Infinitesimal Generators and Kolmogorov Equations

Motivation: why this matters in quant finance

The same diffusion model that simulates a stock path also implies a partial differential equation for expectations. For a Markov state process

dX_t=m(t,X_t)\,dt+\sigma(t,X_t)\,dW_t,

the infinitesimal generator packages the local drift and variance into one operator. That operator is what later appears inside the Feynman-Kac formula and the Black-Scholes PDE.

Without generators, the link between Monte Carlo paths and PDE solvers looks like a coincidence. With generators, both methods are computing the same Markov expectation: one by sampling future paths, the other by evolving a function through time.

Lawler develops this from diffusions in Chapter 3 and then derives the forward density equation from binomial approximations in Chapter 4. The lesson's core question is: what differential operator describes the instantaneous effect of the stochastic dynamics?

The informal idea

For a Markov process, start at $X_0=x$ and look at a smooth test payoff $f(X_t)$ . The generator asks for the first-order rate of change in its expectation:

Lf(x)=\lim_{t\downarrow0}\frac{\mathbb{E}_x[f(X_t)]-f(x)}{t}.

For ordinary deterministic motion, this rate comes only from drift. For a diffusion, Itô's formula says the variance term also contributes at order $dt$ . That is the whole reason the generator contains a second derivative.

There are two directions in time:

The backward equation evolves a conditional expectation as the starting time changes.
The forward equation evolves the density of the state as calendar time moves forward.

They are adjoint views of the same Markov dynamics. The backward equation acts on payoff functions; the forward equation acts on probability densities.

Formal definitions

Generator of a time-homogeneous Markov process. For a Markov process started at $x$ , the generator $L$ acts on suitable test functions $f$ by
$Lf(x)=\lim_{t\downarrow0}\frac{\mathbb{E}_x[f(X_t)]-f(x)}{t}.$

For the diffusion

dX_t=m(X_t)\,dt+\sigma(X_t)\,dW_t,

Lawler derives, using Itô's formula,

Lf(x)=m(x)f'(x)+\frac{1}{2}\sigma^2(x)f''(x).

For a time-inhomogeneous diffusion,

dX_t=m(t,X_t)\,dt+\sigma(t,X_t)\,dW_t,

the time- $t$ generator is

L_t f(x)=m(t,x)f'(x)+\frac{1}{2}\sigma^2(t,x)f''(x).

u(t,x)=\mathbb{E}[F(X_T)\mid X_t=x],

then the backward equation is

\partial_t u(t,x)+L_tu(t,x)=0, \qquad u(T,x)=F(x).

If $p(t,x)$ is the density of a time-homogeneous diffusion with constant volatility $\sigma$ and drift $m(x)$ , Lawler's forward equation is

\partial_t p(t,x)=L^*p(t,x),

where the adjoint operator is

L^*g(x)=-[m(x)g(x)]'+\frac{1}{2}\sigma^2 g''(x).

Key properties

The generator is the drift of $f(X_t)$

Itô's formula gives

df(X_t)=\left[m(X_t)f'(X_t)+\frac{1}{2}\sigma^2(X_t)f''(X_t)\right]dt+\sigma(X_t)f'(X_t)\,dW_t.

The bracketed

dt

coefficient is

Lf(X_t)

. Finance consequence: the pricing PDE is obtained by making the drift of a discounted value process vanish.

The second derivative is not optional

The term

\frac{1}{2}\sigma^2 f''

comes from quadratic variation. Omitting it is equivalent to treating Brownian paths as finite-variation paths. Finance consequence: this is the term that becomes gamma in Black-Scholes-style hedging.

Backward equations act on payoffs

If the target is

u(t,x)=\mathbb{E}[F(X_T)\mid X_t=x]

, the PDE runs backward from the terminal condition

u(T,x)=F(x)

. Finance consequence: option pricing PDEs are terminal-value problems because the payoff is known at maturity.

Forward equations act on densities

The forward equation uses $L^*$ , not $L$ , because it moves probability mass rather than payoff values. For constant volatility,

L^*p=-(mp)'+\frac{1}{2}\sigma^2p''.

Finance consequence: calibration and risk distribution questions often use the forward view, while pricing a single payoff often uses the backward view.

Binomial approximations explain the adjoint

Lawler derives the forward equation by matching a small-time binomial approximation to the diffusion. Expanding the probability of landing at $x$ from neighbouring points produces the $-[mp]'$ term and the $\frac{1}{2}\sigma^2p''$ term.

Worked examples

Example 1: Brownian motion and the heat operator

Let $X_t=\sigma W_t$ . Then $m=0$ and $\sigma$ is constant, so

Lf(x)=\frac{1}{2}\sigma^2 f''(x).

The backward equation for $u(t,x)=\mathbb{E}[F(X_T)\mid X_t=x]$ is

\partial_t u(t,x)+\frac{1}{2}\sigma^2\partial_{xx}u(t,x)=0.

The forward equation for the density $p$ is

\partial_t p(t,x)=\frac{1}{2}\sigma^2\partial_{xx}p(t,x).

The same heat operator appears in both directions because the drift is zero and the operator is self-adjoint in this simple case.

Example 2: generator of geometric Brownian motion

For geometric Brownian motion,

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t,

the generator acting on a smooth function $f$ is

Lf(s)=\mu s f'(s)+\frac{1}{2}\sigma^2s^2f''(s).

For $f(s)=s$ , $Lf(s)=\mu s$ , matching the expected drift of the stock. For $f(s)=\log s$ ,

Lf(s)=\mu-\frac{1}{2}\sigma^2,

which recovers the Itô correction in the log-price. The generator is not an abstract PDE object; it is the drift calculator for transformed states.

Example 3: forward equation with state-dependent drift

Suppose

dX_t=m(X_t)\,dt+\sigma\,dW_t

with constant $\sigma$ and smooth $m$ . Lawler's forward equation is

\partial_t p(t,x)=-[m(x)p(t,x)]'+\frac{1}{2}\sigma^2p''(t,x).

If $m$ is constant, this becomes

\partial_t p=-m\,p_x+\frac{1}{2}\sigma^2p_{xx}.

When $m$ varies with state, the extra $-m'(x)p$ term appears inside $-[mp]'$ . That term is easy to miss if one thinks "forward equation equals backward equation with drift sign flipped."

Common confusions and pitfalls

"The generator is just the drift." Drift is only the first-derivative part. Brownian variance contributes the second-derivative term through quadratic variation.

"Backward and forward equations are the same PDE." They are related, but not interchangeable. The backward equation acts on payoff functions; the forward equation acts on densities through the adjoint $L^*$ .

"The forward equation just changes the sign of the drift." That is only true in special constant-drift cases. With state-dependent drift, Lawler's formula is $L^*p=-(mp)'+\frac{1}{2}\sigma^2p''$ .

"The PDE is separate from the SDE." The PDE is the infinitesimal expression of the SDE's Markov dynamics. Itô's formula is the bridge.

"Regularity is automatic." The generator formula is stated for smooth test functions with bounded derivatives in Lawler's derivation. Payoff kinks and boundary conditions require extra work.

Where this goes next

Feynman-Kac Formula: adds discounting and terminal payoffs to the generator equation.
Black-Scholes PDE: applies the generator of risk-neutral GBM to option values.
Stochastic Differential Equations: supplies the diffusion coefficients that define $L_t$ .
Markov Chains: gives the finite-state analogue, where a transition matrix plays the role of the evolution operator.
Poisson Processes: introduces a jump generator, showing how the operator changes when paths are discontinuous.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 3 §3.5 (Diffusions), Ch. 4 §4.4 (Binomial approximations and forward equations).