Rough Stochastic Differential Equations

• RSDEs are a rigorous framework combining classical Itô SDEs with rough path theory to handle signals rougher than Brownian motion, including jumps.
• The approach integrates stochastic analysis, Malliavin calculus, and tailored numerical methods to ensure robust, pathwise convergence and stability under low regularity.
• Applications span high-dimensional probability, financial modeling, and robust stochastic filtering and control, providing practical insights for simulation and uncertainty quantification.

Rough stochastic differential equations (RSDEs) constitute a rigorously developed framework that unifies classical Itô SDEs with the pathwise theory of rough differential equations (RDEs) à la Lyons, providing tools to analyze, simulate, and control systems driven by signals of arbitrary irregularity—including signals rougher than Brownian motion, as well as signals with jumps. The RSDE framework merges stochastic analysis, rough path integration, Malliavin calculus, pathwise numerical methods, and advanced stochastic filtering into a systematic theory applicable to problems in high-dimensional probability, mathematical finance, engineering, control, and robust statistical learning.

1. Fundamental RSDE Theory and Pathwise Integration

Classical stochastic differential equations,

$dY_t = b_t(Y_t)dt + \sigma_t(Y_t)dB_t\,,$

with $B$ a Brownian motion, admit a well-posed theory under regularity and ellipticity conditions. However, these models become inadequate for processes exhibiting long memory, noise rougher than Brownian (e.g., fractional Brownian motion with $H < 1/2$), or non-semimartingale noise. Lyons' rough path theory generalizes the Riemann–Stieltjes and Itô integrals, constructing deterministic integration theory driven by rough signals, with the solution map continuous in a rough path metric. An RSDE, in its most general form, reads

$$dY_t = b_t(Y_t)\,dt + \sigma_t(Y_t)\,dB_t + f_t(Y_t)\,d\mathbf{X}_t\,,$$

where $\mathbf{X}$ is an (enhanced) rough path, potentially random, deterministic, with or without jumps (Friz et al., 2021).

Key analytical tools:

• Stochastic controlled rough paths: Adapted processes $(Z, Z')$ admitting expansions $\delta Z_{s,t} = Z'_s\,\delta X_{s,t} + R_{s,t}$ with $L^p$-controlled remainders.
• Stochastic sewing lemma: Quantitative control over limits of Riemann sums to define stochastic/rough integrals, ensuring pathwise well-posedness and continuity (Friz et al., 2021).
• Unified fixed-point/regularization arguments: Techniques for strong existence and pathwise uniqueness for SDEs with rough coefficients and without uniform ellipticity, through Fokker–Planck PDEs with weighted Sobolev norms and $L^p$ control over associated densities (Champagnat et al., 2013).

2. RSDEs with Jumps and Discontinuous Signals

The RSDE theory robustly incorporates discontinuous drivers, generalizing the Marcus (canonical) SDE scheme. Signals with jumps (e.g., Lévy processes, general càdlàg semimartingales) are lifted via the canonical or Marcus–Itô rough path, with careful modeling of area/iterated integral contributions (Chevyrev et al., 2017Friz et al., 2017). The solution map is made continuous in a "jump-adapted" rough path metric: $$\sigma_p(X, Z) = \inf_{\lambda \in \Lambda}\, \Big\{\ |\lambda| \vee \|X \circ \lambda - Z\|_{p,[0,T]}\,\Big\}$$ where $\Lambda$ is a class of time changes and $|\lambda|$ the modulus deviation.

Key results:

• Canonical lifts for any multidimensional semimartingale, pathwise Marcus solution characterization, and extension of Burkholder–Davis–Gundy inequalities for rough paths (Chevyrev et al., 2017).
• New limit theorems (law and probability): If $X^n$ converges to $X$ in Skorokhod topology with uniformly controlled variation (UCV), solutions to $dY^n_t = V(Y^n_t)\diamond dX^n_t$ converge in law to $dY_t = V(Y_t)\diamond dX_t$; here, $\diamond$ denotes Marcus integration (Chevyrev et al., 2017).
• Euler- and Milstein-type pathwise convergence of numerical schemes to rough path solutions for general jump processes (Allan et al., 2023).

3. Existence, Uniqueness, and Regularity under Low Regularity and Degeneracy

Strong solutions and pathwise uniqueness are established for SDEs (i) with low regularity drift/diffusion (rough coefficients), (ii) under non-ellipticity, (iii) for kinetic or degenerate diffusions. This is obtained by controlling Sobolev-type quantities weighted by the solution law (from Fokker–Planck) and employing maximal controls: $$\|\sigma\|_{h_1(u)} = \int_0^T \int_{\mathbb{R}^d} \big[ |\nabla\sigma| + M(|\nabla\sigma|)\big]\,u(dx)\,dt$$ with $M$ the Hardy–Littlewood maximal operator, $u$ the solution law (Champagnat et al., 2013).

• In dimension $1$, existence and uniqueness follow with weaker, fractional regularity (e.g., $H^{1/2}$-type norms).
• In kinetic (Langevin) cases with degenerate diffusion, pathwise uniqueness still holds due to control via the symplectic structure in the Fokker–Planck equation.

4. RSDEs in Stochastic Control and Dynamic Programming

The RSDE framework underpins a robust theory of stochastic control driven by both Brownian and rough signals. The value function for a controlled RSDE

$$dY^\eta_t = b(t, Y^\eta_t; \eta_t)\,dt + \sigma(t, Y^\eta_t; \eta_t)\,dB_t + (f, f')(t, Y^\eta_t)\,dX_t$$

is analyzed pathwise as a functional of the rough input $X$, with dynamic programming principle (DPP) and value function regularity stable in the rough path metric (Friz et al., 7 Dec 2024). Crucial advances include:

• The rough HJB equation

$$-d_tv = H(y,t, Dv, D^2 v)\,dt + (f(t, y) \cdot Dv)\,dX$$

with continuity and stability under reduced regularity assumptions on $f$, and the solution map is Lipschitz on bounded sets (Friz et al., 7 Dec 2024).

