Nonlinear Girsanov Theorem

• Nonlinear Girsanov theorem is defined over sublinear expectation spaces, applying a generalized exponential martingale to shift drifts in degenerate G-Brownian settings.
• The approach uses a perturbative method by adding auxiliary Gaussian noise to regain ellipticity before taking the degenerate limit with robust L¹ controls.
• The theorem supports robust pricing and risk-neutral measures, preserving the generator and enforcing dual integrability conditions under model uncertainty.

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 (Liu, 2018).

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 $$\mathbb{E}|\mathcal{E}^{(\varepsilon)}_T - \mathcal{E}_T|\to0$$ (Liu, 2018).

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 (Liu, 2018). 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 (Prato et al., 2023).
• 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 (Privault, 2012).
• 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 (Harang et al., 2017).
• 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 (Hu et al., 2012, Liang et al., 2010).

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 (Liu, 2018).