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.

Formulating the Kolmogorov backward equation

Assume the system state ${\displaystyle X_{t}}$ evolves according to the stochastic differential equation: $${\displaystyle dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t},}$$ where the Kolmogorov backward equation (KBE) is ^[2] : $${\displaystyle -{\frac {\partial }{\partial t}}p(x,t)=\mu (x,t){\frac {\partial }{\partial x}}p(x,t)+{\frac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}}{\partial x^{2}}}p(x,t),}$$ for ${\displaystyle t\leq T}$, with the terminal condition ${\displaystyle p(x,T)=g(x)}$. This is derived via Itō's lemma applied to ${\displaystyle p(x,t)}$, setting the drift term to zero. The Kolmogorov backward equation can also be derived by feyman-kac formula.

Equivalently, for a terminal payoff function ${\displaystyle g(X_{T})}$, define the expected value: $${\displaystyle u(x,t)=\mathbb {E} \left[g(X_{T})\mid X_{t}=x\right],}$$ which satisfies the KBE: $${\displaystyle {\frac {\partial u}{\partial t}}(x,t)+{\mathcal {A}}_{t}u(x,t)=0,}$$ with terminal condition: $${\displaystyle u(x,T)=g(x).}$$ Here, ${\displaystyle {\mathcal {A}}_{t}}$ is the infinitesimal generator acting on ${\displaystyle x}$: $${\displaystyle {\mathcal {A}}_{t}=\mu (x,t){\frac {\partial }{\partial x}}+{\frac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}}{\partial x^{2}}}.}$$

Transition kernel and density

In particular, take the ${\displaystyle g}$ to a indicate function, then for a measurable sets ${\displaystyle A}$, define the transition kernel: $${\displaystyle P(t,x;T,A)=\mathbb {P} \left[X_{T}\in A\mid X_{t}=x\right]=\mathbb {E} \left[\mathbf {1} _{A}(X_{T})\mid X_{t}=x\right],}$$ which solves the KBE: $${\displaystyle {\frac {\partial }{\partial t}}P(t,x;T,A)+{\mathcal {A}}_{t}P(t,x;T,A)=0,}$$ with terminal condition: $${\displaystyle \lim _{t\uparrow T}P(t,x;T,A)=\mathbf {1} _{A}(x).}$$

Under the Lebesgue reference measure, the transition density ${\displaystyle p(t,x;T,y)}$ is: $${\displaystyle p(t,x;T,y){\stackrel {\Delta }{=}}\lim _{A\to \{y\}}{\frac {1}{|A|}}\,\mathbb {P} \left[X_{T}\in A\mid X_{t}=x\right],}$$ satisfying the KBE: $${\displaystyle {\frac {\partial p}{\partial t}}(t,x;T,y)+{\mathcal {A}}_{t}p(t,x;T,y)=0,}$$ with the Dirac delta terminal condition: $${\displaystyle \lim _{t\uparrow T}p(t,x;T,y)=\delta (x-y).}$$

Formulating the Kolmogorov forward equation

The Kolmogorov forward equation (KFE) governs the adjoint evolution: $${\displaystyle {\frac {\partial }{\partial T}}P(t,x;T,y)={\mathcal {A}}_{T}^{*}P(t,x;T,y),}$$ with initial condition: $${\displaystyle \lim _{T\downarrow t}P(t,x;T,y)=\delta (x-y),}$$ where ${\displaystyle {\mathcal {A}}_{T}^{*}}$ is the adjoint of ${\displaystyle {\mathcal {A}}_{T}}$: $${\displaystyle {\mathcal {A}}_{T}^{*}=-{\frac {\partial }{\partial y}}\mu (y,T)+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial y^{2}}}\sigma ^{2}(y,T).}$$ This can be derived by the integration by part and Kolmogorov backward equation.

See also

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

References

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

1. Andrei Kolmogorov, "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (On Analytical Methods in the Theory of Probability), 1931,
2. Risken, H., "The Fokker-Planck equation: Methods of solution and applications", Springer, 1996