23 research outputs found
Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise
We establish weak existence and uniqueness in law for stochastic Volterra
equations (SVEs for short) with completely monotone kernels and non-degenerate
noise under mild regularity assumptions. In particular, our results reveal the
regularization-by-noise effect for SVEs with singular kernels, allowing for
multiplicative noise with H\"{o}lder diffusion coefficients. In order to prove
our results, we reformulate the SVE into an equivalent stochastic evolution
equation (SEE for short) defined on a Gelfand triplet of Hilbert spaces. We
prove weak well-posedness of the SEE using stochastic control arguments, and
then translate it into the original SVE.Comment: 36 page
On the maximum principle for optimal control problems of stochastic Volterra integral equations with delay
In this paper, we prove both necessary and sufficient maximum principles for
infinite horizon discounted control problems of stochastic Volterra integral
equations with finite delay and a convex control domain. The corresponding
adjoint equation is a novel class of infinite horizon anticipated backward
stochastic Volterra integral equations. Our results can be applied to
discounted control problems of stochastic delay differential equations and
fractional stochastic delay differential equations. As an example, we consider
a stochastic linear-quadratic regulator problem for a delayed fractional
system. Based on the maximum principle, we prove the existence and uniqueness
of the optimal control for this concrete example and obtain a new type of
explicit Gaussian state-feedback representation formula for the optimal
control.Comment: 28 page
拡張型後退確率ヴォルテラ積分方程式と時間非整合な再帰的確率制御問題への応用
京都大学新制・課程博士博士(理学)甲第22973号理博第4650号新制||理||1668(附属図書館)京都大学大学院理学研究科数学・数理解析専攻(主査)教授 日野 正訓, 教授 泉 正己, 准教授 矢野 孝次学位規則第4条第1項該当Doctor of ScienceKyoto UniversityDFA
Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations
We introduce a new framework of Markovian lifts of stochastic Volterra
integral equations (SVIEs for short) with completely monotone kernels. We
define the state space of the Markovian lift as a separable Hilbert space which
incorporates the singularity or regularity of the kernel into the definition.
We show that the solution of an SVIE is represented by the solution of a lifted
stochastic evolution equation (SEE for short) defined on the Hilbert space, and
prove the existence, uniqueness and Markov property of the solution of the
lifted SEE. Furthermore, we establish an asymptotic log-Harnack inequality and
some consequent properties for the Markov semigroup associated with the
Markovian lift via the asymptotic coupling method.Comment: 39 page
Linear-quadratic stochastic Volterra controls II: Optimal strategies and Riccati--Volterra equations
In this paper, we study linear-quadratic control problems for stochastic
Volterra integral equations with singular and non-convolution-type
coefficients. The weighting matrices in the cost functional are not assumed to
be non-negative definite. From a new viewpoint, we formulate a framework of
causal feedback strategies. The existence and the uniqueness of a causal
feedback optimal strategy are characterized by means of the corresponding
Riccati--Volterra equation.Comment: 35 page
Linear-quadratic stochastic Volterra controls I: Causal feedback strategies
In this paper, we formulate and investigate the notion of causal feedback
strategies arising in linear-quadratic control problems for stochastic Volterra
integral equations (SVIEs) with singular and non-convolution-type coefficients.
We show that there exists a unique solution, which we call the causal feedback
solution, to the closed-loop system of a controlled SVIE associated with a
causal feedback strategy. Furthermore, introducing two novel equations named a
Type-II extended backward stochastic Volterra integral equation and a
Lyapunov--Volterra equation, we prove a duality principle and a representation
formula for a quadratic functional of controlled SVIEs in the framework of
causal feedback strategies.Comment: 29 page