Discrete Girsanov Transformations

• Discrete Girsanov transformations are change-of-measure techniques for stochastic jump and piecewise-deterministic processes.
• They employ explicit Radon–Nikodym derivatives and exponential martingales to reweight path probabilities and adjust transition intensities.
• Applications include mathematical finance, data-driven modeling, and Markov state models, with rigorous criteria ensuring measure validity.

Discrete Girsanov Transformations are change-of-measure constructions for stochastic processes with discrete, jump, or piecewise-deterministic dynamics. These transformations play a central role in stochastic analysis, probability, mathematical finance, and data-driven modeling, allowing rigorous modification of the probability law governing a process to accommodate controlled perturbations, parameter changes, or external interventions. Girsanov theory yields explicit Radon–Nikodym derivatives that relate the original and altered processes, enabling the computation of reweighted transition probabilities, likelihood ratios, and pathwise observables under the modified dynamics.

1. Mathematical Framework and Typical Examples

Discrete Girsanov transformations generalize the continuous-time Girsanov theorem to settings where the driving process has discontinuous paths, such as jump diffusions, Markov chains, Poisson processes, and Lévy processes. The theory encompasses processes driven by (i) canonical Poisson measures, (ii) finite-state Markov chains, (iii) pure-jump Lévy processes such as symmetric α-stable processes, and (iv) discrete-time Markov or diffusion-type schemes (Wang, 7 Sep 2025, Schilling et al., 2014, Privault, 2012, Korrapati et al., 2024).

For a finite-state, continuous-time Markov chain with generator $q_0(i,j)$, Girsanov transformations can produce a new process with perturbed generator $q_t$ (possibly time-inhomogeneous). For processes built from Poisson random measures, the transformation is applied to the jump structure via compensator modification or random shifts on configuration space, subject to cyclic finite-difference ("quasi-nilpotence") constraints (Privault, 2012). For pure jump Lévy processes, transformations incorporate jump intensity changes via explicit Radon–Nikodym densities (Schilling et al., 2014).

2. Radon–Nikodym Densities and Exponential Martingales

The foundational analytic object in a discrete Girsanov transformation is the Radon–Nikodym derivative (likelihood ratio) between the altered measure $\mathbb{Q}$ and baseline measure $\mathbb{P}$. For a Markov jump process with state space $E$, reference generator $q_0$, and transformed generator $q_t(i,j)$, the likelihood satisfies (Wang, 7 Sep 2025): $$L_t = \frac{d\mathbb{Q}}{d\mathbb{P}}\Bigg|_{\mathcal{F}_t} = \exp\left\{ \sum_{j\in E} \int_0^t \log\left(\frac{q_s(X_{s-}, j)}{q_0(X_{s-}, j)}\right) N^j(ds) - \int_0^t \sum_{j\in E} \bigl[q_s(X_s, j) - q_0(X_s, j)\bigr] ds \right\}$$ where $N^j(ds)$ is the compensated jump measure.

For pure-jump Lévy processes with jump measure $\nu$ and modification kernel $q_t$0, the Doleans–Dade exponential specializes to (Schilling et al., 2014): $q_t$1 This construction yields a new Markov process where the jump intensity is $q_t$2.

In the Poisson configuration space, the density under quasi-nilpotent transformation $q_t$3 with intensity compensation $q_t$4 is given by (Privault, 2012): $q_t$5

For discrete-time diffusions or data-driven Markov models, similar exponential martingale forms apply for the ratio of path probabilities, with drift and/or transition matrix perturbations (Korrapati et al., 2024, Donati et al., 2022).

3. Absolute Continuity, Singularity, and Martingale Criteria

A central concern is the mutual absolute continuity and possible singularity of the transformed path measures. In the finite-state Markov chain regime, the Radon–Nikodym density $q_t$6 is a true martingale (and thus a probability density) if and only if the expected total relative rate change is integrable (Wang, 7 Sep 2025): $q_t$7 and $q_t$8 for all $q_t$9. This condition is analogous to Novikov's and Kazamaki's criteria for exponential martingales.

For pure-jump Lévy processes, the zero–two law established by Schilling–Vondraček states that either the quadratic variation of the martingale additive functional is finite almost surely under $\mathbb{Q}$0 or infinite—yielding strict dichotomy: either mutual absolute continuity or singularity (Schilling et al., 2014).

