TThe Functional Meyer-Tanaka Formula
Yuri F. Saporito ∗ May 8, 2017
Abstract
The functional Itˆo formula, firstly introduced by Bruno Dupire for continuoussemimartingales, might be extended in two directions: different dynamics for theunderlying process and/or weaker assumptions on the regularity of the functional.In this paper, we pursue the former type by proving the functional version of theMeyer-Tanaka Formula. Following the idea of the proof of the classical time-dependent Meyer-Tanaka formula, we study the mollification of functionals andits convergence properties. As an example, we study the running maximum andthe max-martingales of Yor and Obł´oj.
Our goal in this article is to prove the functional extension of the well-known Meyer-Tanaka formula. The theory of functional Itˆo calculus was presented in the seminalpaper [8] and it was further developed and applied to diverse topics, for instance, in[9, 10, 26, 18, 4, 3, 23]. Before proceeding, a remark regarding nomenclature. In thispaper, the adjective classical will always refer to the finite-dimensional Itˆo stochasticcalculus.The Meyer-Tanaka formula is the extension of Itˆo formula to convex functions.More precisely, in the classical case, if f : R −→ R is convex and ( x t ) t ≥ is a continuoussemimartingale, then f ( x t ) = f ( x ) + (cid:90) t f (cid:48) ( x s ) dx s + (cid:90) R L x ( t , y ) d f (cid:48) ( y ) , (1.1)where f (cid:48) is the left-derivative of f and L x ( s , y ) is the local time of the process x at y ; see[19], for example. This formula is easily generalized to functions f that are absolutelycontinuous with derivative of bounded variation, which is equivalent to say that f is thedifference of two convex functions. We would like to remind the reader that the localtime is defined by the limit in probability: L x ( t , y ) = lim ε → + ε (cid:90) t [ y − ε , y + ε ] ( x s ) d (cid:104) x (cid:105) s , ∗ Escola de Matem´atica Aplicada (EMAp), Fundac¸˜ao Get´ulio Vargas (FGV), Rio de Janeiro, Brazil, [email protected] . a r X i v : . [ m a t h . P R ] M a y here (cid:104) x (cid:105) is the quadratic variation of the process x . We are adhering the convention4 ε instead of 2 ε . The random field ( L x ( t , y )) t , y is a.s continuous and increasing in t andc`adl`ag in y . The following extension to time-dependent functions was established in[11]: f ( t , x t ) = f ( , x ) + (cid:90) t ∂ − t f ( s , x s ) ds + (cid:90) t ∂ − x f ( s , x s ) dx s (1.2) + (cid:90) R L x ( t , y ) d y ∂ − x f ( t , y ) − (cid:90) R (cid:90) t L x ( s , y ) d s , y ∂ − x f ( s , y ) , where ∂ − t f and ∂ − x f are the time and space left-derivatives, respectively. It is as-sumed that f is absolutely continuous in each variable, ∂ − t f and ∂ − x f exist, are left-continuous and locally bounded, ∂ − x f is of locally bounded variation in R + × R and ∂ − x f ( , · ) is of locally bounded variation in R . The notation d y and d s , y mean integra-tion with respect to the y variable and the ( s , y ) variables, respectively. We forward thereader to the reference cited above for some other different generalizations of Meyer-Tanaka formula (1.1) and for the precise definition of the Lebesgue-Stieltjes integral (cid:82) R (cid:82) t L x ( s , y ) d s , y ∂ − x f ( s , y ) .Since a functional extension of the Meyer-Tanaka would be inherently time depen-dent, Equation (1.2) is of utmost importance for our goal. However, we will not pursuea functional extension of (1.2) in its full generality of assumptions. It is clear that someof the technical assumptions of the results presented in our work could be weakenedalong the lines of [11], but in order to provide a clear exposition of the subject wewill consider technical assumptions that are general enough to introduce the importanttechniques without adding a cumbersome notation.There