Uncertainty and Probability Continuous Time: g–expectations

Diffusion Models

based on Cheng, R., IMW Working Paper 429

Framework now: Brownian motionW on a filtered probability space

(Ω,F,P 0 ,(Ft)) with the usual conditions

Typical Example: Ambiguous Driftμt(ω)∈[−κ,κ]

P={P:Wis Brownian motion with driftμt(ω)∈[−κ,κ]}

(for time–consistency: stochastic drift important!)

worst case: either +κor−κ, depending on the state

LetEtX= minP∈PEP[X|Ft]

we have the representation

−EtX=−κZtdt+ZtdWt

for some predictable processZ

Knightian expectations solve a backward stochastic differential

equation