Nuclear and Hilbert-Schmidt Operators

Appendix A Nuclear and Hilbert-Schmidt Operators

Let (X,·,·X) and (Y,·,·Y ) be separable Hilbert spaces. Deﬁnition A.1 (Nuclear operator) An operator T ∈ L(Y,X) is said to be nuclear if it can be represented by

∞ Tx = ∑ ajbj,xY , forallx ∈ Y, j=1

{ }⊂ { }⊂ ∞ || || ·|| || < where the two sequences aj X and bj Y are such that ∑j=1 aj X bj Y ∞. The space of all nuclear operators from Y to X is denoted by L1(Y,X).

Proposition A.1 The space L1(Y,X) equipped with the norm

∞ ∞ || || = || || ·|| || = , , ∈ T L1(Y,X) : inf ∑ aj X bj Y : Tx ∑ aj bj x Y x Y j=1 j=1 is a Banach space. Proof See Prévôt and Röckner [1, Proposition B.0.2].

Deﬁnition A.2 Let T ∈ L(Y) and let {ek} be an orthonormal basis of Y.Thenwe deﬁne

∞ trT := ∑Te k,ekY k=1 if the series is convergent. This deﬁnition could depend on the choice of the orthonormal basis. However, note the following result concerning nuclear operators.

© Springer International Publishing Switzerland 2016 369 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Inﬁnite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 370 A Nuclear and Hilbert-Schmidt Operators

Remark A.1 If T ∈ L1(Y) then trT is well deﬁned independently of the choice of the orthonormal basis {ek}. Moreover, we have that

| |≤|| || . trT T L1(Y)

Proof See Prévôt and Röckner [1, Remark B.0.4]. +( ) ( ) Deﬁnition A.3 By L1 Y we denote the subspace L1 Y consisting of all symmetric nonnegative nuclear operators. Deﬁnition A.4 (Hilbert-Schmidt operator) A bounded linear operator T : Y → X is called Hilbert-Schmidt if

∞ 2 ∑ ||Te k|| < ∞ k=1 where {ek} is an orthonormal basis of Y. The space of all Hilbert-Schmidt operators from Y to X is denoted by L2(Y,X). Remark A.2 (i) The deﬁnition of the Hilbert-Schmidt operator and the number

∞ ||T||2 := ∑ ||Te ||2 L2(Y,X) k k=1

does not depend on the choice of the orthonormal basis {ek} and that || || = || ∗|| T L2(Y,X) T L2(X,Y). || || ≤|| || (ii) T L(Y,X) T L2(Y,X). Proof See Prévôt and Röckner [1, Remark B.0.6].

Proposition A.2 Let S,T ∈ L2(Y,X) and let {ek} be an orthonormal basis of Y. If we deﬁne

∞ , = , T S L2(Y,X) : ∑ Sek Te k X k=1 ( ( , ),·,· ) we obtain that L2 Y X L2(Y,X) is a separable Hilbert space. If {fk} is a orthonormal basis of X we get that fj ⊗ek := fjek,·Y , j,k = 1,2,3,... is an orthonormal basis of L2(Y,X). Proof See Prévôt and Röckner [1, Proposition B.0.7]. Proposition A.3 (Square root) Let T ∈ L(Y) be a nonnegative and symmetric operator. Then, there exists exactly one element T1/2 ∈ L(Y) that is nonnegative and symmetric such that

T1/2 ◦ T1/2 = T. A Nuclear and Hilbert-Schmidt Operators 371

1/2 1/2 2 1/2 If trT < ∞, then we have that T ∈ L2(Y) where ||T || = trT and L ◦ T ∈ L2(Y) L2(Y,X)for allL ∈ L(Y,X). Proof See Prévôt and Röckner [1, Proposition 2.3.4]. Appendix B Multivalued Maps

In this appendix, we clarify some elementary notions on multivalued maps. For a detailed overview of this topic, we refer to Aubin and Cellina [1]. Let X and Y be two general sets. We denote the power set of Y by 2Y . A multivalued map F from X to Y is a map that associates with any x ∈ X a (non- necessarily nonempty) subset F(x) ⊂ Y. The subset

D(F) := {x ∈ X|F(x) = 0/} is called the domain of F. Unless otherwise noted the domain of F is assumed to be nonempty. We write F : D(F) ⊂ X → 2Y and say F is a multivalued map on X if Y = X. For a multivalued map we can deﬁne the graph by

G(F) := {[x,y] ∈ X × Y|y ∈ F(x)}.

The graph of F provides a convenient characterization of a multivalued map. Conversely, a nonempty set G ⊂ X × Y deﬁnes a multivalued map by

F(x) := {y ∈ Y|[x,y] ∈ G}.

In that case, G is the graph of F. As usual, the range R(F) ⊂ Y is deﬁned by R(F) := F(x). x∈X

Every multivalued map has an inverse F−1. In general, it is again a multivalued map with the domain D(F−1) := R(F) ⊂ Y and

− − F 1(y) := {x ∈ X|y ∈ F(x)}, ∀y ∈ D(F 1).

© Springer International Publishing Switzerland 2016 373 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Inﬁnite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 374 B Multivalued Maps

The map B : D(B) ⊂ X → 2Y is called an extension of A : D(A) ⊂ X → 2Y if G(A) ⊂ G(B). A single-valued f : X → Y is called a selection of the set-valued map F : X → 2Y if for all x ∈ X, f (x) ∈ F(x). We deﬁne a scalar multiplication for a multivalued map by

(cA)(x) := {cv|v ∈ A(x)}, ∀x ∈ D(cA) := D(A), c ∈ R, and the sum of two multivalued maps A and B by

(A + B)(x)={v + w|v ∈ A(x), w ∈ B(x)} for all x ∈ D(A + B) := D(A) ∩ D(B). Appendix C Maximal Monotone Operators

Let X be a Banach space and X∗ its dual space. Let G(A) denote the graph of the operator A. Proposition C.1 Let A be maximal monotone. Then ∗ (i) A is weakly–strongly closed in X × X ,i.e,if[xn,yn] ∈ G(A), xn → x weakly in ∗ X and yn → y strongly in X ,then[x,y] ∈ G(A). (ii) A−1 is maximal monotone in X∗ × X. (iii) For each x ∈ D(A), A(x) is a closed, convex subset of X∗. Proof See Barbu [1, Section 2.1, Proposition 1.1]. A maximal monotone operator is even weakly–weakly closed as the following proposition shows. ∗ Proposition C.2 Let X be a reﬂexive Banach space and let A : X → 2X be a maximal monotone operator. Let [un,vn] ∈ G(A), n ∈ N, be such that un → u weakly, vn → v weakly, and either

limsup X∗ vn − vm,un − umX ≤ 0 n,m→∞ or

limsup X∗ vn,unX ≤ X∗ v,uX. n→∞

Then [u,v] ∈ G(A). Proof See Barbu [2, Lemma 2.3, p.38] and Barbu [2, Corollary 2.4, p.41]. We will make use of the following characterizations of maximal monotonicity.

