Linear Stochastic Differential Equations

• Linear Stochastic Differential Equations (LSDEs) are defined by state evolution with linear drift and diffusion, driven by Brownian or Lévy processes.
• Their explicit solutions and moment representations enable efficient system identification, filtering, and optimal control in various domains.
• LSDEs underpin applications in control theory, finance, neuroscience, and signal processing with robust analytical and numerical methods.

Linear stochastic differential equations (LSDEs) are a prominent subclass of stochastic differential equations characterized by a state evolution governed by linear drift and possibly linear diffusion coefficients, typically driven by Wiener or more general Lévy processes. These equations constitute the mathematical backbone for the probabilistic modeling of dynamical systems exposed to random perturbations in fields such as control theory, signal processing, neuroscience, mathematical finance, statistical physics, and machine learning. Their analytical tractability, explicit representations for laws and moments, and the connection to core tools in stochastic analysis position LSDEs as central objects in both theory and applications.

1. Definition and Fundamental Structure

A general d-dimensional LSDE has the form

$$dx(t) = [A(t)x(t) + a(t)]dt + \sum_{i=1}^m [B_i(t)x(t) + b_i(t)]dw_i(t),$$

where $A(t) \in \mathbb{R}^{d\times d}$, $B_i(t) \in \mathbb{R}^{d\times d}$, $a(t), b_i(t) \in \mathbb{R}^d$, and $\{w_i(t)\}_{i=1}^{m}$ are mutually independent Wiener (Brownian motion) processes. In the time-homogeneous, additive-noise case, and in the presence of jumps, the formulation generalizes to incorporate Lévy processes: $$dx(t) = Ax(t)dt + h\,dW(t) + \int_{\mathbb{R}^m} v(y)x(t^-) \tilde{N}(dt,dy) + \int_{\mathbb{R}^m} v(y)x(t^-) N(dt,dy),$$ where $W(t)$ is Brownian motion, $N$ is a (possibly compensated) Poisson random measure, and $v(y)$ specifies the (linear) jump structure (León et al., 2012).

The deterministic part (drift) is linear in the state, and the stochastic integration encompasses both Itô (adapted) and (for anticipating setups) Skorohod (anticipating) integration. For Gaussian driving noise and additive structure, LSDEs define Ornstein-Uhlenbeck processes, while Lévy-driven models extend to heavy-tailed, infinite-activity settings ("The Lévy State Space Model" (Godsill et al., 2019)).

2. Analytical Solutions and Explicit Representations

A core strength of LSDEs lies in the analytical accessibility of their solutions:

• State Transition Representation: The explicit solution is

$$x(t) = \Phi(t, t_0) x(t_0) + \int_{t_0}^t \Phi(t, s) a(s) ds + \sum_{i=1}^m \int_{t_0}^t \Phi(t, s) b_i(s) dw_i(s),$$

where $\Phi$ is the fundamental matrix solving $\frac{d}{dt}\Phi(t,s) = A(t)\Phi(t,s)$, $\Phi(s,s) = I$.

• Mean and Second Moment: Modern approaches (Jimenez, 2012) show that the mean $m(t) = \mathbb{E}[x(t)]$ and the second moment $P(t) = \mathbb{E}[x(t)x^\top(t)]$ can be expressed by a single exponential matrix of dimension $d^2+2d+7$ (or smaller for particular cases), simplifying previous, cumbersome representations involving multiple larger exponentials:

$$m(t) = m_0 + L_2 e^{M(t-t_0)} u, \qquad \operatorname{vec}(P(t)) = L_1 e^{M(t-t_0)} u,$$

where $u$ contains initial conditions and $M$ encodes both drift and diffusion structure. This unified form yields substantial computational and storage efficiency in system identification and Kalman filtering.

