For a local martingale Y satisfying dY/Y = dB for some stochastic process , Novikovs condition

For a local martingale Y satisfying dY/Y = θ

dB for some stochastic process θ, Novikov’s condition is that E



exp1 2

T 0

θ

θ dt

 < ∞.

Under this condition, Y is a martingale on [0,T]. Consider Y = MW, where M is an SDF process and W is a self-financing wealth process.

(a) Show that dY/Y = θ
dB, where θ = σ
π −λp −ζ and σ ζ = 0.

(b) Deduce that Novikov’s condition is equivalent to (13.41).

(c) By specializing (13.41), state sufficient conditions for MSi to be a martingale for i = 1,...,n.