Math 2.0

Probability theory and stochastic processes Definition

Kolmogorov backward equation

Builds on Diffusion processInfinitesimal generator of a Markov processWiener processPartial derivativeOrnstein–Uhlenbeck processItô's lemma

Definition

Let b ⁣:ℝn→ℝnb \colon \R^n \to \R^n and σ ⁣:ℝn→ℝn×m\sigma \colon \R^n \to \R^{n \times m} be functions, and let a=σσTa = \sigma \sigma^{T}. Consider the diffusion given by the stochastic differential equation dXt=b(Xt) dt+σ(Xt) dBtdX_t = b(X_t) \, dt + \sigma(X_t) \, dB_t, where BB is an mm-dimensional Brownian motion. Its generator is the differential operator ℒg(x)=∑i=1nbi(x) ∂ig(x)+12∑i,j=1naij(x) ∂i∂jg(x),\mathcal{L} g(x) = \sum_{i=1}^n b_i(x) \, \partial_i g(x) + \frac12 \sum_{i,j=1}^n a_{ij}(x) \, \partial_i \partial_j g(x), with partial derivatives ∂i\partial_i in the variable xx. The Kolmogorov backward equation of the diffusion is the equation ∂tu(t,x)=ℒu(t,x),t>0, x∈ℝn,\partial_t u(t, x) = \mathcal{L} u(t, x), \quad t > 0, \ x \in \R^n, for a function u ⁣:[0,∞)×ℝn→ℝu \colon [0, \infty) \times \R^n \to \R, with an initial condition u(0,x)=f(x)u(0, x) = f(x).

Remarks

Under regularity conditions on bb, σ\sigma and ff, the function u(t,x)=E[f(Xt)∣X0=x]u(t, x) = E[f(X_t) \mid X_0 = x] solves the backward equation. If XX has transition densities p(t,x,y)p(t, x, y), they satisfy ∂tp=ℒxp\partial_t p = \mathcal{L}_x p, where ℒx\mathcal{L}_x acts on the starting point xx. The forward, or Fokker–Planck, equation acts instead on the end point yy. Example: for the Ornstein–Uhlenbeck process dX=−X dt+dBdX = -X \, dt + dB in ℝ\R and f(y)=yf(y) = y, one has u(t,x)=xe−tu(t, x) = x e^{-t}. Check: ∂tu=−xe−t\partial_t u = -x e^{-t} and ℒu=−x ∂xu+12∂xxu=−xe−t\mathcal{L} u = -x \, \partial_x u + \tfrac12 \partial_{xx} u = -x e^{-t}. Non-example: for the same process, v(t,x)=xetv(t, x) = x e^{t} has ∂tv=xet\partial_t v = x e^{t} but ℒv=−xet\mathcal{L} v = -x e^{t}, so it does not solve the equation. The equation follows from Itô's lemma applied to s↦u(t−s,Xs)s \mapsto u(t - s, X_s).

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Diffusion process, concept, no evidence levelDiffusion processconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGeometric Brownian motion, concept, no evidence levelGeometric Brownian mo…conceptInfinitesimal generator of a Markov process, concept, no evidence levelInfinitesimal generat…conceptItô calculus, concept, no evidence levelItô calculusconceptOrnstein–Uhlenbeck process, concept, no evidence levelOrnstein–Uhlenbeck pr…conceptQuadratic variation, concept, no evidence levelQuadratic variationconceptWiener process, concept, no evidence levelWiener processconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptItô's lemma, statement, no evidence levelItô's lemmastatementKolmogorov backward equation, definition, no evidence levelKolmogorov backward e…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Diffusion process concept
  2. Euclidean space object
  3. Function concept
  4. Geometric Brownian motion concept
  5. Infinitesimal generator of a Markov process concept
  6. Itô calculus concept
  7. Ornstein–Uhlenbeck process concept
  8. Quadratic variation concept
  9. Wiener process concept
  10. Function of several variables concept
  11. Partial derivative concept
  12. Itô's lemma statement
  13. Kolmogorov backward equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Kolmogorov_backward_equations_(diffusion)
About this page
Tag284TYTVCC0
URIm2://284TYTVCC0
Kinddefinition
WritingW0 stub
MSC60J60, 35K10, 60H10
DefinesKolmogorov backward equation
UsesDiffusion process, Infinitesimal generator of a Markov process, Wiener process, Partial derivative, Ornstein–Uhlenbeck process, Itô's lemma
Changed2 November 2026

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