Suppose that the continuous-time stochastic process X = (Xt) is defined as Xt = 1 2

t 0

λ2 s ds +

t 0

λs dzs, where z = (zt) is a one-dimensional standard Brownian motion and λ = (λt) is some ‘nice’

stochastic process.

(a) Argue that dXt = 1 2 λ2 t dt + λt dzt.

(b) Suppose that the continuous-time stochastic process ξ = (ξt) is defined as ξt =

exp{−Xt}. Show that dξt = −λtξt dzt.