Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation

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

doi:10.1214/20-aap1645

Cited by 19 publications

(87 citation statements)

References 29 publications

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“…But, even though this method seems appealing, obtaining the required explicit representation of X u in terms of u might be tedious and burdensome. Instead, we will use the approach introduced in [2,8] that is in the same spirit of the one in [12,13,15] mentioned above: we will rewrite the original forward stochastic Volterra integral equation as a stochastic differential equation in a suitable Hilbert space and then apply standard optimization techniques in infinite dimensions (see e.g. [9,11]).…”

Section: Introductionmentioning

confidence: 99%

Optimal control in linear stochastic advertising models with memory

Giordano¹,

Yurchenko-Tytarenko²

2021

Preprint

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This paper deals with a class of optimal control problems which arises in advertising models with Volterra Ornstein-Uhlenbeck process representing the product goodwill. Such choice of the model can be regarded as a stochastic modification of the classical Nerlove-Arrow model that allows to incorporate both presence of uncertainty and empirically observed memory effects such as carryover or distributed forgetting. We present an approach to solve such optimal control problems based on an infinite dimensional lift which allows us to recover Markov properties by formulating a optimization problem equivalent to the original one in a Hilbert space. Such technique, however, requires the Volterra kernel from the forward equation to have a representation of a particular form that may be challenging to obtain in practice. We overcome this issue for Hölder continuous kernels by approximating them with Bernstein polynomials (which turn out to enjoy a simple representation of the required type) and then solving the optimal control problem for the forward process with approximated kernel instead of the original one. The approach is illustrated with simulations.

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Section: Introductionmentioning

confidence: 99%

Optimal control in linear stochastic advertising models with memory

Giordano¹,

Yurchenko-Tytarenko²

2021

Preprint

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“…can be explicitly solved using the Riccati equation (1.1), see [2]. When D = F = 0, the Riccati equation (1.1) also enters in the computation of the Laplace transform of tr T 0 Z ⊤ s QZ s ds , where Z is the d × n-matrix valued Gaussian process…”

Section: Introductionmentioning

confidence: 99%

“…) see[2, Theorem 4.1]. Recall the bold notation G in (2.1) for the integral operator generated by a kernel G. Note that, by virtue of Remark 3.2, Ψ s ∈ L ∞ (µ ⊗ µ), so that Ψ…”

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Integral operator Riccati equations arising in stochastic Volterra control problems

Jaber¹,

Miller²,

Pham³

2019

Preprint

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“…The most prominent example is the rough Heston model as studied in [10,11,12]. We also refer interested readers to [22,26,25,3,4,16] for some recent developments of this model. Extensions to multifactor settings and general Volterra kernels have been studied in [2], where general affine Volterra processes with continuous sample paths have been constructed.…”

mentioning

confidence: 99%

Volterra square-root process: Stationarity and regularity of the law

Friesen¹,

Jin²

2022

Preprint

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The Volterra square-root process on R m + is an affine Volterra process with continuous sample paths. Under a suitable integrability condition on the resolvent of the second kind associated with the Volterra convolution kernel, we establish the existence of limiting distributions. In contrast to the classical square-root diffusion process, here the limiting distributions may depend on the initial state of the process. Our result shows that the non-uniqueness of limiting distributions is closely related to the integrability of the Volterra convolution kernel. Using an extension of the exponential-affine transformation formula we also give the construction of stationary processes associated with the limiting distributions. Finally, we prove that the timemarginals as well as the limiting distributions, when restricted to the interior of the state space R m + , are absolutely continuous with respect to the Lebesgue measure and their densities belong to a weighted Besov space of type B λ 1,∞ .

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Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation

Cited by 19 publications

References 29 publications

Optimal control in linear stochastic advertising models with memory

Optimal control in linear stochastic advertising models with memory

Integral operator Riccati equations arising in stochastic Volterra control problems

Volterra square-root process: Stationarity and regularity of the law

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