Ito Processes
Let W(t), t ≥ ∘⋅ be a Brownian motion on some
probability space (F, P) with filtration F = (f, ),-_-_,0. Recall that for any
Ito process X with decomposition
i i
X: = Xo -+- / Aurlll’u -+- / flurlu
we say the differential of X satisfies the equation
(IX, = AKIN} -+- 9N”
a) Compute the differential (Ipr the process X,
b) Let f(x) be a function and ⋋ be a number such that f(0) = 1, f’(x) is a
bounded function, and f ” (.1′) = ∕∖ f (.1′). For example, f(x) = cos(x), A ⋮ − 1 is such a pair. For such a
pair, show that EUUV, )] ∶ (- ↙⋅∖⋅⊤↙⋅ Hint:
For some constant}, consider the processX, ⋮ ∊⋅⋅↰⋅↾ f(ll ⋅↾ ). What is it’s
Ito process decomposition?