Partial Differential Equations : Wiener Process
Response to the following questions:
1) Suppose: dS = a(S,t)dt + b(S,t)dX,
where dX is a Wiener process. Let f be a function of S and t.
Show that: df = [(∂f/∂S)dS + ( ∂f/∂t+(1/2) +b2(∂2f/∂S2)] dt.
2) Suppose that S satisfies
dS = μSdt + σSdX, 0 ≤ S < ∞,
where μ ≥ 0, σ > 0, and dX is a Wiener process. Let
ξ = S/S+Pm ,
where Pm is a positive constant and the range of ξ is [0,1), if 0 ≤ S < ∞. The stochastic differential equation for ξ is in the form:
d ξ = a(ξ)dt + b(ξ)dX.
Find the concrete expressions for a(ξ) and b(ξ) by Ito’s lemma and show:
{a(0) = 0, and {a(1) = 0,
{b(0) = 0, and {b(1) = 0.