are several other generalizations of the Itˆo formula that could be extended tothe functional framework, for instance, [1, 27, 21, 17, 11, 30, 15, 14, 2]. We will notpursue them here, of course, but we hope that the foundations laid in this work mighthelp in this task.Meyer-Tanaka formula and its generalizations have many interesting applicationsin Finance, as, for instance, [22, 6, 5]. Other applications can be found in the theory ofLocal Volatility of [7], see for example [20].The paper is organized as follows: we finish this introduction with a presentation offunctional Itˆo calculus and we define the mollification of functionals in Section 2. Thisis a very important tool that will be used in Section 3 in order to prove the functionalextension of the Meyer-Tanaka formula. In Section 4, we will apply the theory to therunning maximum to find a pathwise version of a famous identity by Paul L´evy and wewill also study the max-martingales of Yor and Obł´oj in the light of the functional Itˆocalculus. In this section we will present a short review of the functional Itˆo calculus introducedin [8]. The goal is to familiarize the reader with the notation, main definitions andtheorems needed for the results that follow.2he space of c`adl`ag paths in [ , t ] will be denoted by Λ t . For a fixed time horizon T >
0, we define the space of paths as Λ = (cid:91) t ∈ [ , T ] Λ t . We will denote elements of Λ by upper case letters and often the final time of itsdomain will be subscripted, e.g. Y ∈ Λ t ⊂ Λ will be denoted by Y t . The value of Y t at aspecific time will be denoted by lower case letters: y s = Y t ( s ) , for any s ≤ t . Moreover,if a path Y t is fixed, the path Y s , for s ≤ t , will denote the restriction of the path Y t to theinterval [ , s ] .The following important path deformations are always defined in Λ . For Y t ∈ Λ and t ≤ s ≤ T , the flat extension of Y t up to time s ≥ t is defined as Y t , s − t ( u ) = (cid:26) y u , if 0 ≤ u ≤ t , y t , if t ≤ u ≤ s , see Figure 1. For h ∈ R , the bumped path , see Figure 2, is defined by Y ht ( u ) = (cid:26) y u , if 0 ≤ u < t , y t + h , if u = t . b b Figure 1: Flat extension of a path. bb b Figure 2: Bumped path.For any Y t , Z s ∈ Λ , where it is assumed without loss of generality that s ≥ t , weconsider the following metric in Λ , d Λ ( Y t , Z s ) = (cid:107) Y t , s − t − Z s (cid:107) ∞ + | s − t | , where (cid:107) Y t (cid:107) ∞ = sup u ∈ [ , t ] | y u | . One could easily show that ( Λ , d Λ ) is a complete metric space.Additionally, a functional is any function f : Λ −→ R . Continuity with respect to d Λ is defined as the usual definition of continuity in a metric space and is denominated Λ -continuity .For a functional f and a path Y t with t < T , the time functional derivative of f at Y t is defined as ∆ t f ( Y t ) = lim δ t → + f ( Y t , δ t ) − f ( Y t ) δ t , (1.3)3henever this limit exists. The space functional derivative of f at Y t is defined as, ifthe limit exists, ∆ x f ( Y t ) = lim h → f ( Y ht ) − f ( Y t ) h . (1.4)Finally, for any i , j ∈ { } ∪ N ∪ { + ∞ } , a functional f : Λ −→ R is said to belongto C i , j if it is Λ -continuous and it has Λ -continuous derivatives ∆ ( k ) t f and ∆ ( m ) x f , for k = , . . . , i and m = , . . . , j . Here, clearly, ∆ ( k ) t = ∆ t ( ∆ ( k − ) t ) and ∆ ( m ) x = ∆ x ( ∆ ( m − ) x ) .Moreover, we use the notation ∆ xx = ∆ ( ) x .The attentive reader might have noticed that we have not introduced any probabilitynotation so far. We start by fixing a probability space ( Ω , F , P ) . We state now thefunctional Itˆo formula. The proof can be found in [8]. Theorem 1.1 (Functional Itˆo Formula; [8]) . Let x be a continuous semimartingale andf ∈ C , . Then, for any t ∈ [ , T ] ,f ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) t ∆ x f ( X s ) dx s + (cid:90) t ∆ xx f ( X s ) d (cid:104) x (cid:105) s P − a.s. One should notice that the Itˆo formula above is of the same form as the classicalItˆo formula for continuous semimartingale, the only change being the definition of thetime and space functional derivatives given by Equations (1.3) and (1.4). This theoremwas extended in terms of weakening the regularity of f and generalizing the dynamicsof x , see [4, 3, 23]. Here, we will examine a different class of functionals than it wasconsidered in these previous works. We now state the main result of this paper: Theorem (Functional Meyer-Tanaka Formula) . Consider a functional f : Λ −→ R satisfying Hypotheses 3.5 and let x be a continuous semimartingale. Thenf ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) t ∆ − x f ( X s ) dx s (1.5) + (cid:90) R L x ( t , y ) d y ∂ − y f ( X yt − ) − (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y f ( X ys − ) P − a.s. , where X ys − is the path X s with the value at s substituted by y, see Equation (2.9). Notation 1.2. d y φ ( y ) and d s , y φ ( s , y ) denote the Lebesgue-Stieltjes integration withrespect to the integrator φ ( y ) and φ ( s , y ) , respectively.The main example of non-smooth functional to have in mind is the running maxi-mum: m ( Y t ) = sup ≤ s ≤ t y s . (1.6)For more details on this functional, we forward the reader to Section 4.24 Functional Mollification
In this section, we investigate the mollification of functionals. The goal is to create asequence of smooth functionals converging to the original one in various senses. Thistechnique will be used to prove the functional Meyer-Tanaka formula as it is similarlydone in the proof of its classical version.
Definition 2.1.
For any functional f : Λ −→ R , we define F : Λ × R −→ R as F ( Y t , h ) = f ( Y ht ) . (2.1)When denoting functionals, capital letters will be used as above, i.e. it will denote afunction with domain Λ × R where the first variable is the path and the second variableis the bump applied to this path. This notation will be carried out in the remainder ofthe paper. We choose to use this notation to help the analysis of the space functionalderivative of the mollification.A mollifier in R is a positive function ρ : R −→ [ , + ∞ ) such that ρ ∈ C ∞ c ( R ) , thespace of compactly supported smooth functions; (cid:82) R ρ ( z ) dz =
1; and ρ n ( x ) : = n ρ ( nx ) converges to Dirac delta in the sense of distributions. We also refer to the sequence ( ρ n ) n ∈ N as the mollifiers. Notice that ρ n ∈ C ∞ c ( R ) . Definition 2.2.
The sequence of mollified functionals is defined as F n ( Y t , h ) = (cid:90) R ρ n ( h − ξ ) F ( Y t , ξ ) d ξ = (cid:90) R ρ n ( ξ ) F ( Y t , h − ξ ) d ξ . (2.2) Remark 2.3.
This mollification is well-defined as long as the real function F ( Y t , · ) is locally integrable for any path Y t ∈ Λ . See [13], for instance, for details on themollification in the case of real functions. Proposition 2.4.
Suppose f is Λ -continuous. Then F ( Y t , · ) is continuous for eachY t ∈ Λ , F n is well-defined and, as a functional, is infinitely differentiable in space.Moreover, ∆ ( k ) x F n ( Y t , h ) = ∂ ( k ) h F n ( Y t , h ) , where ∂ ( k ) h denotes the k-th derivative with respect to h. This is the main property ofthe mollified functionals.Proof. Notice that since the functional f is Λ -continuous, F ( Y t , · ) is then continu-ous for fixed Y t ∈ Λ , because d Λ ( Y h t , Y h t ) = | h − h | . This implies F ( Y t , · ) is lo-cally integrable, and therefore the mollification F n is well-defined. Notice now that F ( Y zt , h ) = F ( Y t , h + z ) and then F n ( Y zt , h ) = (cid:90) R ρ n ( h − ξ ) F ( Y t , ξ + z ) d ξ (2.3) = (cid:90) R ρ n ( h − ( ξ − z )) F ( Y t , ξ ) d ξ = F n ( Y t , h + z ) . Thus, for any k ∈ N , ∆ ( k ) x F n ( Y t , h ) = ∂ ( k ) h F n ( Y t , h ) .
