Kolmogorov backward equations (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.^[1] 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 ${\displaystyle p_{t}(x)}$ for a system being in state ${\displaystyle x}$ at time ${\displaystyle t,}$ the forward PDE is integrated to obtain ${\displaystyle p_{s}(x)}$ at later times ${\displaystyle s>t.}$ A common case takes the initial value ${\displaystyle p_{t}(x)}$ to be a Dirac delta function centered on the known initial state ${\displaystyle x.}$

The Kolmogorov backward equation is used to estimate the probability of the current system evolving so that it's future state at time ${\displaystyle s>t}$ is given by some fixed probability function ${\displaystyle p_{s}(x).}$ 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 ${\displaystyle B,}$ the target set. Writing the set membership function as ${\displaystyle 1_{B},}$ so that ${\displaystyle 1_{B}(x)=1}$ if ${\displaystyle x\in B}$ and zero otherwise, the backward equation expresses the hit probability ${\displaystyle p_{t}(x)}$ that in the future, the set membership will be sharp, given by ${\displaystyle p_{s}(x)=1_{B}(x)/\Vert B\Vert .}$ Here, ${\displaystyle \Vert B\Vert }$ is just the size of the set ${\displaystyle B,}$ a normalization so that the total probability at time ${\displaystyle s}$ integrates to one.

Kolmogorov backward equation

Let ${\displaystyle \{X_{t}\}_{0\leq t\leq T}}$ be the solution of the stochastic differential equation

${\displaystyle dX_{t}\;=\;\mu {\bigl (}t,X_{t}{\bigr )}\,dt\;+\;\sigma {\bigl (}t,X_{t}{\bigr )}\,dW_{t},\quad 0\;\leq \;t\;\leq \;T,}$

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

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

Then the usual Kolmogorov backward equation for ${\displaystyle p}$ is

${\displaystyle {\frac {\partial p}{\partial t}}(t,x;\,T,y)\;+\;A\,p(t,x;\,T,y)\;=\;0,\quad \lim _{t\to T}\,p(t,x;\,T,y)\;=\;\delta _{y}(x),}$

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

${\displaystyle A\,f(x)\;=\;\sum _{i}\,\mu _{i}(x)\,{\frac {\partial f}{\partial x_{i}}}(x)\;+\;{\frac {1}{2}}\,\sum _{i,j}\,{\bigl [}\sigma (x)\,\sigma (x)^{\mathsf {T}}{\bigr ]}_{ij}\,{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x).}$

Feynman–Kac formula

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

${\displaystyle {\frac {\partial F}{\partial t}}(t,x)\;+\;\mu (t,x)\,{\frac {\partial F}{\partial x}}(t,x)\;+\;{\frac {1}{2}}\,\sigma ^{2}(t,x)\,{\frac {\partial ^{2}F}{\partial x^{2}}}(t,x)\;=\;0,\quad 0\leq t\leq T,\quad F(T,x)\;=\;\Phi (x)}$

and given ${\displaystyle \{X_{t}\}_{0\leq t\leq T},}$ that, just as before, is a solution of

${\displaystyle dX_{t}\;=\;\mu (t,X_{t})\,dt\;+\;\sigma (t,X_{t})\,dW_{t},\quad 0\leq t\leq T,}$

then if the expectation value is finite

${\displaystyle \int _{0}^{T}\,\mathbb {E} \!{\Bigl [}{\bigl (}\sigma (t,X_{t})\,{\frac {\partial F}{\partial x}}(t,X_{t}){\bigr )}^{2}{\Bigr ]}\,dt\;<\;\infty ,}$

then the Feynman–Kac formula is obtained:

${\displaystyle F(t,x)\;=\;\mathbb {E} \!{\bigl [}\;\Phi (X_{T})\,{\big |}\;X_{t}=x{\bigr ]}.}$

Proof. Apply Itô’s formula to ${\displaystyle F(s,X_{s})}$ for ${\displaystyle t\leq s\leq T}$:

${\displaystyle F(T,X_{T})\;=\;F(t,X_{t})\;+\;\int _{t}^{T}\!{\Bigl \{}{\frac {\partial F}{\partial s}}(s,X_{s})\;+\;\mu (s,X_{s})\,{\frac {\partial F}{\partial x}}(s,X_{s})\;+\;{\tfrac {1}{2}}\,\sigma ^{2}(s,X_{s})\,{\frac {\partial ^{2}F}{\partial x^{2}}}(s,X_{s}){\Bigr \}}\,ds\;+\;\int _{t}^{T}\!\sigma (s,X_{s})\,{\frac {\partial F}{\partial x}}(s,X_{s})\,dW_{s}.}$

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

${\displaystyle \mathbb {E} \!{\bigl [}F(T,X_{T})\,{\big |}\;X_{t}=x{\bigr ]}\;=\;F(t,x).}$

