Doss–Sussmann Transformation: SDE Reduction

Updated 19 January 2026

Doss–Sussmann transformation is a method that recasts one-dimensional SDEs and RDEs with state-dependent noise into deterministic ODEs via a flow map inversion, capturing pathwise behavior.
It involves rewriting the SDE in Stratonovich form, constructing a differentiable flow map, and transforming the problem into a deterministic auxiliary ODE governed by the noise sample path.
The technique is widely applied in numerical simulations, SPDE regularization, and studies of rough or fractional diffusions, enhancing error control and stability analysis.

The Doss–Sussmann transformation is a methodology for reducing one-dimensional stochastic differential equations (SDEs) and, more generally, rough differential equations (RDEs) with nonconstant diffusion to deterministic—but noise-parameterized—ordinary differential equations (ODEs) or integral equations. This technique provides a pathwise (trajectory-level) description of the solution, making it central to the analysis, numerical approximation, and stability of SDEs/RDEs, especially for processes with irregular coefficients, state-dependent noise, or rough driving signals. The approach is widely applied in classical SDE theory, model-free pathwise calculus, stochastic partial differential equations, and the numerical simulation of stochastic processes, with notable applications to the Cox–Ingersoll–Ross (CIR) process, rough path equations, stochastic Landau–Lifshitz–Gilbert (LLG) equations, and fractional calculus contexts.

1. Fundamental Construction: From SDE to ODE

The prototypical object is the one-dimensional SDE: $dX_t = a(X_t)\,dt + b(X_t)\,dW_t, \quad X_0 = x,$ with smooth coefficients $a,b$ and $b(x)\neq 0$ on the region of interest. The Doss–Sussmann mechanism proceeds by:

Recasting in Stratonovich form via $A(x) = a(x) - \tfrac12 b(x)b'(x)$ and $B(x) = b(x)$ :

$dX_t = A(X_t)\,dt + B(X_t)\circ dW_t.$

Constructing the flow map $\phi(x,y)$ as the solution to the PDE

$\partial_y\phi(x,y) = B(\phi(x,y)), \quad \phi(x,0)=x,$

which, under smoothness and invertibility conditions on $B$ , defines a local $C^1$ -diffeomorphism in $x$ for each $y$ .

Transforming to a deterministic auxiliary equation by introducing $Y_t=\phi^{-1}(X_t,W_t)$ , which satisfies the ordinary ODE (randomly driven through $W_t$ ):

$dY_t = \frac{A(\phi(Y_t,W_t))}{\partial_x\phi(Y_t,W_t)}\,dt,$

with the solution for $X_t$ then reconstructed via $X_t=\phi(Y_t,W_t)$ (Milstein et al., 2013).

This stepwise reduction is not restricted to classical Itô SDEs; analogous constructions exist for RDEs driven by rough paths, SPDEs, and model-free calculus frameworks.

2. Generalization to Rough and Pathwise Contexts

The transformation is robust beyond the Itô or Stratonovich stochastic calculus. In the context of RDEs, e.g.,

$dy_t = f(y_t)\,dt + g(y_t)\,dx_t,$

where $x_t$ is a $\gamma$ -Hölder path (possibly a rough path with a $p$ -variation lift), the Doss–Sussmann map takes the form: $y_t = \Phi_t(x, z_t),$ where $\Phi_t(x,z)$ is the flow associated with the pure "noise" equation $du = g(u)\,dx$ and $z_t$ solves the deterministic ODE

$\dot z_t = [D_z\Phi_t(x,z_t)]^{-1} f(\Phi_t(x,z_t)).$

Under suitable regularity conditions (bounded derivatives up to order $3$, invertibility of $D_z\Phi_t$ ), this change of variable allows for the analysis of pathwise stability, synchronization, and existence of solutions, even for driving signals far rougher than Brownian motion (Duc et al., 2024, Cao et al., 1 Oct 2025, Richard et al., 2019).

In model-free, game-theoretic frameworks, the same principle applies to equations driven by "typical paths" of finite quadratic variation, utilizing pathwise integration in the sense of Föllmer or Vovk (Łochowski et al., 2021).

