Weak differentiability of Wiener functionals and occupation times

Leão, Dorival; Ohashi, Alberto; Simas, Alexandre B.

doi:10.1016/j.bulsci.2018.07.001

Cited by 6 publications

(6 citation statements)

References 52 publications

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“…Recently, it has been developed a theory of functional stochastic calculus and control applications in a series of studies [18,19,20,21,22]. The main advantage of this approach over other methods lies in the numerically viable description of stochastic systems controlled in continuous time, especially in problems of optimal control, possibly non-Markovian and of high dimension.…”

Section: Classical Results and New Toolsmentioning

confidence: 99%

Stochastic near-optimal control: additive, multiplicative, non-Markovian and applications

Lima,

Ruffino,

Souza

2020

Preprint

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In this survey we present the near-optimal stochastic control problem according to some recent tools in the literature. In particular, we focus on the approach of a discretization of the noise values instead of the canonical time-discretization. This is the so called skeleton structure. This allows to obtain an ǫ-optimal control in non-Markovian systems (the main Theorem). A simple example illustrates the technique. The importance of the approach is emphasised in a final section on open problems related to more geometrical framework and discontinuous noise.

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Section: Classical Results and New Toolsmentioning

confidence: 99%

Stochastic near-optimal control: additive, multiplicative, non-Markovian and applications

Lima,

Ruffino,

Souza

2020

Preprint

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“…• Functional imbedded discrete structures associated with path-dependent functionals under (p, q)-variation regularity. See [35].…”

Section: Pathwise Dynamics Of the Skeleton Let Us Definementioning

confidence: 99%

“…Indeed, the process (4.10) encodes all the possible orthogonal infinitesimal variations of a weakly differentiable process w.r.t the Brownian motion noise. See Section 4.4 for the smooth case and [35] for a universal variational characterization of V X as a functional of local-times.…”

Section: Thenmentioning

confidence: 99%

“…In [35], the authors present a universal variational characterization of the drifts of weakly differentiable processes. It is revealed the asymptotic behavior of (1.2) is always encoded by suitable limits of integral functionals of horizontal-type perturbations and first-order variation w.r.t noise having a two-parameter occupation time process (s, x) → L k,x (s) for D as integrators.…”

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confidence: 99%

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A weak version of path-dependent functional Itô calculus

Leão¹,

Ohashi²,

Simas³

2018

Ann. Probab.

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We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the "sensitivities" of processes, namely derivatives of martingale components and a weak notion of infinitesimal generators, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t Brownian motion driving noise.MSC 2010 subject classifications: Primary 60H07; secondary 60H25

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“…Several of these works make a strong use of the notions of extended convergence, introduced by Aldous [1], as well as that of convergence of filtrations, introduced by Hoover [32] and further developed by Coquet, Mémin, and Mackevičius [16] and Coquet, Mémin, and Słomiński [17], which also plays a major role in the present paper. Let us also mention the recent contributions by Leão, Ohashi and Simas [43,44], who study stability of Wiener functionals under weak convergence of filtration beyond semimartingales, in the context of functional Itō calculus.…”

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confidence: 99%

Stability results for martingale representations: The general case

Papapantoleon¹,

Possamaï²,

Saplaouras³

2019

Trans. Amer. Math. Soc.

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In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi-left-continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems. 1 2 A. PAPAPANTOLEON, D. POSSAMAÏ, AND A. SAPLAOURAS A.3. Young functions 47 A.4. Proof of Corollary 3.4 51 REFERENCES 53 Once the semimartingales considered have more structural properties, other interesting results can be obtained. Barrieu, Cazanave, and El Karoui [6], for instance, and later Barrieu and El Karoui [5] were interested in what they coined "continuous quadratic semimartingales" (see also related articles by Mocha and Westray [54], still in the continuous case, and recent extensions to jump processes by Ngoupeyou [58] and El Karoui, Matoussi, and Ngoupeyou [25]), for which, roughly speaking, the bounded variation process part in the semimartingale X is absolutely continuous with respect to the quadratic variation of the martingale part of X.[5, 6] obtained associated stability results for these processes.An important common feature of the articles mentioned so far, is that they actually only consider the strong framework we described at the beginning of this introduction, in the sense that there is a always a fixed probability space and all processes (meaning here mainly X n and X ∞ ) are adapted to the same fixed filtration G. An important exception is Słomiński [67], where the probability space is fixed, but not the filtration. However, for practical purposes, and especially for the analysis of numerical schemes, it is well-known that the weak framework is also of paramount importance, as illustrated for instance by the famous Donsker theorem. There has thus been a certain number of studies of stability properties for semimartingale or martingale decompositions when the underlying filtration itself is also allowed to change. In that direction, Antonelli and Kohatsu-Higa [3], followed by Coquet, Mackevičius, and Mémin [14, 15], Ma, Protter, San Martín, and Torres [47], Briand, Delyon, and Mémin [9], Briand, Delyon, and Mémin [10], and then Cheridito and Stadje [11], studied such stability properties for continuous backward stochastic differential equations (BSDEs for short), a type of non-linear martingale representation. Mémin [52] looked into the stabi...

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Weak differentiability of Wiener functionals and occupation times

Cited by 6 publications

References 52 publications

Stochastic near-optimal control: additive, multiplicative, non-Markovian and applications

Stochastic near-optimal control: additive, multiplicative, non-Markovian and applications

A weak version of path-dependent functional Itô calculus

Stability results for martingale representations: The general case

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