© Springer International Publishing Switzerland 2016 375 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Inﬁnite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 376 C Maximal Monotone Operators

Theorem C.1 Let X be a reﬂexive Banach space and let A : X → X∗ be a (single- valued) monotone hemicontinuous operator. Then A is maximal monotone in X × X∗. Proof See Barbu [1, Section 2.1, Theorem 1.3]. Theorem C.2 Let X be a reﬂexive Banach space and let A and B be maximal ∗ monotone operators from X to 2X such that

(intD(A)) ∩ D(B) = 0/.

Then A + B is maximal monotone in X × X∗. Proof See Barbu [1, Section 2.1, Theorem 1.5]. Corollary C.1 Let X be a reflexive Banach space, B be a monotone hemicontinuous ∗ operator from X to X∗,andA : X → 2X be a maximal monotone operator. Then A+B is maximal monotone. Proof Apply Theorem C.1 and Theorem C.2. Proposition C.3 Let X be a reflexive Banach space and let A beacoercive, maximal operator from X to X∗.ThenA is surjective, i.e., R(A)=X∗. Proof See Barbu [1, Section 2.1, Theorem 1.2]. We are especially interested in the selection of a maximal monotone operator with respect to its minimal norm: ∗ Definition C.1 The minimal selection A0 : D(A) ⊂ X → 2X of a maximal monotone operator is defined by A0(x) := y ∈ A(x)||y|| = min ||z|| , x ∈ D(A). z∈A(x)

Remark C.1 If X is strictly convex, then A0 is single-valued. 0 0 Proof Let x ∈ D(A). Assume that y1,y2 ∈ A (x), y1 = y2.Deﬁneδ := ||A (x)||.If δ = 0, then y1 = y1 = 0. Thus, δ > 0. By Proposition C.1 iii), A(x) is a closed, convex 1 ( + ) ∈ ( ) || 1 ( + )|| ≥ δ set. Hence 2 y1 y2 A x which implies 2 y1 y2 . On the other hand, fory ˜1 =(1/δ)y1 andy ˜2 =(1/δ)y2,wehave||y˜1|| = ||y˜2|| = 1. Since X is strictly > 1 || + || = 1 || + || convex, it follows that 1 2 y˜1 y˜2 2δ y˜1 y˜2 , which is a contradiction. Hence y1 = y2. Appendix D The Duality Mapping

The duality mapping as a map from X to X∗ represents an important auxiliary tool in the theory of maximal monotone operators on Banach spaces. ∗ Deﬁnition D.1 The duality mapping J : X → 2X is deﬁned by

∗ ∗ ∗ 2 ∗ 2 J(x) := {x ∈ X |X∗ x ,xX = ||x|| = ||x || }∀x ∈ X.

∈ ∗ ∈ ∗ Remark D.1 By the Hahn-Banach Theorem, for every x X there exists x0 X ∗ ∗ ∗ || || = ∗ , = || || = || || such that x0 1andX x0 x X x . Setting u : x x0, it follows that ∗ ∗ , = || ||2 = || |||| || = || ||2 ∈ ( ) ( )= X u x X x x0 x u . Therefore, u J x and, indeed, D J X. The properties of duality mapping are closely related to the convexity of the underlying space. In general, the duality mapping is multivalued. But, we have the following result. Theorem D.1 Let X be a Banach space. If X∗ is strictly convex, then the duality mapping J : X → X∗ is single-valued. Proof See Barbu [1, Chapter 1, Theorem 1.2]. We give now some properties of the duality mapping. Proposition D.1 Let X and X∗ be uniformly convex. Then (i) The duality mapping J : X → X∗ is linearly bounded, 2-coercive, continuous, and odd. (ii) The operator J is bijective and if we identify X∗∗ with X, the inverse operator

J−1 : X∗ → X

is equal to the duality map of the dual space X∗ and single-valued.

© Springer International Publishing Switzerland 2016 377 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Inﬁnite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 378 D The Duality Mapping

(iii) J is strictly monotone, i.e., it is monotone and

X∗ Ju − Jv,u − vX = 0 ⇒ u = v.

Proof See Zeidler [2, Proposition 32.22]. The following fundamental result in the theory of maximal monotone operators due to G. Minty and F. Browder provides a very useful characterization of maximal monotonicity. ∗ Theorem D.2 Let X and X∗ be reﬂexive and strictly convex. Let A : X → 2X be a monotone operator and let J : X → X∗ be the duality mapping of X.ThenA is maximal monotone if and only if, for any γ > 0 (equivalently, for some γ > 0),

R(A + γJ)=X∗

Proof See Barbu [1, Section 2.1, Theorem 1.2]. Corollary D.1 If A is maximal monotone, then μA is maximal monotone for all μ > 0. Proof For a ﬁxed μ > 0, we get Aμ = μA and take x,y ∈ D(A).Forvμ ∈ Aμ (x), there exists v ∈ A(x) such that μv = vμ .SinceA is monotone, for x,y ∈ D(A) and μ μ μ μ μ μ 2 v ∈ Av (x), w = A (y) we have X∗ v − w ,x − yX = μ X∗ v − w,x − yX ≥ 0. By the maximal monotonicity of A and Theorem D.2 with λ := 1, we conclude that R(μA + μJ)=μR(A + J)=X∗. Remark D.2 Let us emphasize that every uniformly convex Banach space is automatically strictly convex and reﬂexive (see Stephan [1, Remark B.8 and Proposition B.9]). Consequently, all results above do also hold for uniformly convex Banach spaces. Appendix E Random Multivalued Operators

Let us introduce the following notion of measurability for multivalued operators from Castaing and Valadier [1]. Deﬁnition E.1 Let (S,S) be a measurable space and (E,E) a polish space. A multivalued operator A : S → 2E is called Effros-measurable if

{x ∈ S|A(x) ∩ G = 0/}∈S for each open set G ⊂ E. Every multivalued Effros-measurable operator A can be characterized as the closure of a countable set of measurable selections, as the next proposition shows. Proposition E.1 Let A be a multivalued operator. Then the following statements are equivalent: (i) The operator A is Effros-measurable. (ii) There exists a sequence {ξn} of measurable functions of A such that

A = {ξn}.

