Jump-Diffusion OM Functionals

• Jump-Diffusion OM Functionals quantify the path-wise probabilities in stochastic processes exhibiting both continuous diffusion and abrupt jump dynamics.
• They leverage advanced methods such as probability-flow and Girsanov transformations to manage nonlocal corrections, especially in infinite-activity regimes.
• Applications include rare-event simulation, variational inference, and financial derivative pricing, with numerical techniques enabling efficient time-discrete approximations.

A jump-diffusion OM (Onsager–Machlup) functional characterizes the path-wise probability of trajectories in stochastic systems exhibiting both continuous (diffusive) and discontinuous (jump) dynamics. These functionals play an essential role in nonequilibrium statistical mechanics, stochastic control, rare-event simulation, and quantification of fluctuations in systems with Lévy or compound Poisson noise. Derivations and practical computation of jump-diffusion OM functionals have historically presented significant challenges due to the inherent complexity induced by jump processes, especially in infinite-activity regimes.

1. Mathematical Formulation and Principle

For a general jump-diffusion process in $\mathbb{R}^d$ of the form

$$dX_t = b(X_{t-})\,dt + \sigma\,dB_t + J_t\,dN_t,\quad X_0 = x_0,$$

where $B_t$ is Brownian motion, $N_t$ is a Poisson process of rate $\lambda(X_{t-})$, and $J_t$ are jump sizes with Lévy measure $\nu_J$, the path distribution is governed by the Onsager–Machlup (OM) functional. The OM functional quantifies, to leading exponential order, the probability of a path remaining close to a given $C^2$ trajectory $x(t)$: $$P\{\|X - x\|_\infty < \epsilon\} \approx C(\epsilon)\,\exp\left(-S_{\mathrm{OM}}[x] + o(1)\right),\quad \epsilon \to 0.$$ For finite-activity jump processes, the OM action admits a closed-form expression: $$S_{\mathrm{OM}}[x] = \frac{1}{2\sigma^2} \int_0^T \|\dot{x}_t - b(x_t) - \ell_J(x_t)\|^2\,dt - \frac{1}{2} \int_0^T \nabla \cdot (b + \ell_J)(x_t)\,dt - \int_0^T \gamma_J(x_t)\,dt,$$ where $\ell_J(x) = \int z\,\nu_J(dz)$ is the mean jump drift and $\gamma_J(x)$ arises from small-jump intensity, with the precise structure determined by the behavior of the Lévy measure near the origin (Huang et al., 2024).

In infinite-activity regimes where $\nu_J$ is singular at zero, only a discrete-time OM functional is generally available, using partitions $\{t_i\}_{i=0}^n$ and increments $x_i - x_{i-1}$, together with explicit nonlocal corrections (Huang et al., 2024).

2. Probability-Flow and Girsanov Approaches

Historically, OM functionals for diffusions rely on path-integral or Girsanov transformations. Extension to jump-diffusions has required new methodologies due to the path-discontinuities induced by jumps. The probability-flow approach rewrites the Lévy–Fokker–Planck equation as a continuity equation for an equivalent pure diffusion process with a modified drift: $$b_{\mathrm{new}}(x, t) = b(x) + \int z\,\frac{p_t(y, x-z)}{p_t(y, x)} \nu_J(dz).$$ This drift ensures that the pure diffusion and the original jump-diffusion share identical one-time marginals. Thus, any path-integral or Girsanov-based OM derivation for the diffusion applies verbatim to the jump-diffusion upon insertion of $b_{\mathrm{new}}$ (Huang et al., 2024).

For finite-activity processes, a short-time asymptotic expansion justifies replacing the jump term by its first-order contribution, yielding a continuous-time OM functional for regular enough coefficients and transition densities. In the infinite-activity case, Truncation and time-discretization techniques yield a path-wise action that is only finite on the time grid (Huang et al., 2024).

