[SOLVED] Amath 562 final exam

1. [40pt] Wt
is a standard Brownian motion.
(a) Find the probability density of W2
t
.
(b) Evaluate the expectation:
E
“Z T
0
W2
t dWt
2
#
.
(c) Show that W3
t − 3tWt
is a martingale.(d) Use Ito’s formula to write the following stochastic process
Xt = e
Wt + t + 2
into the standard form
dXt = µ(t, ω)dt + σ(t, ω)dWt
.2. [40pt] The concept of change of measure in terms of a Radon-Nikodym derivative can be
summarized as in the following diagram:Ω, F, P


fX(x)Ω, F, P˜


˜fX(x)
✲
✲
❄ ❄
X(ω)
X(ω)
dP˜
dP
(ω)(a) Assuming that in the diagram, both probability density functions fX(x) and ˜fX(x)
for a random variable X(ω) are given. Find the RND dP˜
dP
(ω) in terms of the X(ω).(b) In the diagram below, X : Ω → R is a random variable with a smooth probability
density function. A smooth function g(x) : R → R represents the RND
gX(ω)


=
dP˜
dP
(ω).Let us consider a random variable Y (ω) = h
−1X(ω)


, or X(ω) = hY (ω)


, where
h(x) : R → R is a monotonic and smooth function on R and h
−1
is the inverse function.If the random variable Y (ω) under the new measure P˜ has a probability density function
˜fY (x) = fX(x),
find the function h(y).Ω, F, P


fX(x)Ω, F, P˜


˜fX(x)
✲
✲
✑
✑
✑
✑
✑
✑
✑
✑✑✸
❄ ❄
X(ω)
X(ω)
Y (ω)
dP˜
dP
(ω) = g[X(ω)](c) Now consider a probability space (Ω, F, P), and X(ω) =
X1, X2, · · · , Xn)(ω)
is a n-dimensional random variables, whose sccessive differences Xj − Xj−1 are all
conditionally, normally distributed independent random variables:
Xj+1 − Xj ∼ N 
µj+1(Xj ), σ2
j+1(Xj )

.Find the change of measures Z(ω) = dP˜
dP
(ω) such that under the new measure P˜,
Xj+1 − Xj ∼ N
0, σ2
j+1(Xj )

.(d) What is the conditional expectation
EZ|X1, · · · , Xkfor k < n?3. [20pt] Let (X, Y )(t) be an Ito process in R
2
, as the solution to the SDE
(
dX(t) = µt, X, Y

dt + σ
2t, X, Y

dW(t),
dY (t) = θt, X, Y

dt,
in whch µ, σ, and θ are all continuous functions. Find the first and second variations of
Y (t).4. [20pt] Consider SDE
dXt = µ(Xt)dt + σ(Xt)dWt
.
(a) Show that
v(x, t) := E
h
δ(x − Xt)X0 = y
i
,
where δ(t) is the Dirac-δ function, satisfies the partial differential equation



∂v(x, t)
∂t =
∂
2
∂x2

σ
2
(x)
2
v(x, t)

−
∂
∂x

µ(x)v(x, t)

,
v(x, 0) = δ(x − y).(b) Show that
u(x, t) = E
h
φ(Xt)X0 = x
i
satisfies the partial differential equation



∂u(x, t)
∂t =
σ
2
(x)
2
∂
2u(x, t)
∂x2
+ µ(x)
∂u(x, t)
∂x ,
u(x, 0) = φ(x).5. [50pt] We denote Kolmogorov’s backward and forward operators
Lx[u] = σ
2
(x)
2
d
2u(x)
dx
2
+ µ(x)
du(x)
dx
,
L ∗
x
[f] = d
2
dx
2

σ
2
(x)
2
f(x, t)

−
d
dx

µ(x)f(x, t)

.(a) Show that Lx has an alternative expression:
Lx[u] = σ
2
(x)s(x)
2
d
dx

s
−1
(x)
du(x)
dx

, (1)
where s(x) is known as the scale density:
s(x) = exp 
−
Z
2µ(x)
σ
2
(x)
dx

.(b) Give the corresponding expression, as in (1), for L ∗
x
.
(c) Consider the linear partial differential equation (PDE)



∂f(x, t)
∂t = L ∗
xf(x, t)− γ(t, x)f(x, t),
f(x, 0) = ψ(x),
(2)where the operator L ∗
x
is defined above. Express the solution to PDE (2) in terms of the
Ito process Xt
that satisfies
dXt = µ(Xt)dt + σ(Xt)dWt
.
6. [30pt] Consider a two-dimensional SDE
dX1(t) = µ1(X1, X2)dt + σdW1(t),
dX2(t) = µ2(X1, X2)dt + σdW2(t),
whereµ1, µ2


(x) = −∇U(x) and x = (x1, x2),
and W1(t) and W2(t) are two independent standard Brownian motions.(a) Give the generator A and its L
2R
2
, dx


adjoint A∗
.They are also known as Kolmogorov’s
backward and forward operators for the time-homogeneous Ito diffusion:
A[u] = σ
2
2
∇2u + · · · , A
∗
[f] = σ
2
2
∇2
f + · · · .(b) Show that under a proper choice of the weight ρ(x) > 0 for the inner product between
any f, g ∈ L
2
⟨f, g⟩ρ =
Z
R2
ρ(x)f(x)g(x)dx,
A is self-adjoint, i.e.,
⟨f, Ag⟩ρ = ⟨Af, g⟩ρ.You can assume that both f(x) and g(x), and their partial derivatives, go to 0 sufficiently
fast as |x| → ∞.
(c) Similarly, with an alternative choice of the weight for the inner product, A∗
is also
self-adjoint:
⟨f, A
∗
g⟩ρ = ⟨A∗
f, g⟩ρ.
(Hint: This is the 2-d generalization of the material in MLN, Sec. 9.5)7. [30pt] Let Xt ∈ R
2 be a Levy process defined by ´
Xt =
Z t
0
σ(t)dW(t) + Z
R2
zN(t, dz),in which the second term is a compound Poisson process with scalar Poisson random
measure N(t, z, ω) that is independent from the W(t, ω); σ, N ∈ R
1
and W, z ∈ R
2
. If
the Levy measure ´
ν(dz) = EN(1, dz)=
λ
2πη2
exp 
−
z
2
1 + z
2
2
2η
2

dz,
where dz = dz1dz2, and both λ and η are real positive numbers. Find the characteristic
function for Xt
.