$ℒf$ is the drift of $f(X)$.

Equivalently, and this is the definition that does not presuppose Itô’s lemma,

the rate at which the expectation of $f$ starts to move from a state $x$. Itô’s lemma is then the statement that for a diffusion this limit is (3.2), which is a computation rather than a definition, and once done it need not be done again.

Equation (3.3) is written as though it always makes sense, and it does not. The set of $f$ for which the limit exists — uniformly, in the sense the theory requires — is called the domain of the generator, written $𝒟(ℒ)$, and it is a genuine restriction rather than a technicality to be waved past. Two examples, failing in different ways.

Roughness. Take standard Brownian motion and $f(x)=|x|$, and evaluate (3.3) at the origin. Since $𝔼|W_t|=\sqrt{2t/\pi }$,

The limit does not exist, so $|x|$ is not in the domain, and $ℒ|x|$ is not a hard quantity to evaluate at the origin — it does not exist.¹¹1both the closed form and the divergence, checked. This is not a curiosity, because a call payoff ${(S-K)}^{+}$ has exactly this kink at the strike. What saves the pricing equation of chapter 5 is that it never asks for $ℒ$ applied to the payoff. The payoff enters as the terminal condition $V(T,\cdot )={(S-K)}^{+}$, and the equation ${\partial }_{t}V+ℒV=0$ is solved backwards from it. For every the solution is $V(t,\cdot )=P_{T-t}{(S-K)}^{+}$, a bounded function convolved with a Gaussian density, and that is infinitely differentiable however rough the function was: the semigroup smooths instantly, and the kink exists only at the single instant $T$. So $ℒ$ is applied to $V(t,\cdot )$, which is in its domain, and never to the data that started it off.

The one place the kink is still visible is numerical, where the payoff is evaluated on a finite set of points rather than integrated against a density. The grid quadrature measured in chapter 19 wobbles for exactly this reason as its node count changes: the strike falls in a different place between nodes each time, and there is no smoothing left to hide it.

Growth. Now take $f(x)=e^{x^2}$, which is smooth everywhere, so the formal expression (3.2) is perfectly well defined at every point. What fails is the expectation:

So $P_{t}f$ does not exist for $t\ge \frac{1}{2}$, and no amount of care with the limit at zero repairs it.²²2checked, with the exponents combined --- evaluating $e^{x^2}$ and the density separately overflows one and underflows the other. The domain therefore has to control growth as well as smoothness, which is why the natural space to work in is bounded continuous functions rather than all of $C^2$.

The domain is also where boundary behaviour lives.

The coefficients cannot tell these apart: $\mu$ and $\sigma$ are the same function of $x$ in both cases, and the difference is entirely a condition at the boundary — which is to say, a domain.

For a reflecting barrier the condition is $f^{\prime }(0)=0$. A one-sided derivative at the origin would be multiplied by a drift the path is not allowed to follow, and requiring it to vanish is what makes the two sides of the reflection agree.

For an absorbing barrier the process at $0$ does not move, so ${𝔼}^0[f(X_t)]=f(0)$ for every $t$ and (3.3) gives $ℒf(0)=0$: the condition is on $ℒf$, not on $f$. If instead the process is killed at the boundary — sent to a cemetery state where every payoff is worth zero, which is what a knock-out does — the natural space is functions with $f(0)=0$. The two coincide whenever the payoff is worth nothing at the barrier, which in the financial applications it is, and that is why the distinction is easy to miss.

Every statement below is a statement about $f\in 𝒟(ℒ)$, and we will stop saying so.

Dropping does not merely weaken (3.4); it makes it false. Let $X$ be standard Brownian motion, $f(x)=x$, so $ℒf=0$, and let $\tau$ be the first time $X$ reaches $1$. Then $\tau$ is finite almost surely, $f(X_{\tau })=1$ always, and (3.4) would read $1=0+0$.

What has gone wrong is exactly the step above: $𝔼[\tau ]=\mathrm{\infty }$ for that stopping time, the local martingale is not a martingale, and its expectation is not preserved. This is the same failure that appears in chapter 14 as a Riccati solution exploding and in chapter 4 as a doubling strategy.

This converts a search into an equation. A function of the state is a martingale exactly when it is in the kernel of the generator, so hunting for martingales — which is what pricing does — is solving $ℒf=0$.

Chapter 4’s risk neutral measure will be the statement that discounted prices are martingales, which is the statement that they satisfy $ℒf=0$ for the appropriately adjusted $ℒ$. That is the whole of the pricing equation, and it is already visible here.

