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:
)dt.
2) Suppose that S satisfies
μSdt + σSdX, 0 ≤ S < ∞,
where μ ≥ 0, σ > 0, and dX is a Wiener process. Let
,
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:
hш
ж
и
к
м
+ 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.
Mathematics Homework Solutions
#141876
Partial Differential Equations : Wiener Process
Please see the attached file for the fully formatted problems.
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 = dS + ( + b2 )dt.
2) Suppose that S satisfies
dS = μSdt + σSdX, 0 ≤ S < ∞,
where μ ≥ 0, σ > 0, and dX is a Wiener process. Let
ξ = ,
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.
Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach by Duffy. See attached file for full problem description.
Partial Differential Equations and Wiener Processes are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
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