---
title: Nonlinear Girsanov Theorem
url: https://www.emergentmind.com/topics/nonlinear-girsanov-theorem
type: topic
---

# Nonlinear Girsanov Theorem

A nonlinear Girsanov theorem extends the classical Girsanov measure-change to settings where the underlying probability or expectation is neither linear nor absolutely continuous, including sublinear expectation spaces (notably $G$-expectation), stochastic partial differential equations with nonlinear perturbations, and systems with degenerate or path-dependent noise. The resulting formulas identify the law of a drift- or volatility-perturbed process in terms of a generalized exponential martingale or transformation, subject to carefully quantified integrability and structural (often monotonicity or quasi-nilpotence) conditions. Unlike the classical scenario, the nonlinear Girsanov transform often operates at the level of capacities or sublinear expectations, with the Radon–Nikodym derivative and martingale structures replaced by more general objects.

## 1. Nonlinear Girsanov Theorem in $G$-Expectation: The Degenerate Case

The $G$-expectation framework, developed for volatility-uncertain systems, takes place on canonical path space $(\Omega, C_b(\Omega), \mathbb{E})$, where $\Omega = C([0,T];\mathbb{R}^d)$, $B_t(\omega) = \omega_t$ is the coordinate process, and $\mathbb{E}$ is a sublinear expectation with generator $G\colon S(d)\to\mathbb{R}$, monotone and sublinear on $S(d)$, the space of symmetric $d\times d$ matrices. $B$ is a $G$-Brownian motion, possibly degenerate.

Given a progressively measurable $h \in M^2(0,T;\mathbb{R}^d)$, assume both:
- $G$-Novikov’s condition: $\exists\,\varepsilon>0$ such that $\mathbb{E}\exp\left(\frac{1+\varepsilon}{2}\int_0^T\langle h_s,\,d\langle B\rangle_s\,h_s\rangle\right)<\infty$
- Quadratic-integrability: $\exists\,\delta>0$ such that $\mathbb{E}\exp\left(\delta\int_0^T|h_s|^2\,ds\right)<\infty$

The density process is
\[
\mathcal{E}_t = \exp\left( \int_0^t \langle h_s, dB_s\rangle - \frac{1}{2}\int_0^t \langle h_s, d\langle B\rangle_s\,h_s\rangle \right).
\]
Defining the new sublinear expectation $\widetilde{\mathbb{E}}[X] := \mathbb{E}[X\,\mathcal{E}_T]$ on cylinder functionals, one finds that under $\widetilde{\mathbb{E}}$, the process
\[
\widetilde{B}_t = B_t - \int_0^t d\langle B\rangle_s\,h_s,
\]
is again a $G$-Brownian motion with the same generator $G$. This extension is valid even when the generator $G$ is degenerate (i.e., allows zero lower bounds on the possible volatility directions), covering cases unreachable by PDE-based nondegenerate arguments [1808.01825].

## 2. Methodology: Perturbation and Limiting Procedure

In the degenerate regime, the $G$-Brownian motion $B$ may lack uniform ellipticity, rendering classical analytic (PDE regularity) methods inapplicable. The proof thus employs a perturbative approach: the product space $\widetilde{\Omega} = \Omega \times C([0,T];\mathbb{R}^d)$, supporting independent $(B,W)$ (with $W$ a standard Brownian motion), is constructed.

For $\varepsilon>0$, define perturbed process $B^{(\varepsilon)}_t = B_t + \varepsilon W_t$, which possesses a generator $G^{(\varepsilon)}$ that is nondegenerate. The classical (nondegenerate) $G$-Girsanov result applies to $(B^{(\varepsilon)}, h)$. The nonlinear Girsanov theorem is then obtained by proving convergence of expectations of the relevant exponential martingale as $\varepsilon\downarrow 0$, enabled via robust exponential martingale estimates. The proof does not appeal to dominated convergence, but rather relies on explicit $L^1$ control of the density error $\mathbb{E}|\mathcal{E}^{(\varepsilon)}_T - \mathcal{E}_T|\to0$ [1808.01825].

## 3. Key Structure: Sublinear Expectations, Martingale Properties, and Density Processes

The nonlinear Girsanov theorem operates in the absence of a single reference probability measure. The role of the Radon–Nikodym derivative is assumed by the exponential martingale $\mathcal{E}_t$, which is shown to be a symmetric $G$-martingale with $\mathbb{E}[\mathcal{E}_t]=1$. The new transformed process $\widetilde{B}_t$ shifts only by the quadratic variation drift correction, preserving the "law" (generator set $G$) of the $G$-Brownian motion under the new sublinear expectation $\widetilde{\mathbb{E}}$.

