Integral operator Riccati equations arising in stochastic Volterra control problems

Jaber, Eduardo Abi; Miller, Enzo; Pham, Huyên

doi:10.48550/arxiv.1911.01903

Cited by 2 publications

(2 citation statements)

References 11 publications

(17 reference statements)

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“…We mention that for non path-dependent quadratic costs, exact optimal controls for linear state controlled processes driven by FBM are obtained by [26] via solutions of BSDEs driven by FBM and Brownian motion. Infinite-dimensional lifts of linear quadratic control problems driven by non-Markovian stochastic Volterra-type equations (including FBM with H ∈ (0, 1 2 )) have been studied by [27,28]. By using rough path techniques, [14] studies a class of drift controlled differential equations driven by rough paths.…”

Section: Andmentioning

confidence: 99%

Solving non-Markovian Stochastic Control Problems driven by Wiener Functionals

Leão,

Ohashi,

de Souza

2020

Preprint

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In this article, we present a general methodology for stochastic control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main result of this paper is the development of a concrete method for characterizing and computing near-optimal controls for controlled Wiener functionals via a finite-dimensional approximation procedure. The theory does not require a priori functional differentiability assumptions on the value process and ellipticity conditions on the diffusion components. Explicit rates of convergence are provided under rather weak conditions for distinct types of non-Markovian and non-semimartingale states. The analysis is carried out on suitable finite dimensional spaces and it is based on the weak differential structure introduced by [38,39] jointly with measurable selection arguments. The theory is applied to stochastic control problems based on path-dependent SDEs and rough stochastic volatility models, where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for path-dependent SDEs driven by fractional Brownian motion with exponent H ∈ (0, 1) is also discussed.

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Section: Andmentioning

confidence: 99%

Solving non-Markovian Stochastic Control Problems driven by Wiener Functionals

Leão,

Ohashi,

de Souza

2020

Preprint

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“…Similar Riccati equations to that of Γ have appeared in the literature when dealing with convolution kernels of the form (3.2) in the presence of a quadratic structure, see Abi Jaber et al (2019b, Theorem 3.7), Alfonsi and Schied (2013, Theorem 1), Harms and Stefanovits (2019, Lemma 5.4), Cuchiero and Teichmann (2019, Corollary 6.1). A general existence and uniqueness result for more general equations has been recently obtained in Abi Jaber et al (2019c).…”

Section: A Second Representation For Certain Convolution Kernelsmentioning

confidence: 99%

The Laplace transform of the integrated Volterra Wishart process

Jaber

2019

Preprint

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We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore, we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

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Integral operator Riccati equations arising in stochastic Volterra control problems

Cited by 2 publications

References 11 publications

Solving non-Markovian Stochastic Control Problems driven by Wiener Functionals

Solving non-Markovian Stochastic Control Problems driven by Wiener Functionals

The Laplace transform of the integrated Volterra Wishart process

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