The theorem repairing that is the Hille-Yosida theorem.

The difficulty is that $e^{tℒ}$ cannot be defined by its series. Hille-Yosida instead characterises which unbounded operators generate a semigroup, and does it through the resolvent ${(\lambda I-ℒ)}^{-1}$ — an object that is bounded where $ℒ$ is not, because inverting a differential operator is integrating, and integrating smooths. The statement is that $ℒ$ generates a contraction semigroup precisely when it is densely defined, closed, and its resolvent satisfies $\Vert {(\lambda I-ℒ)}^{-1}\Vert \le 1/\lambda$ for every $\lambda >0$.

Two things follow. The exponential notation is a theorem rather than an abbreviation — something has to be checked, and for the diffusions in these notes it holds. And the resolvent, which appears here as a technical device, is the Laplace transform of the semigroup,

which is to say it is the price of a perpetual cashflow discounted at rate $\lambda$. Perpetuities are the objects on which an unbounded generator becomes bounded, and that is not a coincidence: discounting is what makes an infinite horizon finite.

Two things immediately. The first equality in (3.6) is the backward equation and the second is a statement that the generator commutes with its own semigroup, which is why one may evolve either the function or the starting point and get the same answer — the subject of the next section.

The second is a way of reading the model classes to come. If $ℒ$ maps some finite-dimensional family of functions into itself, then $e^{tℒ}$ does too, and computing $P_{t}f$ for $f$ in that family reduces to a finite system of ordinary differential equations. That single observation is why affine models are solvable, and chapter 14 is devoted to it.

Writing ${ℒ}^{\ast }$ as the derivative of a flux makes the forward equation a conservation law: ${\partial }_{t}p=-{\partial }_{x}J$ says probability is neither created nor destroyed, only transported, and $J$ is the rate at which it moves past a point. Chapter 9 makes use of exactly this form.

The boundary terms then have a reading rather than being a nuisance to assume away. On the whole line with $p$ decaying and $f$ bounded, both terms vanish and (3.8) holds as stated. On a bounded interval they do not vanish by themselves, and requiring them to cancel is precisely a boundary condition — $J=0$ at the ends for a reflecting barrier, $p=0$ for an absorbing one. So the pair $(ℒ,{ℒ}^{\ast })$ carries the boundary behaviour in the condition that makes them adjoint, which is the concrete form of the remark above that a domain, not a formula, distinguishes two processes with the same coefficients.

The flux says what the two barriers of the previous section are doing, in one quantity. Integrating the forward equation over $[0,\mathrm{\infty })$,

when $J$ vanishes at infinity. A reflecting barrier has $J(t,0)=0$ and the total is conserved: nothing crosses. An absorbing barrier has , probability leaving through the origin and not returning, so $p$ is the density of the paths that have survived and integrates to less than one. And since the mass lost by time $t$ is exactly $\mathbb{P}(\tau \le t)$, the outward flux at the barrier is the density of the first passage time — the object chapter 16 calls the default density and prices from.

Chapter 14 needs the zero-flux case specifically. Setting $J[p]=0$ pointwise and solving is what produces the stationary density there, and the reason it can be done in one dimension is visible in (3.10): a flux with zero divergence on a line must be constant, and a constant flux that vanishes at either end vanishes everywhere. In two dimensions a divergence-free flux may circulate instead, which is the whole of that chapter’s obstruction to reversibility.

Put (3.2) and (3.8) side by side. The coefficients are the same, the derivatives are the same, and the only difference is that in $ℒ$ they sit outside the derivatives and in ${ℒ}^{\ast }$ they sit inside. Integration by parts is the operation that moves them, and that is the entire content of the relationship.

The difference is not cosmetic, because $\mu$ and $\sigma$ depend on $x$: ${\partial }_x[\mu p]=\mu {\partial }_{x}p+p{\partial }_x\mu$, and the second term is real. Getting it wrong gives an equation that does not conserve probability, and a test makes the failure explicit rather than leaving it as a warning.

An expectation $𝔼[f(X_T)]$ can be computed in two ways. Evolve the payoff backwards to today and pair it with today’s density, or evolve the density forwards to expiry and pair it with the payoff. By (3.7) these agree exactly, and the choice is one of convenience:

Which is more convenient depends on what is being varied. One payoff and many starting points — an option, revalued as the market moves — wants the backward equation. Many payoffs and one starting point — a whole strike surface, today — wants the forward one, since a single forward solve prices every strike at once. That is why chapter 9 reaches for the forward equation and chapter 5 for the backward.