5e would like also to point it out that a particular mollification of the runningmaximum was considered in [8] to derive a pathwise version of the famous formuladue to L´evy: max ≤ s ≤ t x s = x + L x − m ( t , ) , where m is the running maximum process and x is a continuous semimartingale. Thereader is forwarded to [19, Chapter 3 and Chapter 6] for more details on results regard-ing the relations between local time and the running maximum in the Brownian motioncase. Λ -Continuity of the Mollified Functionals and its Derivatives In this section we will study the relation of continuity of f and of its mollification F n .We have already seen that, if F ( Y t , · ) is locally integrable for any given Y t ∈ Λ , then F n ( Y t , · ) is infinitely differentiable in R , and therefore it is also continuous. However,differentiability in the functional sense does not imply Λ -continuity. Hence, it is neces-sary to consider a slightly stronger assumption on the continuity of the functional f inorder to be able to conclude the Λ -continuity of F n . We will thus consider the followingstronger criterion: Definition 2.5.
We say that f is Λ - φ -equicontinuous if there exists φ : R −→ R positiveand locally integrable depending only on f such that ∀ ε > , ∀ Y t ∈ Λ , ∃ δ > d Λ ( Y t , Z s ) < δ ⇒ | F ( Y t , ξ ) − F ( Z s , ξ ) | < εφ ( ξ ) , ∀ ξ ∈ R . (2.4)Notice that Λ - φ -equicontinuity implies that f is Λ -continuous. Moreover, if φ ≡ { F ( · , ξ ) } ξ ∈ R is Λ -equicontinuous.The weakening of this assumption could be pursued, but it is not in the scope ofthis work. Proposition 2.6.
Suppose f is Λ - φ -equicontinuous. Then, for any n ∈ N and h ∈ R ,F n ( · , h ) and ∆ ( k ) x F n ( · , h ) are Λ -continuous, for any k ∈ N .Proof. By Equation (2.2), we see | F n ( Y t , h ) − F n ( Z s , h ) | ≤ (cid:90) R ρ n ( h − ξ ) | F ( Y t , ξ ) − F ( Z s , ξ ) | d ξ . Hence, fixing ε > n ∈ N and h ∈ R , and choosing δ > Λ - φ -equicontinuityof f with ε equals ε (cid:82) R ρ n ( h − ξ ) φ ( ξ ) d ξ , we have, for Y t , Z s ∈ Λ satisfying d Λ ( Y t , Z s ) < δ , | F n ( Y t , h ) − F n ( Z s , h ) | ≤ (cid:90) R ρ n ( h − ξ ) | F ( Y t , ξ ) − F ( Z s , ξ ) | d ξ < ε . F n ( · , h ) is Λ -continuous for any n ∈ N and h ∈ R . Consid-ering now the derivatives of F n , we see ∆ ( k ) x F n ( Y t , h ) = ∂ ( k ) h F n ( Y t , h ) = (cid:90) R ∂ ( k ) h ( ρ n ( h − ξ )) F ( Y t , ξ ) d ξ , and since ∂ ( k ) h ( ρ n ( h − · )) are in C ∞ c ( R ) , the same argument employed above for the Λ -continuity of F n can be used to conclude the Λ -continuity of ∆ ( k ) x F n ( · , h ) . As we have seen, the functional F n is smooth with respect to the space variable. In thissection, we will study the question of the existence of the time functional derivative.Notice that F n ( Y t , δ t , h ) = (cid:90) R ρ n ( h − ξ ) F ( Y t , δ t , ξ ) d ξ . When is F n time functional differentiable as in Equation (1.3)? Definition 2.7.
