This paper investigates Merton's portfolio problem in a rough stochastic environment described by Volterra Heston model. The model has a non-Markovian and nonsemimartingale structure. By considering an auxiliary random process, we solve the portfolio optimization problem with the martingale optimality principle. The optimal strategy is derived in a semi-closed form that depends on the solution of a Riccati-Volterra equation. Numerical studies suggest that investment demand decreases with the roughness of the market. 1 specific example in Abi Jaber et al. (2017). In addition, the rough Heston model becomes a special case of Volterra Heston model under the so-called fractional kernel, K(t) = t H−1/2 Γ(H+1/2) . The structure of characteristic functions in El Euch and Rosenbaum (2019) can be extended to affine Volterra processes using Riccati-Volterra equations as shown in Abi Jaber et al. (2017). Therefore, this paper focuses on the financial market with the Volterra Heston model.We are interested in a question: how does the roughness of the market volatility affect investment demands? We address this by investigating the optimal investment demand with the Merton problem as it is probably the most classic financial economic approach to do so. The literature tends to focus more on the option pricing problems and portfolio optimization under rough volatility models is still at an early stage. However, some recent works do exist (Fouque and Hu, 2018a,b;Bäuerle and Desmettre, 2018;Glasserman and He, 2019;Han and Wong, 2019). The studies on (Fouque and Hu, 2018a,b) consider the expected power utility portfolio maximization with slow or fast varying stochastic factors driven by the fOU processes whereas the fractional Heston model is used in Bäuerle and Desmettre (2018) with the same objective function. Due to some market insights, it is suggested in Glasserman and He (2019) to use roughness as a trading signal. To the best of our knowledge, portfolio selection with Volterra Heston model is firstly studied in Han and Wong (2019) in the context of mean-variance objective.In this paper, we investigate the Merton portfolio problem with an unbounded risk premium which is in contrast to the studies in (Fouque and Hu, 2018a,b) of assuming an essentially bounded risk premium. Therefore, their results are not directly applicable to our problem. Compared with Bäuerle and Desmettre (2018), we allow the correlation between stock and volatility to be non-zero, reflecting the well-known market leverage effect. Although the power utility maximization with the classic Heston model has been studied in Kraft (2005), the non-Markovian and non-semimartingale characteristic in Volterra Heston model prevents the use of the Hamilton-Jacobi-Bellman (HJB) framework for the classical model in Kraft (2005).To overcome the aforementioned difficulty, we apply the martingale optimality principle and construct the Ansatz, which is inspired by the martingale distortion transformation (Zariphopoulou, 2001;Fouque and Hu, 2018a) and the exponential-affine...
