Chapter 9 — Diffusions and Generators

Guiding question

Given an SDE $dX = b(X)\,dt + \sigma(X)\,dW$, is there a single operator that encodes everything about how smooth functions evolve under $X$? How does this operator connect SDEs to partial differential equations?

The infinitesimal generator $\mathcal L$ is the differential operator that describes the local behavior of a diffusion process. It packages Itô's formula into a single operator, and its study bridges stochastic analysis and classical analysis. The generator is the stochastic analogue of the derivative — it measures how fast the expected value of a function changes along the process.

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9.1 The infinitesimal generator: definition

Let $X$ be the strong solution of $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t$. For a smooth function $f:\mathbb R\to\mathbb R$ with compact support, define:

$$\mathcal L f(x) = \lim_{t\downarrow 0}\frac{\mathbb E^x[f(X_t)] - f(x)}{t},$$

where $\mathbb E^x[\cdot] = \mathbb E[\cdot\mid X_0 = x]$.

By Itô's formula, the limit exists and equals:

$$\boxed{\mathcal L f(x) = b(x)f'(x) + \tfrac12\sigma(x)^2 f''(x).}$$

Derivation from Itô's formula

By Itô's formula with $f\in C^2$:

$$f(X_t) = f(x) + \int_0^t [b(X_s)f'(X_s) + \tfrac12\sigma^2(X_s)f''(X_s)]\,ds + \int_0^t\sigma(X_s)f'(X_s)\,dW_s.$$

Taking expectations (the stochastic integral has zero expectation):

$$\mathbb E^x[f(X_t)] = f(x) + \mathbb E^x\!\left[\int_0^t\mathcal L f(X_s)\,ds\right].$$

Dividing by $t$ and taking $t\to 0$ (using $X_s\to x$ and continuity of $\mathcal Lf$):

$$\lim_{t\to 0}\frac{\mathbb E^x[f(X_t)]-f(x)}{t} = \mathcal L f(x).$$

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9.2 Examples of generators

Brownian motion ($b=0$, $\sigma=1$):

$$\mathcal L_{\rm BM} f = \tfrac12 f''.$$

This is $\frac12$ the Laplacian (or the heat operator in one dimension).

Geometric Brownian Motion ($b=\mu x$, $\sigma = \sigma x$):

$$\mathcal L_{\rm GBM} f = \mu x f' + \tfrac12\sigma^2 x^2 f''.$$

Ornstein–Uhlenbeck ($b=\kappa(\theta-x)$, $\sigma=\sigma$):

$$\mathcal L_{\rm OU} f = \kappa(\theta-x)f' + \tfrac12\sigma^2 f''.$$

CIR ($b=\kappa(\theta-x)$, $\sigma=\sigma\sqrt{x}$):

$$\mathcal L_{\rm CIR} f = \kappa(\theta-x)f' + \tfrac12\sigma^2 x f''.$$

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9.3 Dynkin's formula

Dynkin's formula

Let $\tau$ be a stopping time with $\mathbb E^x[\tau]<\infty$ and $f\in\text{Dom}(\mathcal L)$ (the domain of the generator). Then

$$\mathbb E^x[f(X_\tau)] = f(x) + \mathbb E^x\!\left[\int_0^\tau \mathcal L f(X_s)\,ds\right].$$

Proof via optional stopping

The process $M_t = f(X_t) - f(x) - \int_0^t\mathcal L f(X_s)\,ds$ is a local martingale (by Itô's formula). Under appropriate integrability, the optional stopping theorem applies at $\tau$: $\mathbb E^x[M_\tau]=0$, which is Dynkin's formula.

Generator L = ½∂²/∂x² for Brownian motion: Dynkin's formula

t = 0.50

Dynkin's formulaE[W_t² | W₀=x] = x² + t (Lf = 1)

Dynkin's formula: E[f(W_t)|W₀=x] = f(x) + t·Lf(x) + O(t²). The generator L = ½∂² measures the instantaneous rate of change of E[f(W_t)] at t=0. This connects martingale theory to PDEs: if M_t = f(W_t) − ∫₀ᵗ Lf(W_s)ds is a martingale, then Lf = 0 characterizes harmonic functions.

