MATH5965 - Discrete Time Financial Modelling Practice Sheet 4
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MATH5965 - Discrete Time Financial Modelling
exercise
1. Let (Zt)0<t<T be a given adapted, integrable process, and let (Vt)0<t<T be its Snell envelope.
a) Show that if Z is submartingale, then Vt = E [ZT|Ft], and if Z is a supermartingale then Vt = Zt for all t.
b) Let τ be any stopping time taking values in {0, ..., T}. Show that the process (Vt^τ)0<t<T is a supermartingale.
c) Define the random time τ * by
τ * = min{t ∈ {0, ..., T} : Vt = Zt}.
Show that τ * is a stopping time. In addition, show that the process (Vt^τ * )teA0,...,T } is a martingale and, in particular, V0 = E[Zτ * ].
2023-04-22