Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics. Patrick Muldowney

left-parenthesis 4 right-parenthesis EndSet comma StartSet z left-parenthesis 1 right-parenthesis comma z left-parenthesis 2 right-parenthesis comma z left-parenthesis 3 right-parenthesis comma z left-parenthesis 4 right-parenthesis EndSet comma StartSet w left-parenthesis 1 right-parenthesis comma w left-parenthesis 2 right-parenthesis comma w left-parenthesis 3 right-parenthesis comma w left-parenthesis 4 right-parenthesis EndSet comma"/>

      with representing the daily changes in the value of the stockholding. For , the ’tuple

      is a sample path for the processes . Similarly for ’tuples , and processes , , respectively. One of the 16 underlying random transition sequences is no. 7 in Table 2.4:

      With , , the corresponding sample paths

      can be calculated for this particular sequence of share price U‐D transitions, using

for , as in Table 2.5.

Table 2.4 UD sample paths for processes

---

1.	U	U	U	U	10
2.	D	U	U	U	5
3.	U	D	U	U	4
4.	D	D	U	U	1
5.	U	U	D	U	3
6.	D	U	D	U	0
7.	U	D	D	U	‐1
8.	D	D	D	U	‐2
9.	U	U	U	D	2
10.	D	U	U	D	‐1
11.	U	D	U	D	‐2
12.	D	D	U	D	‐3
13.	U	U	D	D	‐3
14.	D	U	D	D