Mathematical Modeling in Finance with Stochastic Processes

122 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES

using the trial function

Wskp =

{

C+Ds ifs≤k

E+Fs ifs > k.

2. Show that

Ws=

S∑− 1

k=1

kWsk

= 2

[

s

S

S∑− 1

k=1

k(S−k)−

∑s−^1

k=1

k(s−k)

]

= 2

[

s

S

[

S(S−1)(S+ 1)

6

]

−

s(s−1)(s+ 1)

6

]

=

s

3

[

S^2 −s^2

]

You will need formulas for

∑N

k=1k and

∑N

k=1k

(^2) or alternatively for
∑N
k=1k(M−k). These are easily found or derived.

3. (a) For the long run average cost

C=

K+ (1/3)rS^3 x(1−x^2 )

S^2 x(S−x)

.

find∂C/∂x.

(b) For the long run average cost

C=

K+ (1/3)rS^3 x(1−x^2 )

S^2 x(1−x)

.

find∂C/∂S.

(c) Find the optimum values ofxandS.

Outside Readings and Links:

1. Milton Friedman: The Purpose of the Federal Reserve system. The
   reaction of the Federal Reserve system at the beginning of the Great
   Depression.