3. General Unified Framework

A general OM functional for jump-diffusions, accounting for both diffusion and jump terms, can be posed as: $$\mathrm{OM}[X, \lambda] = \int_{0}^{T}\!dt\; \left\{ \tfrac{1}{2}\big(\dot{X}_t - F(X_t)\big)^\top D^{-1}(X_t)\big(\dot{X}_t - F(X_t)\big) +\tfrac{1}{2}\nabla\cdot F(X_t) +\sum_{i\neq j}\Big[ \lambda_{ij}(t)\ln\frac{\lambda_{ij}(t)}{r_{ij}(X_t)} -\lambda_{ij}(t)+r_{ij}(X_t) \Big] \right\},$$ where $\lambda_{ij}(t)$ is the empirical jump rate from $i$ to $j$, $F$ is the continuous drift, and $r_{ij}(X_t)$ the nominal jump rates. The jump term is a Kullback–Leibler divergence rate, reflecting the information cost of observing empirical jump fluxes different from the underlying transition rates (Stutzer et al., 6 Aug 2025). This formulation unifies previous approaches and provides a direct calculus for path-wise functionals in general Markovian settings.

4. Extensions to Occupation-Time and Boundary Functionals

Occupation-time OM functionals quantify joint distributions of time spent by a jump-diffusion in prescribed intervals and the terminal value of the process. In the mixed-exponential jump-diffusion (MEJD) model,

$$X_t = X_0 + \mu t + \sigma W_t + \sum_{i=1}^{N_t} Y_i,$$

joint Laplace transforms for occupation times and $X_T$ are characterized by explicit linear algebraic representations: $$w(x) = \sum_{i=1}^{m+1} \omega_i^L e^{\beta_{i,\alpha}(x-h)} - c_L e^{\gamma x},\quad x \leq h,$$ with $\{\omega_i^L\}$ determined as the unique solution to a block-linear system parameterized by the roots of the model's Lévy exponent. These objects yield explicit path-space transforms for step and quantile options in financial mathematics (Aoudia et al., 2016).

Boundary OM functionals, such as two-sided exit-time and location distributions for double-exponential jump-diffusions (Kou processes), reduce to moment generating functions constructed from four roots of the characteristic exponent equations, double integrals over killed-extrema densities, and explicit algebraic normalization (Karnaukh, 2013).

5. Applications and Numerical Implementation

Jump-diffusion OM functionals are crucial in several domains:

• Most-probable transition paths in metastable systems with non-Gaussian noise, by minimizing the OM action (Huang et al., 2024).
• Variational inference and control formulations, using the OM action as a Lagrangian for path-space optimization (Stutzer et al., 6 Aug 2025).
• Rare-event simulation via importance sampling, where the OM functional provides optimal tilts for importance weights (Huang et al., 2024).
• Pricing of complex financial derivatives, including options sensitive to occupation times, barriers, and quantiles. Full path-integral propagators and pricing formulas in jump-diffusion stochastic volatility models are derived via Fourier-space factorization, with jumps incorporated via cumulant generating functions for various jump size distributions (Liang et al., 2010, Aoudia et al., 2016).
• Analytic characterization of joint exit statistics, explicitly for compound Poisson jump-diffusions with double-exponential jumps (Karnaukh, 2013).

Numerically, time-discrete OM functionals permit Euler–Maruyama path sampling with jump-drift corrections and explicit path weights for use in Monte Carlo, variational, or importance sampling algorithms (Huang et al., 2024).

6. Assumptions, Regularity, and Open Challenges

Derivations of OM functionals for jump-diffusions generally require:

• Finite-activity: smoothness and boundedness (e.g., $b, \lambda(x), \nu_J$ are $C^\infty$ with compact support), strictly positive and regular transition densities (Huang et al., 2024).
• Infinite-activity: dominating Lévy measure with regular density near $z=0$, invertibility of Jacobians for jump mappings, and existence of smooth transition semigroups (Huang et al., 2024).

For processes with singular Lévy measures or degenerate diffusion, continuous-time OM functionals may break down, and only their discrete analogs are strictly meaningful. Extension to degenerate, state-dependent, or correlated jump structures remains an area of ongoing research.

7. Connections and Theoretical Significance

The OM functional for jump-diffusion processes provides foundational tools for nonequilibrium fluctuation theory, allowing rigorous derivation of thermodynamic uncertainty and speed-limit relations. Extremal trajectories (i.e., solutions of the Euler–Lagrange equations for the OM–Lagrangian) saturate these inequalities, identifying optimal fluctuation pathways (Stutzer et al., 6 Aug 2025). The unification of path-wise fluctuation calculus for diffusion and jump components clarifies the structural parallels and distinctions between continuous and discrete-state statistical mechanics, directly influencing thermodynamic inference, theory of large deviations, and control of complex stochastic systems.