Martingales and Financial Mathematics in Discrete Time

3Random Walks

This chapter aims to offer an introduction to an interesting example of stochastic process, simple symmetric walk, with minimum formalism. In most bibliographic references, random walks and the classical problem of the gambler’s ruin are treated as applications of the theory of martingales (see Chapter 4) or of Markov chains [BAL 00, BRÉ 99, NOR 98]. An elementary approach is preferred, inspired by [FEL 68], which only requires a knowledge of discrete probabilities and counting. In the next chapter, we will see how certain properties of the random walk can be generalized to the more general class of processes formed by martingales.

Section 3.1 defines the simple symmetric random walk and examines its trajectories. Section 3.2 is dedicated to its long-term behavior. Section 3.3 introduces an initial application in financial mathematics, examining the gambler’s ruin problem. Finally, section 3.4 brings together exercises based on the different concepts examined in this chapter, with the solutions given in Chapter 8.

3.1. Trajectories of the random walk

It must be recalled that a stochastic process is a sequence of random variables indexed by time (cf. Definition 1.18); it makes it possible to model the temporal evolution of a phenomenon, such as the wealth of a gambler or the value of an asset portfolio, for example. The simple symmetric random walk is the simplest model for a discrete-time process.

Throughout the rest of this chapter, we will work on the probability ...