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The Infinitesimal Generator of a Jump-Diffusion Process

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The Formal Theorem

Let Xt X_t be a jump-diffusion process in Rd \mathbb{R}^d governed by the stochastic differential equation dXt=μ(Xt)dt+σ(Xt)dWt+∫Rdγ(Xt−,z)N~(dt,dz) dX_t = \mu(X_t)dt + \sigma(X_t)dW_t + \int_{\mathbb{R}^d} \gamma(X_{t-}, z) \tilde{N}(dt, dz) , where N~ \tilde{N} is a compensated Poisson random measure with intensity measure ν(dz) \nu(dz) . For a sufficiently smooth function f∈Cc2(Rd) f \in C_c^2(\mathbb{R}^d) , the infinitesimal generator L \mathcal{L} acting on f f is defined by Lf(x)=lim⁡t↓0E[f(Xt)∣X0=x]−f(x)t \mathcal{L}f(x) = \lim_{t \downarrow 0} \frac{\mathbb{E}[f(X_t) | X_0 = x] - f(x)}{t} , which yields:
Lf(x)=∑i=1dμi(x)∂f∂xi+12∑i,j=1d(σσT)ij(x)∂2f∂xi∂xj+∫Rd[f(x+γ(x,z))−f(x)−⟨∇f(x),γ(x,z)⟩]ν(dz) \mathcal{L}f(x) = \sum_{i=1}^d \mu_i(x) \frac{\partial f}{\partial x_i} + \frac{1}{2} \sum_{i,j=1}^d (\sigma \sigma^T)_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j} + \int_{\mathbb{R}^d} [f(x + \gamma(x, z)) - f(x) - \langle \nabla f(x), \gamma(x, z) \rangle] \nu(dz)

Analytical Intuition.

Imagine a particle traversing a landscape that is both shaky and prone to sudden teleportation. The L \mathcal{L} operator is our 'local weather forecaster' at any position x x . It decomposes the future evolution of the expected value of f(Xt) f(X_t) into three distinct regimes. First, the drift term μ(x)∇f \mu(x) \nabla f captures the deterministic momentum of the particle. Second, the diffusion term—the Laplacian-like structure involving σσT \sigma \sigma^T —accounts for the 'blurring' effect caused by Brownian fluctuations, smoothing out the function's expectations locally. Finally, the integral term—the jump part—is the most dramatic; it captures the global 'leaps' the process takes. Unlike diffusion, which only 'sees' the immediate neighborhood, the integral term accounts for the possibility of jumping to a distant state x+γ(x,z) x + \gamma(x, z) , weighted by the intensity measure ν(dz) \nu(dz) . By subtracting the gradient term, we effectively 'center' the jumps, ensuring the operator reflects the true infinitesimal change in expectation, essentially turning the complexity of stochastic paths into a clean, deterministic partial integro-differential equation.
CAUTION

Institutional Warning.

Students often struggle with the integral term, specifically why we subtract the gradient inner product. This 'compensation' is strictly necessary to ensure the integral converges and to maintain the martingale property of the resulting process, balancing out the 'drift' induced by the jumps themselves.

Academic Inquiries.

01

Why does the infinitesimal generator include a second-order derivative?

The second-order derivative arises from the Quadratic Variation of the Brownian component (Itô's Lemma), reflecting how diffusion spreads the probability density over time.

02

What happens if the jump measure ν \nu is infinite?

If ∫∣z∣<1∣z∣2ν(dz)<∞ \int_{|z|<1} |z|^2 \nu(dz) < \infty , the process has finite variance, and the generator remains well-defined. If not, the process may exhibit infinite activity, requiring a more nuanced truncation of the jumps.

Standardized References.

  • Definitive Institutional SourceØksendal, B., & Sulem, A., Applied Stochastic Control of Jump Diffusions.

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Institutional Citation

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NICEFA Visual Mathematics. (2026). The Infinitesimal Generator of a Jump-Diffusion Process: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/advanced-stochastic-processes/the-infinitesimal-generator-of-a-jump-diffusion-process

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