Proof See Castaing and Valadier [1, Chapter III] or Molcanov [1, Theorem 2.3].

In the theory of stochastic differential equations with a time-dependent random drift operator, the question of measurability of the resolvent as well as the Yosida approximation is of particular importance. The following proposition generalizes the proof of measurability of the Yosida approximation in Karamolegos and Kravvaritis [1, Theorem 3.2] to the multivalued case.

© Springer International Publishing Switzerland 2016 379 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Inﬁnite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 380 E Random Multivalued Operators

Proposition E.2 Let (Ω,F, μ) be a complete, σ-ﬁnite measure-space, X be a separable uniformly convex Banach space with its dual X∗ and D ⊂ X.Let ∗ A : Ω × D → 2X be an F ⊗ B(X)/B(X∗)-Effros-measurable, maximal monotone operator. Then, the resolvent Jλ : Ω × X → X and the Yosida approximation ∗ ∗ Aλ : Ω × X → X of A are F ⊗ B(X)/B(X)-measurable and F ⊗ B(X)/B(X )– measurable, respectively. For the proof, we need the following result. Proposition E.3 Let (Ω,F, μ) be a complete measure space, X be a separable Banach space, and F : Ω → X a mapping such that G(F) ∈ F × B(X).ThenF is F/B(X)-measurable. Proof See Himmelberg [1, Theorem 3.4]. Proof of Proposition E.2 Let us write A(ω)(·)=A(ω,·), ω ∈ Ω and ﬁx x ∈ X.By Lemma 2.4, we obtain ∗ −1 −1 −1 G(Aλ (·,x)) = (ω,y) ∈ Ω × X |y =(A(ω) + λJ ) x = (ω,y) ∈ Ω × X∗|x ∈ A(ω)−1y + λJ−1y = (ω,y) ∈ Ω × X∗|(x − λJ−1y) ∈ A(ω)−1y = (ω,y) ∈ Ω × X∗|y ∈ A(ω)(x − λJ−1y) = (ω,y) ∈ Ω × X∗|0 ∈ A(ω)(x − λJ−1y) − y .

Since X is reﬂexive, J−1 is the duality mapping from X∗ to X. Since then, J−1 is demicontinuous and X is separable, Pettis Theorem implies that J−1 is B(X∗)/B(X)-measurable. Consequently, the mapping y → x − λJ−1y is B(X∗)/B(X)-measurable. Hence, the mapping (ω,y) → (ω,x − λJ−1y) is F ⊗ B(X∗)/B(X)-measurable. Composing this and A, it follows that (ω,y) → A(ω)(ω,x−λJ−1y−y)−y is F⊗B(X)/B(X)-Effros-measurable. By the deﬁnition ∗ of Effros-measurability we obtain G(Aλ (·,x)) ∈ F ⊗ B(X ). Now Proposition E.3 ∗ implies that Aλ (·,x) is F/B(X )-measurable. Demicontinuity of Aλ in x yields that ∗ Aλ is F ⊗B(X)/B(X )-measurable. The second assertion follows directly by noting −1 that Jλ (ω,x)=x − λJ (Aλ (ω,x)) for (ω,x) ∈ Ω × X. E Random Multivalued Operators 381

Random Inclusions

Let X be a separable Banach space. The existence of solutions y(x) for inclusions of the form

y(x) ∈ A(x), x ∈ X, (E-1) where A is a multivalued operator, has been extensively studied for the deterministic case (see Proposition C.3). However, it is a nontrivial generalization to consider inclusions of type (E-1), where the multivalued operator does not depend on an additional variable ω in a measurable space Ω. The solution x of

y(ω,x(ω)) ∈ A(ω,x(ω)), x ∈ X, ω ∈ Ω, does not necessarily need to be measurable even if there exists an ω-wise solution (see, e.g., Bharucha-Reid [1, Chapter 3], Hans [2], Itoh [1], Kravvaritis [1]). The following result generalizes Kravvaritis [1, Theorem 3.2] by dropping the lower semi-continuity assumption. Proposition E.4 Let (Ω,F, μ) be a complete measure space, X be a separable ∗ uniformly convex Banach space and D ⊂ X.LetA : Ω×D → 2X be an operator such that A(ω,·) is maximal monotone for every ω ∈ Ω, A(·,x) is F-Effros-measurable for every x ∈ D and 0 ∈ A(ω,0) for all ω ∈ Ω.LetL : Ω × X → X∗ be a single- valued bounded, coercive, and maximal monotone operator such that L(·,x) is F - Effros-measurable for every x ∈ X. Then for each F-measurable, bounded operator y(·) ∈ X∗, there exists an F- measurable, bounded operator x(·) ∈ X, such that

y(ω) ∈ A(ω,x(ω)) + L(ω,x(ω)) ∀ω ∈ Ω.

Proof W.l.g. we assume y(ω)=0forallω ∈ Ω. (Otherwise, consider A˜ (ω,·) := A(ω,·)−y(ω).Again,A˜ is maximal monotone and Effros-measurable.) We consider the equation

Aλ (ω,xλ (ω)) + L(ω,xλ (ω)) = 0, ∀ω ∈ Ω, (E-2) where Aλ is the Yosida approximation of A. By Proposition E.2, Aλ is F-measurable. Since Aλ is demicontinuous and maximal monotone, the operator Aλ + L satisﬁes the assumptions of Itoh [1, Theorem 6.2]. Hence, there exists an F-measurable bounded operator xλ (·) ∈ X that solves equation (E-2). The rest of the proof is analogous to Kravvaritis [1, Theorem 3.2]. Remark E.1 Note that in the proof of Kravvaritis [1, Theorem 3.2], the operator A isassumedtobel.s.c. to be able to prove the measurability of the Yosida approximation. However, the lower-semicontinuity of A is obsolete, as Proposition E.2 shows. Bibliographical Notes and Remarks

Stochastic evolution equations in inﬁnite dimensional spaces have been motivated principally by the study of partial differential equations in one-dimensional spaces. In other words, they are abstract formulations of concrete problems such as heat and wave equations, see Pazy [1], Da Prato and Zabczyk [3] and Gawarecki and Mandrekar [1]; reaction diffusion equations, random motion of a string, equation of population genetics and neurophysiology, see Da Prato and Zabczyk [1], to mention only a few. Further examples are provided in Chapter 1. We also refer to Allen [1], Curtain and Pritchard [1], and McKibben [1, 2].

