This exercise verifies the assertion (16.6) regarding the dynamics of an SDF process. Suppose the information in the economy is given by independent Brownian motions B1,...,Bk. Consider a non-dividend-paying asset with price S satisfying dS S = μdt +

k i=1

σi dBi for stochastic processes μ and σi.

(a) Define a stochastic process σ and Brownian motion Bs such that dS S = μdt + σ dBs

.

(b) Consider an SDF process M. We have dM M = −r dt −

k i=1

λi dBi for some stochastic processes λi. Define a stochastic process λ and Brownian motion Bm such that dM M = −r dt − λdBm .

(c) Let ρ denote the correlation process of Bs and Bm. Show that λρ = (μ− r)/σ.

(d) Show that there is a local martingale Z such that dM M = −r dt − μ −r σ
dBs +dZ and (dBs )(dZ) = 0.