Infinitesimal Generator of Affine Processes¶

The infinitesimal generator is the differential operator that encodes the local behavior of a Markov process. For a diffusion, it involves the drift and the second-order diffusion term; for a jump-diffusion, it adds an integral over jump sizes. The defining property of an affine process is that every coefficient appearing in the generator --- drift, diffusion, and jump intensity --- is an affine function of the state. This affine structure is what makes the generator act so simply on exponential functions, producing the functions \(F\) and \(R\) that drive the Riccati ODE system for the characteristic function.

Learning Objectives

By the end of this section, you will be able to:

Write down the infinitesimal generator of a general affine jump-diffusion and identify each component
Verify the affine structure of the drift, diffusion, and jump compensator
Compute the generator applied to exponential-affine functions and derive the functions \(F(u)\) and \(R(u)\)
Explain the domain of the generator and the regularity conditions required
Connect the generator to the Riccati system via the Kolmogorov backward equation

Motivation¶

Why the Generator Matters¶

The transition semigroup \((P_t)_{t \geq 0}\) of a Markov process encodes the entire probabilistic evolution: \(P_t f(x) = \mathbb{E}^x[f(X_t)]\). The infinitesimal generator \(\mathcal{A}\) captures the instantaneous rate of change of this semigroup:

\[ \mathcal{A}f(x) = \lim_{t \downarrow 0} \frac{P_t f(x) - f(x)}{t} \]

Knowing \(\mathcal{A}\) is equivalent to knowing the semigroup (under mild conditions), just as knowing the derivative of a function determines the function up to initial conditions. For pricing and hedging, the generator appears directly in the Kolmogorov backward equation: if \(u(t,x) = \mathbb{E}^x[g(X_T) \mid X_t = x]\), then \(\partial_t u + \mathcal{A}u = 0\). For affine processes, the affine structure of \(\mathcal{A}\) is precisely what converts this PDE into a system of Riccati ODEs.

The General Infinitesimal Generator¶

Diffusion Case¶

For a \(d\)-dimensional diffusion \(dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\) with drift \(\mu : \mathbb{R}^d \to \mathbb{R}^d\) and diffusion matrix \(a(x) = \sigma(x)\sigma(x)^\top\), the generator acts on twice continuously differentiable functions \(f \in C^2(\mathbb{R}^d)\) as

\[ \mathcal{A}f(x) = \sum_{i=1}^d \mu_i(x)\,\frac{\partial f}{\partial x_i}(x) + \frac{1}{2}\sum_{i,j=1}^d a_{ij}(x)\,\frac{\partial^2 f}{\partial x_i \partial x_j}(x) \]

In compact notation:

\[ \mathcal{A}f(x) = \mu(x)^\top \nabla f(x) + \frac{1}{2}\operatorname{tr}\!\bigl[a(x)\,\nabla^2 f(x)\bigr] \]

Jump-Diffusion Case¶

When the process also has jumps with compensator \(m(x, dz)\) (the Levy-type jump measure that may depend on the state), the generator gains an integral term:

Definition: Generator of a Jump-Diffusion

The infinitesimal generator of a jump-diffusion is

\[ \mathcal{A}f(x) = \mu(x)^\top \nabla f(x) + \frac{1}{2}\operatorname{tr}\!\bigl[a(x)\,\nabla^2 f(x)\bigr] + \int_{\mathbb{R}^d \setminus \{0\}} \!\bigl[f(x+z) - f(x) - z^\top \nabla f(x)\,\mathbf{1}_{\{|z| \leq 1\}}\bigr]\,m(x, dz) \]

The three terms represent:

Drift: the deterministic component of the infinitesimal evolution
Diffusion: the second-order contribution from the continuous martingale part
Jumps: the contribution from discontinuous movements, with the truncation function \(\mathbf{1}_{\{|z| \leq 1\}}\) ensuring integrability

Affine Structure of the Generator¶

Affine Coefficients¶

For an affine process on the state space \(D = \mathbb{R}_+^m \times \mathbb{R}^{d-m}\), the generator coefficients are affine functions of the state:

\[ \mu(x) = b_0 + B x \]

\[ a(x) = a_0 + \sum_{i=1}^d \alpha_i\,x^{(i)} \]

\[ m(x, dz) = m_0(dz) + \sum_{i=1}^d x^{(i)}\,m_i(dz) \]

where:

