Snell envelope

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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[edit]

Given a filtered probability space (\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,T]},\mathbb{P}) and an absolutely continuous probability measure \mathbb{Q} \ll \mathbb{P} then an adapted process U = (U_t)_{t \in [0,T]} is the Snell envelope with respect to \mathbb{Q} of the process X = (X_t)_{t \in [0,T]} if

  1. U is a \mathbb{Q}-supermartingale
  2. U dominates X, i.e. U_t \geq X_t \mathbb{Q}-almost surely for all times t \in [0,T]
  3. If V = (V_t)_{t \in [0,T]} is a \mathbb{Q}-supermartingale which dominates X, then V dominates U.[1]

Construction[edit]

Given a (discrete) filtered probability space (\Omega,\mathcal{F},(\mathcal{F}_n)_{n = 0}^N,\mathbb{P}) and an absolutely continuous probability measure \mathbb{Q} \ll \mathbb{P} then the Snell envelope (U_n)_{n = 0}^N with respect to \mathbb{Q} of the process (X_n)_{n = 0}^N is given by the recursive scheme

U_N := X_N,
U_n := X_n \lor \mathbb{E}^{\mathbb{Q}}[U_{n+1} \mid \mathcal{F}_n] for n = N-1,...,0

where \lor is the join.[1]

Application[edit]

  • If X is a discounted American option payoff with Snell envelope U then U_t is the minimal capital requirement to hedge X from time t to the expiration date.[1]

References[edit]

  1. ^ a b c Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467.