Generator of an Itô Diffusion

The Generator

Four pages have gone into turning the solution of a stochastic differential equation into an object worth naming. The Itô diffusion is not one process but a family indexed by the starting point. As a family it forgets the clock, and by the Markov property it restarts from its own present position carrying no memory of the route. The strong Markov property extended the restart to times read off the path itself, and the stopped expectation identity turned Itô's formula into a statement about averages at such times of finite mean. The first three of these are bookkeeping about the law of the process, and none of them computes anything. Ask for the expected value of a function of the diffusion at a fixed time, or for the mean time the diffusion takes to leave a region, and the theory offers no procedure, only the assurance that such quantities are well posed. The fourth is the instrument that will change this, and it is already sharp. What is missing is the object it should be aimed at.

The procedure comes from differentiating at time zero. Fix a function \(f\) on the state space and watch the average \(E^x[ f( X_t ) ]\) as \(t\) decreases to zero. At \(t = 0\) the average is \(f(x)\), since the diffusion starts at \(x\). For small positive \(t\) it has moved off that value by an amount which, if the process is not too wild, should be proportional to \(t\). The proportionality constant is a number attached to the point \(x\), so it is again a function on the state space, and the assignment sending \(f\) to that function is an operator. It is the derivative of the process at the instant of its start, felt through the test function \(f\). Everything the diffusion does in its first instant is recorded there.

The two properties already in hand make it plausible that the operator is a complete description of the process, and not merely its leading term. Time homogeneity makes the first instant of the restarted diffusion the same object as the first instant of the original, and the Markov property makes the restarted diffusion the entire future. What the process does at any later instant is therefore what it does at the first, read from wherever it then happens to be.

That operator is the object this page defines and computes. The definition is immediate. The computation is the substance. The operator turns out to act on compactly supported, twice continuously differentiable functions as a second-order partial differential operator whose coefficients are built from the drift and the noise coefficient of the equation. From that point on, a question about averages of a diffusion is a question about a differential operator, and the two subjects that the track has been developing in parallel become one subject.

The Working Setting Is Several-Component

From here the state carries several coordinates. The state space is \(\mathbb{R}^m\), the driving motion \(\mathbf{w}\) is an \(n\)-dimensional Brownian motion with independent components, the drift is a vector field \(\mathbf{b} : \mathbb{R}^m \to \mathbb{R}^m\), and the noise coefficient is a matrix field \(\sigma : \mathbb{R}^m \to \mathbb{R}^{m \times n}\), both Lipschitz. The equation

\[ d X_t = \mathbf{b}( X_t )\, dt + \sigma( X_t )\, d \mathbf{w}_t , \quad X_0 = x \in \mathbb{R}^m , \]

is read row by row, the \(i\)-th coordinate of the state being driven by the \(i\)-th row of \(\sigma\) against all \(n\) components of the noise. This is the several-component reading recorded earlier in the track. Existence and uniqueness, the clock shift, time homogeneity, and the Markov property were declared to hold in this generality with the same statements and the same proofs. These results were held as declared rather than proved, and the one-dimensional versions remain the proved objects. Nothing on this page repairs that declaration. What this page does instead is keep the debt visible. Each time a proof leans on the declared reading rather than on a theorem proved in several components, the step says so.

The distinction is not pedantry, and the two ingredients of the computation below fall on opposite sides of it. The stopped expectation identity concerns a general Itô process, under hypotheses that the proof below checks one by one, and it rests on theorems proved in several components, so it owes nothing. What is owed lies entirely on the other side: that the solution exists and is unique, that it forgets the clock, that it restarts from its present position. Everything said here about the diffusion as a family is said on the strength of the declaration.

The notation for averages is the start-indexed convention already in force. For a starting point \(x \in \mathbb{R}^m\) and a Borel function \(f : \mathbb{R}^m \to \mathbb{R}\) we write \(E^x[ f( X_t ) ] := \mathbb{E}[ f( X^x_t ) ]\), the superscript naming the start and the expectation being the ordinary one on the underlying probability space.

Two features of the definition are worth pausing over before anything is computed. The limit is one-sided, and it has to be. The diffusion is started at \(x\) at time zero and has no past, so the quotient exists only for \(t \gt 0\) and the limit can only be taken from above. And the integrability clause is our reading of the definition rather than a separate hypothesis. Without it the quotient whose limit is being taken need not exist at all. The condition is automatically satisfied for every function actually used below, since a bounded Borel \(f\) satisfies it at every starting point. The clause is stated because the definition is otherwise silent about which functions may be fed to \(A\), and a silent domain is where unstated assumptions accumulate.

Three observations fix the shape of the object. First, \(A\) is not an operator on a space of functions but a partially defined one, and the domain is part of the data. The two domains recorded above need not coincide, since a limit can exist at one point and fail at another. It is the global domain \(\mathcal{D}_A\) that will serve as the natural class.

Second, on its domain \(A\) is linear. If \(f\) and \(g\) lie in \(\mathcal{D}_A( x )\) and \(\alpha, \beta\) are real, then linearity of the expectation makes the difference quotient of \(\alpha f + \beta g\) the corresponding combination of the two quotients. Hence \(\alpha f + \beta g \in \mathcal{D}_A( x )\) and \(A( \alpha f + \beta g )( x ) = \alpha A f( x ) + \beta A g( x )\). A constant function, whose difference quotient vanishes identically, lies in \(\mathcal{D}_A\) and is annihilated by \(A\). Third, and least comfortably, the definition as it stands exhibits no function in the domain beyond the constants. That the domain contains a rich supply, and that on that supply the operator has a closed form, is exactly what has to be earned.

