stochastic integral

We present a definition of the stochastic integral of a predictable process with respect to a general real-valued semimartingale. In the literature on stochastic calculus there are actually several different different definitions available, often based on specific constructions of the integral in terms of decompositions of the semimartingale into a sum of a local martingalePlanetmathPlanetmath and a finite variation process. The approach taken here is to instead define the integral in terms of its most basic properties — in particular that it satisfies the expected generalizationsPlanetmathPlanetmath of standard non-stochastic integration, namely linearity and dominated convergence. Unlike in standard calculus, where the value of an integral is a real number, here the value is a random variableMathworldPlanetmath.

Many stochastic processesMathworldPlanetmath, such as Brownian motionMathworldPlanetmath, have paths which are nowhere differentiableMathworldPlanetmathPlanetmath and have infiniteMathworldPlanetmathPlanetmath total variationMathworldPlanetmathPlanetmath over finite time intervals. In such cases, standard definitions of integration, such as the Riemann-Stieltjes integral, cannot be used. However, by only considering predictable integrands and by relaxing properties such as dominated convergence to only require convergence in probability, the stochastic integral is a well-defined quantity.

Stochastic integration as described here is sometimes referred to as the Itö or forward integral, in order to distinguish it from the backward and Stratonovich integrals.

Let X be a semimartingale defined with respect to a filtered probability space (Ω,,(t)t+,). Then, for a predictable process ξ, the stochastic integral of ξ with respect to X is a càdlàg process


For each fixed time t, this is a random variable defined on the measurable spaceMathworldPlanetmathPlanetmath (Ω,).

For boundedPlanetmathPlanetmathPlanetmathPlanetmath integrands, the integral satisfies the following properties.

  1. 1.

    (Elementary integrands) For any time T+ and bounded, T-measurable random variable A, a predictable process satisfying ξt=1{t>T}A over t>0 has the integral

    0tξ𝑑X=A1{t>T}(Xt-XT)(almost surely).
  2. 2.

    (Linearity) If α,β are bounded and predictable processes and λ,μ then

    0t(λα+μβ)𝑑X=λ0tα𝑑X+μ0tβ𝑑X(almost surely). (1)
  3. 3.

    (Bounded convergence in probability) If (ξn)n is a sequenceMathworldPlanetmath of predictable processes such that |ξn|1 and ξn0 as n tends to infinityMathworldPlanetmath then,


    in probability as n, for each t0.

These three properties uniquely define the integration for bounded integrands. By linearity, property 1 above is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to stating that the integral agrees with the explicit expression for elementary predictable integrands and then, by bounded convergence, that it agrees with the explicit expression for all simple predictable integrands.

The stochastic integral can be extended to more general unboundedPlanetmathPlanetmath predictable integrands. A predictable process ξ is X-integrable if the set of random variables

{0tα𝑑X:|α||ξ| is bounded and predictable}

is bounded in probability for every t+. The set of all X-integrable processes is sometimes denoted by L1(X). By bounded convergence in probability, this contains all bounded predictable processes, and it is easily shown that the set of X-integrable processes are closed under taking linear combinations. Furthermore, regardless of the specific semimartingale under consideration, every locally bounded predictable process will be X-integrable.

The stochastic integral of arbitrary integrands in L1(X) is the unique extensionPlanetmathPlanetmath from bounded predictable integrands described above such that linearity (1) holds, and the following dominated convergence result holds. If ξ is X-integrable and (ξn)n is a sequence of X-integrable processes such that |ξn||ξ| and ξn0 then


in probability as n, for each t+.

For any semimartingale X and X-integrable process ξ, the integral is sometimes denoted by ξX,


Alternatively, stochastic integrals are often written in differential form. That is,


is equivalent to stating that Yt-Y0=0tξ𝑑X for each t>0.


  • 1 K. Bichteler, Stochastic integration with jumps. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, 2002.
  • 2 Sheng-we He, Jia-gang Wang, Jia-an Yan,Semimartingale theory and stochastic calculus. Kexue Chubanshe (Science Press), CRC Press, 1992.
  • 3 Olav Kallenberg, Foundations of modern probability, Second edition. Probability and its Applications. Springer-Verlag, 2002.
  • 4 Philip E. Protter, Stochastic integration and differential equations. Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. Springer-Verlag, 2004.
  • 5 L.C.G. Rogers & David Williams, Diffusions, Markov processes, and martingalesMathworldPlanetmath. Vol. 2. Itô calculus. Reprint of the second edition. Cambridge Mathematical Library. Cambridge University Press, 2000.
Title stochastic integral
Canonical name StochasticIntegral
Date of creation 2013-03-22 18:36:45
Last modified on 2013-03-22 18:36:45
Owner gel (22282)
Last modified by gel (22282)
Numerical id 9
Author gel (22282)
Entry type Definition
Classification msc 60G07
Classification msc 60H10
Classification msc 60H05
Synonym stochastic integration
Related topic ItoIntegral
Related topic PropertiesOfXIntegrableProcesses
Related topic StochasticIntegrationAsALimitOfRiemannSums