Doss–Sussmann Transformation

• Doss–Sussmann transformation is a variable change method that converts stochastic and rough differential equations into deterministic ordinary differential equations with random inputs.
• It facilitates rigorous analysis by decoupling irregular noise, making it applicable to fractional Brownian motion, non-smooth coefficients, and systems where classical Itô calculus fails.
• The technique underpins high-precision numerical schemes, stability proofs, and synchronization analysis, bridging classical stochastic analysis with modern rough path theory.

The Doss–Sussmann transformation is a change of variables method in the analysis of stochastic differential equations (SDEs) and rough differential equations (RDEs), which allows the “removal” or “absorption” of stochastic noise so that the evolution of the system becomes deterministic, or is reducible to an ordinary differential equation (ODE) with random coefficients. Originally introduced by Doss and further exploited by Sussmann, the transformation plays a critical role in problems where the driving noise is irregular (e.g., fractional Brownian motion or generic rough signals) and classical Itô calculus does not apply, as well as for equations with non-Lipschitz, non-smooth, or state-dependent coefficients. The transformation is also foundational in constructing strong approximations, proving existence and uniqueness, analyzing stability, and establishing synchronization properties in stochastic dynamical systems.

1. Mathematical Framework of the Doss–Sussmann Transformation

The core principle is to represent the solution of a stochastic (or rough) differential equation

$dX_t = b(X_t)dt + \sigma(X_t) dB^H_t$

(where $B^H$ is fractional Brownian motion with Hurst index $H$ and $b,\sigma$ satisfy appropriate regularity bounds) in transformed coordinates by a composite function

$X_t = h(Y_t, B^H_t),$

where $Y_t$ solves a deterministic ordinary differential equation (ODE) that is parametrized by the path of the noise and $h$ solves

$$\frac{\partial h}{\partial x_2}(x_1, x_2) = \sigma\left(h(x_1, x_2)\right), \quad h(x_1, 0) = x_1,$$

with equivalent integral form

$h(x, y) = x + \int_0^y \sigma(h(x, s)) ds.$

The ODE satisfied by $Y_t$ is then

$$Y'_t = \left(\frac{\partial h}{\partial x_1}(Y_t, B^H_t)\right)^{-1} b(h(Y_t, B^H_t)), \quad Y_0 = x_0$$

with

$$\frac{\partial h}{\partial x_1}(x, y) = \exp\left(\int_0^y \sigma'(h(x, s)) ds\right).$$

This representation “decouples” the randomness from the state-dependent coefficients, recasting the stochastic problem in terms of deterministic flows and fixed sample paths.

2. Application Domains: Fractional SDEs, SPDEs, and RDEs

In stochastic analysis, particularly for equations driven by non-semimartingale signals such as fractional Brownian motion (FBM), the Doss–Sussmann transformation enables the reformulation of SDEs and RDEs when classic Itô calculus fails. For FBM-driven SDEs ($dX_t = b(X_t) dt + \sigma(X_t) dB^H_t$), this approach is vital since FBM exhibits long-range dependence and non-Markovian, non-martingale properties, making pathwise definitions and strong approximations central to analysis (Garzón et al., 2011, Araya et al., 2019).

In backward doubly stochastic differential equations (BDSDEs) involving both standard and fractional Brownian motions, the transformation “absorbs” the fractional part—yielding a BSDE driven solely by Brownian motion—with the solution reconstruction given by

$U_t = n(t, Y_t) = a(Y_t, B_t),$

where $a$ solves $\partial_z a(y, z) = g(a(y, z)),\ a(y, 0)=y$ and $Y_t = h(U_t, B_t)$ is its inverse (Jing, 2011).

For rough differential equations (RDEs) and reflected RDEs with unbounded drift, the Doss–Sussmann technique translates the system into an ODE for a new variable $Z$

$$Y_t = U_{t,0}(Z_t),\quad Z_t = y_0 + \int_0^t J_{0\leftarrow s} b(U_{s,0}(Z_s)) ds,$$

where $U_{t,0}$ is the flow for the driftless equation and $J_{0\leftarrow s}$ the inverse Jacobian (crucial for managing irregular coefficients and enforcing boundary conditions via penalization strategies) (Richard et al., 2019).

3. Numerical Schemes and Approximation Theory

The Doss–Sussmann representation underpins high-precision numerical schemes for SDEs/RDEs with rough noise. The archetypal strategy is as follows:

• Strong Approximation of Noise: Approximate the driving fractional Brownian motion using transport processes (e.g., via the Mandelbrot–van Ness formula):

$$P\left[\sup_{t\in[0,T]} |B^H_t - B^n_t| > C n^{-1/2+\beta} (\log n)^{5/2} \right] = o(n^{-q})$$

where $\beta$ is a regularity parameter (Garzón et al., 2011).

• Discretized Euler Schemes: Approximate the deterministic components (auxiliary process $Y$ and function $h$) using Euler or Taylor expansions on adaptive partitions, with step sizes tuned to the noise regularity. For example, first-order Taylor approximation for “flow” functions like $\phi(a, u)$:

$$\phi_\ell(z, u_{k}) \approx \phi_\ell(z, u_{k-1}) + \sigma(\phi_\ell(z, u_{k-1})) (u_{k} - u_{k-1})$$

The process $Y$ is discretized with coefficients depending on the approximated $\phi$ and the realizations of the noise.

