Stochastic analysis

By I. Shigekawa. 2004

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Non-degeneracy of Wiener functionals arising from rough differential equations (2007, background): “The concept is standard [3, 22, 24, 28], as is the notion of a weakly non-degenerate R-valued Wiener functional φ which has the desirable property that the image measure φ∗μ is absolutely continuous with respect to Lebesgue measure on R.”
Strong convergence for the Euler-Maruyama approximation of stochastic differential equations with discontinuous coefficients (2016, unknown): “|≥Kn i ≤ P Zs 0 σ(X(n) ηn(u) )dWu ≥ Kn − x0 + Zs 0 b(X(n) ηn(u) )du ≤ P Zs 0 σ(X(n) ηn(u) )dWu ≥ Kn −kbk∞T−|x0| . Since h R· 0 σ(X (n) ηn(s) )dWsit ≤ σ2Talmost surely, from Proposition 6.8 of [23] and the inequality (a−b)2 ≥ a2/2 −b2 for any a,b∈ R,we have P sup 0≤t≤T Zt 0 σ(X(n) ηn(s) )dWs ≥ Kn”
Wiener integrals, Malliavin calculus and covariance measure structure (2006, unknown): “the result. 32 7.2 Continuityof theSkorohod integralprocess It is possible to connect previous objects with the classical Wiener analysis on an abstract Wiener space, related to Hilbert spaces H, see [30, 39]. In the classical theory the Malliavin gradient (derivative) ∇ and the divergence operator ”