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ONE-DIMENSIONAL RANDOM WALKS

ONE-DIMENSIONAL RANDOM WALKS1. SIM P LERA NDO MWA LKDefinition walkon the integersZwith step distributionFand initial statex Zis a sequenceSnof RANDOM variables whose increments are independent, identically distributedrandom variables iwith common distributionF, that is,(1)Sn=x+n i=1 definition extends in an obvious way to RANDOM WALKS on thed dimensional integer lat-ticeZd: the increments are then randomd RANDOM walkonZdis the particularcase where the step distribution is the uniform distribution on the 2dnearest neighbors of theorigin; in one dimension, this is theRademacher-12distribution, the distribution that puts mass1/2 at each of the two values 1. The moves of a simple RANDOM walk in 1D are determined byindependent fair coin tosses: For each Head, jump one to the right; for each Tail, jump one tothe s RANDOM walk describes (among other things) the fluctuations in aspeculator s wealth when he/she is fully invested in a risky asset whose value jumps by either 1 in each time period.

Solve the gambler’s ruin problem for p q random walk by setting up and solving a difference equation. (Reformulate the difference equation as a matrix equation, and use this to represent the solution as a matrix multiplication. To get a simple formula …

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