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. 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. Given a filtered probability space ( Ω , F , ( F t ) t ∈ [ 0 , T ] , P ) {displaystyle (Omega ,{mathcal {F}},({mathcal {F}}_{t})_{tin },mathbb {P} )} and an absolutely continuous probability measure Q ≪ P {displaystyle mathbb {Q} ll mathbb {P} } then an adapted process U = ( U t ) t ∈ [ 0 , T ] {displaystyle U=(U_{t})_{tin }} is the Snell envelope with respect to Q {displaystyle mathbb {Q} } of the process X = ( X t ) t ∈ [ 0 , T ] {displaystyle X=(X_{t})_{tin }} if Given a (discrete) filtered probability space ( Ω , F , ( F n ) n = 0 N , P ) {displaystyle (Omega ,{mathcal {F}},({mathcal {F}}_{n})_{n=0}^{N},mathbb {P} )} and an absolutely continuous probability measure Q ≪ P {displaystyle mathbb {Q} ll mathbb {P} } then the Snell envelope ( U n ) n = 0 N {displaystyle (U_{n})_{n=0}^{N}} with respect to Q {displaystyle mathbb {Q} } of the process ( X n ) n = 0 N {displaystyle (X_{n})_{n=0}^{N}} is given by the recursive scheme where ∨ {displaystyle lor } is the join (in this case equal to the maximum of the two random variables).