--- title: "Kolmogorov equations" firstLetter: "K" publishDate: 2021-11-14 categories: - Mathematics - Statistics date: 2021-11-13T21:05:30+01:00 draft: false markup: pandoc --- # Kolmogorov equations Consider the following general [Itō diffusion](/know/concept/ito-calculus/) $X_t \in \mathbb{R}$, 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](/know/concept/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](/know/concept/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](/know/concept/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](/know/concept/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 $\phi(t, y)$: $$\begin{aligned} \boxed{ \pdv{\phi}{t} = \hat{L}{}^\dagger \phi = - \pdv{y}(f \phi) + \frac{1}{2} \pdv[2]{y}(g^2 \phi) } \end{aligned}$$ ## References 1. U.H. Thygesen, *Lecture notes on diffusions and stochastic differential equations*, 2021, Polyteknisk Kompendie.