We say a functional f is h-time functional differentiable if F ( · , h ) istime functional differentiable for every h ∈ R , i.e. ∆ t F ( Y t , h ) = lim δ t → + F ( Y t , δ t , h ) − F ( Y t , h ) δ t = lim δ t → + f (( Y t , δ t ) h ) − f ( Y ht ) δ t , (2.5)exists for every Y t ∈ Λ and h ∈ R .We are then ready to answer the previous question: Proposition 2.8.
If f is h-time functional differentiable, then ∆ t F n exists, ∆ t F n ( Y t , h ) = ( ∆ t F ) n ( Y t , h ) , for any Y t ∈ Λ and h ∈ R , and ∆ ( k ) x ∆ t F n ( Y t , h ) = ∂ ( k ) h ( ∆ t F ) n ( Y t , h ) . Moreover, if ∆ t F is Λ - φ -equicontinuous, then ∆ t F n ( · , h ) is Λ -continuous, and hence in C , ∞ .Proof. For fixed Y t ∈ Λ and ξ ∈ R , define ψ ( δ t ) = F ( Y t , δ t , ξ ) . By Definition 2.7, ψ ∈ C ( R + ) and therefore, ∆ t F n ( Y t , h ) = (cid:90) R ρ n ( h − ξ ) ∆ t F ( Y t , ξ ) d ξ = ( ∆ t F ) n ( Y t , ξ ) . Since ∆ t F is Λ - φ -equicontinuous and ∆ t F n ( Y t , h ) = ( ∆ t F ) n ( Y t , ξ ) , by Proposition 2.6,we conclude that ∆ t F n is Λ -continuous. 7 emark 2.9. We would like to point out the similarity of the Equation (2.5) and thelimit characterization of the Lie bracket given in [18, Lemma 3.2]: [ ∆ t , ∆ x ] f ( Y t ) = lim δ t → + h → f (( Y t , δ t ) h ) − f (( Y ht ) t , δ t ) h δ t . However, it is obvious that Definition 2.7 does not require the functional f to be locallyweakly path-dependent ( [ ∆ t , ∆ x ] f =
0, as defined in [18]). Definition 2.7 is indeed just atechnicality and encompasses many interesting functionals. For example, the runningintegral ( f ( Y t ) = (cid:82) t y s ds ) satisfies Assumption 2.7 and it is not locally weakly path-dependent. We will not pursue this here, but it is important to mention two different mollificationpossibilities: (i) Time Mollification : F n ( Y t , δ t ) = (cid:90) R ρ n ( δ t − η ) f ( Y t , η ) d η . (2.6) (ii) Joint Mollification : F n ( Y t , δ t , h ) = (cid:90) R (cid:90) R ρ n ( h − ξ ) ρ n ( δ t − η ) f (( Y t , η ) ξ ) d ξ d η . (2.7)An obvious issue with the joint mollification is the choice between f (( Y t , η ) ξ ) and f (( Y ξ t ) t , η ) ; both would be initially valid choices. This is not a problem when we restrictourselves to the path-independent case: f ( Y t ) = h ( t , y t ) . However, as it was noted in[18], the different ordering of bump and flat extension is a very important aspect of thefunctional Itˆo calculus.Additionally, as it happened in the aforesaid reference in a different circumstance,the Lie bracket of the operators ∆ t and ∆ x would probably play an important role if thejoint mollification were used. We will now derive some integration by parts computations that will be useful later inthe proof of the functional Meyer-Tanaka formula.First some definitions. For any Y t ∈ Λ and y ∈ R , Y yt − ( u ) = (cid:26) y u , if 0 ≤ u < t , y , if u = t . (2.8)Notice that Y yt − = Y y − y t t ∈ Λ t and it is different than ( Y t − ) y = Y y − y t + y t − t . Moreover,define F ( Y t , y ) = f ( Y yt − ) . (2.9)8he definition of the function F above serves two purposes. Firstly, alleviates no-tation. Secondly, it helps us take derivatives with respect to the Y t and the last value y separately. Capital calligraphic letters will always be used as above meaning that itwill denote a function with domain Λ × R where the first variable is the path and thesecond variable is the value will replace the last value of