Kasper Larsen – Rutgers University
Date/Time/Location
Thursday, October 1, 2026, 12:10 pm; Hill Center 705
Strict SDE Comparison for Cusp Coefficients and Counterexamples
We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation dX = sigma(X)dB, with initial condition x in an interval I, for sigma positive and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in the local Sobolev space W1p, for p between 1 and 2. We illustrate using sigma of x equal to 1 plus the absolute value of x to the power beta, for beta between 0 and 1, and show that strict comparison holds if and only if beta is at least one half. We give examples showing that neither local W1p regularity for p between 1 and 2 nor Hölder regularity of order beta between one half and 1 is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.