The maneuver of differentiating a family at time zero has appeared on this curriculum before. A smooth flow on a manifold has an infinitesimal generator, obtained by taking the velocity at time zero of the trajectory through each point, and the result is a vector field. Here the same maneuver is applied to a family of random trajectories, and it is applied to the average of a test function rather than to the trajectory itself, since a Brownian path has no velocity to take. The two constructions agree in spirit and differ in output. A flow returns a tangent vector at each point, which acts on functions as a first-order derivative along the flow, whereas the diffusion returns an operator on functions directly. The computation below settles whether the noise leaves a trace on that operator beyond the first-order term the flow would contribute, and what that trace is.

The route is short, because the hard work was done before the generator was named. The identity already in hand expresses the average of a compactly supported \(C^2\) function of an Itô process, at a stopping time of finite mean, as its initial value plus the average of a second-order expression accumulated along the path. Specializing the stopping time to a fixed deterministic time and dividing by it produces the difference quotient of the definition, and the limit can then be taken.

Computing the Generator

The diffusion is an Itô process with a particular pair of coefficients. It satisfies

\[ X_t = x + \int_0^t \mathbf{b}( X_s )\, ds + \int_0^t \sigma( X_s )\, d \mathbf{w}_s , \]

so the general coefficients \(\mathbf{u}\) and \(\mathbf{v}\) of the identity become \(\mathbf{b}( X_s )\) and \(\sigma( X_s )\), evaluated along the path rather than prescribed independently of it. Substituting them into the identity and taking the stopping time to be a fixed time is the entire computation. What has to be checked is that the hypotheses survive the substitution, and what has to be supplied is the limit that the definition of the generator asks for and the identity does not perform.

Two Operators and What the Theorem Does Not Say

The right-hand side of the theorem is a differential operator in its own right, defined by the same formula on any function that is twice differentiable, with no probability in sight. It deserves a name of its own, because from here on the two operators must be kept apart.

In this notation the theorem says that \(C_0^2( \mathbb{R}^m ) \subseteq \mathcal{D}_A\) and that \(A\) and \(L\) agree there. Both halves are informative and neither is more than it says. The theorem does not assert that \(\mathcal{D}_A\) equals \(C_0^2\). The domain is larger, since the nonzero constants already lie in it, and how much larger is a question the definition of \(A\) does not settle. Nor does it assert that \(A\) and \(L\) agree wherever both make sense. What it provides is a supply of functions on which the probabilistic operator is computable, and that supply is rich enough to determine everything the track will ask of \(A\).

The second-order coefficients deserve a look. They form the matrix \(a = \sigma \sigma^\top\), which is symmetric, and for any vector \(\xi\) satisfies \(\xi^\top a\, \xi = | \sigma^\top \xi |^2 \geq 0\), so \(a\) is positive semidefinite at every point. The operator \(L\) is therefore elliptic in the weak sense, and non-degenerate exactly where \(\sigma\) has full row rank. Degeneracy is not pathology here but the normal state of affairs. Whenever the noise has fewer components than the state, \(a\) is singular everywhere. Along the degenerate directions the noise injects no randomness directly. The state need not stay deterministic along them, however, since the drift can couple them to coordinates the noise does reach and carry randomness in.

One feature of the formula should be stated plainly, because it is the shape of everything that follows. The noise enters only through \(a\), never through \(\sigma\) itself. Two diffusions built with different noise coefficients but the same \(a\) have the same differential operator, and so generators that agree on \(C_0^2( \mathbb{R}^m )\). The direction each component of the driving motion pushed in is information the operator has thrown away. Nothing on this page decides the sharper question of whether two such diffusions must in fact have the same law. What is decided is that the operator, and everything built from the operator alone, carries no trace of how the noise was wired.

Two Diffusions and Their Operators

Two instances show what the formula delivers. Both have coefficients simple enough that the computation is a matter of reading entries off a matrix, and both produce operators that were on this curriculum long before any stochastic process was.

The factor of one half records a convention made much earlier. Brownian motion was normalized so that \(\mathbf{w}_t\) has variance \(t\) in each component, and the second-order coefficient of the generator is half that variance rate. Had the motion been scaled differently the constant would move with it. The heat kernel on the line makes the agreement visible. At diffusivity \(k = \tfrac{1}{2}\) its formula collapses to \(( 2 \pi t )^{-1/2} \exp( - x^2 / 2 t )\), which is precisely the density of \(w_t\). The operator computed here and the equation that kernel solves carry the same constant because they describe the same process.

This example is where the declared several-component reading is most heavily spent. The first example needed it for the definition of the object and for the moment bound inside the theorem, and its explicit solution makes even that bound immediate. The space-time extension was proved to be a diffusion only in that reading, and the theorem of this page was applied to it as such. The computation itself is a matrix multiplication and owes nothing.

The example also repays the trouble of having introduced the enlarged state at all. A derivative in the clock direction has appeared in the generator, yet nothing about the construction has become time-dependent. The first coordinate is a coordinate of the state like any other, the process moving along it at unit speed, and \(\partial_\tau\) is a spatial derivative of the enlarged problem. That is the mechanism by which a question with time in it becomes a question about a time-homogeneous object. The sign \(\varepsilon\) decides whether the clock of the enlarged state runs with the equation or against it, so the two operators above are two operators and not one written twice.