Search results
Results from the WOW.Com Content Network
The Stratonovich integral or Fisk–Stratonovich integral of a semimartingale against another semimartingale Y can be defined in terms of the Itô integral as := + [,], where [X, Y] t c denotes the optional quadratic covariation of the continuous parts of X and Y, which is the optional quadratic covariation minus the jumps of the processes and , i.e.
The stochastic integral is a càdlàg process. Furthermore, it is a semimartingale. The discontinuities of the stochastic integral are given by the jumps of the integrator multiplied by the integrand. The jump of a càdlàg process at a time t is X t − X t−, and is often denoted by ΔX t. With this notation, Δ(H · X) = H ΔX. A particular ...
In stochastic processes, the Stratonovich integral or Fisk–Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in ...
In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals.One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals.
Itô pioneered the theory of stochastic integration and stochastic differential equations, now known as Itô calculus. Its basic concept is the Itô integral, and among the most important results is a change of variable formula known as Itô's lemma (also known as the Itô formula).
Stochastic Integral. Proc. Imperial Acad. Tokyo 20, 519–524. This is the paper with the Ito Formula; Online; Kiyosi Itô (1951). On stochastic differential equations. Memoirs, American Mathematical Society 4, 1–51. Online; Bernt Øksendal (2000). Stochastic Differential Equations. An Introduction with Applications, 5th edition, corrected ...
The calculus has been applied to stochastic partial differential equations as well. The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.
Kuo has focused his research on subjects of Theory of Stochastic Integration, White Noise Theory and Infinite Dimensional Analysis. [25] He has published several research papers on stochastic differential equations featuring adapted integrands and a range of initial conditions , particularly focusing on their examination within the framework of ...