  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.
