Mathematical Modeling in Finance with Stochastic Processes

3.4. A STOCHASTIC PROCESS MODEL OF CASH MANAGEMENT 117

3.3 Several typical cycles in a model of the reserve requirement.

where (^1) {Tj=k}is theindicator random variablewhere
(^1) {Tj=k}=

{

1 Tj=k

0 Tj 6 =k.

Note that the inner sum is a random sum, since it depends on the length of
the cycleN, which is cycle dependent.
Then using first-step analysisWsksatisfies the equations

Wsk=δsk+

1

2

Ws− 1 ,k+

1

2

Ws+1,k

with boundary conditionsW 0 k=WSk= 0. The termδskis theKronecker
delta

δsk=

{

1 ifk=s

0 ifk 6 =s.

The explanation of this equation is very similar to the derivation of the
equation for the expected duration of the coin-tossing game. The terms
1
2 Ws−^1 ,k+

1
2 Ws+1,karise from the standard first-step analysis or expectation-
by-conditioning argument forWsk. The non-homogeneous term in the prior