance · Mathematics · #34C29 #60F15 #60L20 #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2401.05038
openalex publication_date 2024/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider again the fast-slow motions setups in the continuous time \frac dXN(t)dt=N1/2 \sig(XN(t))(ξ(tN))+b(XN(t)), t∈ [0,T] and the discrete time XN((n+1)/N)=XN(n/N)+N-1/2\sig(XN(n/N))ξ(n)+N-1b(XN(n/N)), n=0,1,...,[TN] where \sig and b are smooth matrix and vector functions, respectively, ξ is a centered vector stationary stochastic process with weak dependence in time and N is a big parameter. We obtain estimates for the almost sure approximations of the process XN by certain diffusion process \Sig. In \citeFK and in other recent papers concerning similar setups the results were obtained relying fully on the rough paths theory. Here we derive our probabilistic results as corollaries of quite general deterministic estimates which are obtained with all details provided following somewhat ideology of the rough paths theory but not relying on this theory per se which should allow a more general readership to follow complete arguments.