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Sub- and Super-solution Approach to Accuracy Analysis of Portfolio Optimization Asymptotics in Multiscale Stochastic Factor Market

The problem of portfolio optimization when stochastic factors drive returns and volatilities has been studied in previous works by the authors. In particular, they proposed asymptotic approximations for value functions and optimal strategies in the regime where these factors are running on both slow and fast timescales. However, the rigorous justification of the accuracy of these approximations has been limited to power utilities and a single factor. In this paper, we provide an accurate analysis for cases with general utility functions and two timescale factors by constructing sub- and super-solutions to the fully nonlinear problem so that their difference is at the desired level of accuracy. This approach will be valuable in various related stochastic control problems.


Introduction
The study of the portfolio optimization problem in continuous time has a long history dating back to [14,15]. Specifically, an agent aims to maximize her expected utility of the terminal wealth when the investment is divided between risky assets and a riskless asset, with particular types of utility functions. In Merton's seminal work, the risky assets are assumed to follow the Black-Scholes model, where the expected returns and volatilities are constant. Since then, the problem has been studied extensively in various settings and levels of generality, including considering transaction costs [4,10], stochastic volatility models for risky assets (see, for instance [9], and references therein), and various trading constraints (see [16] for a survey).
In this paper, we consider the portfolio optimization problem in a stochastic environment, where both the expected return µ and volatility σ of a stock price S are driven by two factors Y t and Z t evidenced in empirical study [8]: Here W is a standard Brownian motion. The two factors are characterized by the small parameters ε and δ, the fast timescale being represented by ε, and the slow timescale by 1/δ. Along this direction, results have been developed in [9] about asymptotic expansions of the optimal trading strategy and the maximized utility when the utility function is general, as both ε and δ tend to zero. However, the accuracy of approximation was rigorously justified only in the cases with power utilities and one stochastic factor, when a distortion transformation of Hopf-Cole type introduced in [17] enables a reduction to a linear PDE problem. This is only possible in the case of multiple factors in special cases; see [1]. After that, several attempts have been made to partially justify these expansions in more general cases, e.g., in [5,12,6]; see also [11] for a comprehensive review. The technique of sub-and super-solutions to prove the accuracy of asymptotic approximations is used in [2] for a model of power utility maximization with two nearly correlated factors corresponding to a regular perturbation problem, and in [3] for optimal investment under stochastic volatility and transaction costs. The contribution of this paper is to rigorously justify the heuristic expansion provided in [9]. The methodology presented here can be adapted to the derivation of accuracy results in various contexts, as in [7] for instance, where fast mean-reversion is shown in data and the corresponding control problem appears as a Ricatti equation with fast mean-reverting random coefficients. Proof of accuracy of an approximate solution to this singular perturbation problem was obtained by constructing sub-and super-solutions.
We shall construct two functions close to the value function of the problem. These two functions act as lower and upper bounds, namely sub-and super-solutions of this problem. By requiring certain properties for each function and, most importantly, by constructing their difference to reach the desired order, one can show that the asymptotic approximations derived in [9] are rigorous under general utility. The requirement for the sub-solution is rather relaxed, and we shall work with a particular zeroth-order strategy. Since the supersolution acts as an upper bound of the problem, we need to ensure the property holds for all admissible strategies.
The rest of this paper is organized as follows. In Section 2, we first introduce the Merton problem in a multiscale stochastic environment and the associated Hamilton-Jacobi-Bellman (HJB) equation, and we describe the main result of the paper, Theorem 2.1. In Section 2.1, we briefly explain our approach which consists in constructing sub-and supersolutions so that their difference is at the desired level. The asymptotic approximations derived in [9] are reviewed in Section 2.2. Section 2.3 summarizes standing assumptions in this paper and some preliminary estimates which facilitate the asymptotic analysis in Section 3. Section 3 is dedicated to the construction of the sub-and super-solutions, thus completing the proof of Theorem 2.1. We make concluding remarks in Section 4.

