1. Review Taylor's Theorem from Calculus.

2. Brownian Motion: (3 points)
Let (Wt )t≥0 and (Zt )t≥0 be two independent standard Brownian motions. Use the definition of Brownian
motion to show whether or not the stochastic processes (Si )t

0 , i = 1, 2, 3, 4, are also standard Brownian
motions, where:

(a) For fixed 0 ≤ ρ ≤ 1, (b) For fixed 0 ≤ ρ ≤ 1,

(c) Time inversion:

t = ρ Wt + (1 - ρ) Zt . St = ρ Wt + p1 - ρ2 Zt .
(d) Brownian scaling: (c > 0 fixed)

S3

t
S4
c2

3. n-th central moment of a Normal random variable:
Let X be a Normal random variable with expected value µ = 0 and variance σ2 > 0, i.e. X ∼ N 0, σ2 . Calculate the expected value of X n , E [X n ] , n = 3, 4, and 5.

4. Expected value and variance of the log-Normal distribution:
Let X be a Normal random variable with expected value µ ∈ R and variance σ2 > 0, i.e. X ∼ N µ, σ2 , and let Y = exp {X }. Determine E[Y ] and V ar[Y ].

5. Martingales:
Let X be a random variable on (?, F , P ) with E [|X |] < ∞, and let (Wt )t≥0 be a standard Brownian motion.
Use the definition of a martingale to show whether or not the stochastic processes (Si )t

0 , i = 1, 2, 3, are
martingales, where:

(a) S1 = E [ X | Ft ] , where (F )

t≥0
is an (arbitrary) filtration with Ft ⊆ F ∀t ≥ 0.
(b) S2 = Wt , i.e. Brownian motion is a martingale. (c) S3 = W 2 - t.
t t

6. Path of a Brownian motion:
Use Excel or another spreadsheet software to simulate a (discretized) sample path of a standard Brownian motion over the interval [0, 1] using 250 equidistant time steps. (Just hand in a print of the resulting path.)