Robustness of Stochastic Optimal Control to Approximate Diffusion Models under Several Cost Evaluation Criteria

In stochastic control applications, typically only an ideal model is assumed, or learned from available incomplete data, based on which an optimal control is designed and then applied to the actual system. This gives rise to the problem of performance loss due to the mismatch between the actual system and the assumed system. A robustness problem in this context is to show that the error due to mismatch decreases to zero as the assumed system approaches the actual system. With this motivation, in this article, our goal is to study the continuity and robustness properties of finite horizon and infinite horizon discounted/ergodic cost problems for a large class of multidimensional controlled diffusions. We note that the problems of existence, uniqueness and verification of optimality of stationary Markov policies have been studied extensively in literature see e.g., [Bor-book], [HP09-book] (finite horizon) [BS86], [BB96] (discounted cost) [AA12], [AA13], [BG88I], [BG90b] (ergodic cost) and references therein. For a book-length exposition of this topic see e.g., [ABG-book].

In more explicit terms, here is the problem that we will study. For a precise statement please see Section 2.3. Suppose that our true model is represented as $(X,c)$ (see, e.g., Eq. 2.1), where $X$ is the true system model (representing the system evolution model via the drift and diffusion terms) and $c$ is the associated running cost function, and let $(X_{n},c_{n})$ (see, e.g., Eq. 2.15) be a sequence of approximating models $X_{n}$ with associated running cost functions $c_{n}$, such that as $n\to\infty$ approximating models $X_{n}$ converge to the true model $X$ in some sense to be precisely stated. Suppose that for each choice of control policy $U$ the associated total cost in true/approximating models are ${\mathcal{J}}(c,U),$ ${\mathcal{J}}_{n}(c_{n},U)$ respectively. The objective of the controller is to minimize the total cost over all possible admissible policies. If the optimal control policies of the true/approximating models are $v^{*}$, $v^{*n}$ respectively, the performance loss due to mismatch is given by $|{\mathcal{J}}(c,v^{*})-{\mathcal{J}}(c,v^{*n})|$. Thus the robustness problem in this context is to show that $|{\mathcal{J}}(c,v^{*})-{\mathcal{J}}(c,v^{*n})|\to 0$ as $n\to\infty$ . See Section 2.3. In this sense, our paper can be viewed as a continuous-time counterpart of the setting studied in [KY-20], [KRY-20].

This problem is of major practical importance and, accordingly, there have been many studies. Most of the existing works in this direction are concerned with the discrete time Markov decision process, see for instance [KY-20], [KRY-20], [BJP02], [KV16], [NG05] [SX15], and references therein.

We should note that the term robustness has various interpretations, contexts and solution methods. A common approach to robustness in the literature has been to design controllers that work sufficiently well for all possible uncertain systems under some structured constraints, such as $H_{\infty}$ norm bounded perturbations (see [basbern], [zhou1996robust]). For such problems, the design for robust controllers has often been developed through a game theoretic formulation where the minimizer is the controller and the maximizer is the uncertainty. In [DJP00], [jacobson1973optimal] the authors established the connections of this formulation to risk sensitive control. Using Legendre-type transforms, relative entropy constraints came in to the literature to probabilistically model the uncertainties, see e.g. [dai1996connections, Eqn. (4)] or [DJP00, Eqns. (2)-(3)]. Here, one selects a nominal system which satisfies a relative entropy bound between the actual measure and the nominal measure, solves a risk sensitive optimal control problem, and this solution value provides an upper bound for the original system performance. Therefore, a common approach in robust stochastic control has been to consider all models which satisfy certain bounds in terms of relative entropy pseudo-distance (or Kullback-Leibler divergence); see e.g. [DJP00, dai1996connections, dupuis2000kernel, boel2002robustness] among others. In order to quantify the uncertainty in the system models, other than the relative entropy pseudo-distance, various other metrics/criterion have also been used in the literature. In [tzortzis2015dynamic], for discrete time controlled models, the authors have studied a min-max formulation for robust control where the one-stage transition kernel belongs to a ball under the total variation metric for each state action pair. For distributionally robust stochastic optimization problems, it is assumed that the underlying probability measure of the system lies within an ambiguity set and a worst case single-stage optimization is made considering the probability measures in the ambiguity set. To construct ambiguity sets, [blanchet2016], [esfahani2015] use the Wasserstein metric, [erdogan2005] uses the Prokhorov metric which metrizes the weak topology, [sun2015] uses the total variation distance, and [lam2016] works with relative entropy. For fully observed finite state-action space models with uncertain transition probabilities, the authors in [iyengar2005robust], [nilim2005robust] have studied robust dynamic programming approaches through a min-max formulation. Similar work with model uncertainty includes [oksendal2014forward], [benavoli2011robust], [xu_mannor]. In the economics literature related work has been done in [hansen2001robust], [gossner2008entropy].