Substitute ${\displaystyle F(T,X_{T})=\Phi (X_{T})}$ to conclude

${\displaystyle F(t,x)\;=\;\mathbb {E} \!{\bigl [}\;\Phi (X_{T})\,{\big |}\;X_{t}=x{\bigr ]}.}$

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

${\displaystyle dX_{t}\;=\;\mu (t,X_{t})\,dt\;+\;\sigma (t,X_{t})\,dW_{t}.}$

For any set ${\displaystyle B}$, define

${\displaystyle p_{B}(t,x;\,T)\;\triangleq \;\mathbb {P} \!{\bigl [}X_{T}\in B\,\mid \,X_{t}=x{\bigr ]}\;=\;\mathbb {E} \!{\bigl [}\mathbf {1} _{B}(X_{T})\,{\big |}\;X_{t}=x{\bigr ]}.}$

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

${\displaystyle {\frac {\partial p_{B}}{\partial t}}(t,x;\,T)\;+\;A\,p_{B}(t,x;\,T)\;=\;0,\quad p_{B}(T,x;\,T)\;=\;\mathbf {1} _{B}(x),}$

where

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

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

${\displaystyle p(t,x;\,T,y)\;\triangleq \;\lim _{B\to y}\,{\frac {1}{|B|}}\,\mathbb {P} \!{\bigl [}X_{T}\in B\,\mid \,X_{t}=x{\bigr ]}.}$

Then

${\displaystyle {\frac {\partial p}{\partial t}}(t,x;\,T,y)\;+\;A\,p(t,x;\,T,y)\;=\;0,\quad p(t,x;\,T,y)\;\to \;\delta _{y}(x)\quad {\text{as }}t\;\to \;T.}$

Derivation of the forward Kolmogorov equation

The Kolmogorov forward equation is

${\displaystyle {\frac {\partial }{\partial T}}\,p{\bigl (}t,x;\,T,y{\bigr )}\;=\;A^{*}\!{\bigl [}p{\bigl (}t,x;\,T,y{\bigr )}{\bigr ]},\quad \lim _{T\to t}\,p(t,x;\,T,y)\;=\;\delta _{y}(x).}$

For ${\displaystyle T>r>t}$, the Markov property implies

${\displaystyle p(t,x;\,T,y)\;=\;\int _{-\infty }^{\infty }p{\bigl (}t,x;\,r,z{\bigr )}\,p{\bigl (}r,z;\,T,y{\bigr )}\,dz.}$

Differentiate both sides w.r.t. ${\displaystyle r}$:

${\displaystyle 0\;=\;\int _{-\infty }^{\infty }{\Bigl [}{\frac {\partial }{\partial r}}\,p{\bigl (}t,x;\,r,z{\bigr )}\,\cdot \,p{\bigl (}r,z;\,T,y{\bigr )}\;+\;p{\bigl (}t,x;\,r,z{\bigr )}\,\cdot \,{\frac {\partial }{\partial r}}\,p{\bigl (}r,z;\,T,y{\bigr )}{\Bigr ]}\,dz.}$

From the backward Kolmogorov equation:

${\displaystyle {\frac {\partial }{\partial r}}\,p{\bigl (}r,z;\,T,y{\bigr )}\;=\;-\,A\,p{\bigl (}r,z;\,T,y{\bigr )}.}$

Substitute into the integral:

${\displaystyle 0\;=\;\int _{-\infty }^{\infty }{\Bigl [}{\frac {\partial }{\partial r}}\,p{\bigl (}t,x;\,r,z{\bigr )}\,\cdot \,p{\bigl (}r,z;\,T,y{\bigr )}\;-\;p{\bigl (}t,x;\,r,z{\bigr )}\,\cdot \,A\,p{\bigl (}r,z;\,T,y{\bigr )}{\Bigr ]}\,dz.}$

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

${\displaystyle \int _{-\infty }^{\infty }{\bigl [}{\frac {\partial }{\partial r}}\,p{\bigl (}t,x;\,r,z{\bigr )}\;-\;A^{*}\,p{\bigl (}t,x;\,r,z{\bigr )}{\bigr ]}\,p{\bigl (}r,z;\,T,y{\bigr )}\,dz\;=\;0.}$

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

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

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

${\displaystyle {\frac {\partial }{\partial T}}\,p{\bigl (}t,x;\,T,y{\bigr )}\;=\;A^{*}\!{\bigl [}p{\bigl (}t,x;\,T,y{\bigr )}{\bigr ]},\quad \lim _{T\to t}\,p(t,x;\,T,y)\;=\;\delta _{y}(x).}$

Finally,

${\displaystyle A^{*}\,g(x)\;=\;-\sum _{i}\,{\frac {\partial }{\partial x_{i}}}{\bigl [}\mu _{i}(x)\,g(x){\bigr ]}\;+\;{\frac {1}{2}}\,\sum _{i,j}\,{\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}{\Bigl [}{\bigl (}\sigma (x)\,\sigma (x)^{\mathsf {T}}{\bigr )}_{ij}\,g(x){\Bigr ]}.}$

See also

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