Merton Problem under Multiscale Stochastic Environment
We consider the utility maximization problem on the finite horizon [0, T ] with general terminal utility U, where the (single) underlying asset S t is driven by two factors: one fast mean-reverting Y t , and one slowly varying Z t : Here W, W Y , W Z are standard Brownian motions on a filtered probability space (Ω, F , P) that are correlated: where |ρ 1 | < 1, |ρ 2 | < 1, |ρ 12 | < 1 and 1 + 2ρ 1 ρ 2 ρ 12 − ρ 2 1 − ρ 2 2 − ρ 2 12 > 0 to ensure the positive definiteness of the covariance matrix of (W, W Y , W Z ). The time scales of Y t and Z t are described by the small positive parameters ε and δ respectively. We shall assume that the process Y t is ergodic and has a unique invariant distribution Φ which is independent of ε as its infinitesimal generator is of the form ε −1 L y . Further assumptions on the model parameters µ, σ, b, a, c, g will be list in Section 2.3. In particular, they will imply that the system (2.1)-(2.3) has a unique strong solution.
Let X π t be a wealth process with π t being the dollar amount invested in the underlying asset at time t and the remaining held in a money market earning interest at rate r. Under the self-financing assumption, and without loss of generality, assuming r = 0, the wealth process follows: We are interested in the utility maximization of the terminal wealth: for all admissible strategies π (see Definition 2.4), and where U(·) is a general utility function satisfying Assumption 2.3.
Restricting the problem to Markovian strategies of the form π(t, x, y, z), we denote by V ε,δ (t, x, y, z) the value function (2.5) As described in [9], the Hamilton-Jacobi-Bellman (HJB) PDE associated to this problem is, sup π Q π [V ε,δ ](t, x, y, z) = 0, V ε,δ (T, x, y, z) = U(x), (2.6) where Q π = ∂ t + L π and L π is the infinitesimal generator of (X π t , Y t , Z t ) for any Markovian strategy π(t, x, y, z). Specifically, We stress the fact that, in general, it is not known if the PDE (2.6) admits a classical solution.
The formal asymptotic expansion in the regime where (ε, δ) are both small, performed in [9,Section 4] (1,0) and v (0,1) are functions of (t, x, z) given here in Section 2.2. The main result of this paper is to justify this asymptotic expansion, i.e., to prove the following theorem.

Methodology
The method for proving Theorem 2.1 is to construct two functions V ± (t, x, y, z) as suband super-solutions, whose asymptotic expanded terms of the orders up to O( 1) . Specifically, we aim to find a function V − for the subsolution such that, for all (x, y, z) and sufficiently small (ε, δ), the following requirements are satisfied: (R1) The function value V − (T, x, y, z) is dominated by U(x); (R2) The process V − (t, X π (0) t , Y t , Z t ) along a zero-order strategy, denoted by π (0) , is a submartingale.
Thus, by the definition of V ε,δ (t, x, y, z), (R1) and (R2), one can obtain: Then, we aim to find a function V + for the super-solution such that, for all (x, y, z) and sufficiently small (ε, δ), the following requirements are satisfied: (R3) The function value V + (T, x, y, z) dominates U(x); x, y, z) exists and is non-positive; (R5) The Itô integrals are true martingales, for any admissible π.
Then, as in the argument used for (2.9), one can deduce: Taking the supremum over all admissible π on both sides of (2.10) gives V ε,δ (t, x, y, z) ≤ V + (t, x, y, z). Combining (2.9) and (2.10) gives In fact, the next order terms after can be replaced by O(ε + δ) and (2.8) follows. Our choice of V ± takes the following form: where (N A , N B , N C ) are functions of (t, x, z), and (F, G, H) are functions of (t, x, y, z).
The intuition for such form is the following: (a) functions w (2,0) , w (3,0) , w (2,1) are added to eliminate terms of O(1), O( √ ε) and O( √ δ) when applying the operator Q π to V ± ; (b) (N A , N B , N C ) and (F, G, H) helps to fulfill (R2) and (R4); (c) the coefficient 2T − t is for (R1) and (R3). In Section 3, we shall show how these functions are determined and why they can be chosen as functions of particular variables such that the requirements (R1)-(R5) are satisfied. In the rest of this section, we briefly review the existing derivations of v (0) , v (1,0) , v (0,1) , the definition of w (2,0) , w (3,0) , w (2,1) , the standing assumptions in this paper, and some preliminary estimates.

