Abstract
This mini-course deals with the following inter-connected topics:
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Brownian Motion and Martingales.
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Stochastic orders, or comparison of distributions: First-degree stochastic dominance, or stochastic inequality. Second-degree stochastic dominance, or (super-) Martingale Dilation.
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Stopping times in Brownian Motion and Skorokhod embeddings. The Azéma — Yor stopping time.
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The distribution of the first time Brownian Motion differs from its cumulative maximum by some fixed amount.
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Optimal stopping when the sampling cost is linear in time and the reward upon stopping is a non-decreasing function of the cumulative maximum. This can be viewed as pricing and management of a type of look-back American put option. The case of linear reward function was studied by Dubins & Schwarz [7].
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References
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Meilijson, I. (2003). Stochastic Orders and Stopping Times in Brownian Motion. In: Picco, P., San Martin, J. (eds) From Classical to Modern Probability. Progress in Probability, vol 54. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8053-4_6
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DOI: https://doi.org/10.1007/978-3-0348-8053-4_6
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