\(b_0 \in \mathbb{R}^d\) and \(B \in \mathbb{R}^{d \times d}\) define the affine drift
\(a_0 \in \mathbb{R}^{d \times d}\) (symmetric, positive semi-definite) and \(\alpha_i \in \mathbb{R}^{d \times d}\) (symmetric, positive semi-definite) define the affine diffusion
\(m_0(dz)\) and \(m_i(dz)\) are Levy measures on \(\mathbb{R}^d \setminus \{0\}\) defining the affine jump structure

Theorem: Generator of an Affine Process

The infinitesimal generator of an affine process takes the form

\[ \mathcal{A}f(x) = (b_0 + Bx)^\top \nabla f(x) + \frac{1}{2}\operatorname{tr}\!\left[\left(a_0 + \sum_{i=1}^d \alpha_i\,x^{(i)}\right)\nabla^2 f(x)\right] + \int_{\mathbb{R}^d \setminus \{0\}} \bigl[f(x+z) - f(x) - h(z)^\top \nabla f(x)\bigr]\left(m_0(dz) + \sum_{i=1}^d x^{(i)}\,m_i(dz)\right) \]

where \(h(z) = z\,\mathbf{1}_{\{|z| \leq 1\}}\) is the truncation function.

The affine dependence on \(x\) in every term is the structural property that distinguishes affine processes from general Markov processes.

Action on Exponential Functions¶

The Key Computation¶

The characteristic function of an affine process has the exponential-affine form precisely because the generator acts on exponential functions in a particularly simple way. This computation is the technical heart of the affine theory.

Proposition: Generator Applied to Exponentials

For \(e_u(x) = e^{u^\top x}\) with \(u \in \mathbb{C}^d\) in the appropriate domain, the generator satisfies

\[ \mathcal{A}e_u(x) = \bigl[F(u) + R(u)^\top x\bigr]\,e_u(x) \]

where

\[ F(u) = b_0^\top u + \frac{1}{2}u^\top a_0\,u + \int_{\mathbb{R}^d \setminus \{0\}} \bigl(e^{u^\top z} - 1 - u^\top h(z)\bigr)\,m_0(dz) \]

\[ R_i(u) = (Bu)_i + \frac{1}{2}u^\top \alpha_i\,u + \int_{\mathbb{R}^d \setminus \{0\}} \bigl(e^{u^\top z} - 1 - u^\top h(z)\bigr)\,m_i(dz) \]

for \(i = 1, \ldots, d\).

Proof. Applying the generator to \(e_u(x)\):

Drift term:

\[ (b_0 + Bx)^\top \nabla e_u(x) = (b_0 + Bx)^\top u\,e_u(x) = \bigl(b_0^\top u + (Bx)^\top u\bigr)\,e_u(x) \]

Diffusion term:

\[ \frac{1}{2}\operatorname{tr}\!\left[\left(a_0 + \sum_i \alpha_i x^{(i)}\right) u\,u^\top\right] e_u(x) = \frac{1}{2}\left(u^\top a_0\,u + \sum_i u^\top \alpha_i\,u\,x^{(i)}\right) e_u(x) \]

Jump term:

\[ \int \bigl[e^{u^\top(x+z)} - e^{u^\top x} - h(z)^\top u\,e^{u^\top x}\bigr]\left(m_0(dz) + \sum_i x^{(i)}m_i(dz)\right) \]

\[ = e_u(x)\int\bigl(e^{u^\top z} - 1 - u^\top h(z)\bigr)\left(m_0(dz) + \sum_i x^{(i)}m_i(dz)\right) \]

Collecting terms independent of \(x\) into \(F(u)\) and terms proportional to \(x^{(i)}\) into \(R_i(u)\) gives the result. \(\square\)

Connection to the Riccati System¶

The functions \(F\) and \(R\) are precisely the right-hand sides of the Riccati ODEs for the characteristic function. If \(\phi(\tau, u)\) and \(\psi(\tau, u)\) satisfy

\[ \frac{d\psi}{d\tau} = R(\psi), \qquad \frac{d\phi}{d\tau} = F(\psi) \]

with \(\psi(0, u) = u\) and \(\phi(0, u) = 0\), then

\[ \mathbb{E}\!\left[e^{u^\top X_T} \mid X_t = x\right] = \exp\!\bigl(\phi(\tau, u) + \psi(\tau, u)^\top x\bigr) \]

This follows because the process \(M_s = \exp(\phi(T-s, u) + \psi(T-s, u)^\top X_s)\) is a martingale on \([t, T]\), which requires \(\partial_s M_s + \mathcal{A}M_s = 0\) --- and the Riccati system is precisely the condition for this cancellation.