3. Extensions to Generalized Noises and Anticipating Settings

Lévy-driven LSDEs

LSDEs driven by Lévy processes accommodate jumps and can exhibit infinite variance, self-similarity, and heavy-tails, modeling phenomena from finance to neural spiking. Explicitly,

$$dx(t) = Ax(t)dt + h\,dW(t) + \int_{0<|y|\leq1} v(y)x(t^-) \tilde{N}(dt,dy) + \int_{|y|>1} v(y)x(t^-) N(dt,dy)$$

captures both small and large jumps (León et al., 2012). The associated solution inherits the structure of the jump decomposition, often expressed via shot-noise series: $$x(t) = e^{At}x(0) + \int_0^t e^{A(t-u)}h\,dW(u) + \sum_i H(\Gamma_i, U_i)I(V_i \leq t),$$ with shot noise terms $H$ parameterized for, e.g., $\alpha$-stable processes (Godsill et al., 2019).

Anticipating Coefficients

Allowing coefficients and initial data to be non-adapted (anticipating) requires analysis beyond Itô calculus. Anticipative Girsanov transformations on both Wiener and Lévy spaces facilitate existence and uniqueness results for LSDEs in this generality. Fundamental solutions employ measure transformations of the form

$$L_{s,t}(\omega) = \exp\left\{\int_s^t a_r(A_{r,t}\omega)\delta W_r - \frac{1}{2}\int_s^t |D_\cdot a_r(A_{r,t}\omega)|^2 dr \right\}$$

to yield explicit, transformed forms even in the presence of non-adapted coefficients (León et al., 2012).

4. Moment Stability and Delay: The Role of Characteristic Functions

Second moment boundedness (mean-square stability) for linear stochastic delay differential equations is captured via a characteristic function $H(\lambda)$ constructed in Laplace space. For an $N$-dimensional LSDE with a single discrete delay, the boundedness of the second moment matrix $M(t)$ is equivalent to the negativity of the real parts of all roots of $H$: $H(\lambda) = \det[I - D(\lambda)]$ where $D(\lambda)$ incorporates both drift/delay and noise coefficients through Laplace transforms of the deterministic propagator (Wang et al., 2012). In two-dimensional, decoupled systems, $H$ simplifies and allows for explicit calculation of moment stability domains, paralleling ODE stability via eigenvalue analysis.

5. Optimal Control: Riccati Equations and Feedback

In LSDE-driven control problems, especially Linear-Quadratic (LQ) settings, optimal feedback controls are derived by solving Riccati-type differential equations. In mean-field settings, the forward-backward SDE (MF–FBSDE) system is decoupled via an affine ansatz ($$Y(t) = P(t)(X(t)-\mathbb{E}[X(t)]) + \Pi(t)\mathbb{E}[X(t)]$$) leading to two Riccati ODEs for $P$, $\Pi$. Under uniform positive-definiteness in the cost functional's quadratic terms, unique solvability is ensured, and the optimal regulator is constructed in feedback form (Yong, 2011, Yong, 2013).

A table summarizing Riccati structure across variants:

Problem Type	Riccati Equation Structure	Feedback Representation
Classic LQ Control	Symmetric, autonomous	$u=-R^{-1}B^\top P x$
Mean-field LQ (open-loop equilibrium)	Asymmetric, coupled (state and expectation)	State and mean feedback
Mean-field LQ (closed-loop, time-consistent)	Symmetric via multi-person game construction	State-feedback equilibrium

6. Decorrelation, Noise-Induced Mixing, and Cut-off Phenomena

The phenomenon of abrupt decorrelation formalizes how LSDEs—especially Ornstein-Uhlenbeck processes and their time-inhomogeneous generalizations—can lose memory of the initial state extremely rapidly ("cut-off"). Consider a family of SDEs indexed by small noise parameter ε, with deterministic part decaying as $e^{Q t}$. For $t_\varepsilon = |\ln \varepsilon|/\theta$, where $\theta$ is the drift rate, the statistical dependence between $x(0)$ and $x(t)$ collapses from maximal to negligible across a narrow transition window: $$\lim_{\epsilon \to 0} d^{(\epsilon)}(c t_\varepsilon) = \begin{cases} M & \text{if } 0 < c < 1\ 0 & \text{if } c > 1 \end{cases}$$ for diverse statistical distances $d^{(\epsilon)}(\cdot)$ (Kullback–Leibler, Wasserstein, total variation). This formalism parallels the cut-off phenomenon in Markovian mixing theory, with explicit decorrelation profiles calculable for LSDEs (López et al., 20 Sep 2025).

