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| Main Author: | |
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| Format: | Preprint |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1802.06760 |
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Table of Contents:
- We consider SDEs of the form $dX_t = |f(X_t)|/t^γ dt+1/t^γ dB_t$, where $f(x)$ behaves comparably to $|x|^k$ in a neighborhood of the origin, for $k\in [1,\infty)$. We show that there exists a threshold value $:=\tildeγ$ for $γ$, depending on $k$, such that when $γ\in (1/2, \tildeγ)$ then $\mathbb{P}(X_n\rightarrow 0) = 0$, and for the rest of the permissible values $\mathbb{P}(X_n\rightarrow 0)>0$. The previous results extend for discrete processes that satisfy $X_{n+1}-X_n = f(X_n)/n^γ+Y_n/n^γ$. Here, $Y_{n+1}$ are martingale differences that are a.s. bounded. This result shows that for a function $F$, whose second derivative at degenerate saddle points is of polynomial order, it is always possible to escape saddle points via the iteration $X_{n+1}-X_n =F'(X_n)/n^γ+Y_n/n^γ$ for a suitable choice of $γ$.