Domain of the Generator¶

Core Domain¶

The generator \(\mathcal{A}\) is defined on a domain \(\mathcal{D}(\mathcal{A}) \subset C_0(\mathbb{R}^d)\) (continuous functions vanishing at infinity) that is dense in \(C_0(\mathbb{R}^d)\). For affine diffusions (no jumps), the domain contains \(C_c^2(\mathbb{R}^d)\) (twice continuously differentiable functions with compact support).

For affine jump-diffusions, the domain must additionally satisfy integrability conditions with respect to the jump measures \(m_0\) and \(m_i\): the function \(f\) must be such that the integral term in the generator converges.

Extended Generator¶

In practice, the exponential functions \(e_u(x) = e^{u^\top x}\) do not belong to \(C_0(\mathbb{R}^d)\) (they grow without bound). The computation above uses the extended generator, which acts on a larger class of functions satisfying appropriate growth conditions. The extended generator coincides with \(\mathcal{A}\) on the core domain and satisfies the same formula on exponential functions provided the moments \(\int |z|^2\,(m_0(dz) + \sum_i |x^{(i)}|\,m_i(dz))\) are finite.

Growth Conditions

The exponential function \(e_u\) belongs to the domain of the extended generator only when \(u\) lies in the set \(\mathcal{U} = \{u \in \mathbb{C}^d : \mathbb{E}[e^{\operatorname{Re}(u)^\top X_t}] < \infty\}\). For CIR-type components (\(x^{(i)} \geq 0\)), this restricts \(\operatorname{Re}(u_i) \leq 0\) for the corresponding coordinates.

Example: CIR Process¶

Consider the one-dimensional CIR process

\[ dX_t = \kappa(\theta - X_t)\,dt + \xi\sqrt{X_t}\,dW_t \]

on \(D = \mathbb{R}_+\), with no jumps. The affine coefficients are:

\(b_0 = \kappa\theta\), \(B = -\kappa\) (scalar)
\(a_0 = 0\), \(\alpha_1 = \xi^2\)
\(m_0 = m_1 = 0\) (no jumps)

The generator is

\[ \mathcal{A}f(x) = \kappa(\theta - x)\,f'(x) + \frac{1}{2}\xi^2 x\,f''(x) \]

Applying to \(e_u(x) = e^{ux}\):

\[ \mathcal{A}e^{ux} = \left[\kappa\theta\,u + \left(-\kappa u + \frac{1}{2}\xi^2 u^2\right)x\right]e^{ux} \]

Reading off:

\[ F(u) = \kappa\theta\,u, \qquad R(u) = -\kappa u + \frac{1}{2}\xi^2 u^2 \]

The Riccati equation \(\psi' = R(\psi) = -\kappa\psi + \frac{1}{2}\xi^2\psi^2\) is the familiar CIR Riccati equation, whose solution gives the CIR characteristic function and bond price formula.

Example: Affine Jump-Diffusion¶

Consider a one-dimensional affine process with constant-intensity compound Poisson jumps:

\[ dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t + dJ_t \]

where \(J_t = \sum_{k=1}^{N_t} Z_k\) is a compound Poisson process with intensity \(\lambda\) and jump size distribution \(\nu(dz)\). Here the jumps are state-independent, so \(m_0(dz) = \lambda\,\nu(dz)\) and \(m_1 = 0\).

The generator is

\[ \mathcal{A}f(x) = \kappa(\theta - x)f'(x) + \frac{1}{2}\sigma^2 f''(x) + \lambda\int_{\mathbb{R} \setminus \{0\}}\bigl[f(x+z) - f(x)\bigr]\,\nu(dz) \]

where the truncation function is absorbed since \(\int |z|\,\nu(dz) < \infty\) for compound Poisson processes. Applying to \(e^{ux}\):

\[ F(u) = \kappa\theta\,u + \frac{1}{2}\sigma^2 u^2 + \lambda\bigl(\hat{\nu}(u) - 1\bigr) \]

\[ R(u) = -\kappa u \]

where \(\hat{\nu}(u) = \int e^{uz}\,\nu(dz)\) is the moment generating function of the jump size distribution. The jump contribution \(\lambda(\hat{\nu}(u) - 1)\) adds to the \(F\) function because the jump intensity is state-independent.

