Parameter learning: stochastic optimal control approach with reinforcement learning

Zhou (2023) commented that ”We strive to seek optimality, but often find ourselves trapped in bad optimal, solutions that are either local optimizers, or are too rigid to leave any room for errors”. Indeed, exploration through randomization brings a way to break this curse of optimality. In this present study, we try to balance the model based and model free methods, that is we aim to use the reinforcement learning structure to explore the unknown parameters appeared in the model based method.

Recently, Wang et al. (2020) first considered the reinforcement learning in continuous time and spaces, and an exploratory formulation for the dynamics equation of the state and an entropy-regularized reward function were introduced. Then, Wang and Zhou (2020) solved the continuous time mean variance portfolio selection problem under the reinforcement learning stochastic optimal control framework. Tang et al. (2022) considered the exploratory Hamilton-Jacobi-Bellman (HJB) equation formulated by Wang et al. (2020) in the context of reinforcement learning in continuous time, and established the well-posedness and regularity of the viscosity solution to the HJB equation. Gao et al. (2022) considered the temperature control problem for Langevin diffusions in the context of nonconvex optimization under the regularized exploratory formulation developed by Wang et al. (2020). Differing from the continuous-time entropy-regularized reinforcement learning problem of Wang et al. (2020), Han et al. (2023) proposed a Choquet regularizer to measure and manage the level of the exploration for reinforcement learning.

For the classical stochastic optimal control theory and related applications, we refer the readers to monographs Fleming and Rishel (1975); Yong and Zhou (1999); Fleming and Soner (2006). For the recursive stochastic optimal control problem and related dynamic programming principle, see Pardoux and Peng (1990); Peng (1990, 1992). Well-known that the classical optimal control theory is applied to solve the mean-variance model, see Zhou and Li (2000); Basak and Chabakauri (2010); Björk et al. (2014); Dai et al. (2021). When we consider the applications of the classical optimal control problem, for example in the mean-variance investment portfolio problem, we first need to estimate the parameters appeared in the model. Some statistic methods could be employed such as moment estimation, maximum likelihood estimation which heavily depend on the stringent assumptions on the observed samples. Furthermore, based on the observations (historical data), it is difficult to estimate the time varying parameter appeared in the model based problems. The reinforcement learning algorithm has been widely used to study optimization, engineering and finance, etc. In particular, Wang et al. (2020) first introduced the reinforcement learning in continuous time stochastic optimal control problem.

However, it is important to find a better estimation for the parameter appeared in the model. Based on the estimation of the parameter, we can do empirical and time trend analysis. Therefore, in this study, we first show how to use a stochastic optimal control approach with reinforcement learning structure to learn the unknown parameters appeared in the dynamic equation. We consider the following stochastic optimal control problem with unknown deterministic parameter $\beta(\cdot)$,

where $b,\sigma$ are given deterministic function, $W(\cdot)$ is a Brownian motion, and $u(\cdot)$ is an input control. In the input control $u(\cdot)$, we can observe the output state $X^{u}(\cdot)$. In the classical mean-variance investment portfolio framework, the parameter $\beta(\cdot)$ in (1.1) can be used to describe the mean and volatility of risky asset. Thus, it is useful to obtain the estimation of the unknown parameter $\beta(\cdot)$.

Now, we show the details of the stochastic optimal control approach with reinforcement learning structure for learning the unknown parameters. Step 1: Based on equation (1.1), by choosing an appropriate cost functional, we obtain a feedback optimal control $u^{*}(\cdot)$ which contains the unknown parameter $\beta(\cdot)$, denote by $u^{*}_{s}=u^{*}(\beta_{s},s,X^{*}_{s}),\ t\leq s\leq T$. Step 2: We take place the unknown parameter $\beta(\cdot)$ in the optimal control $u^{*}(\cdot)$ as a new deterministic control $\rho(\cdot)$, and the input control is denoted by $u^{\rho}_{s}=u^{\rho}(\rho_{s},s,X^{\rho}_{s}),\ t\leq s\leq T$. Step 3: We put the new control $u^{\rho}(\cdot)$ into equation (1.1) and the related cost functional, and establish a new optimal control problem. Indeed, the optimal control of the new exploratory control problem can be used to estimate the unknown parameters. Therefore, we consider the linear stochastic differential equation case where the drift or diffusion term contains the unknown parameter, and the general case where both the drift and diffusion terms contain unknown parameters. For the above cases, we find that the optimal density function can be used to learn the unknown parameters.