The robustness formulation we study has been considered in [KY-20], [KRY-20] for discrete-time models, where the authors studied both continuity of value functions as transition kernel models converge, as well as the robustness problem where an optimal control designed for an incorrect approximate model is applied to a true model and the mismatch term is studied. The solution approach is fundamentally different in the continuous-time analysis we present in this paper. In a related study [Dean18], the author studied the optimal control of systems with unknown dynamics for a linear quadratic regulator setup and proposes an algorithm to learn the system from observed data with quantitative convergence bounds. The author in [Lan81, Theorem 5.1] has considered fully observed discrete time controlled models and established continuity results for approximate models and gives a set convergence result for sets of optimal control actions, this set convergence result is inconclusive for robustness without further assumptions on the true system model (for more details see [KY-20]). For fully observed MDPs, [muller1997does] studied continuity of the value function under a general metric defined as the integral probability metric, which captures both the total variation metric or the Kantorovich metric with different setups (which is not weaker than the metrics leading to weak convergence). A recent study on game problems along a similar theme is presented in [subramanian2021robustness].

For control problems of MDPs with standard Borel spaces, the approximation methods through quantization, which lead to finite models, can be viewed as approximations of transition kernels, but this interpretation requires caution: indeed, [SaYuLi17, arruda2012, arruda2013], among many others, study approximation methods for MDPs where the convergence of approximate models is satisfied in a particularly constructed fashion. Reference [SaYuLi17] presents a construction for the approximate models through quantizing the actual model with continuous spaces (leading to a finite space model), which allows for continuity and robustness results with only a weak continuity assumption on the true transition kernel which, in turn, leads to the weak convergence of the approximate models. For both fully observed and partially observed models, a detailed analysis of approximation methods for continuous state and action spaces can be found in [SaLiYuSpringer] .

The literature on robustness of stochastic optimal control for continuous time system seems to be rather limited; see e.g., [GL99], [LJE15] [hansen2001robust] . In [GL99] the authors have considered the problem of controlling a system whose dynamics are given by a stochastic differential equation (SDE) whose coefficients are known only up to a certain degree of accuracy. For the finite horizon reward maximization problem, using the technique of contractive operators, [GL99] has obtained upper bounds of performance loss due to mismatch (or, “robustness index”) and has shown by an example that the robustness index may behave abnormally even if we have the convergence of the value functions. The associated discounted payoff maximization problem has been studied in [LJE15], where using a Lyapunov type stability assumption the authors have studied the robustness problem via a game theoretic formulation. For controlled diffusion models, the authors in [hansen2001robust] described the links between the max-min expected utility theory and the applications of robust-control theory, in analogy with some of the papers on discrete-time noted above adopting a min-max formulation. Along a further direction, for controlled diffusions, via the Maximum Principle technique, [PDPB02a], [PDPB02b], [PDPB02c] have established the robustness of optimal controls for the finite horizon payoff criterion.

In a recent comprehensive work [RZ21], the authors have studied the robustness of feedback relaxed controls for a continuous time stochastic exit time problem. Under sufficient smoothness assumptions on the coefficients (i.e, uniform Lipschitz continuity on the diffusion coefficients and uniform Hölder continuity on the discount factor and payoff function on a fixed bounded domain) they have established that a regularized control problem admits a Hölder continuous optimal feedback control and also they have shown that both the value function and the feedback control of the regularized control problem are Lipschitz stable with respect to parameter perturbations when the action space is finite. It is known that the optimal control obtained form the HJB equation (i.e. the argmin function) in general is unstable with respect to perturbations of coefficients; in practice, this would result in numerical instability of learning algorithms (as noted in [RZ21]).