Summary¶

The infinitesimal generator of an affine process has three components --- drift, diffusion, and jumps --- each affine in the state variable \(x\). The key computation is the action of the generator on exponential functions: \(\mathcal{A}e^{u^\top x} = [F(u) + R(u)^\top x]\,e^{u^\top x}\), where \(F\) collects the state-independent contributions and \(R\) collects the state-dependent ones. The functions \(F\) and \(R\) are precisely the right-hand sides of the Riccati ODE system that governs the characteristic function. This connection --- from the generator to exponentials to Riccati equations --- is the fundamental mechanism that makes affine processes analytically tractable.

Exercises¶

Exercise 1. For the Vasicek model \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t\), write down the infinitesimal generator \(\mathcal{A}\) and compute \(\mathcal{A}f\) for \(f(x) = x^2\). Verify that \(\mathcal{A}f(x) = 2\kappa(\theta - x)x + \sigma^2\), and explain why this result does not have the form \([F(u) + R(u)x]\,f(x)\) (i.e., polynomial test functions do not simplify in the same way as exponentials).

Solution to Exercise 1

The Vasicek model \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t\) has drift \(\mu(x) = \kappa(\theta - x)\) and constant diffusion \(a(x) = \sigma^2\). The generator is

\[ \mathcal{A}f(x) = \kappa(\theta - x)\,f'(x) + \frac{1}{2}\sigma^2\,f''(x) \]

For \(f(x) = x^2\), we have \(f'(x) = 2x\) and \(f''(x) = 2\), so

\[ \mathcal{A}(x^2) = \kappa(\theta - x)\cdot 2x + \frac{1}{2}\sigma^2 \cdot 2 = 2\kappa(\theta - x)x + \sigma^2 \]

This expression is a polynomial in \(x\) (specifically \(2\kappa\theta x - 2\kappa x^2 + \sigma^2\)), not of the form \([F(u) + R(u)\,x]\cdot x^2\). For the exponential \(e^{ux}\), the generator produces \([\cdots]\,e^{ux}\) because every derivative of \(e^{ux}\) is proportional to \(e^{ux}\) itself. For the polynomial \(x^2\), each derivative lowers the degree, so the generator mixes different powers of \(x\) and the result cannot be factored as a scalar-affine function of \(x\) times \(f(x)\). This is why the exponential basis is special for affine processes: it is an eigenfunction of the generator up to affine dependence on the state.

Exercise 2. For the CIR process, compute \(\mathcal{A}e^{ux}\) directly by evaluating \(\kappa(\theta - x)(ue^{ux}) + \frac{1}{2}\xi^2 x(u^2 e^{ux})\). Collect terms to identify \(F(u) = \kappa\theta u\) and \(R(u) = -\kappa u + \frac{1}{2}\xi^2 u^2\). Verify that these match the Riccati ODE right-hand sides.

Solution to Exercise 2

The CIR generator applied to \(e^{ux}\) is

\[ \mathcal{A}e^{ux} = \kappa(\theta - x)(ue^{ux}) + \frac{1}{2}\xi^2 x(u^2 e^{ux}) \]

Expanding:

\[ \mathcal{A}e^{ux} = \left[\kappa\theta\,u - \kappa u\,x + \frac{1}{2}\xi^2 u^2 x\right]e^{ux} \]

Collecting the constant term (independent of \(x\)) and the term proportional to \(x\):

\[ \mathcal{A}e^{ux} = \left[\kappa\theta\,u + \left(-\kappa u + \frac{1}{2}\xi^2 u^2\right)x\right]e^{ux} \]

Reading off: \(F(u) = \kappa\theta\,u\) and \(R(u) = -\kappa u + \frac{1}{2}\xi^2 u^2\). These are the right-hand sides of the Riccati system: \(\phi'(\tau) = F(\psi(\tau))\) and \(\psi'(\tau) = R(\psi(\tau)) = -\kappa\psi + \frac{1}{2}\xi^2\psi^2\), confirming that the generator action on exponentials directly produces the Riccati equations.

Exercise 3. Consider a two-dimensional affine process where \(X_t^{(1)}\) follows a CIR process and \(X_t^{(2)}\) follows a Vasicek process, with no cross-diffusion. Write down the generator \(\mathcal{A}\) and compute \(\mathcal{A}e^{u_1 x_1 + u_2 x_2}\). Show that \(F\) and \(R = (R_1, R_2)\) decompose into independent scalar contributions.

