Introduction to Probability and Stochastic Processes with Applications

CHAPTER 11

STOCHASTIC CALCULUS

Stochastic calculus plays an essential role in modern mathematical finance and risk management. The objective of this chapter is to develop conceptual ideas of stochastic calculus in order to provide a motivational framework. This chapter presents an informal introduction to martingales, Brownian motion, and stochastic calculus. Martingales were first defined by Paul Lévy (1886–1971). The mathematical theory of martingales has been developed by American mathematician Joseph Doob (1910–2004). We begin with the basic notions of martingales and its properties.

11.1   MARTINGALES

The martingale is a strategy in a roulette game in which, if a player loses a round of play, then he doubles his bet in the following games so that if he wins he would recover from his previous losses. Since it is true that a large losing sequence is a rare event, if the player continues to play, it is possible for the player to win, and thus this is apparently a good strategy. However, the player could run out of funds as the game progresses, and therefore the player cannot recover the losses he has previously accumulated. One must also take into account the fact that casinos impose betting limits.

Formally, suppose that a player starts a game in which he wins or loses with the same probability of . The player starts betting a single monetary unit. The strategy is progressive ...