Stability/continuity of solutions of PDEs with respect coefficient perturbations is a significant mathematical and practical question in PDE theory (see e.g. [WLS01], [SI72]). The continuity results established in this paper (see Theorems 3.3, 4.3, 4.8) will provide sufficient conditions which ensure stability of solutions of semilinear elliptic PDEs (HJB equations) in the whole space ${\mathds{R}^{d}}$.

Our robustness results also will be useful to the study of the robust optimal investment problems for local volatility models, e.g. given in [AS08, Remark 2.1] (also, see [KT12], [BDD20]) .

When the system noise is not given by a Wiener process, but it is given by a general wide bandwidth noise (or, a more general discontinuous martingales [LRT00]), the controlled process becomes a non-Markovian process even under stationary Markov policies. The general method of studying optimal control problem for such a system is to find suitable Markovian processes which approximate the non-Markovian process (see, [K90], [KR87], [KR87a], [KR88]). For wide bandwidth noise driven controlled systems [K90], [KR87], [KR87a], [KR88], diffusion approximation techniques were used to study stochastic optimal control problems. The results described in this paper are complementary to the above mentioned works on the diffusion approximation of wide bandwidth noise driven systems.

Contributions and main results. In the present paper, our aim is to study the continuity and robustness properties for a general class of controlled diffusion processes in ${\mathds{R}^{d}}$ for both infinite horizon discounted/ ergodic costs, where the action space is a (general) compact metric space. As in [KY-20], [KRY-20], in order to establish our desired robustness results we will use the continuity result as an intermediate step. For the discounted cost case, we will establish our results following a direct approach (under a relatively weaker set of assumptions on the diffusion coefficients, i.e., locally Lipschitz continuous coefficients). Using the results on existence and uniqueness of solutions of the associated discounted Hamilton Jacobi Bellman (HJB) equation and the complete characterization of (discounted) optimal policies in the space of stationary Markov policies (see [ABG-book, Theorem 3.5.6]), we first establish the continuity of value functions. Then utilizing this continuity of value functions, we derive a robustness result. The analysis of ergodic cost (or long-run expected average cost) is somewhat more involved. To the best of our knowledge there is no work on continuity and robustness properties of optimal controls for the ergodic cost criterion in the existing literature (for the discrete-time setup, see [KRY-20]). We have studied these ergodic cost problems under two sets of assumptions: In the first case, we assume that our running cost function satisfies a near-monotone type structural assumption (see, eq. Eq. 4.1, Assumption (A6)), and in the second case we assume Lyapunov type stability assumptions on the dynamics of the system (see Assumption (A7)) .

One of the major issues in analyzing the robustness of ergodic optimal controls under the near-monotone hypothesis is the non-uniqueness/restricted uniqueness of solutions of the associated HJB equation (see, [ABG-book, Example 3.8.3], [AA13]). It is shown in [ABG-book, Example 3.8.3] that the ergodic HJB equation may admit uncountably many solutions. Considering this, in [AA13, Theorem 1.1] the author has established the uniqueness of compatible solution pairs (see [AA13, Definition 1.1]). Exploiting this uniqueness result, under a suitable tightness assumption (on a certain set of invariant measures) we will establish the desired robustness result. Under the Lyapunov type stability assumption it is known that the ergodic HJB equation admits a unique solution in a certain class of functions, also the complete characterization of ergodic optimal control is known (see [ABG-book, Theorem 3.7.11] and [ABG-book, Theorem 3.7.12]) . Utilizing this characterization of optimal controls, we derive the robustness properties of ergodic optimal controls under a Lyapunov stability assumption.

We also emphasize the duality between the PDE approach vs. a probabilistic flow approach to study robustness. The PDE approach presents a very general and conclusive, yet concise and unified, approach for several cost criteria (notably, a probabilistic approach via Dynkin’s lemma would require separate arguments for discounted infinite-horizon and average cost infinite-horizon criteria) and such a unified approach had not been considered earlier, to our knowledge.