Chapter 1 The linear stochastic heat equation (1.1) is picked from Ichikawa [2], its abstract formulation (1.3), as well as its explicit solution. The semilinear stochastic heat equation (1.2) is from Ichikawa [3] and also its abstract representation given in equation (1.4). We refer to Bao, Mao, and Yuan [1] for a study on hybrid stochastic heat equations. The electric circuit problem given in Section 1.2 is originally from Chukwu [1] wherein a ﬁgure of such a circuit is also available. See also Govindan [8] for a study of such circuits and for the consideration of abstract equation (1.9). In Section 1.3, the interacting particle system (1.10) is taken from Ahmed and Ding [1] that motivates the study of McKean-Vlasov equation (1.11). See also Kurtz and Xiong [1]. We refer to McKean [1] and Gärtner [1] for the so-called McKean-Vlasov theory. The problem of lumped control system (1.12) considered in Section 1.4 is basically from Hernández and Henriquez [1]. The uncontrolled neutral stochastic partial functional differential equation (1.14) is from Govindan [1]. The hyperbolic equation (1.15) and the corresponding Volterra integrodifferential equation (1.16) given in Section 1.5 are from Miller and Wheeler [1]. The study on stochastic linear integrodifferential equation (1.17) was initiated in Kannan and Bharucha- Reid [1]. This study was later generalized to semilinear stochastic integrodifferential equations (1.18) in Govindan [3]. The stock and the option price dynamics given in Section 1.6 was originally proposed by Merton [1] and the stochastic evolution equation with Poisson jumps considered in equation (1.21) is from Luo and Liu

© Springer International Publishing Switzerland 2016 383 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Inﬁnite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 384 Bibliographical Notes and Remarks

[1] and Yuan and Bao [1]. We refer to Cont and Tandov [1], Dufﬁe, Pan, and Singleton [1], Gemam and Roncoroni [1], Kou [1], and Platen and Bruti-Liberati [1] for modeling with jump processes.

Chapter 2 This chapter provides the necessary mathematical background to read the book and also includes some fundamental results for the sake of completeness. However, Section 2.2 is an exception to this. It begins with the basics from the semigroup theory in Section 2.1 including the fundamental Hille-Yosida theorem given in Theorem 2.4, Yosida approximations and their consequences. The whole material including Section 2.1.1 is taken from Pazy [1]andAhmed[1]. See also the classical references Hille [1] and Hille and Phillips [1]. Section 2.1.2 is concerned with Yosida approximations of maximal monotone operators which is essentially picked from Barbu [1] and Stephan [1]. Section 2.2 is really very interesting as it presents a relation, in some sense, between the Yosida approximations and the central limit theorem. The material from the whole section, particularly, Section 2.2.1 on optimal convergence rate for Yosida approximations followed by asymptotic expansions for Yosida approximations in Section 2.2.2 are taken from Vilkiene [2, 3]. See also Vilkiene [1, 4]. This is presented for an interested reader. This section can be skipped without losing continuity from reading the book further. Section 2.3 on the theory of evolution operators is taken essentially from Curtain and Pritchard [1], Tudor [1], and Govindan [2, 8]. In the next Section 2.4, we give a brief background on analysis and probability including stochastic processes from various excellent texts, papers, and recent doctoral theses, namely, Arnold [1], Bharucha-Reid [1], Joshi and Bose [1], Joshi and Govindan [1], Da Prato and Zabczyk [1], Dunford and Schwartz [1], Ichikawa [3], Ikeda and Watanabe [1], Khasminskii [1], Prévôt and Röckner [1], Stephan [1], Knoche [1], Liu and Stephan [1], Barbu [1, 2], Peszat and Zabczyk [1], Protter [1], Albeverio and Rüdiger [1], Bichteler [1], Yosida [2], and Hans [1], among others. Section 2.5 on stochastic calculus is standard. Section 2.5.1 on Itô stochastic integral with respect to a Q-Wiener process is taken from Da Prato and Zabczyk [1]. However, the properties given in Propositions 2.14 and 2.15 are from Ichikawa [3]. We refer to Funaki [1] as well. Section 2.5.2 on Itô stochastic integral with respect to a cylindrical Wiener process is picked from Prévôt and Röckner [1]. In Section 2.5.3, we discuss stochastic integral with respect to a compensated Poisson measure and is taken from the interesting doctoral thesis of Knoche [1], Albeverio and Rüdiger [1], Liu and Stephan [1], Peszat and Zabczyk [1] and Stephan [1]. We also refer to the recent book by Mandrekar and Rüdiger [1]. The next Section 2.5.4 on Itô’s formulas in various settings are picked from Da Prato and Zabczyk [1], Ichikawa [3], Gawarecki and Mandrekar [1], Mao and Yuan [1], and Peszat and Zabczyk [1]. We refer to Da Prato, Jentzen, and Röckner [1] for a mild Itô formula. It is also interesting to mention here a recent book by Ahmed [8] which is mainly concerned with the distribution theory on a Wiener space and provides solution to a much larger class of random processes than those covered by Itô processes. Bibliographical Notes and Remarks 385

In Section 2.6, material on stochastic Fubini theorem is picked from Da Prato and Zabczyk [1], Luo and Liu [1], and Ichikawa [3]. Section 2.7 on stochastic convolution integrals is taken from Da Prato and Zabczyk [1, 2]. The Burkholder type inequalities given in Section 2.8 are from Tubaro [1], Luo and Liu [1], Brez- niak, Hausenblas, and Zhu [1], Marinelli, Prévôt, and Röckner [1], and Kallenberg [1]. See also Liu [1] and Stephan [1]. Lastly, Section 2.9 is from Altman [1], Padgett [1], Padgett and Rao [1], and Govindan [2, 9]. We also refer to Kuo [1].