3. Applications: Numerical Methods, SPDEs, and Singular Coefficient Regimes

(a) Simulation and Uniform Approximation (CIR Process)

In the Cox–Ingersoll–Ross process, with $dV_t = \kappa(\theta-V_t)\,dt + \sigma\sqrt{V_t}\,dW_t$ , the Doss–Sussmann (after a Lamperti transformation $U_t = \sqrt{V_t}$ ) yields a representation: $U(t) = Y(t) + (\sigma/2)W(t),$ where $Y$ satisfies a deterministic (random-coefficient) ODE: $\frac{dY}{dt} = \frac{\alpha}{Y + (\sigma/2)W(t)} - \frac{\kappa}{2}[Y + (\sigma/2)W(t)],$ permitting accurate simulation and giving a uniform pathwise error bound superior to time-discretized Euler/Maruyama schemes (Milstein et al., 2013).

(b) Regularization of SPDEs: Stochastic LLG Equation

For the Stratonovich-form stochastic Landau–Lifshitz–Gilbert PDE with multi-dimensional noise, the Doss–Sussmann transform, via a coefficient orthogonalizing transformation $Z_t$ , eliminates all stochastic integrals from the transformed equation, yielding a deterministic PDE with random coefficients and promoting absolute continuity of solutions in time (Goldys et al., 2017).

(c) Fractional Brownian Motion and Non-Markovian Settings

For SDEs driven by fractional Brownian motions (fBM) $B^H$ with Hurst index $H\in(1/4,1)$ , the method underpins numerical schemes with explicit convergence rates. For $H\in(1/4,1/2)$ , the Doss–Sussmann representation allows ODE-based discretizations with error rate $O(n^{-2H+\rho})$ (Araya et al., 2019). For coupled dissipative SDEs under fBM, this approach yields direct synchronization proofs at the pathwise level (Cao et al., 1 Oct 2025).

4. Structural Properties, Well-Posedness, and Limitations

The rigorous application of the Doss–Sussmann transformation relies on:

Invertibility and Regularity of the Flow: $b$ (or $g$ ) must be $C^1$ (or $C^3$ in the rough path setting) and nonvanishing; the resulting flow $\phi(x,y)$ (or $\Phi_t(x,z)$ ) must be a $C^1$ -diffeomorphism.
Growth and Regularity in the Drift: $a$ (or $f$ ) and $b$ must satisfy conditions ensuring global existence and uniqueness of solutions for the ODE satisfied by the auxiliary process $Y$ or $Z$ .
Compatibility with Generalized Drift and Non-smooth Dispersion: With minimal regularity—finite first variation of $\log s(\cdot)$ and measurable finite-variation drifts—the pathwise Lamperti plus Doss–Sussmann strategy still applies (Karatzas et al., 2013).
Dimension and Extension: For multi-dimensional diffusions, invertibility of the multi-parameter flow and regularity of the Jacobian are essential and nontrivial.

5. Impact on Analysis, Approximation, and Pathwise Representations

By reducing the stochastic component to a deterministic ODE driven by sample paths, the Doss–Sussmann device enables:

Uniform simulation and strong pathwise error control for state-dependent SDEs (notably for diffusion coefficients with degeneracies, e.g., CIR near zero) (Milstein et al., 2013).
Transformation of SDEs/SPDEs with non-smooth or singular coefficients to deterministic integral equations, generalizing well beyond classical settings (e.g., equations with generalized drift, local time, and non-smooth coefficients) (Karatzas et al., 2013).
Foundation for pathwise existence and uniqueness in rough, model-free, and non-Lipschitz settings (Łochowski et al., 2021, Cao et al., 1 Oct 2025).
Enabling sharp numerical analysis and discretization for SDEs under highly irregular (fractional or rough) noise, extending to weak and strong convergence bounds (Araya et al., 2019).
Reduction of stochastic PDEs to deterministic evolution equations with random coefficients, facilitating existence theory and weak/martingale solution constructions (Goldys et al., 2017).

The Doss–Sussmann method is a direct generalization of the Lamperti transform, which linearizes the diffusion coefficient via a coordinate change. In many settings, Doss–Sussmann is utilized after a Lamperti-type reduction: first regularize the diffusion, then "freeze out" the stochasticity for the remaining nonlinear effects (Karatzas et al., 2013). In the context of backward SDEs driven by mixed standard and fractional Brownian motions, the transform removes the fractional component, reducing the system to a pure $W$ -driven BSDE, making available the classical stochastic analysis toolkit (Jing, 2011).

The approach is also the basis for model-free calculus under Vovk/Föllmer outer measure, solving SDEs on typical (non-probabilistic) paths, and is compatible with penalization and reflection theory for RDEs (e.g., for one-dimensional reflected rough equations) (Richard et al., 2019).

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