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9.4 Harmonic functions

A function $f$ is $\mathcal L$-harmonic on a domain $D$ if $\mathcal L f = 0$ on $D$.

Mean value property for diffusions

If $\mathcal L f = 0$ on $D$ and $\tau_D = \inf\{t: X_t\notin D\}$ is the first exit time from $D$ (with $\mathbb E^x[\tau_D]<\infty$), then

$$f(x) = \mathbb E^x[f(X_{\tau_D})].$$

The value at any interior point equals the expected boundary value.

Proof

By Dynkin's formula: $\mathbb E^x[f(X_{\tau_D})] = f(x) + \mathbb E^x[\int_0^{\tau_D}\mathcal L f(X_s)\,ds] = f(x)$, since $\mathcal L f = 0$.

Diffusions average over their domain

For ordinary Brownian motion, $\mathcal L f = \frac12f''=0$ means $f$ is linear (in 1D). The mean value property says: the value of a linear function at any point equals its average on any symmetric interval. For multidimensional BM, harmonic functions satisfy the classical mean value property of harmonic analysis.

For diffusions with drift and varying diffusion, "harmonic" is with respect to the weighted operator $b\cdot\nabla + \frac12\sigma^2\Delta$ — the natural geometry determined by the SDE.

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9.5 The Kolmogorov equations

The generator connects SDEs to two classical PDEs.

Forward (Fokker–Planck) equation. The probability density $p(t,y;0,x) = \partial_y\mathbb P^x(X_t\le y)$ satisfies the PDE (written for the $y$-variable):

$$\partial_t p = \mathcal L^* p,$$

where $\mathcal L^*$ is the adjoint operator:

$$\mathcal L^* p = -\partial_y(b(y)p) + \tfrac12\partial_{yy}(\sigma^2(y)p).$$

This describes how the probability mass flows forward in time.

Backward (Kolmogorov backward) equation. The function $u(t,x) = \mathbb E^x[f(X_T)\mid X_t = x]$ satisfies the PDE (written for the $x$-variable):

$$\partial_t u + \mathcal L_x u = 0,\qquad u(T,x) = f(x).$$

The subscript $x$ means $\mathcal L$ acts on the spatial variable $x$, with $t$ as a parameter.

Forward vs backward: two perspectives on the same dynamics

The forward equation tracks the evolving probability distribution of $X_t$. The backward equation tracks the conditional expectation of a functional, running time backward from the terminal condition.

In option pricing:

• Forward equation = time-evolution of the stock price density (used in local volatility calibration).
• Backward equation = the option price PDE (Black–Scholes, Feynman–Kac).

Both encode the same SDE; the difference is which variable (current state vs future payoff) is being tracked.

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9.6 Generators in $d$ dimensions

For the $d$-dimensional SDE $d\mathbf X = \mathbf b(\mathbf X)\,dt + \boldsymbol\sigma(\mathbf X)\,d\mathbf W$, the generator is:

$$\mathcal L f(\mathbf x) = \sum_i b_i(\mathbf x)\,\partial_i f(\mathbf x) + \tfrac12\sum_{i,j}(\sigma\sigma^\top)_{ij}(\mathbf x)\,\partial_{ij}f(\mathbf x).$$

The matrix $A = \sigma\sigma^\top$ is the diffusion matrix. The generator is:

$$\mathcal L = \mathbf b\cdot\nabla + \tfrac12\text{tr}(A\,\nabla^2).$$

For $A = I$ and $\mathbf b = 0$: $\mathcal L = \frac12\Delta$ (half-Laplacian).

