[SOLVED] Amath 562 homework assignment 4

1. Professor Matt Lorig’s notes, exercises 8.5
2. The Ornstein-Uhlenbeck process, defined by time-homogeneous linear SDE
dX(t) = −µX(t)dt + σdW(t), X(0) = x0,
in which σ and µ > 0 are two constants, has its Kolmogorov forward equation
∂
∂tΓ(x0;t, x) = σ
2
2
∂
2
∂x2
Γ(x0;t, x) + ∂
∂x

µxΓ(x0;t, x)

, (1)
with the initial condition Γ(x0; 0, x) = δ(x − x0).(a) Show that the solution to the linear PDE (1) has a Gaussian form and find the solution.
(b) What is the limit of
lim
t→∞
Γ(x0;t, x)?
(c) Find E[X(t)] and V[X(t)].(d) You note that E[X(t)] is the same as the solution to the ODE dx
dt = −µx, which is
obtained when σ = 0. Is this result true for a nonlinear SDE?3. The time-independent solution to a Kolmogorov forward equation gives a stationary probability
density function for the Ito process dXt = µ(Xt)dt + σ(Xt)dW(t):
−
∂
∂x

µ(x)f(x)

+
1
2
∂
2
∂x2

σ
2
(x)f(x)

= 0.This is a linear, second-order ODE. We assume that both µ(x) and σ(x) satisfy the
conditions required to have a solution f(x) on the entire R. Find the expression for
the general solution. There are two constants of integration, which should be determined
according to appropriate probabilistic reasoning.4. Professor Matt Lorig’s notes, exercises 9.35. Consider a continuous-time (n+1)-state Markov process X(t), X ∈ S = {0, 1, 2, · · · , n},
with transition rates
g(i, j) = 1
dt
P

X(t + dt) = j|X(t) = i, j ̸= i.Let state 0 be an absorbing state, e.g., all g(0, j) = 0, 1 ≤ j ≤ n. Let τk be a hitting
time:
τk := inf 
t ≥ 0 : X(t) = 0, X(0) = k.
(a) Show that
X
1≤k≤n
g(j, k)E[τk] = −1.
(b) Derive a system of equations relating E[τ
2
k
] to E[τj
], 1 ≤ j, k ≤ n.(c) Now if both states 0 and n are absorbing, let uk be the probability of X(t), starting
with X(0) = k, being absorbed into state 0 and 1 − uk be the probability being absorbed
into state n. Derive a system of equations for uk.