the path. We will keep thisnotation through out the paper.We start by noticing that, for any function q : R −→ R regular enough for thecomputations to follow, the subsequent identity is obviously true: (cid:90) R (cid:18) (cid:90) t F ( Y s , y ) d s q ( s , y ) (cid:19) dy = (cid:90) R F ( Y t , y ) q ( t , y ) dy − (cid:90) R (cid:18) (cid:90) t ∂ t F ( Y s , y ) q ( s , y ) ds (cid:19) dy , where F is given by Equation (2.9) and ∂ t F ( Y s , y ) = lim u → s F ( Y s , y ) − F ( Y u , y ) s − u , the usual time derivative of a function. Let us now verify that this derivative existsunder certain regularity assumptions. Notice that F ( Y s , y ) does not depend on the lastvalue of the path Y s , and hence ∆ x F ( Y s , y ) = ∆ xx F ( Y s , y ) =
0. So, if f satisfies Def-inition 2.7, ∆ t F ( Y s , y ) exists. Assuming Lambda -continuity of F ( · , y ) and ∆ t F ( · , y ) implies that F ( · , y ) ∈ C , . Then, one can show, by the functional Itˆo formula, Theo-rem 1.1, that for any continuous semimartingale x , F ( X s , y ) = F ( X u , y ) + (cid:90) tu ∆ t F ( X r , y ) dr , which implies that ∂ t F ( X s , y ) = ∆ t F ( X s , y ) , ∀ s ≥ , P − a.s.Moreover, F ( Y t , y + h ) = F ( Y yt − , h ) = f ( Y y + ht − ) , (2.10)and then ∂ ( k ) y F ( Y t , y ) = ∂ ( k ) h F ( Y ys − , ) = ∆ ( k ) x f ( Y ys − ) . Before proceeding, we would like to comment on the commutation of ∂ t and ∂ y . Itis well-known now that ∆ t and ∆ x do not commute. However, we do not experiencea similar problem here. ∂ t and ∂ y do commute: ∂ y ∂ t F ( Y t , y ) = ∂ t ∂ y F ( Y t , y ) , as onecan easily verify by direct computation and assuming these derivatives exist and arecontinuous. The reason is that in the definition of F ( Y t , y ) it is implied that the bump y happens always at the end of the path Y t . Therefore, there is no ambiguity in the orderof the time perturbation and the bump that we experience in the case of ∆ x and ∆ t .9f g ∈ C c ( R ) and ∂ yy F ( Y t , y ) exists, then (cid:90) R g ( y ) ∆ xx f ( Y yt − ) dy = (cid:90) R g ( y ) ∂ yy F ( Y t , y ) dy (2.11) = − (cid:90) R g (cid:48) ( y ) ∂ y F ( Y t , y ) dy Furthermore, if we consider q : R −→ R smooth with compact support and assume ∂ t ∂ yy F ( Y t , y ) exists, we find (cid:90) t (cid:90) R ∂ t ∆ xx f ( Y ys − ) q ( s , y ) dyds = (cid:90) t (cid:90) R ∂ t ∂ yy F ( Y s , y ) q ( s , y ) dyds = (cid:90) t (cid:90) R ∂ y ∂ t ∂ y F ( Y s , y ) q ( s , y ) dyds = − (cid:90) t (cid:90) R ∂ t ∂ y F ( Y s , y ) ∂ y q ( s , y ) dyds = − (cid:90) R ∂ y F ( Y s , y ) ∂ y q ( s , y ) (cid:12)(cid:12)(cid:12)(cid:12) t dy + (cid:90) t (cid:90) R ∂ y F ( Y s , y ) ∂ sy q ( s , y ) dyds , where ψ ( s , y ) (cid:12)(cid:12)(cid:12)(cid:12) t = ψ ( t , y ) − ψ ( , y ) . The local time of the process x at level y , denoted by L x ( s , y ) , is defined as the limit inprobability: L x ( t , y ) = lim ε → + ε (cid:90) t [ y − ε , y + ε ] ( x s ) d (cid:104) x (cid:105) s . A very important identity related to the local time is the occupation times formula ,[29, Corollary 1.6, Chapter VI], which says that if ϕ : R + × R −→ R is bounded andmeasurable, then (cid:90) t ϕ ( s , x s ) d (cid:104) x (cid:105) s = (cid:90) R (cid:18) (cid:90) t ϕ ( s , y ) d s L x ( s , y ) (cid:19) dy , ∀ t ≥ , P − a.s. (3.1)The following extension of the occupation time formula will be fundamental in thefollowing, see [29, Exercise 1.15, Chapter VI]. Lemma 3.1.