Thus, the main results of this article can be roughly described as follows.

• •

  For discounted cost criterion: We establish continuity of value functions and provide sufficient conditions which ensure robustness/stability of optimal controls designed under model uncertainties.

• •

  For ergodic cost criterion: Under two different sets of assumptions ((i) where the running cost is near-monotone or (ii) where a Lyapunov stability condition holds) we establish the continuity of value functions and exploiting the continuity results we derive the robustness/stability of ergodic optimal controls designed for approximate models applied to actual systems.

• •

  For finite horizon cost criterion: Under uniform boundedness assumptions on the drift term and diffusion matrices (of the true and approximating models), we establish continuity of value functions. Then exploiting the continuity result we prove the robustness/stability of optimal controls designed under model uncertainties.

• •

  For cost up to an exit time: Similar to the above criteria, under a mild set of assumptions we first establish the continuity of value functions and then using the continuity results we establish the robustness/stability of optimal controls designed under model uncertainties.

We will see that compared with the discrete-time counterpart of this problem studied in [KY-20] (discounted cost) and [KRY-20] (average cost), where value iteration methods were crucially used, in our analysis here we will develop rather direct arguments, with strong implications, utilizing regularity properties of value functions: In the discrete-time setup, these properties need to be established via tedious arguments whereas the continuous-time theory allows for the use of regularity properties of solutions to PDEs. Nonetheless, we will see that continuous convergence in control actions of models and cost functions is a unifying condition for continuity and robustness properties in both the discrete-time setup studied in [KY-20] (discounted cost) and [KRY-20] (average cost) and our current paper. Compared to [RZ21], in addition to the infinite horizon criteria we study, we note that the perturbations we consider do not involve only coefficient/parameter variations, i.e., we consider functional perturbations, and the action space we consider is uncountable, though we do not establish the Lipschitz property of control policies, unlike [RZ21].

The rest of the paper is organized as follows. Section 2 introduces the the problem setup and summarizes the notation. Section 3 is devoted to the analysis of robustness of optimal controls for discounted cost criterion. In Section 4 we provide the analysis of robustness of ergodic optimal control under two different sets of hypotheses (i) near-monotonicity (ii) Lyapunov stability. For the finite horizon cost criterion the robustness problem is analyzed in Section 5. The robustness problem for optimal controls up to an exit time is considered in Section 6.

Let $\mathbb{U}$ be a compact metric space and $\mathrm{V}=\mathscr{P}(\mathbb{U})$ be the space of probability measures on $\mathbb{U}$ with topology of weak convergence. Let

be given functions. We consider a stochastic optimal control problem whose state is evolving according to a controlled diffusion process given by the solution of the following stochastic differential equation (SDE)

Where

• •

  $W$ is a $d$-dimensional standard Wiener process, defined on a complete probability space $(\Omega,{\mathfrak{F}},\mathbb{P})$.

• •

  We extend the drift term $b:{\mathds{R}^{d}}\times\mathrm{V}\to{\mathds{R}^{d}}$ as follows:

  $$b(x,\mathrm{v})=\int_{\mathbb{U}}b(x,\zeta)\mathrm{v}(\mathrm{d}\zeta),$$

  for $\mathrm{v}\in\mathrm{V}$.

• •

  $U$ is a $\mathrm{V}$ valued process satisfying the following non-anticipativity condition: for $s<t\,,$ $W_{t}-W_{s}$ is independent of

  $${\mathfrak{F}}_{s}:=\,\,\mbox{the completion of}\,\,\,\sigma(X_{0},U_{r},W_{r}:r\leq s)\,\,\,\mbox{relative to}\,\,({\mathfrak{F}},\mathbb{P})\,.$$

The process $U$ is called an admissible control, and the set of all admissible controls is denoted by $\mathfrak{U}$ (see, [BG90]).

To ensure existence and uniqueness of strong solutions of Eq. 2.1, we impose the following assumptions on the drift $b$ and the diffusion matrix $\upsigma$ .