Solution to Exercise 3

Let \(X_t^{(1)}\) follow the CIR process \(dX_t^{(1)} = \kappa_1(\theta_1 - X_t^{(1)})\,dt + \xi_1\sqrt{X_t^{(1)}}\,dW_t^{(1)}\) and \(X_t^{(2)}\) follow the Vasicek process \(dX_t^{(2)} = \kappa_2(\theta_2 - X_t^{(2)})\,dt + \sigma_2\,dW_t^{(2)}\), with \(W^{(1)}\) and \(W^{(2)}\) independent. The generator is

\[ \mathcal{A}f(x_1, x_2) = \kappa_1(\theta_1 - x_1)\frac{\partial f}{\partial x_1} + \kappa_2(\theta_2 - x_2)\frac{\partial f}{\partial x_2} + \frac{1}{2}\xi_1^2 x_1\frac{\partial^2 f}{\partial x_1^2} + \frac{1}{2}\sigma_2^2\frac{\partial^2 f}{\partial x_2^2} \]

For \(f(x_1, x_2) = e^{u_1 x_1 + u_2 x_2}\), we compute \(\partial f/\partial x_i = u_i f\), \(\partial^2 f/\partial x_i^2 = u_i^2 f\), and there is no cross-derivative term. Substituting:

\[ \mathcal{A}e^{u_1 x_1 + u_2 x_2} = \left[\kappa_1(\theta_1 - x_1)u_1 + \kappa_2(\theta_2 - x_2)u_2 + \frac{1}{2}\xi_1^2 x_1 u_1^2 + \frac{1}{2}\sigma_2^2 u_2^2\right]e^{u_1 x_1 + u_2 x_2} \]

Collecting constant terms and terms proportional to \(x_1\) and \(x_2\):

\[ F(u_1, u_2) = \kappa_1\theta_1 u_1 + \kappa_2\theta_2 u_2 + \frac{1}{2}\sigma_2^2 u_2^2 \]

\[ R_1(u_1, u_2) = -\kappa_1 u_1 + \frac{1}{2}\xi_1^2 u_1^2 \]

\[ R_2(u_1, u_2) = -\kappa_2 u_2 \]

Observe that \(R_1\) depends only on \(u_1\) and \(R_2\) depends only on \(u_2\). Similarly, \(F\) decomposes as \(F = F^{(1)}(u_1) + F^{(2)}(u_2)\) where \(F^{(1)}(u_1) = \kappa_1\theta_1 u_1\) and \(F^{(2)}(u_2) = \kappa_2\theta_2 u_2 + \frac{1}{2}\sigma_2^2 u_2^2\). This decomposition holds because there is no cross-diffusion: the Riccati ODEs for \(\psi_1\) and \(\psi_2\) decouple into independent scalar equations, and \(\phi\) is the sum of two independent integrals.

Exercise 4. For the affine jump-diffusion \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t + dJ_t\) with compound Poisson jumps of intensity \(\lambda\) and exponential jump sizes with parameter \(\eta\), compute \(\hat{\nu}(u) = \int_0^\infty e^{uz}\eta e^{-\eta z}\,dz\) and identify the domain of \(u\) for which the integral converges. Write down \(F(u)\) incorporating the jump term \(\lambda(\hat{\nu}(u) - 1)\).

Solution to Exercise 4

The jump sizes are exponentially distributed with parameter \(\eta\), so \(\nu(dz) = \eta e^{-\eta z}\mathbf{1}_{z > 0}\,dz\). The moment generating function is

\[ \hat{\nu}(u) = \int_0^\infty e^{uz}\eta e^{-\eta z}\,dz = \eta\int_0^\infty e^{(u - \eta)z}\,dz = \frac{\eta}{\eta - u} \]

The integral converges if and only if \(\operatorname{Re}(u - \eta) < 0\), that is, \(\operatorname{Re}(u) < \eta\). For real \(u\), the domain is \(u < \eta\).

Substituting into the expression for \(F(u)\):

\[ F(u) = \kappa\theta\,u + \frac{1}{2}\sigma^2 u^2 + \lambda\!\left(\frac{\eta}{\eta - u} - 1\right) = \kappa\theta\,u + \frac{1}{2}\sigma^2 u^2 + \frac{\lambda u}{\eta - u} \]

valid for \(\operatorname{Re}(u) < \eta\), and \(R(u) = -\kappa u\) as before (since the jumps are state-independent).