Multiscale asymptotic expansions
Generally, closed-form solutions are barely available for HJB equations. In our setup, we do not even know if V ε,δ solves (2.6) in the viscosity sense. In [9], a first-order expansion of V ε,δ around small (ε, δ) is formally derived via singular and regular perturbation techniques. Since the formulas of these terms and the equations they satisfy play an important role in proving our main theorem, we summarize them below for readers' convenience. For detailed derivations, we refer the readers to [9, Section 4] and [6, Sections 2 and 3].
The combined expansion in slow and fast scales of V ε,δ is of the following form: where the superscript of v corresponds to the powers in √ ε and √ δ and where v (0,0) has been rewritten as v (0) . To precisely give the equations which identify these terms, we introduce the following notations, following [9]. Denote by · the average with respect to the ε-independent invariant distribution Φ of Y : g = g(y) Φ(dy), and by M(t, x; λ) the solution to the classical Merton PDE where µ and σ are constants: We define the associate risk-tolerance function and the differential operators: We denote the "square-averaged" Sharpe ratio λ(z) = λ 2 (·, z) , and the version of D k (λ) that will be used in the sequel is D k (λ) = R(t, x; λ(z)) k ∂ k x . We shall use D k for brevity (omitting the argument λ). We also define the averaged Sharpe ratio: λ(z) = λ(·, z) . Now we are ready to present the formulations of v (0) , v (1,0) and v (0,1) .
Since it possesses a unique solution (see [6, Proposition 2.2]), we have v (0) (t, x, z) = M(t, x; λ(z)). (2.12) (ii) The first order correction in the fast variable v (1,0) is defined as the solution to the linear PDE: where B(z) = λ(·, z)a(·)∂ y θ(·, z) , and θ is a solution of the Poisson equation (2.14) (iii) The first order correction in the slow variable v (0,1) is defined as the solution to the linear PDE: The z-derivatives of the leading order term v (0) and the risk-tolerance function R satisfy x; λ(z)).
In the sequel, when deriving the concrete formula of V ± , we shall choose w (2,0) , w (3,0) and w (2,1) to be the corresponding terms in the expansion of V ε,δ with C i (t, x, z) ≡ 0, i = 1, 2, 3. That is, we choose

Model assumptions and preliminary estimates
We first make precise the regularity assumptions on the utility function U, on the risk tolerance −U ′ /U ′′ , and on the inverse marginal utility (U ′ ) (−1) . They will be satisfied by mixtures of power utilities or sums of inverse marginal power utilities for instance, and we refer to Appendix A in [5] for further details. The advantage of these mixtures is that the Arrow-Pratt risk aversion (−xU ′′ /U ′ ) is wealth dependent as opposed to constant for pure power utilities.
Assumption 2.3. We make the following assumptions on the utility function U(x): 1. U(x) is C 9 (0, ∞), strictly increasing, and strictly concave and satisfies the following conditions (Inada and asymptotic elasticity): , and there exists K ∈ R + , such that for x ≥ 0, and 2 ≤ i ≤ 7, 3. Define the inverse function of the marginal utility U ′ (x) as I : R + → R + , I(y) = U ′(−1) (y), and assume that, for some positive α, I(y) satisfies the polynomial growth condition: Now, we make precise the definition of a Markovian admissible strategy.
Definition 2.4 (Admissibility). A Markovian strategy π is admissible if X π t stays positive a.s. for all t ∈ [0, T ] and Next, we make the following technical assumptions on the model parameters and the various quantities appearing in our expansion. In particular, it involves several Poisson equations for which we assume that the solutions are bounded.
1. For any starting points (s, y, z) and fixed (ε, δ), the system of SDEs (2.1)-(2.2)-(2.3) has a unique strong solution (S t , Y t , Z t ). The process Y is ergodic and has a unique invariant distribution Φ (independent of ε).
With all the notations and assumption introduced, we obtain the following propositions by lengthy but straightforward calculations. We omit the proofs here and refer to [5,6]. Proposition 2.6. Under the above assumptions, the functions v (i,j) and w (i,j) , i + j > 0 (cf. (2.14), (2.15), (2.17), (2.18) and (2.19)) satisfy v (i,j) ≤ h(y, z)D 1 v (0) , with a bounded function h(y, z), where h(y, z) denotes a bounded function in y and z, and may vary from case to case. Similar inequalities hold for w (i,j) . In particular, one has with a bounded function h(z).

Proof of the Main Theorem
Recall the sub-and super-solution we shall construct are of the form This section is dedicated to identify the terms (N A , N B , N C ) which will be functions of (t, x, z) and (F, G, H) which will be functions of (t, x, y, z) in V ± .

Proof. From (2.4) and (3.2), we have
Using that 0 ≤ R(t, x; λ(z)) ≤ K 0 x, and the boundedness of λ, one deduces that X π (0) t is a proper exponential and stays positive. Moreover, it has p th -moments for any p ∈ N, uniformly in t ∈ [0, T ]. Next we observe that the condition (2.20) applied to π (0) reduces to the condition (2.21) after using the boundedness of λ. Thus it suffices to verify (2.21) when π = π (0) . To this end, we recall the H-transform used in [13].

