Mathematical Modeling in Finance with Stochastic Processes

120 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES

nto find the average cost

Expected total cost in cyclei =K+E[Ri],

but we have another expression for the expectationE[Ri],

Expected opportunity cost =E[Ri] =r

s

3

[

S^2 −s^2

]

.

Likewise the total length ofncycles isN 1 +···+Nn. Divide bynto find
the average length,

Expected length =

N 1 +···+Nn

n

=s(S−s).

These expected values allow us to calculate the average costs

Long run average cost, dollars per week =

K+E[Ri]

E[Ni]

.

ThenE[Ri] =rWsandE[Ni] =s(S−s). Therefore

Long run average cost, dollars per week =

K+ (1/3)rs(S^2 −s^2 )

s(S−s)

.

Simplify the analysis by settingx=s/Sso that the expression of interest is

Long run average cost =

K+ (1/3)rS^3 x(1−x^2 )

S^2 x(1−x)

.

Remark.Aside from being a good thing to non-dimensionalize the model
as much as possible, it also appears that optimizing the original long run
cost average in the original variablesS andsis messy and difficult. This
of course would not be known until you had tried it. However, knowing the
optimization is difficult in variablessandSadditionally motivates making
the transformation to the non-dimensional ratiox=s/S.

Now we have a function in two variables that we wish to optimize. Take
the partial derivatives with respect toxandSand set them equal to 0, then
solve, to find the critical points.
The results are that

xopt=

1

3

Sopt= 3

(

3 K

4 r

)^13

.