The thesis deals with the mirror and synchronous couplings of geometric Brownian motions, the policy improvement (or iteration) algorithm in completely continuous settings, and an application where the latter is applied to the former. First we investigate whether the mirror and synchronous couplings of Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We prove (via Bellman's principle) that this is indeed the case in the infinite horizon and ergodic average problems, but not necessarily in the finite horizon and exponential efficiency problems, for which we characterise when the two couplings are suboptimal. Then we describe the policy improvement algorithm for controlled diffusion processes in the framework of the discounted infinite horizon problem, both in one and several dimensions. Under some assumptions on the data of the problem, we prove that the algorithm yields a sequence of Markov policies such that its accumulation point is an optimal policy, and that the corresponding payoff functions converge monotonically to the value function. We use no discretisation procedures at any stage. We show that a large class of data satisfies the assumptions, and an example implemented in Matlab demonstrates that the convergence is numerically fast. Next we study the policy improvement algorithm for continuous finite horizon problem. We obtain analogous results as for the infinite horizon problem. Finally we apply the algorithm to a certain sequence of data to approximate the value function of the (partially unsolved) finite horizon problem for geometric Brownian motions.