The methodology produces the following pair of expectations and processes:

| Object                                       | Definition (under $G$-expectation)       | Meaning                          |
|-----------------------------------------------|------------------------------------------|----------------------------------|
| Density $\mathcal{E}_t$                       | $\exp \left(\int_0^t h_s dB_s - \frac12\int_0^t h_s d\langle B\rangle_s h_s \right)$ | Change of expectation weight     |
| Process under new expectation $\widetilde{B}$ | $B_t - \langle B,h\cdot\rangle_t$        | G-Brownian motion under new law  |
| Expectation $\widetilde{\mathbb{E}}$          | $\mathbb{E}[X \mathcal{E}_T]$            | Tilted sublinear expectation     |

For classical Brownian motion, this maps to the exponential martingale density and drift-transformed Wiener process. In the $G$-framework, the same structure emerges, but under the sublinear capacity and with $G$-Brownian increments replacing Gaussian increments.

## 4. Applications and Generalizations

The degenerate nonlinear Girsanov formula enables drift-shift changes for a broad class of volatility-uncertain and degenerate processes. One explicit example is the one-dimensional degenerate $G$:
\[
G(a)=\frac12\overline\sigma^2 a^+ \qquad (\underline\sigma^2=0),
\]
for which the nonlinear Girsanov theorem still yields valid risk-neutral change-of-measure formulas, enabling risk-neutral pricing and control under extreme model uncertainty. Analogues of these transformations appear in $G$-BSDE theory, with applications to pricing, hedging, and analysis under model ambiguity [1808.01825]. The mathematical machinery underlying this result, notably the perturbation argument and sublinear expectation calculus, is foundational for stochastic analysis beyond the linear, measure-based regime.

## 5. Comparison with Other Nonlinear Girsanov Formulations

The nonlinear Girsanov mechanism in the degenerate $G$-Brownian case is closely related to, but distinct from, other nonlinear measure-change results in different frameworks:
- In the mild Girsanov formula for semilinear SPDEs, the law of the nonlinear evolution is expressed in terms of a nonlinear shift on a Gaussian reference measure (via the Ramer theorem), with the density incorporating the energy and divergence of the shift [2308.04184].
- In probabilistic frameworks for anticipating Poisson transformations, a nonlinear Girsanov identity arises for transformations satisfying a strong quasi-nilpotence condition, with the density constructed by an explicit combinatorial argument [1205.5158].
- For SDEs driven by multifractional Brownian motion, the nonlinear Girsanov formula involves the multifractional derivative of the shift and controls the law of the nonlinear process via the corresponding Doleans-Dade exponential [1706.07387].
- In BSDE and $G$-BSDE contexts, the nonlinear Girsanov transformation impacts both the law of the driving signal and the structure of the backward equation, yielding transformed dynamics and predictable corrections [1212.5403, 1011.3228].

These generalizations exploit either the path-space structure (Itô-map), sublinear expectation theory, or combinatorial identities, but all bypass the need for absolute continuity between reference and target measures.

## 6. Distinctive Features and Technical Advances in the Degenerate $G$-Case

The most salient nonlinear aspects of the $G$-expectation Girsanov formula are:

- **No uniform ellipticity required**: The new proof applies even when the law allows zero volatility in some directions, extending the theory to degenerate $G$-Brownian motion.

- **Perturbative proof technique**: Rather than appealing to analytic PDE regularity, the argument introduces an auxiliary (Gaussian) noise to temporarily regain ellipticity, then passes to the degenerate limit with robust $L^1$ controls.

- **Two simultaneous integrability conditions**: Both the $G$-Novikov-type exponential integrability along the quadratic variation and the quadratic-exponential integrability are required, reflecting the interplay between the sublinear drift and the possible singularity of the quadratic variation.

- **Preservation of the generator**: Under the nonlinear transform, the generator $G$ is unchanged, so all volatility-uncertainty information is preserved under the shift.

This combination makes the nonlinear Girsanov theorem for degenerate $G$-Brownian motion foundational for model-uncertainty analysis, robust pricing, and pathwise constructions of stochastic analysis under ambiguity [1808.01825].

Source: https://www.emergentmind.com/topics/nonlinear-girsanov-theorem