The same logic determines when Girsanov identities hold for Poisson measures under random transformations, depending on the quasi-nilpotence and integrability conditions (Privault, 2012).

4. Algorithmic Reweighting in Empirical or Discretized Models

Discrete Girsanov transformations offer practical computational strategies for reweighting path ensembles, as seen in Markov State Models (MSMs) and generative modeling (Donati et al., 2022, Korrapati et al., 2024). In MSMs, the empirically estimated transition matrix under a perturbed potential $\mathbb{Q}$1 is obtained by reweighting short reference trajectories via trajectory-wise Girsanov weights: $\mathbb{Q}$2 and updating transition counts accordingly before normalization.

In generative probabilistic models using discrete-time SDEs, the discrete Radon–Nikodym derivative is used to bound KL divergences and total variation distances between the approximate (score-based reverse) and reference (forward noising) distributions. The Girsanov formula determines the path likelihood ratio and facilitates analytic error control (Korrapati et al., 2024).

5. Special Constructions for Poisson and Jump Processes

The Girsanov theorem extends to Poisson random measures and anticipative transformations, provided the shift map $\mathbb{Q}$3 is quasi-nilpotent, significantly simplifying the form of the Radon–Nikodym derivative. For a Poisson process with compensator $\mathbb{Q}$4, the discrete Girsanov theorem asserts that under suitable integrability, the transformed process has modified compensator $\mathbb{Q}$5, and the change-of-measure density is explicitly given by a stochastic exponential and product over jump times (Privault, 2012): $\mathbb{Q}$6 This machinery underpins invariance statements for Poisson measures under intensity-preserving random translations and figures in combinatorial moment identities such as those involving Bell and Charlier polynomials.

6. Limitations, Regularity, and Sample Path Coverage

The validity of discrete Girsanov transformations depends critically on several regularity and overlap conditions. For empirical/algorithmic applications, the following limitations apply (Donati et al., 2022):

• If the perturbation $\mathbb{Q}$7 is large or the lag time is long, weight variance can become excessive, causing numerical instability.
• Reference path samples must cover all regions of the perturbed state space; otherwise, the reweighting fails due to lack of absolute continuity.
• Recording pathwise sufficient statistics (e.g., increments of noise or jump events) may require nontrivial modifications at the simulation or data acquisition level.
• For time-inhomogeneous or high-dimensional cases, effective sample size should be monitored to ensure meaningful results.

For processes with varying jump intensities or under random, possibly non-adapted transformations, further care is needed to check the quasi-nilpotence, integrability, and invertibility conditions necessary for validity of the discrete Girsanov identity (Privault, 2012, Schilling et al., 2014).

7. Connections and Comparisons with Continuous (Diffusion) Girsanov Theory

While in diffusion theory (Brownian motion, SDEs driven by Wiener processes) the Girsanov transform modifies the drift, in the discrete or jump setting it acts via local changes in the jump (or transition) intensities (Wang, 7 Sep 2025). Both frameworks utilize exponential martingale criteria, martingale problem uniqueness, and pathwise likelihood ratios. In the Poisson/Lévy regime, cyclic finite-difference conditions play a role analogous to trace-class or Fredholm-determinant constraints in the Wiener case, but combinatorial simplifications often eliminate determinants in the jump framework.

The purely discontinuous Girsanov transformation specializes and sharpens the contrast with continuous Girsanov theory—while both admit dichotomy (absolute continuity/singularity), the criteria for finiteness of Radon–Nikodym densities are scale-sensitive and reflect the discrete or jump intensity landscape.

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Key References:

• R.L. Schilling & Z. Vondraček, "Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process" (Schilling et al., 2014)
• E. Privault, "Girsanov identities for Poisson measures under quasi-nilpotent transformations" (Privault, 2012)
• H. Wang, "Martingale Problem and Quadratic Family" (Wang, 7 Sep 2025)
• "Discrete vs. Continuous Trade-offs for Generative Models" (Korrapati et al., 2024)
• "A review of Girsanov Reweighting and of Square Root Approximation for building molecular Markov State Models" (Donati et al., 2022)