• Pontryagin maximum principle (PMP) in rough stochastic settings: The first-order optimality conditions reduce to forward–backward RSDEs without requiring traditional FBSDEs (Lew, 10 Feb 2025). Indirect shooting methods based on the adjoint process (linear backward RSDE) are shown to be efficient and converge rapidly for stochastic optimal control tasks.

5. Malliavin Calculus, Densities, and Hypoellipticity in RSDEs

A comprehensive Malliavin calculus for RSDEs shows that:

• The law of the solution at time $t$ is absolutely continuous under standard ellipticity or Hörmander bracket-generating conditions, and its density is smooth if coefficients are smooth/bounded and the Hörmander condition holds (Bugini et al., 19 Feb 2024).
• Malliavin derivatives of the solution satisfy linear RSDEs, whose analysis relies on weighted norm a priori estimates and robust stability theory.
• The reduced Malliavin matrix,

$\gamma_{X_t} = J_t C_t J_t^\top\,,$

where $J_t$ is the Jacobian (solution of a linear RSDE) and $$C_t = \int_0^t I_s \sigma(X_s) \sigma(X_s)^\top I_s^\top ds$$, governs invertibility and thus density smoothness.

These findings have ramifications in filtering, stochastic sampling (particle filters, MCMC), and mean-field models.

6. Numerical Methods: Pathwise Convergence and Discretization Schemes

Pathwise numerical methods for RSDEs have been established:

• Euler and Milstein schemes converge in $p$-variation to the solution under a pathwise criterion, “Property RIE”, for the driving signal and adapted partitions exhausting jump times (Allan et al., 2023).
• Property (RIE) ensures the left-point Riemann sums of increments and their iterated integrals converge, ensuring the discrete Euler–Maruyama scheme converges almost surely for Brownian motion, Itô processes, Lévy processes, Young semimartingales, and typical price paths.
• Semi-implicit Taylor and Runge–Kutta schemes: Implicitness in the drift alongside explicit rough expansions of the diffusion improve stability and convergence, especially for stiff RDEs/Ito SDEs driven by rough noise (Riedel et al., 2020Redmann et al., 2020). The convergence rate is determined by the regularity of the driver (e.g., $h^{1/p}$ for the implicit Euler applied to a $1/p$-Hölder path).

7. Filtering, Conditional Laws, and RSDEs with Randomization

RSDEs serve as the core of robust nonlinear stochastic filtering:

• Pathwise (deterministic) rough filters are defined by RSDEs driven by the lifted observation signal, yielding deterministic (rough) Kallianpur–Striebel, Zakai, and Kushner–Stratonovich equations which, upon randomization, coincide with the classical stochastic counterparts (Bugini et al., 15 Sep 2025).
• Upon replacing the rough observation path in the RSDE filtering problem by the Itô lift of the observation semimartingale, the rough and classical filters coincide in law, yielding robust, dimension-independent well-posedness results (Bugini et al., 15 Sep 2025).
• Robust filtering for jump-diffusions: The conditional law (filter) can be expressed as a continuous function of the observed rough path lift with respect to the rough path p-variation or Skorokhod metric (including in the presence of jumps), ensuring stable numerical and learning-based approximations (Allan et al., 8 Jul 2025).
• The randomization of rough signals (random rough paths) provides equivalence between conditional distributions in "doubly stochastic" Ito processes and deterministic RSDEs parameterized by the noise realization (Friz et al., 9 Mar 2025).

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Summary Table: RSDE Regularity and Application Requirements

Aspect	Key Regularity/Assumption	Main Consequence
Existence/Uniqueness	Weighted Sobolev, $L^p$ Fokker-Planck	Pathwise uniqueness for rough/degenerate coefficients (Champagnat et al., 2013)
Jumps/Càdlàg Drivers	Marcus/Itô lift, UCV condition	Pathwise solution, Wong–Zakai and limit theorems (Chevyrev et al., 2017)
Filtering	Rough path metrics, randomization	Robust filter, continuous in rough path (Bugini et al., 15 Sep 2025Allan et al., 8 Jul 2025)
Control/PMP/DPP	Pathwise RSDEs, rough HJB stability	DPP validity, rough PMP, indirect methods (Friz et al., 7 Dec 2024Lew, 10 Feb 2025)
Malliavin Calculus	Smooth coefficients, Hörmander	Density smoothness for solutions (Bugini et al., 19 Feb 2024)
Numerics/Euler Schemes	Property (RIE), explicit rough integrals	Pathwise (strong) convergence in $p$-variation (Allan et al., 2023)

8. Perspectives and Future Directions

The RSDE framework, synthesizing advancements in pathwise stochastic analysis, filtered stochastic control, and high regularity analysis, enables:

• Stability and robustness results crucial for model validation and uncertainty quantification in high-dimensional stochastic filtering and control.
• Pathwise, deterministic representations for fundamentally probabilistic objects (filters, value functions) that are numerically stable and fit for learning-based methods.
• Extension to rough-drift mean-field models, partially observed control, (hypo)elliptic RSDEs, and high-dimensional degenerate systems.

The modular, pathwise nature of RSDEs positions the framework as a unifying paradigm for probabilistic analysis, control theory, and numerical simulation under low regularity, non-Markovian, and non-semimartingale conditions. This yields broad applicability for both theoretical probabilistic research and algorithmic/statistical approaches to stochastic systems.