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., 2 Sep 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., 2 Sep 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., 2 Sep 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., 2 Sep 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., 2 Sep 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., 2 Sep 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., 2 Sep 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., 2 Sep 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., 2 Sep 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.