• Composed Solution: The final approximate solution is assembled as $X^n_t = h^n(Y_t^n, B^n_t)$ or variants such as $X^{(n)}_t = \phi_\ell(Y_t^{(m)}, B_t)$ (Araya et al., 2019).
• Error Analysis: Under suitable regularity assumptions on $b$ and $\sigma$, the convergence rate is typically algebraic in the step size with explicit dependence on the Hurst index:

$$\sup_t |X_t - X_t^n| \leq C n^{-1/2 + \beta + \delta} (\log n)^{5/2}$$

or more generically $n^{-2H+\rho}$ for $H \in (1/4, 1/2),\ \rho > 0$ (Araya et al., 2019).

This analysis demonstrates the utility of the transformation for rigorous pathwise control, often outperforming standard or even “exact” simulation at discrete time grids; for instance, the DS-based simulation of the CIR process guarantees uniform error bounds, whereas naive grid-based schemes may yield arbitrarily large errors between grid points (Milstein et al., 2013).

4. Theoretical and Analytical Implications

The Doss–Sussmann transformation facilitates the proof of strong (almost sure) convergence, existence and uniqueness, exponential stability, and synchronization in a variety of stochastic or rough systems:

• Stability: By converting an RDE with noise into an ODE with modified drift, classical Lyapunov methods can be applied for stability analysis. Combined with stopping time arguments to localize the roughness, one obtains pathwise exponential stability under small noise regimes (Duc et al., 10 Oct 2024).
• Synchronization: For coupled dissipative SDEs driven by multiplicative fractional noise, the transformation recasts the coupled system in “Z–coordinates,” where synchronization becomes a problem of forced contraction in random ODEs, with pathwise synchronization results attainable even under nonlinear noise (Cao et al., 1 Oct 2025).
• Generalized Drift and Dispersion: In systems where coefficients are non-smooth or not absolutely continuous, the transformation separates the “rough” stochastic input so the solution can be analyzed path-by-path, enabling comparison principles, existence of densities, and pathwise analysis even for irregular data (Karatzas et al., 2013, Richard et al., 2019).
• Infinite-dimensional Analysis: In Banach manifold settings, the approach generalizes to so-called “$\ell^1$-orbits,” where concatenations of flows controlled by summable time-durations yield weak Banach submanifolds and define the geometry of attainable sets for infinite-dimensional control systems (Lathuille et al., 2011).

5. Extensions to SPDEs, PDEs, and Quantum Systems

The Doss–Sussmann technique extends beyond SDEs/RDEs to the analysis of stochastic partial differential equations (SPDEs), PDEs with stochastic coefficients, and even quantum mechanical systems:

• Backward Doubly Stochastic Differential Equations (BDSDEs): By absorbing the fractional noise component, one can reformulate BDSDEs as BSDEs, unlocking powerful existence and regularity theory and providing probabilistic representations for viscosity solutions of SPDEs (Jing, 2011).
• Stochastic Landau–Lifshitz–Gilbert Equation: For multi-dimensional SPDEs, the technique permits the removal of stochastic integrals from the time derivative, transforming the equation into a deterministic PDE (with random coefficients), supporting unconditionally convergent numerical schemes and weak martingale solution existence proofs (Goldys et al., 2017).
• Schrödinger Equation with Doss Potentials: The transformation allows a complex scaling in the path integral formulation, yielding fundamental solutions for Hamiltonians with super-quadratic polynomial potentials that are smooth and “live” outside the usual $L^2$ theory; the resulting solution is differentiable even when standard self-adjoint extensions are not (Grothaus et al., 2015).

6. Practical Implementation and Limitations

While the Doss–Sussmann transformation is theoretically robust, implementing it in practice requires careful consideration of the regularity and invertibility of flow maps, accuracy in the Jacobian computation, management of boundary effects (e.g., in reflected equations), and the stability of numerical discretization especially for near-zero or degenerate regimes. In cases where coefficients vanish or approach zero, the transformation may necessitate finer grid discretizations and auxiliary approximation methodologies to maintain error control (Garzón et al., 2011, Milstein et al., 2013).

Furthermore, transport process approximations for noise, while theoretically sound, may not be computationally optimal and could be replaced by alternatives depending on the application context (Garzón et al., 2011).

7. Summary and Connections to Modern Stochastic Analysis

The Doss–Sussmann transformation is a foundational technique enabling deterministic (or pathwise) analysis in stochastic dynamics subjected to rough, fractional, or degenerate noise. It underlies both theoretical advances (existence, uniqueness, regularity, stability, synchronization, pathwise control) and practical methodologies (strong simulations, approximation schemes, PDE solution representations). The approach is broadly connected with Lamperti-type transformations, Wong–Zakai approximations, pathwise Itô and Tanaka formulas, and rough path theory, providing a bridge between classical stochastic analysis tools and modern, non-semimartingale frameworks.

Its wide applicability across analysis, numerical approximation, control, infinite-dimensional geometry, and quantum mechanics highlights its central role in the contemporary paper of stochastic, rough, and deterministic evolution equations.