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

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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 StartSet c left-parenthesis 1 right-parenthesis comma c left-parenthesis 2 right-parenthesis comma c left-parenthesis 3 right-parenthesis comma c left-parenthesis 4 right-parenthesis EndSet representing the daily changes in the value of the stockholding. For s equals 1 comma 2 comma 3 comma 4, the ’tuple

x equals left-parenthesis x left-parenthesis s right-parenthesis right-parenthesis Subscript s equals 1 comma 2 comma 3 comma 4 Baseline equals left-parenthesis x left-parenthesis 1 right-parenthesis comma x left-parenthesis 2 right-parenthesis comma x left-parenthesis 3 right-parenthesis comma x left-parenthesis 4 right-parenthesis right-parenthesis

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

normal upper U normal upper D normal upper D normal upper U

      With x left-parenthesis 0 right-parenthesis equals 10, z left-parenthesis 0 right-parenthesis equals 1, the corresponding sample paths

left-parenthesis x left-parenthesis s right-parenthesis right-parenthesis comma left-parenthesis z left-parenthesis s right-parenthesis right-parenthesis comma left-parenthesis w left-parenthesis s right-parenthesis right-parenthesis

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

c left-parenthesis s right-parenthesis equals z left-parenthesis s minus 1 right-parenthesis left-parenthesis x left-parenthesis s right-parenthesis minus x left-parenthesis s minus 1 right-parenthesis right-parenthesis and w left-parenthesis s right-parenthesis equals w left-parenthesis s minus 1 right-parenthesis plus c left-parenthesis s right-parenthesis

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s equals 1 s equals 2 s equals 3 s equals 4 w left-parenthesis 4 right-parenthesis
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