Three Essays on Stopping

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ / σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).

P-SV wave diffusion in fluid-saturated medium

Propagating P-SV waves in the fluid-saturated mediums are categorized to fall into two distinct groups: insoluble and soluble mediums. These waves are known as surface Rayleigh waves. By introducing these waves with slowness in accordance to Snell Law, the diffusive and rotating waves are obtained. The results bear out that the propagating P-SV waves in soluble medium share similar diffusion characteristic as of insoluble medium while the discussions on fluid density in the mediums show that high density fluid promotes diffusive characteristic while low density fluid endorses nondiffusive P-SV waves. There is a compressed zone during the propagation of P-SV waves in medium saturated with high density fluids.