[...]... applications of the Malliavin calculus have been found, including partial information optimal control, insider trading and, more generally, anticipative stochastic calculus At the same time Malliavin calculus was extended from the original setting of Brownian motion to more general L´vy processes This extensions were at e G.D Nunno et al., Malliavin Calculus for L´vy Processes with Applications e to. .. accustomed with classical analysis and Itˆ stochastic integration o may find (2.3) to be just a formal definition for an operator, which can hardly be matched with the general meaning of integral The purpose of the two following sections is to motivate Definition 2.2, showing that the operator (2.3) is a meaningful stochastic integral having strong links with the Itˆ stochastic o integral itself In the forthcoming... we base the interpretation of the Malliavin derviative as a stochastic gradient, while in the discontinuous case, the Malliavin derivative is actually a difference operator How to use this book It is the purpose of this book to give an introductory presentation of the theory of Malliavin calculus and its applications, mainly to finance For pedagogical reasons, and also to make the reading easier and the... regard it as a derivative with respect to the random parameter ω For this to make sense, one needs some mathematical structure on the space Ω In the original approach used by Malliavin, for the Brownian motion case, Ω is represented as the Wiener space C0 ([0, T ]) of continuous functions ω : [0, T ] −→ R with ω(0) = 0, equipped with the uniform topology In this book we prefer to use the representation... flexible, the book is divided into two parts: Part I The Continuous Case: Brownian Motion Part II The Discontinuous Case: Pure Jump L´vy Processes e In both parts the emphasis is on the topics that are most central for the applications to finance The results are illustrated throughout with examples In addition, each chapter ends with exercises Solutions to some selection of exercises, with varying level of detail,... respectively, Ft = lim Fu := u t Fu u>t See, for example, [128] or [206] G.D Nunno et al., Malliavin Calculus for L´vy Processes with Applications e to Finance, c Springer-Verlag Berlin Heidelberg 2009 7 8 1 The Wiener–Itˆ Chaos Expansion o Definition 1.1 A real function g : [0, T ]n → R is called symmetric if g(tσ1 , , tσn ) = g(t1 , , tn ) (1.2) for all permutations σ = (σ 1 , , σ n ) of (1,... , tn , tn+1 ) = fn (t1 , , tn , t) := fn,t (t1 , , tn ) G.D Nunno et al., Malliavin Calculus for L´vy Processes with Applications e to Finance, c Springer-Verlag Berlin Heidelberg 2009 19 20 2 The Skorohod Integral and we may regard fn as a function of n + 1 variables Since this function is symmetric with respect to its first n variables, its symmetrization fn is given by fn (t1 , , tn+1 )... 263 15 The Forward Integral 265 15.1 Definition of Forward Integral and its Relation with the Skorohod Integral 265 15.2 Itˆ Formula for Forward and Skorohod Integrals 268 o 15.3 Exercises 272 16 Applications to Stochastic Control: Partial and Inside Information ... of view of applications it is important also to be able to find the integrand ϕ more explicitly This can be achieved, for example, by the Clark–Ocone formula (see Chap 4), which says that, under some suitable conditions, ϕ(t) = E[Dt F |Ft ], 0 ≤ t ≤ T, where Dt F is the Malliavin derivative of F We discuss this topic later in the book However, for certain random variables F it is possible to find ϕ directly,... representation formula [46, 47] in terms of the Malliavin derivative This remarkable result later became known as the Clark– Ocone formula Sometimes also called Clark–Haussmann–Ocone formula in view of the contribution of Haussmann in 1979, see [97] In 1991, Ocone and Karatzas [173] applied this result to finance They proved that the Clark– Ocone formula can be used to obtain explicit formulae for replicating . lim ut F u :=  u>t F u . See, for example, [128] or [206]. G.D. Nunno et al., Malliavin Calculus for L´evy Processes with Applications 7 to Finance, c  Springer-Verlag. Introduction to L´evy Processes 159 9.1 Basics on L´evyProcesses 159 9.2 The ItˆoFormula 163 9.3 The Itˆo Representation Theorem for Pure Jump L´evyProcesses