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.Andrei Kolmogorov, "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (On Analytical Methods in the Theory of Probability), 1931

/ref> 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 $\_$ be the solution of the stochastic differential equation

:$dX_t \;=\; \mu\bigl(t, X_t\bigr)\,dt \;+\; \sigma\bigl(t, X_t\bigr)\,dW_t, \quad 0 \;\le\; t \;\le\; T,$

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

:$p(t,x;\,T,y) \;=\; \frac, \quad t < T.$

Then the usual Kolmogorov backward equation for $p$ is

:$\frac(t, x;\,T, y) \;+\; A\, p(t, x;\,T, y) \;=\; 0, \quad \lim_\, p(t,x;\,T,y) \;=\; \delta_(x),$

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

:
A\,f(x)
\;=\;
\sum_\,\mu_(x)\,\frac(x)
\;+\;
\frac12\,\sum_\,
\bigl sigma(x)\,\sigma(x)^\bigr\,
\frac(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

:$\frac(t,x) \;+\; \mu(t,x)\,\frac(t,x) \;+\; \frac\,\sigma^2(t,x)\,\frac(t,x) \;=\; 0, \quad 0 \le t \le T, \quad F(T,x) \;=\; \Phi(x)$

and given $\_,$ that, just as before, is a solution of

:$dX_t \;=\; \mu(t, X_t)\,dt \;+\; \sigma(t, X_t)\,dW_t, \quad 0 \le t \le T,$

then if the expectation value is finite

:
\int_^\,
\mathbb\!\Bigl \bigl(\sigma(t, X_t)\,\frac(t, X_t)\bigr)^2
\Bigr,
dt
\;<\;\infty,

then the Feynman–Kac formula is obtained:

:
F(t,x)
\;=\;
\mathbb\!\bigl \;X_t = x\bigr

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

:$F(T, X_T) \;=\; F(t, X_t) \;+\; \int_^\!\Bigl\\,ds \;+\; \int_^\!\sigma(s, X_s)\,\frac(s, X_s)\,dW_s.$

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

:
\mathbb\!\bigl \;X_t=x\bigr\;=\;
F(t, x).

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

:
F(t,x)
\;=\;
\mathbb\!\bigl \;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

:$dX_t \;=\; \mu(t, X_t)\,dt \;+\; \sigma(t, X_t)\,dW_t.$

For any set $B$, define

:
p_B(t, x;\,T)
\;\triangleq\;
\mathbb\!\bigl _T \in B \,\mid\, X_t = x\bigr\;=\;
\mathbb\!\bigl \;X_t = x\bigr

By Feynman–Kac (under integrability conditions), taking $\Phi=\mathbf_B$, then

:$\frac(t, x;\,T) \;+\; A\,p_B(t, x;\,T) \;=\;0, \quad p_B(T, x;\,T) \;=\;\mathbf_B(x),$

where

:$A\,f(t, x) \;=\; \mu(t, x)\,\frac(t, x) \;+\; \tfrac12\,\sigma^2(t, x)\,\frac(t, x).$

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\;
\lim_\,\frac\,\mathbb\!\bigl _T \in B\,\mid\,X_t = x\bigr

Then

:$\frac(t, x;\,T, y) \;+\; A\,p(t, x;\,T, y) \;=\;0, \quad p(t, x;\,T, y) \;\to\; \delta_y(x) \quad \text t\;\to\;T.$

Derivation of the forward Kolmogorov equation

The Kolmogorov forward equation is

:
\frac\,p\bigl(t, x;\,T, y\bigr)
\;=\;
A^\!\bigl \bigl(t, x;\,T, y\bigr)\bigr
\quad
\lim_\,p(t,x;\,T,y)
\;=\;
\delta_(x).

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

:$p(t, x;\,T, y) \;=\; \int_^ p\bigl(t, x;\,r, z\bigr)\, p\bigl(r, z;\,T, y\bigr) \,dz.$

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

:
0
\;=\;
\int_^
\Bigl \frac\,p\bigl(t, x;\,r, z\bigr)\,\cdot\,p\bigl(r, z;\,T, y\bigr)
\;+\;
p\bigl(t, x;\,r, z\bigr)\,\cdot\,
\frac\,p\bigl(r, z;\,T, y\bigr)
\Bigr,dz.

From the backward Kolmogorov equation:

:$\frac\,p\bigl(r, z;\,T, y\bigr) \;=\; -\,A\,p\bigl(r, z;\,T, y\bigr).$

Substitute into the integral:

:
0
\;=\;
\int_^
\Bigl \frac\,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 $A^$:

:
\int_^
\bigl \frac\,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 $p(r,z;\,T,y)$ can be arbitrary, the bracket must vanish:

:
\frac\,p\bigl(t, x;\,r,z\bigr)
\;=\;
A^\bigl \bigl(t, x;\,r,z\bigr)\bigr

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

:
\frac\,p\bigl(t, x;\,T, y\bigr)
\;=\;
A^\!\bigl \bigl(t, x;\,T, y\bigr)\bigr
\quad
\lim_\,p(t,x;\,T,y)
\;=\;
\delta_(x).

Finally,

:
A^\,g(x)
\;=\;
-\sum_\,\frac
\bigl mu_(x)\,g(x)\bigr\;+\;
\frac12\,
\sum_\,
\frac
\Bigl \bigl(\sigma(x)\,\sigma(x)^\bigr)_\,g(x)
\Bigr

See also

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

References

*
{{reflist

Parabolic partial differential equations
Stochastic differential equations