IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes
Consider the random walk S_n = X_1 + ... + X_n, where the X_i are independent and identically distributed with P(X_i = +1) = 0.6 and P(X_i = -1) = 0.4, and S_0 = 0. Which statement is correct?
S_n is a Markov chain with E[S_n] = 0.2n. The next value depends only on the current value plus an independent step, and each step has mean 0.6 minus 0.4, which is 0.2, so the mean grows linearly.
- AS_n is not Markov because it depends on all past steps
- BS_n is a Markov chain, and E[S_n] = 0.2nCorrect
- CS_n is a Markov chain, and E[S_n] = 0.6n
- DS_n is a Markov chain with a finite state space
- S_n has stationary increments, so E[S_n] = 0 for all n
Explanation
Given S_n, the next value is S_n plus an independent step, so the process is Markov. E[X_i] = 0.6 - 0.4 = 0.2, so E[S_n] = 0.2n. The state space is all integers, which is infinite, and the mean is not zero.
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