Random Periodic Mean-Square Exponential Stability

• The paper introduces a stability condition for linear SDEs with random periodic coefficients, ensuring controlled mean-square exponential decay.
• It establishes an equivalence between the stability criterion and the solvability of a Lyapunov-type BSDE, underscoring uniform controllability bounds.
• The study demonstrates practical verification through numerical methods and a one-dimensional example, emphasizing its importance in ergodic linear-quadratic optimal control.

A random periodic mean-square exponentially stable condition is a mathematical property within stochastic analysis and ergodic optimal control, providing a stability criterion for solutions to linear stochastic differential equations (SDEs) with coefficients that display random periodicity. This condition ensures that, under the influence of both deterministic periodicity and randomness (such as Brownian motion), the system exhibits a controlled mean-square growth rate and satisfies quantitative controllability bounds uniformly over periods. The random periodic mean-square exponentially stable condition underpins foundational results in control theory involving random periodic environments, notably in the context of ergodic linear-quadratic optimal control problems with random periodic coefficients (Wu et al., 13 Jan 2026).

1. Formal Definition

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space equipped with a one-dimensional Brownian motion $W$ and its completed natural filtration $\{\mathcal{F}_t\}$. Consider the class $\mathcal{B}_\tau(\mathbb{R}^{d\times d})$ of $\tau$-random-periodic, essentially bounded matrix-valued processes, where $f \in \mathcal{B}_\tau(\mathbb{R}^{d\times d})$ satisfies

$$f_{t+\tau}(\omega) = (\theta_\tau \circ f_t)(\omega),\quad\forall\,t\ge0,$$

with $\theta_\tau$ the Wiener shift operator.

The homogeneous linear SDE of interest is

$$d\Phi_t = A_t\,\Phi_t\,dt + C_t\,\Phi_t\,dW_t, \qquad \Phi_0 = I_d,$$

with $A,C \in \mathcal{B}_\tau(\mathbb{R}^{d\times d})$. The pair $[A, C]$ is said to be $\tau$-random-periodic mean-square exponentially stable if there exist constants $\beta, \lambda, \delta>0$ such that:

1. Mean-square exponential decay:

$$\mathbb{E}\bigl\|\Phi_t\bigr\|^2 \le \beta e^{-\lambda t}, \qquad t\ge0,$$

where $\|\Phi\|^2 = \operatorname{Tr}(\Phi^\top \Phi)$.

2. Uniform controllability-Gramian bound:

$$\inf_{r \in [0, \tau)} \mathbb{E}\left[\int_r^\infty (\Phi_s \Phi_r^{-1})^\top (\Phi_s \Phi_r^{-1})\,ds \mid \mathcal{F}_r \right] \succeq \delta\,I_d \quad \text{a.s.}$$

2. Random Periodicity and Wiener Shift

Random periodicity captures the notion that process statistics repeat modulo a period $\tau$, but under a shifted randomness (Wiener shift). The Wiener shift $\theta_\tau$ is defined by $\theta_\tau W_s = W_{s+\tau} - W_\tau$, modeling the invariance of Brownian motion increments and ensuring that for a random period $\tau$, the future dynamics reflect both time periodicity and the randomness propagated from previous intervals.

For matrix-valued processes $A$ and $C$, random periodicity ensures that both drift and volatility coefficients maintain consistent statistical structure across periods, precisely matching the SDE setting required for the random periodic mean-square exponent-ially stable condition.

3. Characterization via Backward Stochastic Differential Equations

The random periodic mean-square exponentially stable condition can be characterized via the solvability of a Lyapunov-type matrix-valued backward SDE (BSDE). Given any $\Lambda \in \mathcal{B}_\tau(\mathcal{S}^d)$ with $\Lambda_t \succeq 0$, consider

$$\begin{cases} dK_t &= -\bigl[K_tA_t + A_t^\top K_t + C_t^\top K_t C_t + L_t C_t + C_t^\top L_t + \Lambda_t\bigr]\,dt + L_t\,dW_t, \ \lim_{t \to \infty} K_t &= 0 \quad \text{(in a suitable sense)}. \end{cases}$$

The equivalence result is as follows: $[A, C]$ is $\tau$-random-periodic mean-square exponentially stable if and only if, for each $\Lambda_t \succeq 0$, this BSDE admits a unique $\tau$-random-periodic, bounded, positive semidefinite solution $(K, L)$. If $\Lambda_t \succ \alpha I$ (uniformly positive definite), $K_t$ is uniformly positive definite. With $\Lambda_t = I_d$, $K$ serves as a random-periodic Lyapunov function (Wu et al., 13 Jan 2026).