Exercise 5. The extended generator allows \(\mathcal{A}\) to act on exponential functions that do not vanish at infinity. Explain why the standard generator defined on \(C_0(\mathbb{R}^d)\) cannot handle \(e_u(x) = e^{u^\top x}\), and state the growth condition that \(u\) must satisfy for \(e_u\) to belong to the domain of the extended generator when \(X_t\) has CIR-type components on \(\mathbb{R}_+\).

Solution to Exercise 5

The standard generator is defined on \(C_0(\mathbb{R}^d)\), the space of continuous functions vanishing at infinity: \(\lim_{|x| \to \infty} f(x) = 0\). The exponential function \(e_u(x) = e^{u^\top x}\) does not vanish at infinity --- for any \(u\) with \(\operatorname{Re}(u) \neq 0\), \(|e_u(x)| = e^{\operatorname{Re}(u)^\top x} \to \infty\) along the direction of \(\operatorname{Re}(u)\). Therefore \(e_u \notin C_0(\mathbb{R}^d)\), and the standard generator cannot be applied to it.

The extended generator relaxes the \(C_0\) requirement and acts on functions in a larger domain, provided appropriate moment conditions hold. For \(e_u\) to belong to the domain of the extended generator, we need \(\mathbb{E}^x[|e_u(X_t)|] < \infty\) for all \(t\) in some neighborhood of zero, which requires

\[ u \in \mathcal{U} = \{u \in \mathbb{C}^d : \mathbb{E}[e^{\operatorname{Re}(u)^\top X_t}] < \infty\} \]

For CIR-type components with \(X_t^{(i)} \geq 0\), the moment generating function \(\mathbb{E}[e^{v X_t^{(i)}}]\) is finite only when \(v\) is below a critical threshold. Specifically, since \(X_t^{(i)} \geq 0\), the condition requires \(\operatorname{Re}(u_i) \leq 0\) for each CIR-type coordinate \(i = 1, \ldots, m\). For the Gaussian components \(i = m+1, \ldots, d\), the exponential moments are finite for all \(\operatorname{Re}(u_i) \in \mathbb{R}\).

Exercise 6. Using the generator definition \(\mathcal{A}f(x) = \lim_{t \downarrow 0}\frac{P_t f(x) - f(x)}{t}\), verify \(\mathcal{A}e^{ux} = [F(u) + R(u)x]e^{ux}\) for the OU process by first computing \(P_t e^{ux} = \mathbb{E}[e^{uX_t} \mid X_0 = x]\) using the known Gaussian transition density, then differentiating at \(t = 0\).

Solution to Exercise 6

The OU process is \(dX_t = -\kappa X_t\,dt + \sigma\,dW_t\), so conditional on \(X_0 = x\), \(X_t\) is Gaussian with mean \(xe^{-\kappa t}\) and variance \(\frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa t})\). The semigroup applied to \(e^{ux}\) is

\[ P_t e^{ux} = \mathbb{E}[e^{uX_t} \mid X_0 = x] = \exp\!\left(u x e^{-\kappa t} + \frac{u^2\sigma^2}{4\kappa}(1 - e^{-2\kappa t})\right) \]

using the moment generating function of a Gaussian. Now compute the generator:

\[ \mathcal{A}e^{ux} = \lim_{t \downarrow 0}\frac{P_t e^{ux} - e^{ux}}{t} = e^{ux}\lim_{t \downarrow 0}\frac{\exp\!\left(ux(e^{-\kappa t} - 1) + \frac{u^2\sigma^2}{4\kappa}(1 - e^{-2\kappa t})\right) - 1}{t} \]

As \(t \to 0\), \(e^{-\kappa t} - 1 \approx -\kappa t\) and \(1 - e^{-2\kappa t} \approx 2\kappa t\), so the exponent is approximately \(-\kappa u x\,t + \frac{1}{2}\sigma^2 u^2 t\). Using \(\lim_{t \to 0}(e^{ct} - 1)/t = c\):

\[ \mathcal{A}e^{ux} = e^{ux}\left(-\kappa u x + \frac{1}{2}\sigma^2 u^2\right) = \left[\frac{1}{2}\sigma^2 u^2 + (-\kappa u)x\right]e^{ux} \]

Reading off: \(F(u) = \frac{1}{2}\sigma^2 u^2\) and \(R(u) = -\kappa u\), which matches the OU affine coefficients \(b_0 = 0\), \(B = -\kappa\), \(a_0 = \sigma^2\), \(\alpha_1 = 0\).