Non-Gaussian Translation Processes: Theory & Applications

• Non-Gaussian translation processes are constructed by mapping Brownian motion through quantile transformations to achieve specified non-Gaussian marginals, such as skewed or heavy-tailed distributions.
• The methodology employs percentile-matching techniques with drift corrections and discrete approximations to preserve self-similarity and replicate target distributions in simulations.
• These processes are applied in financial mathematics, stochastic simulation, and time series analysis, enhancing models for asset returns and non-standard error distributions.

Non-Gaussian translation processes refer to stochastic processes constructed to have prescribed non-Gaussian marginal distributions, often while retaining some aspects of standard diffusion processes (such as self-similarity or covariance structure). These models address the limitations of classical Gaussian-based diffusion and time series models by enabling the explicit modeling of skewness, heavy tails, and other deviations from normality frequently observed in empirical data. The methodologies, theory, and simulation results summarized below provide a rigorous and flexible foundation for such processes, with applications spanning financial mathematics, time series, stochastic modeling, and simulation.

1. Mathematical Construction of Non-Gaussian Translation Processes

The canonical construction is based on mapping a Brownian motion—or more generally, a Gaussian process—through a percentile-matching transformation that imparts the desired non-Gaussian marginals. Let $\{B_t\}_{t \geq 0}$ denote standard Brownian motion on a filtered probability space and $F$ be a specified absolutely continuous cumulative distribution function (cdf) with mean zero and variance one. Define the translation process by

$$Z_t(\omega) = \sqrt{t}\, F^{-1}\left(\Phi\left(\frac{B_t(\omega)}{\sqrt{t}}\right)\right), \quad t > 0, \ \omega \in \Omega,$$

where $\Phi$ is the standard normal cdf and $F^{-1}$ its inverse. At each time $t$, this transformation maps the value of the Brownian motion to the corresponding quantile of $F$, scaled to have variance growing linearly in $t$.

This construction ensures that for any $t > 0$, the marginal distribution of $Z_t$ is $F(\cdot\,/\sqrt{t})$. If $F$ is normal, $Z_t$ reduces to Brownian motion; for non-Gaussian $F$ (e.g., Student's $t$, asymmetric Laplace, EGB2), the path increments inherit non-Gaussian structure (e.g., skewness, kurtosis).

A discrete approximation is provided by

$$Z_{k+\Delta t} = Z_k + h(Y_k) \sqrt{\Delta t} \, \xi_k,$$

with $\xi_k \in \{-1, 1\}$ equiprobable, $Y_k$ the current state, and $$h(y) = \varphi\left(y/\sqrt{t}\right) / f\left(F^{-1}\left(\Phi(y/\sqrt{t})\right)\right)$$, where $\varphi$ is the standard normal density and $f$ the density corresponding to $F$. This expression is interpreted as a transformed random walk, converging to the continuous translation process.

2. Probabilistic Properties

• Marginals: For any fixed $t > 0$, $Z_t$ has cdf $F(\cdot / \sqrt{t})$.
• Moments: $E[Z_t] = 0$, $Var[Z_t] = t$ (provided $F$ is standardized).
• Martingale and Increment Structure: Except in the Gaussian case, $Z_t$ is not a martingale. The conditional expectation

$$E[Z_t\,|\,Z_s] = \sqrt{t}\, F^{-1}\left(\Phi\left(B_s/\sqrt{t}\right)\right) + \frac{1}{2} h'(B_s)(t-s) + O((t-s)^{3/2})$$

includes a drift correction $h'(B_s)$. Increment distributions are non-stationary and not independent.

• Self-similarity: The process is 1/2-self-similar: $(1/\sqrt{c}) Z_{ct} \stackrel{d}{=} Z_t$.
• Copula Structure: The dependence structure is constructed via a copula induced by the Brownian motion, distinguishing this framework from marginal transformation via independent draws.