4. Analytical Proof Structure

The proof of the equivalence between the stability condition and the Lyapunov BSDE is two-pronged:

• Stability Implies Lyapunov BSDE Solvability: The BSDE is solved on $[0, \tau]$ with a terminal condition $K_\tau = M$ as an unknown. Applying Itô’s formula to $\Phi_s^\top K_s \Phi_s$ leads to a fixed-point equation for $M$, leveraging the exponential mean-square decay and contraction principles to ensure uniqueness and boundedness of the periodic solution.
• Lyapunov BSDE Solvability Implies Stability: Given a positive definite $K$ from the BSDE, applying Itô’s formula to $\Phi_t^\top K_t \Phi_t$ yields a negative-definite drift term proportional to $\mathbb{E}\|\Phi_t\|^2$, from which Grönwall's inequality establishes exponential decay. The controllability-Gramian bound follows by a similar conditional expectation argument utilizing the Lyapunov function (Wu et al., 13 Jan 2026).

5. Computation and Practical Verification

In practice, verifying $\tau$-random-periodic mean-square exponential stability proceeds by numerically solving the matrix-valued Riccati-type backward SDE with $\Lambda_t = I$ over one period and checking the positivity of the solution $K_t$. This computation typically involves numerical iteration to match terminal conditions and integrate the BSDE on $[0,\tau]$. Alternatively, one may synthesize a periodic feedback law $\Theta_t$ ensuring that the closed-loop SDE with new coefficients $(A_t+B_t\Theta_t,\, C_t+D_t\Theta_t)$ achieves mean-square exponential stability, thus transferring stability from the feedback-closed system to the open-loop pair $[A,C;B,D]$ (Wu et al., 13 Jan 2026).

6. Illustrative One-Dimensional Example

For the scalar case, consider

$$d\Phi_t = a_t\,\Phi_t\,dt + c_t\,\Phi_t\,dW_t, \qquad \Phi_0=1,$$

with $a, c \in \mathcal{B}_\tau(\mathbb{R})$ and $-\mu_2 \le 2a_t + c_t^2 \le -\mu_1 < 0$. It follows that

$$\Phi_t = \exp\left(\int_0^t [a_s - \tfrac{1}{2} c_s^2] ds + \int_0^t c_s\,dW_s\right)$$

with mean-square bound

$$\mathbb{E}|\Phi_t|^2 = \exp\left(\int_0^t [2a_s + c_s^2] ds\right) \le e^{-\mu_1 t},$$

and Gramian estimate

$$\mathbb{E}\left[ \int_r^\infty (\Phi_s/\Phi_r)^2 ds \Big| \mathcal{F}_r \right] \ge 1/\mu_2.$$

This demonstrates the $\tau$-random-periodic mean-square exponential stability of $[a(\cdot), c(\cdot)]$ (Wu et al., 13 Jan 2026).

7. Context and Significance in Ergodic Optimal Control

The random periodic mean-square exponentially stable condition forms a critical technical foundation for analyzing ergodic linear-quadratic optimal control problems with random periodic coefficients. It guarantees well-posedness of the state equation over infinite horizons and supports the reduction of an infinite-horizon ergodic cost functional to an equivalent cost computed over a single period. The equivalence result involving the Lyapunov BSDE ensures the existence and uniqueness of random periodic solutions to stochastic Riccati equations, which are instrumental in constructing explicit closed-loop optimal control laws. This condition enables the extension of classical stability and Riccati theory to settings with both stochasticity and random periodicity in dynamics and cost structure (Wu et al., 13 Jan 2026).