Empirical studies indicate the presence of multi-scales in the volatility of underlying assets: a fastscale on the order of days and a slow-scale on the order of months. In our previous works, we have studied the portfolio optimization problem in a Markovian setting under each single scale, the slow one in [Fouque and Hu, SIAM J. Control Optim., 55 (2017), 1990-2023, and the fast one in [Hu, Proceedings of the 2018 IEEE CDC, 5771-5776, 2018]. This paper is dedicated to the analysis when the two scales coexist in a Markovian setting. We study the terminal wealth utility maximization problem when the volatility is driven by both fast-and slow-scale factors. We first propose a zerothorder strategy, and rigorously establish the first order approximation of the associated problem value. This is done by analyzing the corresponding linear partial differential equation (PDE) via regular and singular perturbation techniques, as in the single-scale cases. Then, we show the asymptotic optimality of our proposed strategy by comparing its performance to admissible strategies of a specific form. Interestingly, we highlight that a pure PDE approach does not work in the multi-scale case and, instead, we use the so-called epsilon-martingale decomposition. This completes the analysis of portfolio optimization in both fast mean-reverting and slowly-varying Markovian stochastic environments.

Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non‐Markovian) fractional stochastic environment (for all values of the Hurst index H∈(0,1)). We rigorously establish a first‐order approximation of the optimal value, when the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein–Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth‐ order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.

Abstract. Motivated by the problem of estimating optimal feedback policy maps in stochastic control applications, we propose and analyze sequential design methods for ranking several response surfaces. Namely, given L ≥ 2 response surfaces over a continuous input space X , the aim is to efficiently find the index of the minimal response across the entire X . The response surfaces are not known and have to be noisily sampled one-at-a-time, requiring joint experimental design both in space and response-index dimensions.To generate sequential design heuristics we investigate Bayesian stepwise uncertainty reduction approaches, as well as sampling based on posterior classification complexity. We also make connections between our continuous-input formulation and the discrete framework of pure regret in multi-armed bandits. To model the response surfaces we utilize kriging metamodels. Several numerical examples using both synthetic data and an epidemics control problem are provided to illustrate our approach and the efficacy of respective adaptive designs.Key words. sequential design, response surface modeling, stochastic kriging, sequential uncertainty reduction, expected improvement 1. Introduction. A central step in stochastic control problems concerns estimating expected costs-to-go that are used to approximate the optimal feedback control. In simulation approaches to this question, costs-to-go are sampled by generating trajectories of the stochastic system and then regressed against current system state. The resulting Q-values are finally ranked to find the action that minimizes expected costs.When simulation is expensive, computational efficiency and experimental design become important. Sequential strategies rephrase learning the costs-to-go as another dynamic program, with actions corresponding to the sampling decisions. In this article, we explore a Bayesian formulation of this sequential design problem. The ranking objective imposes a novel loss function which mixes classification and regression criteria. Moreover, the presence of multiple stochastic samplers (one for each possible action) and a continuous input space necessitates development of targeted response surface methodologies. In particular, a major innovation is modeling in parallel the spatial correlation within each Q-value, while utilizing a multi-armed bandit perspective for picking which sampler to call next.To obtain a tractable approximation of the Q-values, we advocate the use of Gaussian process metamodels, viewing the latent response surfaces as realizations of a Gaussian random field. Consequently, the ranking criterion is formulated in terms of the posterior uncertainty about each Q-value. Thus, we connect metamodel uncertainty to the sampling decisions, akin to the discretestate frameworks of ranking-and-selection and multi-armed bandits. Our work brings forth a new link between emulation of stochastic simulators and stochastic control, offering a new class of approximate dynamic programming algorithms.

Empirical studies indicate the existence of long range dependence in the volatility of the underlying asset. This feature can be captured by modeling its return and volatility using functions of a stationary fractional Ornstein-Uhlenbeck (fOU) process with Hurst index H ∈ ( 1 2 , 1). In this paper, we analyze the nonlinear optimal portfolio allocation problem under this model and in the regime where the fOU process is fast mean-reverting. We first consider the case of power utility, and rigorously give first order approximations of the value and the optimal strategy by a martingale distortion transformation. We also establish the asymptotic optimality in all admissible controls of a zeroth order trading strategy. Then, we consider the case with general utility functions using the epsilon-martingale decomposition technique, and we obtain similar asymptotic optimality results within a specific family of admissible strategies..Here, ǫ is a small parameter to make the process Y ǫ,H t fast-varying and its natural time scale to be of order ǫ (that is, its mean-reversion time scale proportional to ǫ), and W (H) t is a fractional Brownian motion

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