Kolmogorov backward equations (diffusion)

Short description: Partial differential equations describing diffusion

The Kolmogorov backward equation (KBE) and its adjoint, the Kolmogorov forward equation, are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorov in 1931. Later it was realized that the forward equation was already known to physicists under the name Fokker–Planck equation; the KBE on the other hand was new.

Overview

The Kolmogorov forward equation is used to evolve the state of a system forward in time. Given an initial probability distribution $p_t(x)$ for a system being in state $x$ at time $t,$ the forward PDE is integrated to obtain $p_s(x)$ at later times $s>t.$ A common case takes the initial value $p_t(x)$ to be a Dirac delta function centered on the known initial state $x.$

The Kolmogorov backward equation is used to estimate the probability of the current system evolving so that it's future state at time $s>t$ is given by some fixed probability function $p_s(x).$ That is, the probability distribution in the future is given as a boundary condition, and the backwards PDE is integrated backwards in time.

A common boundary condition is to ask that the future state is contained in some subset of states $B,$ the target set. Writing the set membership function as $1_B,$ so that $1_B(x)=1$ if $x\in B$ and zero otherwise, the backward equation expresses the hit probability $p_t(x)$ that in the future, the set membership will be sharp, given by $p_s(x)=1_B(x)/\Vert B\Vert .$ Here, $\Vert B\Vert$ is just the size of the set $B,$ a normalization so that the total probability at time $s$ integrates to one.

Kolmogorov backward equation

Let $\{X_t{\}}_{0\le t\le T}$ be the solution of the stochastic differential equation

$d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{W}_t,0\le t\le T,$

where $W_t$ is a (possibly multi-dimensional) Wiener process (Brownian motion), $\mu$ is the drift coefficient, and $\sigma$ is related to the diffusion coefficient $D$ as $D={\sigma }^2/2.$ Define the transition density (or fundamental solution) $p(t,x;T,y)$ by

$p(t,x;T,y)=\frac{\mathbb{\mathbb{P}}[X_T\in dy\mid X_t=x]}{dy},t<T.$

Then the usual Kolmogorov backward equation for $p$ is

$\frac{\partial p}{\partial t}(t,x;T,y)+Ap(t,x;T,y)=0,\underset{t\to T}{\lim }p(t,x;T,y)={\delta }_y(x),$

where ${\delta }_y(x)$ is the Dirac delta in $x$ centered at $y$, and $A$ is the infinitesimal generator of the diffusion:

$Af(x)=\sum _i{\mu }_i(x)\frac{\partial f}{\partial x_i}(x)+\frac{1}{2}\sum _{i,j}[\sigma (x)\sigma (x{)}^{\mathsf{T}}{]}_{ij}\frac{{\partial }^{2}f}{\partial x_i\partial x_j}(x).$

Feynman–Kac formula

The backward Kolmogorov equation can be used to derive the Feynman–Kac formula. Given a function $F$ that satisfies the boundary value problem

$\frac{\partial F}{\partial t}(t,x)+\mu (t,x)\frac{\partial F}{\partial x}(t,x)+\frac{1}{2}{\sigma }^2(t,x)\frac{{\partial }^{2}F}{\partial x^2}(t,x)=0,0\le t\le T,F(T,x)=\mathrm{\Phi }(x)$

and given $\{X_t{\}}_{0\le t\le T},$ that, just as before, is a solution of

$d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{W}_t,0\le t\le T,$

then if the expectation value is finite

$\underset{0}{\overset{T}{\int }}\mathbb{𝔼}\left[\right(\sigma (t,X_t)\frac{\partial F}{\partial x}(t,X_t){)}^2]dt<\mathrm{\infty },$

then the Feynman–Kac formula is obtained:

$F(t,x)=\mathbb{𝔼}[\mathrm{\Phi }(X_T)|X_t=x].$

Proof. Apply Itô’s formula to $F(s,X_s)$ for $t\le s\le T$:

$F(T,X_T)=F(t,X_t)+\underset{t}{\overset{T}{\int }}\{\frac{\partial F}{\partial s}(s,X_s)+\mu (s,X_s)\frac{\partial F}{\partial x}(s,X_s)+\frac{1}{2}{\sigma }^2(s,X_s)\frac{{\partial }^{2}F}{\partial x^2}(s,X_s)\}ds+\underset{t}{\overset{T}{\int }}\sigma (s,X_s)\frac{\partial F}{\partial x}(s,X_s)d{W}_s.$

