TOPICS
Search

Stochastic Convolution


A stochastic convolution is a stochastic process of the form

 (S⋄Phi)(t)=int_0^tS(t-s)Phi(s)dW(s).

Here H and U are Hilbert spaces, S(t) is a strongly continuous semigroup of bounded linear operators on H, W(t) is a U-valued Wiener process, and Phi(t) is a predictable stochastic process of suitable linear operators from U to H for which the stochastic integral exists. It gives the random forcing term in the integral formulation of a stochastic differential equation on H.


See also

Convolution, Semigroup, Stochastic Integral, Wiener Process

Explore with Wolfram|Alpha

References

Da Prato, G. and Zabczyk, J. Stochastic Equations in Infinite Dimensions. Cambridge, England: Cambridge University Press, 1992.

Cite this as:

Weisstein, Eric W. "Stochastic Convolution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StochasticConvolution.html

Subject classifications