In this present paper, we focus on develop the theory of a stochastic optimal control approach with reinforcement learning structure to learn the unknown parameters appeared in the equation (1.1). For the application of this new theory, we leave it as the future work. However, we refer the reader to the following references for the study of the algorithm of the optimal control. A unified framework to study policy evaluation and the associated temporal difference methods in continuous time was investigated in Jia and Zhou (2022a). Jia and Zhou (2022b) studied the policy gradient for reinforcement learning in continuous time and space under the regularized exploratory formulation developed by Wang et al. (2020). Dong (2022) studied the optimal stopping problem under the exploratory framework, and established the related HJB equation and algorithm. Furthermore, Jia and Zhou (2022c) introduced a q-learning in continuous time and space. For the policy evaluation, we follow Doya (2000) for learning the value function. An introduction of reinforcement learning could be found in Sutton and Barto (2018), and the deep learning and related topic see Goodfellow et al. (2016).

The main contributions of this study are threefold:

(i). To learn the unknown parameters appeared in the dynamic equation of the state, we first develop a stochastic optimal control approach with reinforcement learning structure.

(ii). By choosing an appropriate cost functional, based on a classical optimal feedback control, we translate the original optimal control problem into a new control problem, in which the related optimal control is used to estimate unknown parameter. Based on the obtained estimations of the parameters, we can do empirical analysis for a given model.

(iii). We consider the linear stochastic differential equation case where the drift or diffusion term contains unknown parameter, and the general case where both the drift and diffusion terms contain unknown parameters. We show that the optimal density function satisfies a Gaussian distribution which can be used to learn the unknown parameters.

The remainder of this study is organized as follows. In section 2, we formulate the stochastic optimal control approach with reinforcement learning structure, and show how to learn the unknown parameter. Then, we investigate the exploratory HJB equation for the stochastic optimal control approach in Section 3. In Section 4, we consider the problem of the Linear SDE case, where the drift or diffusion term is allowed to contain the unknown parameter. Furthermore, in Section 5, we general the model in Section 4 to the case where both the drift and diffusion terms contain the unknown parameters. Then, we give an example to verify the main results. Finally, we conclude this study in Section 6.

Given a probability space $(\Omega,\mathcal{F},P)$, Brownian motion $\{W_{t}\}_{t\geq 0}$, and filtration $\{\mathcal{F}_{t}\}_{t\geq 0}$ generated by $\{W_{t}\}_{t\geq 0}$, where $\mathcal{F}=\mathcal{F}_{T}$ with a given terminal time $T$. We introduce the following stochastic differential equation with the deterministic unknown parameter $\beta_{s}\in\mathbb{R},\ t\leq s\leq T$, and control $u(\cdot)\in\mathcal{U}[t,T]$,

where $\mathcal{U}[t,T]$ is the set of all progressing measurable, square integrate processes on $[t,T]$, and the processes take value in a Euclidean space. Based on the state $\{X^{u}_{s}\}_{t\leq s\leq T}$, we consider the running and terminal cost functional,

and the target is to minimize $J(t,x;u(\cdot))$ over $u(\cdot)\in\mathcal{U}[t,T]$, that is

The conditions on functions $b,\sigma,f$ and $\Phi$ in equations (2.1) and (2.2) will be given later. Under mild conditions, we can show that there exists a feedback optimal control $u^{*}_{s}=u^{*}(\beta_{s},s,X^{*}_{s}),\ t\leq s\leq T$ (see Yong and Zhou (1999) for further details), where $X^{*}$ is the optimal state with the optimal control $u^{*}$.

Now, we give the assumptions of this study. The functions $b,\sigma,f$ and $\Phi$ are continuous on all the variables.

In practical analysis, we always assume the form of the functions $b,\sigma,f$ and $\Phi$ and leave room for estimating the unknown parameter $\beta(\cdot)$. Some classical statistics estimations are implemented, for example moment estimation, maximum likelihood estimation, etc. However, these estimation methods are heavily based on the observations. In this study, we aim to develop a new estimations method based on the explicit feedback control $u^{*}(\cdot)$.