Generator of GBM in finance

For GBM $dS = rS\,dt + \sigma S\,dW$ (risk-neutral), the generator is:

$\mathcal L_{\rm BS} = rS\,\partial_S + \tfrac12\sigma^2 S^2\,\partial_{SS}.$

The Black–Scholes PDE is $\partial_t f + \mathcal L_{\rm BS} f = rf$, i.e.,

$(\partial_t + \mathcal L_{\rm BS} - r)f = 0.$

The operator $\mathcal L_{\rm BS} - r$ is the Black–Scholes operator: its null space consists of correctly priced option prices. Adding $r$ on the right reflects discounting at the risk-free rate.

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9.7 Problems

Problem 1: Generator of OU

For $dX = -\kappa X\,dt + \sigma\,dW$, compute $\mathcal L(x^2)$ and verify via Dynkin that $\mathbb E^x[X_t^2] = x^2 e^{-2\kappa t} + \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t})$.

Problem 2: Harmonic function for BM

Find all functions $f$ on $(a,b)$ with $\mathcal L_{\rm BM}f = \frac12 f'' = 0$. What does the mean value property say for BM on $(a,b)$?

Problem 3: CIR generator applied to bond price

For CIR with $\mathcal L f = \kappa(\theta-x)f' + \frac12\sigma^2 xf''$, show that the function $f(x) = e^{-Bx}$ (for constant $B$) satisfies $\mathcal L f = (\kappa\theta B - \frac12\sigma^2 B^2 x + \kappa Bx - \kappa\theta B)\cdots$ — work out the full calculation as a step toward Vasicek bond pricing.

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9.8 Selected solutions

Solution 1

$\mathcal L(x^2) = -\kappa x\cdot 2x + \frac12\sigma^2\cdot 2 = -2\kappa x^2 + \sigma^2$.

By Dynkin: $\frac{d}{dt}\mathbb E^x[X_t^2] = \mathbb E^x[\mathcal L(X_t^2)] = -2\kappa\mathbb E^x[X_t^2] + \sigma^2$.

This is an ODE $\dot v = -2\kappa v + \sigma^2$ with solution $v(t) = \frac{\sigma^2}{2\kappa} + (x^2 - \frac{\sigma^2}{2\kappa})e^{-2\kappa t}$. This matches the known variance formula.

Solution 2

$f''=0$ implies $f(x) = ax + b$ (linear functions). The mean value property: for BM starting at $x\in(a,b)$, $f(x) = \mathbb E^x[f(X_{\tau_{ab}})]$, where $\tau_{ab}$ is the first exit time from $(a,b)$. For $f$ linear, the Brownian paths "average" to reach the boundary at $a$ or $b$ proportionally: $f(x) = f(a)\mathbb P^x(X_{\tau}=a) + f(b)\mathbb P^x(X_{\tau}=b)$. With $\mathbb P^x(X_\tau=b) = (x-a)/(b-a)$, this gives $f(x) = f(a)\frac{b-x}{b-a} + f(b)\frac{x-a}{b-a}$, which is consistent with linearity.

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9.9 What this chapter gives you

Concept	Formula
Infinitesimal generator	$\mathcal L f = b f' + \frac12\sigma^2 f''$
Dynkin's formula	$\mathbb E^x[f(X_\tau)] = f(x) + \mathbb E^x[\int_0^\tau \mathcal L f\,ds]$
Harmonic functions	$\mathcal Lf = 0 \Rightarrow f(x) = \mathbb E^x[f(X_{\tau_D})]$
Forward equation	$\partial_t p = \mathcal L^*p$ (density evolution)
Backward equation	$\partial_t u + \mathcal Lu = 0$, $u(T,\cdot) = g$
Black–Scholes operator	$\mathcal L_{\rm BS} = rS\partial_S + \frac12\sigma^2 S^2\partial_{SS}$

The next chapter derives the Feynman–Kac formula, which makes the correspondence between conditional expectations and PDEs precise and quantitative.