3. Diffusion Model and Stochastic Calculus

The process $Z_t$ is used as the driving noise in SDEs of the form

$dX_t = \alpha(X_t, t)\,dt + \sigma(X_t, t)\,dZ_t,$

with $dZ_t$ interpreted through Itô's lemma for the transformation $g(b, t) = \sqrt{t}\, F^{-1}(\Phi(b/\sqrt{t}))$ applied to $B_t$: $$dZ_t = \left[\frac{\partial g}{\partial t} + \frac{1}{2}\frac{\partial^2 g}{\partial b^2}\right]dt + \frac{\partial g}{\partial b} dB_t.$$ Thus, the SDE in terms of $B_t$ is

$$dX_t = [\alpha(X_t, t) + r(B_t, t)] dt + \sigma(X_t, t) h(B_t, t)dB_t,$$

with $r(B_t, t)$ the drift correction and $h(B_t, t)$ the local scale adjustment. For practical purposes, a simplified SDE omits $r(B_t, t)$ when the drift is negligible, resulting in a time-changed diffusion with stochastic volatility driven by $h$.

Existence and uniqueness of solutions are established using Lipschitz conditions and the Banach fixed-point theorem in the adapted-path space $C([0, T]; L^2(\Omega))$.

4. Approximation, Error Bounds, and Path Properties

An important practical aspect is the error incurred by omitting the drift term $r(B_t, t)$: $$E_t = Z_t - \widetilde{Z}_t = \int_0^t r(B_s, s) ds.$$ As $E[r(B_s, s)^2] = O(1/s)$, the mean squared error accumulates as $O(T \log T)$, so the root mean square error is $O(\sqrt{T \log T})$, which is asymptotically negligible relative to the $O(\sqrt{T})$ scaling of $Z_t$.

The process preserves self-similarity but generally lacks stationary increments or the Markov property (outside the Gaussian case), distinguishing it from classical Brownian motion and Lévy processes.

5. Numerical Simulations and Recovery of Target Marginals

Simulations for a range of target distributions (Student's $t$ with varying degrees of freedom, asymmetric Laplace with variable skewness parameter, and EGB2) demonstrate that the copula transformation and its discrete-time random walk approximation accurately reproduce the target marginal distributions for each $t$. Both the exact and simplified forms recover heavy tails and skewness as specified by $F$.

An explicit error analysis shows that omitting the drift term results in a small discrepancy between target and empirical marginal distributions, as confirmed by plotting the simulation results: the drift-corrected scheme (red line) matches the prescribed marginal, while the simplified scheme (orange line) remains close.

6. Applications and Theoretical Significance

• Financial Mathematics: Direct modeling of asset returns exhibiting skewness and heavy tails beyond the Black-Scholes paradigm; robust SDE models with prespecified marginal behavior.
• Stochastic Simulation: Fast generation of non-Gaussian noise processes with prescribed marginal features and continuous-path properties, useful for Monte Carlo studies.
• Time Series Analysis: Flexible error models for processes whose innovations or measurement noise are non-Gaussian, avoiding reliance on Lévy flights or jump diffusions.

The construction provides an alternative to processes with jumps (Lévy) or long memory (fractional Brownian motion), giving explicit control over both shape (via $F$) and scaling/self-similarity (inherited from $B_t$). Because the pathwise structure is inherited from Brownian motion but transformed through $F^{-1} \circ \Phi$, the resulting processes are continuous but with richer marginal behavior—a feature not generally available in Lévy-based models.

7. Extensions and Open Questions

• Further drift corrections: Higher-order asymptotic corrections to the drift term could further improve the fidelity of the process to target marginals.
• Generalizations to multivariate or infinite-dimensional settings: While the core construction is one-dimensional, extension to vector-valued processes via copula approaches or joint transformations is plausible.
• Nonparametric estimation and inference: With the explicit form relating observed marginals to underlying Brownian motion, likelihood-based inference or nonparametric estimation for the target $F$ becomes tractable.

Summary Table: Core Features

Property	Brownian Motion	Non-Gaussian Translation Process
Marginals	Normal, mean 0, var t	Prescribed $F$ scaled by $\sqrt{t}$
Martingale	Yes	No (unless $F$ is normal)
Stationary increments	Yes	No
Self-similarity	Yes, H=0.5	Yes, H=0.5 (inherited)
Path type	Continuous	Continuous

The non-Gaussian translation process construction thus offers a mathematically rigorous and practically tractable framework for continuous-path stochastic modeling with arbitrary marginal distributions, explicitly bridging the gap between classical diffusion theory and the distributional complexity required in real-world applications (Richardson et al., 5 Aug 2025).