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This thesis is motivated by the problem of modelling and analysing real share price data. Various approaches and models are considered. One approach is to consider a random walk on a discrete-time Markov chain perturbed by Gaussian noise as a model for real share price data. To implement this model a numerical algorithm is constructed to treat the NP hard Emdedding problem. A second approach to modelling share price data is to consider a random walk on the lamplighter group perturbed by Gaussian noise. This class of problems leads to interesting theoretical questions about asymptotic behaviour of random stochastic matrices. In particular, we found an asymptotic expression for the L 2 error between two independent random stochastic matrices. We apply a variety of statistical and modelling techniques to justify the models including traditional econometric transforms, regression and MLE techniques, EM algorithms, and Monte Carlo methods such as random search.