Chapter 3 The theory on linear stochastic evolution equations from Section 3.1 is taken from Ichikawa [2] but for the Itô’s formula given in Theorem 3.1 which is deduced from Ichikawa [3]. To the best of our knowledge, this is the ﬁrst work on stochastic differential equations wherein the Yosida approximations have been introduced to study an optimal control problem. This was the beginning of using such approximations in the study on stochastic evolution equations in inﬁnite dimensions for various applications. In the next Section 3.2, we present the pioneering work again by Ichikawa [3] on semilinear stochastic evolution equation extending his own earlier study given in Section 3.1. We introduce the concepts of strong and mild solutions and also show that a strong solution is always a mild solution. This forms the basis for the rest of the chapter as well as for the book. Yosida approximations are introduced for this class of semilinear equations with an aim to study exponential stability of mild solutions. It is interesting to point out that Theorem 3.4 (see Ichikawa [3]) plays a pivotal role in the stability analysis in Chapter 5 and the introduction of Yosida approximations. Theorem 3.5 is motivated by the Hille-Yosida theorem. See also Gawarecki and Mandrekar [1, pp. 212– 213]. The classical work of Ichikawa [2, 3] on Yosida approximations of stochastic differential equations given in Sections 3.1 and 3.2 has also been dealt with by Liu [2]. See also Gawarecki and Mandrekar [1] and McKibben [2] for the work of Ichikawa [3]. We refer to Cox [1] for space approximations of semilinear stochastic evolution equations using Yosida approximations. The study presented in Section 3.2 is generalized to semilinear stochastic evolution equations with a constant delay in Section 3.3.1 (see Govindan and Ahmed [1]). In particular, Yosida approximations are introduced for the delay system under consideration which will be crucial later on in Chapter 5. The results presented in Section 3.3.1 are natural generalizations of the classical work of Ichikawa [3] presented in Section 3.2. See Liu [2, Section 4.4]. Section 3.3.2 studies strong solutions of stochastic evolution equations with a variable delay and the details are picked from Caraballo and Real [1]. We refer to Bao, Truman, and Yuan [2], Govindan [4, 7], and Liu [2] for a related study. This subsection uses variational approach to establish strong solutions. The main results, namely, Theorems 3.7 and 3.8, shall be useful later in the next Section 3.3.3 which deals with a more general semilinear stochastic evolution equation with a variable delay studied in Liu [1]. Here, mild solutions of such delay systems are studied. Yosida approximating system is also introduced to such stochastic equations and is shown to have strong solutions in Theorem 3.10 using arguments from Theorem 3.8. Note that 386 Bibliographical Notes and Remarks

Theorem 3.10 and the other results are picked from Liu [1]. For another interesting Carathéodory approximation procedure of mild solutions of the equation considered in Section 3.3.3, see Liu [3]. In Section 3.4, McKean-Vlasov stochastic evolution equations are studied. Section 3.4.1 considers such equations with an additive noise. The results presented here are picked from Ahmed and Ding [1]. These results are then generalized to such stochastic equations with a multiplicative diffusion in Govindan and Ahmed [3]. All the results given in Section 3.4.2 are picked from this paper including the Itô’s formula given in Lemma 3.2 and the Corollary 3.2. Section 3.5 presents an interesting class of neutral stochastic partial functional differential equations introduced in Govindan [1]. See Luo [1] for a subsequent work. The results presented here in this section are taken from Govindan [5]. This class of equations, for instance, includes the equation studied earlier in Section 3.3.1 as a particular case. We refer to Govindan [6] for some generalization and also to Wu [1] for an extensive study on deterministic partial functional differential equations. In Section 3.6, we consider another class of equations, namely, stochastic integrodifferential equations. In Section 3.6.1, we first study linear stochastic integrodifferential equations. All the results presented here are taken from Kannan and Bharucha-Reid [1]. This study was later generalized to semilinear stochastic integrodifferential equations by Govindan [3]. Some of the results obtained in this paper are given in Section 3.6.2. In Section 3.7, we deal with yet another important class of equations called multivalued stochastic partial differential equations driven by Wiener processes. The goal here is to study the existence and uniqueness of a solution to such equations. To do so, Yosida approximations play a key role. The main result of this section is given in Theorem 3.24. The whole work is picked from the interesting doctoral thesis of Stephan [1] which is exclusively dedicated to Yosida approximations of multivalued stochastic partial differential equations. We also refer to Pettersson [1] for some earlier study. All the stochastic equations studied so far dealt with only time-invariant systems, that is, the operator A was time-invariant. In Section 3.8, we consider semilinear stochastic evolution equations wherein the linear operator A(t) is now time-varying which is assumed to generate an almost strong evolution operator U(t,s). The results presented here are from the doctoral thesis of the author (see Govindan [9]). This part of the thesis was later published in Govindan [2]. The results presented here, in some sense, generalize the classical work of Ichikawa in Section 3.2. One of the main results, Theorem 3.29, shows the existence and uniqueness of a strong solution to Yosida approximating system introduced to such stochastic equations. An interested reader can refer to Hu [1] for a study on forward-backward SDEs with monotone and continuous coefficients in finite dimensions. Section 3.9 deals with stochastic evolution equations that do not possess the conventional mild or strong solutions. This motivates the introduction of a relaxed or a generalized solution and its existence and uniqueness are shown in Theorem 3.30. Note that Yosida approximations play a vital role in the proof of this existence result. This result is then applied to a forward Kolmogorov equation in Corollary 3.6. Bibliographical Notes and Remarks 387

Both these results are picked from Ahmed [6]. This section is then generalized in Section 3.10 wherein stochastic evolution equations are now driven by stochastic vector measures. The results presented in this section are originally obtained by Ahmed [5]. The rest of the chapter is devoted to controlled stochastic differential equations. In Section 3.11.1, measure-valued McKean-vlasov stochastic evolution equations are studied. This class of equations is related to McKean-Vlasov equations studied earlier in Section 3.4. All the results presented here are taken from Ahmed [3]. The main result presented in Theorem 3.33 employs Lemma 3.6 whose proof uses Yosida approximations. The last Subsection 3.11.2 deals with stochastic evolution equations with partially observed relaxed controls. The existence and uniqueness result Theorem 3.34, the continuity of the solution with respect to the control given in Theorem 3.35, and the existence of an optimal control in Theorem 3.36 are all picked from Ahmed [4]. The crucial Lemma 3.8 that uses Yosida approximations to develop necessary conditions of optimality later on in Chapter 6 is also from Ahmed [4].