• (A1)

  Local Lipschitz continuity: The function $\upsigma\,=\,\bigl{[}\upsigma^{ij}\bigr{]}\colon\mathds{R}^{d}\to\mathds{R}^{d\times d}$, $b\colon{\mathds{R}^{d}}\times\mathbb{U}\to{\mathds{R}^{d}}$ are locally Lipschitz continuous in $x$ (uniformly with respect to the control action for $b$). In particular, for some constant $C_{R}>0$ depending on $R>0$, we have

  $$\lvert b(x,\zeta)-b(y,\zeta)\rvert^{2}+\lVert\upsigma(x)-\upsigma(y)\rVert^{2}\,\leq\,C_{R}\,\lvert x-y\rvert^{2}$$

  for all $x,y\in{\mathscr{B}}_{R}$ and $\zeta\in\mathbb{U}$, where $\lVert\upsigma\rVert:=\sqrt{\operatorname*{Tr}(\upsigma\upsigma^{\mathsf{T}})}$ . Also, we are assuming that $b$ is jointly continuous in $(x,\zeta)$.

• (A2)

  Affine growth condition: $b$ and $\upsigma$ satisfy a global growth condition of the form

  $$\sup_{\zeta\in\mathbb{U}}\,\langle b(x,\zeta),x\rangle^{+}+\lVert\upsigma(x)\rVert^{2}\,\leq\,C_{0}\bigl{(}1+\lvert x\rvert^{2}\bigr{)}\qquad\forall\,x\in\mathds{R}^{d},$$

  for some constant $C_{0}>0$.

• (A3)

  Nondegeneracy: For each $R>0$, it holds that

  $$\sum_{i,j=1}^{d}a^{ij}(x)z_{i}z_{j}\,\geq\,C^{-1}_{R}\lvert z\rvert^{2}\qquad\forall\,x\in{\mathscr{B}}_{R}\,,$$

  and for all $z=(z_{1},\dotsc,z_{d})^{\mathsf{T}}\in\mathds{R}^{d}$, where $a:=\frac{1}{2}\upsigma\upsigma^{\mathsf{T}}$.

By a Markov control we mean an admissible control of the form $U_{t}=v(t,X_{t})$ for some Borel measurable function $v:\mathds{R}_{+}\times{\mathds{R}^{d}}\to\mathrm{V}$. The space of all Markov controls is denoted by $\mathfrak{U}_{\mathsf{m}}$ . If the function $v$ is independent of $t$, then $U$ or by an abuse of notation $v$ itself is called a stationary Markov control. The set of all stationary Markov controls is denoted by $\mathfrak{U}_{\mathsf{sm}}$. From [ABG-book, Section 2.4], we have that the set $\mathfrak{U}_{\mathsf{sm}}$ is metrizable with compact metric under the following topology: A sequence $v_{n}\to v$ in $\mathfrak{U}_{\mathsf{sm}}$ if and only if

for all $f\in L^{1}({\mathds{R}^{d}})\cap L^{2}({\mathds{R}^{d}})$ and $g\in{\mathcal{C}}_{b}({\mathds{R}^{d}}\times\mathbb{U})$ (for more details, see [ABG-book, Lemma 2.4.1]) . It is well known that under the hypotheses (A1)–(A3), for any admissible control Eq. 2.1 has a unique strong solution [ABG-book, Theorem 2.2.4], and under any stationary Markov strategy Eq. 2.1 has unique strong solution which is a strong Feller (therefore strong Markov) process [ABG-book, Theorem 2.2.12].

In this section we analyze the robustness of optimal controls for discounted cost criterion. From [ABG-book, Theorem 3.5.6], we have the following characterization of the optimal $\alpha$-discounted cost $V_{\alpha}$ .

As in Theorem 3.1, following [ABG-book, Theorem 3.5.6], for each approximating model we have the following complete characterization of an optimal policy, which is in the space of stationary Markov policies.

In the next theorem, we prove that $V_{\alpha}^{n}(x)$ converges to $V_{\alpha}(x)$ as $n\to\infty$ for all $x\in{\mathds{R}^{d}}$ . This result will be useful in establishing the robustness of discounted optimal controls.