Snell envelope


The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating a stochastic process. The Snell envelope is named after James Laurie Snell.

Definition

Given a filtered probability space and an absolutely continuous probability measure then an adapted process is the Snell envelope with respect to of the process if
  1. is a -supermartingale
  2. dominates, i.e. -almost surely for all times
  3. If is a -supermartingale which dominates, then dominates.

    Construction

Given a filtered probability space and an absolutely continuous probability measure then the Snell envelope with respect to of the process is given by the recursive scheme
where is the join.

Application