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Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions. (English) Zbl 0777.60051
This work is a brief variant of the doctoral thesis of the author, and it is a presentation of her results published in detail before in 1991-92. The famous theorem of E. Wong and M. Zakai proved in 1965, states that under some usual conditions the solutions \(x_ n\) of the “approximate” ordinary differential equations \[ dx_ n(t)=m(x_ n(t),t)dt+\sigma(x_ n(t),t)dw_ n(t), \qquad x_ n(a)=x_ a, \] where \(w_ n(t)\) are regular approximations of the Wiener process \(w(t)\), converges as \(n\to \infty\) to a process that does not satisfy the same original equation but the “corrected” equation \[ dy(t)=m(y(t),t)dt+{1\over 2} \sigma(y(t),t) {{\partial\sigma(y(t),t)} \over {\partial y}}+\sigma(y(t),t)dw(t), \quad y(a)=x_ a. \] It is known that the second term of the right-hand side (“Itô correction term”) does not appear if one uses Stratonovich integrals instead of Itô integrals.
There exists a various work up to the present time of numerous mathematicians devoting extensions of Wong-Zakai theorem in two directions such as for more general driven processes and for equations with coefficients depending on the trajectories of solutions. It turns out that only recent results of the author concern stochastic delay equations.
Thus, Chapter 1 of this work contains a brief survey on this as well as other approximation methods which do not give any correction term. Chapter 2 presents again with proof the author’s theorem of Wong-Zakai type for the stochastic delay equations (systems) (with “delay” \(X_ t(\theta,w)=X(t+\theta,w))\) \[ X^ i(t,w)=X^ i_ 0(w)+\int^ t_ 0 b^ i(X_ s(\cdot,w))ds+ \sum^ m_{p=1}\int^ t_ 0 \sigma^{ip}(X_ s(\cdot,w)) dw^ p(s) \quad (i=1,\dots,d), \] where the \(m\)-dimensional Wiener process \(B(t,w)=w(t)\) is approximated by the standard piecewise linear approximations. This result has been published in [Probab. Math. Stat. 12, No. 2, 319-334 (1991; Zbl 0774.60056)].
Chapter 3 presents again with proof [under some weaker assumptions than in the author’s paper, Stochastic Anal. Appl. 10, No. 4, 471-500 (1992; Zbl 0754.60060)] the theorem of this type for stochastic evolution equation in Hilbert spaces of the type \[ dz(t)=Az(t)dt+ C(z(t))dt+B(z(t))dw(t), \qquad z(0)=z_ 0. \] Chapter 4 shows that the theorem of Chapter 3 contains as special case the theorem of Chapter 2 with an appropriate strongly continuous semigroup. Chapter 5 presents the extension of the known relations between the Itô and Stratonovich integrals to the case of Hilbert spaces. Chapter 6 indicates some interesting research problems as well as possible applications to some other concrete equations.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K50 Stochastic functional-differential equations
60H05 Stochastic integrals
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35G20 Nonlinear higher-order PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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