Math 2.0

Probability theory and stochastic processes Definition

Diffusion process

Builds on Wiener processMarkov propertyInfinitesimal generator of a Markov processItô's lemma

Definition

Let d,m≥1d, m \ge 1, let b ⁣:ℝd→ℝdb \colon \R^d \to \R^d and σ ⁣:ℝd→ℝd×m\sigma \colon \R^d \to \R^{d \times m} be Lipschitz continuous, and let WW be an mm-dimensional Wiener process. The diffusion process (Itô diffusion) with drift bb and dispersion σ\sigma started at x∈ℝdx \in \R^d is the unique continuous adapted solution (Xt)t≥0(X_t)_{t \ge 0} of dXt=b(Xt) dt+σ(Xt) dWt,X0=x.dX_t = b(X_t)\, dt + \sigma(X_t)\, dW_t, \qquad X_0 = x. The matrix a=σσ𝖳a = \sigma \sigma^{\mathsf{T}} is its diffusion matrix.

Remarks

The process is a strong Markov process, and by Itô's lemma its generator acts on twice continuously differentiable functions ff with compact support by Lf(x)=12∑i,j=1daij(x)∂2f∂xi∂xj(x)+∑i=1dbi(x)∂f∂xi(x).Lf(x) = \frac12 \sum_{i,j=1}^d a_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j}(x) + \sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x). Example: d=m=1d = m = 1, b(x)=−θxb(x) = -\theta x with θ>0\theta > 0 and σ(x)=1\sigma(x) = 1 give the Ornstein–Uhlenbeck process; both coefficients are Lipschitz, and Lf=12f′′−θxf′Lf = \tfrac12 f'' - \theta x f'. Non-example: the Poisson process is a Markov process but not a diffusion, because its paths jump. In a wider sense, a diffusion is any strong Markov process with continuous paths; the Itô diffusions above form the main class.

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Euclidean 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 calculusconceptMarkov property, concept, no evidence levelMarkov propertyconceptQuadratic 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 lemmastatementDiffusion process, definition, no evidence levelDiffusion processdefinition
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  1. Euclidean space object
  2. Function concept
  3. Geometric Brownian motion concept
  4. Infinitesimal generator of a Markov process concept
  5. Itô calculus concept
  6. Markov property concept
  7. Quadratic variation concept
  8. Wiener process concept
  9. Function of several variables concept
  10. Partial derivative concept
  11. Itô's lemma statement
  12. Diffusion process definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Diffusion_process
About this page
TagGVMM9GC7CE
URIm2://GVMM9GC7CE
Kinddefinition
WritingW0 stub
MSC60J60, 60H10
DefinesDiffusion process
UsesWiener process, Markov property, Infinitesimal generator of a Markov process, Itô's lemma
Changed2 November 2026

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