Categories: Mathematics, Statistics, Stochastic analysis.

# Kolmogorov equations

Consider the following general Itō diffusion $$X_t \in \mathbb{R}$$, which is assumed to satisfy the conditions for unique existence on the entire time axis:

\begin{aligned} \dd{X}_t = f(X_t, t) \dd{t} + g(X_t, t) \dd{B_t} \end{aligned}

Let $$\mathcal{F}_t$$ be the filtration to which $$X_t$$ is adapted, then we define $$Y_s$$ as shown below, namely as the conditional expectation of $$h(X_t)$$, for an arbitrary bounded function $$h(x)$$, given the information $$\mathcal{F}_s$$ available at time $$s \le t$$. Because $$X_t$$ is a Markov process, $$Y_s$$ must be $$X_s$$-measurable, so it is a function $$k$$ of $$X_s$$ and $$s$$:

\begin{aligned} Y_s \equiv \mathbf{E}[h(X_t) | \mathcal{F}_s] = \mathbf{E}[h(X_t) | X_s] = k(X_s, s) \end{aligned}

Consequently, we can apply Itō’s lemma to find $$\dd{Y_s}$$ in terms of $$k$$, $$f$$ and $$g$$:

\begin{aligned} \dd{Y_s} &= \bigg( \pdv{k}{s} + \pdv{k}{x} f + \frac{1}{2} \pdv[2]{k}{x} g^2 \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s} \\ &= \bigg( \pdv{k}{s} + \hat{L} k \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s} \end{aligned}

Where we have defined the linear operator $$\hat{L}$$ to have the following action on $$k$$:

\begin{aligned} \hat{L} k \equiv \pdv{k}{x} f + \frac{1}{2} \pdv[2]{k}{x} g^2 \end{aligned}

At this point, we need to realize that $$Y_s$$ is a martingale with respect to $$\mathcal{F}_s$$, since $$Y_s$$ is $$\mathcal{F}_s$$-adapted and finite, and it satisfies the martingale property, for $$r \le s \le t$$:

\begin{aligned} \mathbf{E}[Y_s | \mathcal{F}_r] = \mathbf{E}\Big[ \mathbf{E}[h(X_t) | \mathcal{F}_s] \Big| \mathcal{F}_r \Big] = \mathbf{E}\big[ h(X_t) \big| \mathcal{F}_r \big] = Y_r \end{aligned}

Where we used the tower property of conditional expectations, because $$\mathcal{F}_r \subset \mathcal{F}_s$$. However, an Itō diffusion can only be a martingale if its drift term (the one containing $$\dd{s}$$) vanishes, so, looking at $$\dd{Y_s}$$, we must demand that:

\begin{aligned} \pdv{k}{s} + \hat{L} k = 0 \end{aligned}

Because $$k(X_s, s)$$ is a Markov process, we can write it with a transition density $$p(s, X_s; t, X_t)$$, where in this case $$s$$ and $$X_s$$ are given initial conditions, $$t$$ is a parameter, and the terminal state $$X_t$$ is a random variable. We thus have:

\begin{aligned} k(x, s) = \int_{-\infty}^\infty p(s, x; t, y) \: h(y) \dd{y} \end{aligned}

We insert this into the equation that we just derived for $$k$$, yielding:

\begin{aligned} 0 = \int_{-\infty}^\infty \!\! \Big( \pdv{s} p(s, x; t, y) + \hat{L} p(s, x; t, y) \Big) h(y) \dd{y} \end{aligned}

Because $$h$$ is arbitrary, and this must be satisfied for all $$h$$, the transition density $$p$$ fulfills:

\begin{aligned} 0 = \pdv{s} p(s, x; t, y) + \hat{L} p(s, x; t, y) \end{aligned}

Here, $$t$$ is a known parameter and $$y$$ is a “known” integration variable, leaving only $$s$$ and $$x$$ as free variables for us to choose. We therefore define the likelihood function $$\psi(s, x)$$, which gives the likelihood of an initial condition $$(s, x)$$ given that the terminal condition is $$(t, y)$$:

\begin{aligned} \boxed{ \psi(s, x) \equiv p(s, x; t, y) } \end{aligned}

And from the above derivation, we conclude that $$\psi$$ satisfies the following PDE, known as the backward Kolmogorov equation:

\begin{aligned} \boxed{ - \pdv{\psi}{s} = \hat{L} \psi = f \pdv{\psi}{x} + \frac{1}{2} g^2 \pdv[2]{\psi}{x} } \end{aligned}

