This thesis investigates Merton's portfolio problem under two different rough Heston models, which have a non-Markovian structure. The motivation behind this choice of problem is due to the recent discovery and success of rough volatility processes. The optimisation problem is solved from two different approaches: firstly by considering an auxiliary random process, which solves the optimisation problem with the martingale optimality principle, and secondly, by a finite dimensional approximation of the volatility process which casts the problem into its classical stochastic control framework. In addition, we show how classical results from Merton's portfolio optimisation problem can be used to help motivate the construction of the solution in both cases. The optimal strategy under both approaches is then derived in a semi-closed form, and comparisons between the results made. The approaches discussed in this thesis, combined with the historical works on the distortion transformation, provide a strong foundation to build models capable of handling increasing complexity demanded by the ever growing financial market.
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