Abstract
Among the dominant notions from the theory of stochastic processes used to develop the theory of stochastic differential equations are those of martingales, stop** times, and the Markov property.
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Notes
- 1.
Throughout, BCPT refers to Bhattacharya and Waymire (2016), A Basic Course in Probability Theory.
- 2.
See Billingsley (1968), p. 110.
- 3.
See Billingsley (1968), p.121.
- 4.
See Billingsley (1968), pp. 113–116.
- 5.
See Bhattacharya and Waymire (2021), Proposition 5.1.
- 6.
- 7.
See BCPT, p.59.
- 8.
See BCPT, Proposition 3.7.
- 9.
See BCPT Theorems 3.11, 3.12.
- 10.
See BCPT, Proposition 3.8.
- 11.
See BCPT, Theorem 3.8.
- 12.
See BCPT, Theorem 3.2.
- 13.
See BCPT, p.9.
- 14.
See BCPT, Proposition 1.4.
- 15.
See BCPT, Theorem 3.4.
References
Bhattacharya R, Waymire E (2016) A basic course in probability theory. Springer, New York. Errata: https://sites.science.oregonstate.edu/~waymire/
Bhattacharya R, Waymire E (2021) Random walk, brownian motion, and martingales. Graduate texts in mathematics. Springer, New York
Billingsley P (1968) Convergence of probability measures. Wiley, New York
Crump KS (1975) On point processes having an order statistic structure. Sankhya Indian J Statist Ser A 37:396–404
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Bhattacharya, R., Waymire, E. (2023). A Review of Martingales, Stop** Times, and the Markov Property. In: Continuous Parameter Markov Processes and Stochastic Differential Equations. Graduate Texts in Mathematics, vol 299. Springer, Cham. https://doi.org/10.1007/978-3-031-33296-8_1
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