Kolmogorov backward equations (diffusion)

The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) 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.

Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state x of the system at time t (namely a probability distribution ${\displaystyle p_{t}(x)}$); we want to know the probability distribution of the state at a later time ${\displaystyle s>t}$. The adjective 'forward' refers to the fact that ${\displaystyle p_{t}(x)}$ serves as the initial condition and the PDE is integrated forward in time (in the common case where the initial state is known exactly, ${\displaystyle p_{t}(x)}$ is a Dirac delta function centered on the known initial state).

The Kolmogorov backward equation on the other hand is useful when we are interested at time t in whether at a future time s the system will be in a given subset of states B, sometimes called the target set. The target is described by a given function ${\displaystyle u_{s}(x)}$ which is equal to 1 if state x is in the target set at time s, and zero otherwise. In other words, ${\displaystyle u_{s}(x)=1_{B}}$, the indicator function for the set B. We want to know for every state x at time ${\displaystyle t,\ (t<s)}$ what is the probability of ending up in the target set at time s (sometimes called the hit probability). In this case ${\displaystyle u_{s}(x)}$ serves as the final condition of the PDE, which is integrated backward in time, from s to t.

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) Brownian motion, ${\displaystyle \mu }$ is the drift coefficient, and ${\displaystyle \sigma }$ is the diffusion coefficient. 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

Assume that the function ${\displaystyle F}$ solves the boundary value problem

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

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

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

where ${\displaystyle \{W_{t}\}_{t\geq 0}}$ is standard Brownian motion under the measure ${\displaystyle \mathbb {P} }$. If

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

then

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

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

We use the Feynman–Kac representation 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), if we take ${\displaystyle \Phi =\mathbf {1} _{B}}$, 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