Chapter 4 In this chapter we consider Yosida approximations of stochastic differential equations with Poisson jumps. To begin with, Section 4.1 deals with stochastic partial differential delay equations with Poisson jumps. The notions of strong and mild solutions are first introduced for such stochastic equations. Theorem 4.1 is the first main result of the section and it proves the existence and uniqueness of a mild solution. Yosida approximations are then introduced for such systems and it is shown in Theorem 4.2 that Yosida approximating system has a strong solution. This latter result is important to prove stability in distribution in Chapter 5. The results presented here are drawn from Bao, Truman, and Yuan [1]. In Section 4.2 we present some results from Luo and Liu [1] on stochastic functional differential equations with Markovian switchings driven by Lévy martingales. The concepts of strong and mild solutions are introduced for this stochastic equation. It is then shown in Proposition 4.3 that a mild solution together with some additional assumptions is also a strong solution. Theorem 4.3 yields the existence and uniqueness of a mild solution of this class of equations. Yosida approximations are then introduced for such equations and it is shown in Theorem 4.4 that the Yosida approximating system has a strong solution which will be crucial later on in Chapter 5 on stability analysis. Section 4.3 considers switchings diffusion processes with Poisson jumps. The equation is of the type considered earlier in Section 4.2 without delay. But, now the equation has time-varying coefficients. The first main result on existence and uniqueness of a mild solution is stated and proved in Theorem 4.5. Yosida approximating system is then introduced to this equation and its existence of a strong solution is established in Theorem 4.6. Both these results are picked from Yuan and Bao [1]. Theorem 4.6 is important to study exponential stability in Chapter 5. Lastly, in Section 4.4, we consider multivalued stochastic partial differential equations driven by Poisson jumps. Section 4.4.1 considers such equations with Poisson noise. The main result of this section is stated in Theorem 4.7 which 388 Bibliographical Notes and Remarks proves the existence and uniqueness of a solution. Yosida approximations are then introduced followed by an Itô’s formula in Theorem 4.9 without proof. The results considered here are all picked from Liu and Stephan [1]. Section 4.4.2 presents an application to stochastic porous media equation which is again picked from Liu and Stephan [1]. In Section 4.4.3, multivalued stochastic equations driven by Poisson noise with a general drift term is introduced briefly from Stephan [1]. The details can be found there.

Chapter 5 This chapter studies stochastic stability as applications of Chapters 3 and 4. We begin with classical work of Ichikawa (1982). In Section 5.1.1, we consider exponential stability of moments in Theorem 5.1 and Corollary 5.1 of mild solutions of semilinear stochastic evolution equations; and the continuity of sample paths of mild solutions is established in Theorem 5.2 in Section 5.1.2. Using these, almost sure exponential stability of sample paths is obtained in Theorem 5.3 in Section 5.1.3. All these sections are taken from Ichikawa [3]. In Theorem 5.4 stability in distribution of mild solutions of semilinear stochastic evolution equations is proved in Section 5.1.4. This subsection is picked from Bao, Hou, and Yuan [1]. We refer to Ichikawa [4] for a related study. Section 5.2 deals with exponential stabilizability problems of semilinear stochastic evolution equations. In Theorem 5.5, it is proved that quadratic moments of mild solutions is exponentially stabilizable with a constant decay in Section 5.2.1. This section is picked from Ahmed [1, Section 7.2]. We also refer to Taniguchi [2] for a related study. In the next subsection, Section 5.2.2, the classical work of Ichikawa [3] is slightly generalized with time-varying coefﬁcients. Theorem 5.6 establishes the existence and uniqueness of a mild solution of such stochastic evolution equations. Yosida approximations are then introduced for such equations and it is shown in Theorem 5.7 that the Yosida approximating system has a strong solution. These results as well as Theorems 5.8 and 5.9 are taken from Govindan and Ahmed [2]. See also Liu [2]. The motivating Example 5.5, Deﬁnition 5.4, and Theorem 5.10 including the Example 5.6 are picked from Liu [2]. The main result Theorem 5.11 followed by the Example 5.7 and 5.8 are again from Govindan and Ahmed [2]. We also refer to McKibben [2]. All the results on polynomial stability of mild solutions of semilinear stochastic evolution equations with a variable delay including the interesting examples from Section 5.3.1 are taken from Liu [1]. The Section 5.3.2 on stability in distribution of mild solutions of stochastic partial differential delay equations with Poisson jumps is from Bao, Truman, and Yuan [1] but for the Itô’s formula given in Theorem 5.14 which is taken from Luo and Liu [1]. Theorem 5.16 of the Razumikhin type in Section 5.4 is stated from Liu [2, Theorem 4.4.1]. At this stage, it is important to mention Razumikhin [1, 2] for a pioneering contribution to deterministic differential equations with a delay. The main result on exponential state feedback exponential stabilizability of mild solutions of stochastic evolution equations with a delay given in Theorem 5.17 and the Example 5.13 are from Govindan and Ahmed [1]. In Section 5.5.1, Theorem 5.18 on weak convergence of induced probability measures of mild solutions of McKean-Vlasov equation is from Ahmed and Ding Bibliographical Notes and Remarks 389

[1]. The whole Section 5.5.2 is picked from Govindan and Ahmed [3]. Theorem 5.21 from Section 5.6 on weak convergence of induced probability measures of Yosida approximating mild solutions of neutral stochastic partial differential equations is taken from Govindan [5]. In Section 5.7, the exponential stability of mild solutions of stochastic integrodifferential equations given in Theorem 5.22 is taken from Govindan [3] and also the Example 5.16. Note that Proposition 5.2 on weak convergence of induced probability measures is from Kannan and Bharucha-Reid [1] while Proposition 5.3 is taken from Govindan [3]. Section 5.8 is on exponential stability of mild solutions of stochastic evolution equations with Markovian switchings driven by Lévy martingales. The whole Section 5.8.1 is taken from Luo and Liu [1]. The only main result from Section 5.8.2 is from Yuan and Bao [1]. For more details on Remark 5.8, see Svishchuk and Kazmerchuk [1]. Lastly, Section 5.9 on exponential stability of mild solutions of time-varying stochastic evolution equations is from Govindan [2, 9].

Chapter 6 The last chapter is on applications of Yosida approximations from Chapter 3 to stochastic optimal control problems. In Section 6.1, we consider a regulator problem, in other words, optimal control over a finite time horizon for a linear stochastic evolution equation. Using the optimality Lemma 6.1 that exploits a feedback control law, the main optimality result is given in Theorem 6.1. Both these results are taken from Ichikawa [2]. We refer to Borkar and Govindan [1] for a study on optimal control problem for semilinear stochastic evolution equations. In the next Section 6.2, a periodic control problem governed by a time- varying stochastic evolution equation is considered. The first main result, namely, Theorem 6.2 is from Da Prato [1] that yields a unique θ-periodic solution of equation (6.21) and also yields an optimal control that minimizes the cost functional (6.23) subject to a deterministic evolution equation (6.24). Next, Theorem 6.3 gives sufficient conditions under which the Riccati equation (6.20) has a unique θ- periodic solution and that the evolution operator UL(Q) is exponentially stable. Under the hypothesis of Theorem 6.3 together with some additional assumptions, the main result of the section on optimal periodic control is given in Theorem 6.4. The whole section is picked from Tudor [1]. Two optimal control problems for measure-valued McKean-Vlasov evolution equations are presented in Section 6.3. The first problem proposed is a terminal control problem. This problem is resolved in Theorem 6.6. Problem 2 is on maximizing a cost functional. A solution to this problem is obtained in Theorem 6.7. An interesting Example 6.2 on mobile communication is then given. This material is taken from Ahmed [3]. Next, in Section 6.4, a Bolza problem is considered. The necessary conditions for optimal relaxed controls and the corresponding solution are presented in Theorem 6.8. This result is picked from Ahmed [4]. Lastly, in Section 6.5, many optimal feedback control problems associated with stochastic evolution equation driven by stochastic vector measures are considered. The first problem posed is the Lagrange problem. Since the equation has no pathwise solution, it has been reformulated in terms of measure solutions. Using the result 390 Bibliographical Notes and Remarks