It should be note that we can observe the value of the state $X^{u}(\cdot)$ with a given control $u(\cdot)$, but not the unknown parameter $\beta(\cdot)$. In the following, we develop the model used to estimate the parameter $\beta(\cdot)$. Since we don’t know the value of the parameter $\beta(\cdot)$, we take place $\beta(\cdot)$ as a new deterministic control process $\rho(\cdot)$ in the feedback optimal control $u^{*}(\beta_{s},s,X^{*}_{s})$, and denotes as $u^{\rho}_{s}=u^{*}(\rho_{s},s,X^{\rho}_{s})$, where $X^{\rho}_{s}$ satisfies the following stochastic differential equation,

where $\hat{b}(\beta_{s},X_{s}^{\rho},\rho_{s})={b}(\beta_{s},X_{s}^{\rho},u^{\rho}_{s})$ and $\hat{\sigma}(\beta_{s},X_{s}^{\rho},\rho_{s})={\sigma}(\beta_{s},X_{s}^{\rho},u^{\rho}_{s})$. Based on the above given assumptions, we have the following unique results for the solution of equation (2.3).

Furthermore, we rewrite the cost functional (2.2) as follows,

where $\rho_{s}\in\mathbb{R}$, $\hat{f}(X_{s}^{\rho},\rho_{s})=f(X_{s}^{\rho},u^{\rho}_{s})$, and $\hat{\Phi}(X_{T}^{\rho})={\Phi}(X_{T}^{\rho})$, $t\leq s\leq T$. We denote $\mathcal{D}[t,T]$={all deterministic piecewise continuous functions on $[t,T]$}. Obviously, we have the following equivalence between the cost functional $J(t,x;u(\cdot))$ and $\hat{J}(t,x;\rho(\cdot))$.

From equation (2.3), we can observe the value of the state $X^{\rho}(\cdot)$ based on an input control $\rho(\cdot)$. Following the idea investigated in Wang et al. (2020), we use reinforcement learning (RL) structure to learn the optimal control $\rho^{*}(\cdot)$. Motivated with the law of large numbers of samples of state under the density function $\pi(\cdot)$, Wang et al. (2020) introduced the following exploratory dynamic equation for state $X^{\pi}(\cdot)$,

where $\pi_{s}$ is the density function of the input control $\rho_{s}$,

and

for $t\leq s\leq T$.

It should be note that, the observations of the exploratory state $X^{\pi}(\cdot)$ should satisfy the equation (2.3). Based on this fact, we can prove that $X^{\pi}(\cdot)$ satisfies equation (2.5).

Furthermore, in the framework of exploratory, we rewrite the cost functional (2.4) as follows,

where the term

denotes the Shanon’s differential entropy used to measure the level of exploratory, and $\lambda>0$ is the temperature constant balanced the exploitation and exploration. We denote all the admissible density functions control on $\mathcal{D}[t,T]$ as $\mathcal{A}[t,T]$. Thus, our target is to minimum the cost functional (2.16) over $\mathcal{A}[t,T]$.

Now, we consider equation (2.5) with cost functional (2.16). Similar with the manner in Wang et al. (2020), employing the classical dynamic programming principle, one obtains that

and the related Hamilton-Jacobi-Bellman (HJB) equation is

with terminal condition $v(T,x)=\hat{\Phi}(x)$.

It is worth noting that, equation (3.1) has an optimal control $\pi^{*}_{t}(\cdot)$ which satisfies

Letting $\lambda\to 0$, $\pi^{*}_{t}(\cdot)$ reduces to a Dirac measure, where

and $\rho_{t}^{*}$ solves the following equation of $\rho$,

By Theorem 2.2, we have that equation (3.3) has at least one solution $\rho_{t}^{*}=\beta_{t}$, and thus

Furthermore, the function of $\rho$, $\mathcal{L}(t,x,\beta_{t},\rho)$ takes the minimum value at $\beta_{t}$, where

Since $\lambda>0$, thus $\pi^{*}_{t}(\rho)$ takes the maximum value at $\beta_{t}$. The above observations motive us to find the optimal estimation for the parameter $\beta(\cdot)$ from the optimal density function $\pi^{*}(\cdot)$.

Based on the above analysis, we conclude the following results.

In the following section, we consider a linear stochastic differential equation (SDE) model to verify the results given in Theorem 3.1, where the drift or diffusion term contains unknown parameter.

We now consider the problem of the Linear SDE case, where the drift or diffusion term is allowed to contain the unknown parameter. We first study the diffusion term with unknown parameter case, and then investigate the drift term with unknown parameter.