Moving on, we can define the traditional probability density function $$\phi(t, y)$$ from the transition density $$p$$, by fixing the initial $$(s, x)$$ and leaving the terminal $$(t, y)$$ free:

\begin{aligned} \boxed{ \phi(t, y) \equiv p(s, x; t, y) } \end{aligned}

With this in mind, for $$(s, x) = (0, X_0)$$, the unconditional expectation $$\mathbf{E}[Y_t]$$ (i.e. the conditional expectation without information) will be constant in time, because $$Y_t$$ is a martingale:

\begin{aligned} \mathbf{E}[Y_t] = \mathbf{E}[k(X_t, t)] = \int_{-\infty}^\infty k(y, t) \: \phi(t, y) \dd{y} = \braket{k}{\phi} = \mathrm{const} \end{aligned}

This integral has the form of an inner product, so we switch to Dirac notation. We differentiate with respect to $$t$$, and use the backward equation $$\pdv*{k}{t} + \hat{L} k = 0$$:

\begin{aligned} 0 = \pdv{t} \braket{k}{\phi} = \braket{k}{\pdv{\phi}{t}} + \braket{\pdv{k}{t}}{\phi} = \braket{k}{\pdv{\phi}{t}} - \braket{\hat{L} k}{\phi} = \braket{k}{\pdv{\phi}{t} - \hat{L}{}^\dagger \phi} \end{aligned}

Where $$\hat{L}{}^\dagger$$ is by definition the adjoint operator of $$\hat{L}$$, which we calculate using partial integration, where all boundary terms vanish thanks to the existence of $$X_t$$; in other words, $$X_t$$ cannot reach infinity at any finite $$t$$, so the integrand must decay to zero for $$|y| \to \infty$$:

\begin{aligned} \braket{\hat{L} k}{\phi} &= \int_{-\infty}^\infty \pdv{k}{y} f \phi + \frac{1}{2} \pdv[2]{k}{y} g^2 \phi \dd{y} \\ &= \bigg[ k f \phi + \frac{1}{2} \pdv{k}{y} g^2 \phi \bigg]_{-\infty}^\infty - \int_{-\infty}^\infty k \pdv{y}(f \phi) + \frac{1}{2} \pdv{k}{y} \pdv{y}(g^2 \phi) \dd{y} \\ &= \bigg[ -\frac{1}{2} k g^2 \phi \bigg]_{-\infty}^\infty + \int_{-\infty}^\infty - k \pdv{y}(f \phi) + \frac{1}{2} k \pdv[2]{y}(g^2 \phi) \dd{y} \\ &= \int_{-\infty}^\infty k \: \big( \hat{L}{}^\dagger \phi \big) \dd{y} = \braket{k}{\hat{L}{}^\dagger \phi} \end{aligned}

Since $$k$$ is arbitrary, and $$\pdv*{\braket{k}{\phi}}{t} = 0$$ for all $$k$$, we thus arrive at the forward Kolmogorov equation, describing the evolution of the probability density $$\phi(t, y)$$:

\begin{aligned} \boxed{ \pdv{\phi}{t} = \hat{L}{}^\dagger \phi = - \pdv{y}(f \phi) + \frac{1}{2} \pdv[2]{y}(g^2 \phi) } \end{aligned}

This can be rewritten in a way that highlights the connection between Itō diffusions and physical diffusion, if we define the diffusivity $$D$$, advection $$u$$, and probability flux $$J$$:

\begin{aligned} D \equiv \frac{1}{2} g^2 \qquad \quad u = f - \pdv{D}{x} \qquad \quad J \equiv u \phi - D \pdv{\phi}{x} \end{aligned}

Such that the forward Kolmogorov equation takes the following conservative form, so called because it looks like a physical continuity equation:

\begin{aligned} \boxed{ \pdv{\phi}{t} = - \pdv{J}{x} = - \pdv{x} \Big( u \phi - D \pdv{\phi}{x} \Big) } \end{aligned}

Note that if $$u = 0$$, then this reduces to Fick’s second law. The backward Kolmogorov equation can also be rewritten analogously, although it is less noteworthy:

\begin{aligned} \boxed{ - \pdv{\psi}{t} = u \pdv{\psi}{x} + \pdv{x} \Big( D \pdv{\psi}{x} \Big) } \end{aligned}

Notice that the diffusivity term looks the same in both the forward and backward equations; we say that diffusion is self-adjoint.

## References

1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.