7. Numerical Schemes and Computational Methods

LSDEs admit robust and efficient numerical methods, vital for system identification, simulation, and filtering:

• Finite Difference Schemes: Explicit and implicit-explicit discretizations yield convergence rates of order $O(h)$ in space and $O(\tau^{1/2})$ in time for parabolic SIDEs, with LSDEs as a special case (Dareiotis et al., 2013). Error bounds are derived using discrete Itô calculus and energy estimates in Sobolev spaces.
• Parallel-in-Time Integration: SParareal (Stochastic Parareal) algorithms introduce time parallelization and stochastic corrections. Compared to classical Parareal, SParareal achieves linear convergence and improved efficiency for both linear and nonlinear SDEs, maintaining mean-square stability by careful parameter choice (Wang et al., 18 Feb 2025).
• Kalman Filtering and Particle Methods: Closed-form moment evolution supports efficient Kalman filtering. For heavy-tailed LSDEs, shot-noise representations combined with particle filtering (e.g., Rao-Blackwellised SMC) support inference even with irregular data and latent states (Godsill et al., 2019).

8. Identification, Parameter Estimation, and Learning

• Bayesian Inference and Sparse Identification: The evolution equations for mean and covariance support quasi-likelihood maximum likelihood estimators (QMLE), notably under partial observation (hidden Ornstein-Uhlenbeck drivers) (Kurisaki, 2022). Asymptotic normality and consistency hold under high-frequency observation.
• Compressive Sensing Approaches: Using a parameterization of drift and diffusion terms as linear combinations from a function library, coefficients can be identified via Bayesian regression with Laplacian (sparse-promoting) priors, yielding interpretable LSDE models. Automatic Threshold Sparse Bayesian Learning (ATSBL) improves robustness and parsimony, especially in high-dimensional, noisy data regimes (Huang et al., 2022).
• Learning via Neural Laplace: The Neural Laplace framework learns LSDEs not in the time domain but via the Laplace transform of trajectories, exploiting the closed-form expectation when available (as in Geometric Brownian Motion, $E[F(s)]=X_0/(s-\mu)$). This approach unifies learning across ODEs and SDEs, with competitive empirical accuracy and advantages in stability and flexibility (Carrel, 7 Jun 2024).

• "A simplified derivation of the Linear Noise Approximation" (Wallace, 2010)
• "A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations" (Yong, 2011)
• "Anticipating Linear Stochastic Differential Equations Driven by a Lévy Process" (León et al., 2012)
• "Simplified formulas for the mean and variance of linear stochastic differential equations" (Jimenez, 2012)
• "Second Moment Boundedness of Linear Stochastic Delay Differential Equations" (Wang et al., 2012)
• "Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations --- Time-Consistent Solutions" (Yong, 2013)
• "Finite Difference Schemes for Linear Stochastic Integro-Differential Equations" (Dareiotis et al., 2013)
• "On Classical Solutions of Linear Stochastic Integro-Differential Equations" (Leahy et al., 2014)
• "Exact Controllability of Linear Stochastic Differential Equations and Related Problems" (Wang et al., 2016)
• "The Lévy State Space Model" (Godsill et al., 2019)
• "Sparse inference and active learning of stochastic differential equations from data" (Huang et al., 2022)
• "Parameter estimation for ergodic linear SDEs from partial and discrete observations" (Kurisaki, 2022)
• "Neural Laplace for learning Stochastic Differential Equations" (Carrel, 7 Jun 2024)
• "Stochastic Differential Equations models for Least-Squares Stochastic Gradient Descent" (Schertzer et al., 2 Jul 2024)
• "Stochastic Parareal Algorithm for Stochastic Differential Equations" (Wang et al., 18 Feb 2025)
• "Abrupt decorrelation for linear stochastic differential equations" (López et al., 20 Sep 2025)