Chaos expansion methods in Malliavin calculus: A survey of recent results

Seleši, Dora; Pilipović, Stevan; Levajković, Tijana

doi:10.30755/nsjom.dans14.03

Cited by 12 publications

(21 citation statements)

References 44 publications

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“…Detailed analysis of domain and range of operators of the Malliavin calculus in spaces of stochastic distributions can be found in [43].…”

Section: Stochastic Integration and Wick Multiplicationmentioning

confidence: 99%

“…Classes of elliptic and evolution stochastic differential equations (SDEs) that involve operators of the Malliavin calculus within white noise framework were recently studed in [46,40,44,38,48,58]. In [42,43] it was proved that the Malliavin derivative indicates the rate of change in time between the ordinary product and the Wick product. In this paper, we consider stochastic optimal control problems with stochastic perturbations given in an integral form.…”

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confidence: 99%

“…Since U is a separable Hilbert space, the spaces [43]. Moreover, an H-valued Brownian motion is denoted by (W t ) t≥0 .…”

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confidence: 99%

“…In particular, for the orthogonal polynomial basis of L 2 (Ω) we have H α ♦H β = H α+β , for α, β ∈ I. Whenever F , G and F ♦G are integrable it holds E(F ♦G) = E(F ) • E(G), without independence requirement [25,43]. The ordinary product F • G of random variables F, G ∈ L 2 (Ω) is defined by using the multiplication formula…”

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confidence: 99%

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The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

Levajković¹,

Mena²,

Tuffaha³

2016

EECT

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We consider the stochastic linear quadratic optimal control problem for state equations of the Itô-Skorokhod type, where the dynamics are driven by strongly continuous semigroup. We provide a numerical framework for solving the control problem using a polynomial chaos expansion approach in white noise setting. After applying polynomial chaos expansion to the state equation, we obtain a system of infinitely many deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of deterministic linear quadratic regulator problems. Solving these control problems, we find optimal coefficients for the state and the control. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to a direct approach. Moreover, we apply our result to a fully stochastic problem, in which the state, control and observation operators can be random, and we also consider an extension to state equations with memory noise.

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“…Detailed analysis of domain and range of operators of the Malliavin calculus in spaces of stochastic distributions can be found in [43].…”

Section: Stochastic Integration and Wick Multiplicationmentioning

confidence: 99%

mentioning

confidence: 99%

“…Since U is a separable Hilbert space, the spaces [43]. Moreover, an H-valued Brownian motion is denoted by (W t ) t≥0 .…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

Levajković¹,

Mena²,

Tuffaha³

2016

EECT

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“…In [11] we proved that the Malliavin derivative indicates the rate of change in time between the ordinary product and the Wick product, i.e. hÁW t 2 hSW t ¼ DðhÞ holds.…”

Section: Introductionmentioning

confidence: 97%

Fundamental equations with higher order Malliavin operators

2015

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We consider three fundamental equations of the Malliavin calculus: the equation involving the Malliavin derivative, the Skorokhod integral and the Ornstein -Uhlenbeck operator of order k, k [ N. These equations provide a complete characterization of the domain and range of the aforementioned operators. Applying the chaos expansion method in white noise spaces we solve these equations and obtain an explicit form of the solutions in the space of Kondratiev generalized stochastic processes.

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