For any bounded measurable function ψ : R + × Ω × R −→ R , (cid:90) t ψ ( s , ω , x s ) d (cid:104) x (cid:105) s = (cid:90) R (cid:18) (cid:90) t ψ ( s , ω , y ) d s L x ( s , y ) (cid:19) dy , ∀ t ≥ , P − a.s. (3.2)10 roof. By Equation (3.1), for any ϕ : R + × R −→ R bounded and measurable, thereexists Ω ϕ ∈ F with P ( Ω ϕ ) = ω ∈ Ω ϕ , (cid:90) t ϕ ( s , x s ( ω )) d (cid:104) x (cid:105) ( ω ) s = (cid:90) R (cid:18) (cid:90) t ϕ ( s , y ) d s L x ( ω )( s , y ) (cid:19) dy , ∀ t ≥ , where (cid:104) x (cid:105) ( ω ) and L x ( ω ) are the realizations of the quadratic variation and the localtime, respectively. Moreover, since ψ ( · , ω , · ) is bounded and measurable, it can beuniformly approximated by simple functions of the form: ψ n ( t , ω , y ) = b n ∑ k = a k , n ( t , y ) A k , n ( ω ) . Define now Ω ψ = (cid:84) + ∞ n = (cid:84) b n k = Ω k , n , where Ω k , n is defined as Ω ϕ for ϕ = a k , n . Therefore, P ( Ω ψ ) = R + × R , we find, for ω ∈ Ω ψ , (cid:90) t ψ n ( s , ω , x s ( ω )) d (cid:104) x ( ω ) (cid:105) s = b n ∑ k = A k , n ( ω ) (cid:90) t a k , n ( s , x s ( ω )) d (cid:104) x ( ω ) (cid:105) s = b n ∑ k = A k , n ( ω ) (cid:90) R (cid:18) (cid:90) t a k , n ( s , y ) d s L x ( ω )( s , y ) (cid:19) dy = (cid:90) R (cid:18) (cid:90) t ψ n ( s , ω , y ) d s L x ( ω )( s , y ) (cid:19) dy . Letting n → + ∞ and using the uniformity of the convergence ψ n → ψ , we have foundthe desired result.For a given functional f , we would like to apply the proposition above to ψ f ( s , ω , y ) = F ( X s ( ω ) , y ) , where F is defined in Equation (2.9). Then, for every functional f suchthat ψ f above is bounded and measurable, we have (cid:90) t f ( X s ) d (cid:104) x (cid:105) s = (cid:90) R (cid:18) (cid:90) t F ( X s , y ) d s L x ( s , y ) (cid:19) dy . (3.3) Example 3.2 (Running Integral) . Consider the running integral functional f ( Y t ) = (cid:82) t y u du . We clearly have F ( Y s , y ) = f ( Y s ) and moreover, we find • (cid:90) t f ( X s ) d (cid:104) x (cid:105) s = (cid:90) t (cid:18) (cid:90) s x u du (cid:19) d (cid:104) x (cid:105) s = (cid:90) t ( (cid:104) x (cid:105) t − (cid:104) x (cid:105) u ) x u du , • (cid:90) R (cid:18) (cid:90) t F ( X s , y ) d s L x ( s , y ) (cid:19) dy = (cid:90) R (cid:18) (cid:90) t (cid:18) (cid:90) s y u du (cid:19) d s L x ( s , y ) (cid:19) dy = (cid:90) t (cid:18) (cid:90) R L x ( t , y ) dy − (cid:90) R L x ( u , y ) dy (cid:19) x u du . Therefore, since (cid:104) x (cid:105) t = (cid:90) R L x ( t , y ) dy , we verify Equation (3.3) for this particular example.11 .2 Convergence Properties The idea behind the proof of the classical Meyer-Tanaka formula (see [19] and [11],for example) is to apply Itˆo formula to the smooth mollification of the function inconsideration, let n go to infinity to approximate the original function and then analyzethe limit of all the terms of the Itˆo formula. Having this strategy in mind, in this sectionwe will investigate the convergence of certain quantities that will be important whenproving the functional Meyer-Tanaka formula.We firstly define the functional f n : Λ −→ R as f n ( Y t ) = F n ( Y t , ) = (cid:90) R ρ n ( ξ ) F ( Y t , − ξ ) d ξ , (3.4)where F n is the mollification of F given in Equation (2.2). Remark 3.3.