Because $F$ solves the PDE, the first integral is zero. Taking conditional expectation and using the martingale property of the Itô integral gives

$\mathbb{𝔼}[F(T,X_T)|X_t=x]=F(t,x).$

Substitute $F(T,X_T)=\mathrm{\Phi }(X_T)$ to conclude

$F(t,x)=\mathbb{𝔼}[\mathrm{\Phi }(X_T)|X_t=x].$

Derivation of the backward Kolmogorov equation

The Feynman–Kac representation can be used to find the PDE solved by the transition densities of solutions to SDEs. Suppose

$d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{W}_t.$

For any set $B$, define

$p_B(t,x;T)\triangleq \mathbb{\mathbb{P}}[X_T\in B\mid X_t=x]=\mathbb{𝔼}[{\mathbf{𝟏}}_B(X_T)|X_t=x].$

By Feynman–Kac (under integrability conditions), taking $\mathrm{\Phi }={\mathbf{𝟏}}_B$, then

$\frac{\partial p_B}{\partial t}(t,x;T)+A{p}_B(t,x;T)=0,p_B(T,x;T)={\mathbf{𝟏}}_B(x),$

where

$Af(t,x)=\mu (t,x)\frac{\partial f}{\partial x}(t,x)+\frac{1}{2}{\sigma }^2(t,x)\frac{{\partial }^{2}f}{\partial x^2}(t,x).$

Assuming Lebesgue measure as the reference, write $|B|$ for its measure. The transition density $p(t,x;T,y)$ is

$p(t,x;T,y)\triangleq \underset{B\to y}{\lim }\frac{1}{|B|}\mathbb{\mathbb{P}}[X_T\in B\mid X_t=x].$

Then

$\frac{\partial p}{\partial t}(t,x;T,y)+Ap(t,x;T,y)=0,p(t,x;T,y)\to {\delta }_y(x)\text{as }t\to T.$

Derivation of the forward Kolmogorov equation

The Kolmogorov forward equation is

$\frac{\partial }{\partial T}p(t,x;T,y)=A^{*}\left[p\right(t,x;T,y\left)\right],\underset{T\to t}{\lim }p(t,x;T,y)={\delta }_y(x).$

For $T>r>t$, the Markov property implies

$p(t,x;T,y)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}p(t,x;r,z)p(r,z;T,y)dz.$

Differentiate both sides w.r.t. $r$:

$0=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left[\frac{\partial }{\partial r}p\right(t,x;r,z)\cdot p(r,z;T,y)+p(t,x;r,z)\cdot \frac{\partial }{\partial r}p(r,z;T,y\left)\right]dz.$

From the backward Kolmogorov equation:

$\frac{\partial }{\partial r}p(r,z;T,y)=-Ap(r,z;T,y).$

Substitute into the integral:

$0=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left[\frac{\partial }{\partial r}p\right(t,x;r,z)\cdot p(r,z;T,y)-p(t,x;r,z)\cdot Ap(r,z;T,y\left)\right]dz.$

By definition of the adjoint operator $A^{*}$:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left[\frac{\partial }{\partial r}p\right(t,x;r,z)-A^{*}p(t,x;r,z\left)\right]p(r,z;T,y)dz=0.$

Since $p(r,z;T,y)$ can be arbitrary, the bracket must vanish:

$\frac{\partial }{\partial r}p(t,x;r,z)=A^{*}\left[p\right(t,x;r,z\left)\right].$

Relabel $r\to T$ and $z\to y$, yielding the forward Kolmogorov equation:

$\frac{\partial }{\partial T}p(t,x;T,y)=A^{*}\left[p\right(t,x;T,y\left)\right],\underset{T\to t}{\lim }p(t,x;T,y)={\delta }_y(x).$

Finally,

$A^{*}g(x)=-\sum _i\frac{\partial }{\partial x_i}[{\mu }_i(x)g(x)]+\frac{1}{2}\sum _{i,j}\frac{{\partial }^2}{\partial x_i\partial x_j}\left[\right(\sigma (x)\sigma (x{)}^{\mathsf{T}}{)}_{ij}g(x)].$

See also

• Feynman–Kac formula
• Fokker–Planck equation
• Kolmogorov equations

References

• Etheridge, A. (2002). A Course in Financial Calculus. Cambridge University Press.