Uncertainty and Probability Optimal Stopping under g–expectations: Theory

Worst–Case Priors

Drift ambiguity

Vis ag–supermartingale

from the Doob–Meyer–Peng decomposition

−dVt=g(t,Zt)dt−ZtdWt+dAt

for some increasing processA

=−κ|Zt|dt−ZtdWt+dAt

Girsanov: =−ZtdWt∗+dAtwith kernelκsgn(Zt)

Theorem (Duality forκ–ambiguity)

There exists a probability measure P∗∈Pκsuch that

Vt= ess supτ≥tEt(Xτ) = ess supτ≥tE∗[Xτ|Ft]. In particular:

maxτ Pmin∈PκEP[Xτ|Ft] = minP∈Pκmaxτ EP[Xτ|Ft]