Lemma 6.4 on the continuous dependence of the solution on the control, the control problem is taken up in Theorem 6.9. Then a Lagrange Problem 2 is formulated and is then solved in Theorem 6.10. Another interesting problem is a maximization Problem 3. This problem is resolved in Theorem 6.11. A fourth interesting problem is the minimization or maximization of a suitable cost functional. This problem is shown to have a solution in Theorem 6.12. As special cases, two more problems are considered. The ﬁfth problem is closely related to Problem 4 considered in Theorem 6.12. This problem involves minimizing a cost functional which is resolved in Theorem 6.13. Another interesting control problem is maximizing a cost functional which is equivalent to a tracking a diffuse moving target. This problem is shown to have a solution in Theorem 6.14. All these results are taken from Ahmed [5].

Appendices Appendix A on nuclear and Hilbert-Schmidt operators is picked from Da Prato and Zabczyk [1] and Prévôt and Röckner [1]. Some elementary notions on multivalued maps in Appendix B is taken from Aubin and Cellina [1]. Appendix C deals with the basics from maximal monotone operators which is picked from Barbu [1, 2]. Duality mapping is discussed in Appendix D which is taken from Barbu [1] and Zeidler [2]. Lastly, basic theory on random multivalued operators and random inclusions is given in Appendix E and is picked from Castaing and Valadier [1], Karamolegos and Kravvaritis [1], Molcanov [1], Himmelberg [1], Bharucha-Reid [1], Kravvaritis [1], and Itoh [1]. We also refer to Stephan [1] as a quick reference. Bibliography

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Symbols Burkholder type inequality, 62, 90, 95, 152, C0-semigroup, 12 216 differentiable semigroup, 27 for a Poisson integral, 63 exponentially stable, 13, 98, 185 of contractions, 13 pseudo-contraction, 13, 204, 214, 219, 317 C uniformly bounded, 13 càdlàg, 36 Cauchy, 91, 117, 255, 294 Cauchy’s formula, 33, 118 A central limit theorem, 21 a version, 35 Chapman-Kolmogorov equation, 249 adapted process, 35 Chebyshev’s inequality, 251, 289, 341 admissible control, 334, 350 closed, 17 analytic semigroup, 6 weakly closed, 350 Ascoli-Arzella theorem, 367 closed convex hull, 19 asymptotic expansion, 21, 29 coercive, 20, 87, 137, 222, 238 asymptotic expansions, 27 compact set, 170 attainable set, 350 compact support, 167, 179 control law, 337 converges strongly, 33 B converges weakly, 33 Banach fixed point theorem, 100, 108 counting Poisson random measure, 10 Banach-Alaoglu theorem, 141, 164 covariance, 34 Bellman-Gronwall lemma, 32, 95, 119, 128, joint covariance, 35 134, 149, 159, 188, 227, 233 Bochner integrable, 34 Bolza problem, 356 D Borel-Cantelli Lemma, 119, 248, 275, 341 Davis’s inequality, 226 bounded, 137, 222 demicontinuous, 17, 19, 381 bounded holomorphic semigroup, 22 dense, 184 bounded linear operators, 11, 15, 44 detectability, 338 bounded stochastic integral contractor, 67, 154 Dhumels formula, 195 is regular, 156 differentiable, 22 Burgers type equation, 282 Dirichlet boundary condition, 263, 269, 354

© Springer International Publishing Switzerland 2016 403 T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 404 Index dissipative, 187 Gelfand triple, 45, 135 duality mapping, 17, 377 graph of the operator A,17 Dunford-Pettis theorem, 160 dynamic programming method, 334 H Hölder’s inequality, 61, 62, 115, 148, 317 E Hahn-Banach theorem, 348 electric circuit, 3 Hausdorff space, 161 empirical measure, 5 heat equation, 1 empirical measure-valued process, 5 controlled stochastic heat equation, 260, ergodic property of Markov chains, 327 298 Euler beam equation, 303 stochastic heat equation, 2, 70, 248, 310 Euler’s approximations, 21 hemicontinuous, 32 evolution operator hereditary control, 6 almost strong, 31, 152 Hilbert-Schmidt operators, 46, 135, 170, 370 mild, 30 Hille-Yosida Theorem, 13, 16, 25, 82 quasi, 31 hyperbolic equation, 7 strong, 31, 338 expectation, 34 conditional, 35, 177 I exponential martingale inequality with jumps, improper Riemann integral, 14 324 increasing process, 42 exponential stability independence, 35 of moments of SEEs, 242 infinitesimal generator of the semigroup, 12 of moments of stochastic integral operator, 153 integrodifferentiable equations, integro-differential identity, 23, 25 309 interacting particle system , 5 of moments of time-varying SEEs, 331 isometric property, 229 of sample paths of McKean-Vlasov isometrically isomorphic, 185 equations, 302 Itô stochastic integral, 45 of sample paths of SEEs, 249 w.r.t. a Q-Wiener process , 46 of sample paths of SEEs with Markovian w.r.t. a cylindrical Wiener process, 50 switching driven by Lévy martingales, Itô’s formula, 54, 71, 106, 140, 224, 242, 273, 316, 322 309, 315, 335 exponential stabilizability for a Q-Wiener process, 54 of moments of SEEs, 259 for a compensated Poisson process, 56 of moments of SEEs with delay, 298 for a cylindrical Wiener process, 55 iterations, 109, 114 F factorization method, 60 J Fatou’s lemma, 266 Jordan decomposition, 98 feedback control, 6, 182, 259, 334 PID feedback control, 6 filtration, 39, 42 K right-continuous filtration, 42 Kirchhoff’s law, 3 forward Kolmogorov equation, 167, 179 Fréchet derivative, 32, 65, 106, 162, 242, 309, 334 L Fubini’s theorem, 142, 209, 364 Lévy martingales, 211 Lévy process, 43 Lévy-Itô decomposition, 44 G Lévy-Khinchine formula, 43 Gateaux derivative, 357 Lagrange problem, 361 Gaussian law, 37 law of large numbers, 345 Index 405

