The Snell envelope, used in
stochastics
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
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
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
and an
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
probability measure then an
adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every re ...
is the Snell envelope with respect to
of the process
if
#
is a
-supermartingale
#
dominates
, i.e.
-
almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...
for all times