We consider SDEs of the form dX_t = |X_t|^k/t^gamma dt+1/t^gamma dB_t, for a fixed k in [1,infty). We find the values of gamma in (1/2,1] such that X_t will not converge to the origin with probability 1. Furthermore, we can show that for the rest of the permissible values the process will converge to the origin with some positive probability. The previous results extend for processes that satisfy dX_t = f(X_t)/t^gamma dt+1/t^gamma dB_t, when f(x) is comparable to |x|^k in a neighborhood of the origin. As it is expected, similar results are true for discrete processes satisfying X_{n+1} - X_n =f(X_n)/n^gamma+Y_{n+1}/n^gamma. Here, Y_{n+1} are martingale differences that are almost surely bounded and satisfy E(Y_{n+1}^2| F_n )>delta>0.

### Probability and Combinatorics

Tuesday, December 5, 2017 - 3:00pm

#### Kostis Karatapanis

University of Pennsylvania