Consequently, equation (3.5) is bounded by
which is finite as we have shown that X π (0) t has bounded moments of any other. Therefore, π (0) is admissible.
For the derivations presented below, the submartingality requirement (R2) will enable us to pin down certain formulas for (N A , N B , N C ) and (F, G, H) up to some constants (C A , C B , C C ). Then with sufficiently large choices of (C A , C B , C C ) and sufficiently small (ε, δ), (R1) is fulfilled.
For the item (R2-1) we first require that I ε , I εδ and I δ are strictly positive. This will enable us to determine the forms of (N A , N B , N C ) up to some constants and the formulas of (F, G, H). Precisely speaking, the form of N A is determined by the necessary conditions I ε > 0, and F is the solution to I ε − I ε = 0. N B , N C , G, H are determined in a similar manner, and we present the detailed computations for the pair (N A , F ) as follows.
where B 1 (z) is define in (2.22). Note that all terms in (3.7) except the last one are bounded by a multiple of D 1 v (0) by Assumption 2.5 and Proposition 2.6. Therefore, we can choose Then the last term in (3.7) becomes Thus, the choice (3.8) indeed does the job of making I ε positive for sufficiently large C A . We next derive the formula for F . Identifying F as the solution to I ε − I ε = 0, we have which yields a formula for F : By the Vega-Gamma relations (2.16), Assumption 2.5 and Proposition 2.6, it suffices to choose a large N B so that I δ is strictly positive. For identifying G from I δ − I δ = 0 , we first write down and then obtain for G: For terms of O( √ εδ), a similar derivation yields and where θ 6 , θ 7 and θ 8 solve the Poisson equations (2.25), (2.26) and (2.27), respectively. The next step is to ensure that Q π (0) [V − ] ≥ 0, i.e. the higher order terms in (3.6) are indeed negligible and can be dominated by terms of O(ε+δ). By straightforward but tedious calculation, with Assumption 2.5, and Propositions 2.6 and 2.7 we can verify that all terms higher than O(ε + δ) can be bounded f (y, z, C i )D 1 v (0) , where the function f (y, z, C i ) is bounded in (y, z) and linear in C i , for i = A, B, C. This is because (F, G, H) contribute to higher order terms and their formulas contain linear functions in (C A , C B , C C ). On the other hand, we have I ε = I ε ≥f (y, z) , which are the coefficients at order ε, δ and √ εδ. The functionf (y, z) is also bounded in (y, z) and may vary from case to case, but free of (C A , C B , C C ). More precisely, one has the following Therefore, one can first choose ε < ε ′ and δ < δ ′ such that the coefficients of C i are positive, then for C i > C ′ i , i = A, B, C, Q π (0) [V − ] is always non-negative. We now take care of (R2-2) . With our choice of N i = C i D 1 v (0) for i = A, B, C and choice of (F, G, H) (cf. equations (3.9), (3.10), (3.11)), we observe that terms in V − x π (0) σ = λD 1 V − are all of the form h(y, z)Dv (0) , with D being the following operators: Under model assumptions, and with Propositions (2.6) and (2.7) we deduce that h(y, z) is always bounded and Dv (0) can be bounded by a multiple of D 1 v (0) . Therefore, for t 0 V − x π (0) σ dW t being a martingale, we essentially require that t 0 D 1 v (0) dW t is square integrable, which is fulfilled by Lemma 3.1 and (2.21). Repeating this argument with similar calculation for  − ε 2 F (T, x, y, z) − ε 3/2 √ δH(T, x, y, z) − εδG(T, x, y, z), which reads as Since all θ i and g are bounded functions and with Proposition 2.6, we can again first choose ε < ε ′′ and δ < δ ′′ such that the coefficients of C i are negative, then for C i > C ′′ i , i = A, B, C the above inequality hold.

Super-solution
Similar to the derivation in Section 3.1, we shall first take care of (R4). This will allow us to derive the form of N i in V + which are also given by With such forms, the requirement (R5) is shown as a consequence of Definition 2.4, and (R3) will be satisfied with sufficient large choices of C i and sufficiently small (ε, δ).
As in the sub-solution case, we first show that with the choice (3.13), I ε , I δ and I εδ can be strictly negative. Then by letting (F, G, H) be the solution of (I ε , I δ , I εδ ) − ( I ε , I δ , I εδ ) = 0, we have terms at O(ε + δ) are strictly negative. The computation and reasoning are very similar to the sub-solution case, thus we omit here and summarize the results.
Regarding O(ε) terms, one has I ε = −N A + L t,x (λ)w (2,0) + ρ 1 aλD 1 w (3) and f (y, z) is bounded in (y, z) and at most linearly growth in C i . Thus, one can first

Conclusion
This paper provides the accuracy analysis of asymptotics for the portfolio optimization problem with general utility functions and two (fast and slow) stochastic factors. This sets up the theoretical foundation of using asymptotic expansion to derive approximations for value functions and optimal strategies in the regime where these factors are running on both slow and fast timescales. Specifically, we construct the sub-and super-solutions to the fully nonlinear problem so that their difference is at the desired level of accuracy.
In the present context, the fast varying factor requires a careful treatment of the singular perturbation for a fully nonlinear equation. Moreover, the proofs presented here can be adapted to justify other derivations of accuracy in various contexts as in [7] for instance.