Abstract
This chapter is devoted to the presentation of the \(L^2\) theory for the existence and uniqueness of mild solutions for a class of second-order infinite-dimensional HJB equations in Hilbert spaces through a perturbation approach.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Hypothesis 4.76.
- 2.
In Sect. 1.6 we define the semigroup directly on all the functions of \(B_b(H)\). The arguments of the present chapter are more transparent if we start by defining the semigroup only on \(C_b(H)\). Since it will be extended (Proposition 5.9) to \(L^p(H,\overline{\mathcal {B}}, m)\), and (Lemma 5.10), for any \(\phi \in L^p(H,\overline{\mathcal {B}}, m)\), \(P_t [\phi ](x) = \mathbb {E} \phi (X(t;x))\), the two approaches are equivalent.
- 3.
For this reason, since we are interested in a definition that also works when the operator \(D_Q\) is non-closable, we do not work in the space \(W^{1,2}(H,{{m}})\) defined (see e.g. Chap. 9, p. 196 of [179]) as the linear space of all functions \(\phi \in L^2(H,\overline{\mathcal {B}},{{m}})\) such that \(D\phi \in L^2(H,\overline{\mathcal {B}},{{m}} ;H)\).
- 4.
In particular, in Propositions 4.61 and 6.10 we work in \(L^p_{\mathcal {P}}(\Omega ; C([0,T], H))\), while here we use \(\mathcal {H}_2^{\mu _0}(0,T;H)\). Indeed, in the mentioned propositions it is assumed that \(\mathrm{Tr}\left[ e^{sA}Q e^{sA^{*}}\right] \le C_\beta s^{-2\beta }\) for some \(\beta \in [0,1/2)\) and \(C_\beta >0\).
- 5.
Following the notation we use for HJB equations throughout the book, in the first line of (5.49) we only explicitly mention the dependence on t and x of the functions \(F_0\) and l while we do not do so for \(u_t\), \(D_Qu\) and \(\mathcal {A}u\).
- 6.
In Chap. 4 and in Appendix B, when the transition semigroup reduces to the Ornstein–Uhlenbeck case, the notation \(R_t\) is used. In this section, and in the proof of Theorem 5.41, we keep the notation \(P_t\) even for the Ornstein–Uhlenbeck case because the semigroup plays exactly the same role, from the perspective of the \(L^2\) approach to the HJB equation, as the semigroup \(P_t\) in Sect. 5.3 and, differently from Chap. 4 and Appendix B, the two semigroups never appear at the same time, so there is no possibility of confusion.
- 7.
For uniqueness results the reader is referred to the review [443] and the references there.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Fabbri, G., Gozzi, F., Święch, A. (2017). Mild Solutions in L2 Spaces. In: Stochastic Optimal Control in Infinite Dimension. Probability Theory and Stochastic Modelling, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-319-53067-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-53067-3_5
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53066-6
Online ISBN: 978-3-319-53067-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)