Optimal stopping for diffusion with discontinuous coefficients

We show through examples that simple discontinuity of the coefficients of a diffusion produce non connected optimal stopping regions for simple payoffs.

Our examples comprise the broken drift diffusion (i.e. a diffusion with piecewise constant drift coefficient changing at zero) and a linear payoff and the oscillating brownian motion (i.e. a diffusion with piecewise constant diffusion coefficient changing at zero) with a quadratic payoff.

Both examples show the same behaviour. This is joint work with Paavo Salminen.

References

1) Optimal stopping of Brownian motion with broken drift. Ernesto Mordecki Paavo Salminen
High Frequency. Pages: 113-120. First Published: 11 April 2019

2) Optimal stopping of oscillating Brownian motion. Ernesto Mordecki, Paavo Salminen.
arXiv:1903.01457 [math.PR]