In: Advanced Math

Here are four questions:

1. Prove a standard Brownian motion is Gaussian process .

2. Prove a Brownian bridge is Gaussian process.

3. Prove Ornstein-Uhlenbeck process is Gaussian

4. Prove the position process is Gaussian.

Please provide as many detail as you can, thanks.

Let Bt be a 1-dimensional Brownian
motion and let c > 0 be a constant. Prove
that
is also a Brownian motion.

Please elaborate on the differences and the relationships
between Brownian motion, Wiener process and Levy process, and also
the characteristics of each.
Thank you so much!

If X(t), t ≥ 0 is a Brownian motion process with drift parameter
μ and variance parameter σ2 for which X(0)=0, show that
-X(t), t ≥ 0 is a Brownian motion process with drift parameter -μ
and variance parameter σ2.

If X(t), t>=0 is a Brownian motion process with drift mu and
variance sigma squared for which X(0)=0, show that -X(t), t>=0
is a Brownian Motion process with drift negative mu and variance
sigma squared.

Let {X(t), t >=0} be a Brownian motion with drift coefficient
μ and variance
parameter σ^2. What is the joint density function of X(s) and
X(t), s < t?

Question 1 Consider a dividend-paying stock whose
price follows a geometric Brownian motion (GBM) of the form: dS =
(u - )Sedt +os dz (a) Using Ito's lemma, write the stochastic
process that is followed by Y= 5. (b) Your derivation in (a) should
show that Y, also follows a GBM of the form dy, = (u - q*)Yedt +
o*Yedz. What are q* and o* as functions of u, q and o? (c) Consider
a derivative that pays off...

Here are returns and standard deviations for four
investments.
Return (%)
Standard Deviation (%)
Treasury bills
5.0
0
Stock P
11.5
16
Stock Q
14.5
28
Stock R
24.0
27
Calculate the standard deviations of the following
portfolios.
a. 50% in Treasury bills, 50% in stock P.
(Enter your answer as a percent rounded to 2 decimal
places.)
a.Standart Deviation:?%
b. 50% each in Q and R, assuming the shares
have: (Do not round intermediate calculations. Enter your
answers as...

Let {W(t),t≥0} be a standard Brownian motion and let
M(t)=max0≤s≤tW(s). Find P(M(9)≥3).

These questions tail off of each other so I will post the four
parts here:
1.
ABC Inc., began the year with 10,000 units in stock but finished
with 5,000 units. It produced 45,000 units for the period. Its
selling price is $12 per unit, variable manufacturing cost is $5
per unit, and variable selling is $3 per unit. Fixed manufacturing
and selling costs are $100,000 and $72,000 respectively.
The firm notes that variable cost per unit (both mfg and...

The last four years of returns for a stock are as shown
here:
Years
1
2
3
4
Return
-4.2%
+28.3%
+12.2%
+3.9%
a. What is the average annual return?
b. What is the variance of the stock's returns?
c. What is the standard deviation of the stock's returns?

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