Lebesgue dominated convergence theorem, 81, multivalued map, 373 101, 128, 134, 145, 159, 217, 243, 310 multivalued operator, 17 Levy’s Theorem, 37 linear growth condition, 93, 100, 108, 113, 204, 222, 241 N Lipschitz condition, 67, 92, 93, 101, 108, 113, necessary conditions of optimality, 356 136, 204, 222, 238, 241 Newton-Leibnitz formula, 24 one-sided, 137 non-decreasing process, 36, 237 local martingale property, 49 nuclear operator, 34, 369 lower semicontinuous, 200, 353 lumped control systems, 6 Lyapunov exponent, 263, 271, 313 O Lyapunov function, 266 obstacle avoidance problem, 365 Lyapunov functional, 281 optimal control, 345, 350, 353, 354, 362, 364, 365 optimal convergence rate, 26 M optimal error bound, 21 Markov chain, 56 optimality lemma, 334 irreducible, 218 option price dynamics, 9 Markov inequality, 341 Ornstein-Uhlenbeck process, 185 Markov property, 293, 339 Ornstein-Uhlenbeck semigroup, 183, 184 Markovian switching, 211 orthonormal basis, 40, 105, 369 martingale, 36 semimartingale, 357 submartingale, 36 Mazur theorem, 142 P McKean-Vlasov theory, 5 Poincaré’s inequality, 328 measurable, 35 point function, 41 Borel, 83, 113, 193 Poisson integral, 53 Effros, 379 Poisson jumps, 203, 295 progressively, 36, 136, 142, 146, 223 Poisson noise, 221 progressively Effros, 136, 237 Poisson point process, 41 strongly, 114 Poisson process, 9 measurable selection, 379 Poisson random measure, 40 measurable space, 33 compensated Poisson random measure, 10, measure 42 characteristic, 10, 42 Polish space, 379 Dirac delta, 124, 164, 167 polynomial decay, 271 Dirac delta measure, 175 polynomial nonlinearities, 162 intensity, 43 predictable process, 47, 60 invariant, 183, 337 probability distribution, 5, 97, 108 Lévy, 317 probability measure, 33 Lebesgue, 42, 58, 197 complete, 33 stochastic vector, 171 probability measure space, 33 metric, 98, 110 Fréchet metric, 123 Hausdorff metric, 366 Q Prohorov metric, 366 quasi-generator, 31 minimal selection, 145 quasi-leftcontinuous, 42 mobile communication, 354 monotone, 17 maximal monotone, 17, 19, 136, 221, 238, R 375 Radon-Nikodym derivatives, 238 strictly, 378 Radon-Nikodym property, 160 406 Index

Radon-Nikodym theorem, 238 ofSEEswithdelaydrivenbyPoisson random contractor, 64 jumps, 294 random evolution equation, 124 state dependent noise, 69 random inclusion, 381 stochastic convolution integrals, 59 random multivalued operator, 379 stochastic differential equations random operator, 44 multivalued SDEs, 20 random variable, 33 stochastic differential equations with Poisson Bochner, 33 jumps, 10 Pettis, 34 stochastic evolution equation, 2 Razumikhin type theorem, 296 stochastic evolution equations with delay, 4 Razumikhin-Lyapunov function, 321 stochastic evolution equations with variable regularity, 194 delay, 92 regulator problem, 335 stochastic Fubini theorem, 58, 60, 209 relaxed control, 197 for Poisson integral, 58 resolvent of A,13 stochastic integrodifferential equation, 8 of a maximal monotone operator A,18 stochastic partial differential equations, 4 resolvent set of A,13 neutral stochastic partial differential Riccati equation, 335, 341 equation, 6 Riesz isomorphism, 234 stochastic porous media equations, 233 Riesz theorem, 362 stochastic process, 35 robustness in stability stock price dynamics, 9 with a constant decay, 258 stopping times, 44, 225, 228, 254, 291 with a general decay, 263 strongly continuous semigroup, 12 subnet, 164 switching diffusion processes, 218 S sample path continuity, 243 a modiﬁcation, 245 T sample path stability target set, 365 with a general decay, 272 Taylor series, 23, 326 Skorokold extension, 180 terminal control problem, 352 Sobolev embedding, 234 theory of lifting, 160 Sobolev space, 234 tight, 251, 308 solution topological compactiﬁcation, 161 classical, 170 of Stone-Cech, 161 generalized, 162, 173 trace, 35 mild, 70, 75, 83, 93, 99, 106, 108, 113, 129, transition semigroup, 183 153, 184, 205, 213, 219, 242, 261, 333 Tychonoff space, 161 of a multivalued equation with Poisson noise, 222 of a multivalued equation with white noise, U 136 uniformly continuous semigroup, 12, 15 of multivalued equation driven by Poisson uniformly continuous semigroup of noise with a general drift term, 237 contractions, 15, 22 periodic solution, 341 uniformly convex, 17 strong, 69, 74, 86, 93, 94, 99, 104, 105, 129, upper semicontinuous, 353, 364 135, 153, 157, 204, 213 weak, 170, 186 square bracket, 53 V square root, 370 variance, 34 stabilizability, 338 variational method, 86 stable in distribution, 249 Volterra integrodifferential equation, 7 of SEEs, 256 Volterra series, 68 Index 407

W of multivalued SPDEs driven by Poisson wave equation, 269 noise, 223 weak convergence of induced probability of multivalued SPDEs driven by Poisson measures noise with a general drift term, 239 of McKean-Vlasov equations, 301 of neutral SPDEs, 119 of neutral SPDEs, 305 of SDEs with delay with Markovian weak limit, 145 switching driven by Lévy processes, weakly compact subset, 350 215 Wiener process, 2, 4 of SEEs driven by stochastic vector Q-Wiener process, 38 measures, 175 cylindrical Wiener process, 5, 40, 97, 182 of SEEs with constant delay, 85 of SEEs with delay with Poisson jumps, 208 Y of SEEs with time-varying coefﬁcients, Yosida approximation, 15, 60, 163, 200, 235, 262 251, 290, 338 of SEEs with variable delay, 94 of measure-valued McKean-Vlasov of semilinear stochastic evolution evolution equation, 186 equations, 79 of a periodic control problem, 346 of semilinear stochastic integrodifferential of Itô stochastic integrodifferential equations, 132 equation, 134 of stochastic evolution equations, 72 of linear stochastic integrodifferential of switching diffusion processes with equation, 126 Poisson Jumps, 220 of McKean-Vlasov equation, 104 of time-varying stochastic evolution of McKean-Vlasov equation with equations, 157 multiplicative diffusion, 111 of a multivalued operator, 17 of multivalued SDEs with white noise, 138 Young’s inequality, 61, 152, 225, 226