**Aims & Learning Objectives:**
**Aims: **To develop and apply the theory of probability and stochastic processes to examples from finance and economics.
**Objectives: **
At the end of the course, students should be able to
* formulate mathematically, and then solve, dynamic programming problems
* price an option on a stock modelled by a log of a random walk
* perform simple calculations involving properties of Brownian motion.
**Content: ** Dynamic programming: Markov decision processes, Bellman equation; examples including consumption/investment, bid acceptance, optimal stopping. Infinite horizon problems; discounted programming, the Howard Improvement Lemma, negative and positive programming, simple examples and counter-examples.
Option pricing for random walks: Arbitrage pricing theory, prices and discounted prices as Martingales, hedging.
Brownian motion: Introduction to Brownian motion, definition and simple properties.
Exponential Brownian motion as the model for a stock price, the Black-Scholes formula.
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