In what follows, we will explicitly use the fact that the mollifier ρ hascompact support. Without loss of generality, we may assume that its support is inside [ ρ min , ρ max ] , where ρ min < ρ max > Proposition 3.4.
Assume ∂ − h F ( Y t , · ) exists. The following facts hold true:1. For each Y t ∈ Λ , if F ( Y t , · ) and ∂ − h F ( Y t , · ) are continuous at 0, then lim n → + ∞ f n ( Y t ) = f ( Y t ) , lim n → + ∞ ∆ x f n ( Y t ) = lim n → + ∞ ∂ h F n ( Y t , ) = ∂ − h F ( Y t , ) = ∆ − x f ( Y t ) .
2. If f satisfies Definition 2.7, we have lim n → + ∞ ∆ t f n ( Y t ) = ∆ t f ( Y t ) , lim n → + ∞ (cid:90) t ∆ t f n ( Y s ) ds = (cid:90) t ∆ t f ( Y s ) ds , for any Y t ∈ Λ .3. If ( ∂ − h F ( X s , − h )) s ∈ [ , T ] is bounded in h ∈ [ ρ min , ρ max ] by an x-integrable process,then lim n → + ∞ (cid:90) t ∆ x f n ( X s ) dx s = (cid:90) t ∆ − x f ( X s ) dx s u.c.p. , where u.c.p. means uniformly on compacts in probability.Proof.
1. It follows easily from standard results in mollification theory.12. The first limit follows from Proposition 2.8. Moreover, one can easily notice (cid:90) t ∆ t f n ( Y s ) ds = (cid:90) t (cid:90) R ρ n ( − ξ ) ∆ t F ( Y s , ξ ) d ξ ds = (cid:90) R ρ n ( − ξ ) (cid:18) (cid:90) t ∆ t F ( Y s , ξ ) ds (cid:19) d ξ . Therefore, lim n → + ∞ (cid:90) t ∆ t f n ( Y s ) ds = (cid:90) t ∆ t F ( Y s , ) ds = (cid:90) t ∆ t f ( Y s ) ds .
3. Notice that ∆ x f n ( Y t ) = (cid:90) ρ max ρ min ρ ( ξ ) ∂ − h F (cid:18) Y t , − ξ n (cid:19) d ξ . (3.5)The boundedness assumptions means there exists an x -integrable process ( ψ s ) s ∈ [ , T ] such that max h ∈ [ ρ min , ρ max ] (cid:12)(cid:12) ∂ − h F ( X s , − h ) (cid:12)(cid:12) ≤ ψ s . Hence, | ∆ x f n ( Y t ) | ≤ (cid:90) ρ max ρ min ρ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ − h F (cid:18) Y t , − ξ n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d ξ ≤ (cid:90) ρ max ρ min ρ ( ξ ) ψ s d ξ = ψ s . Therefore, by the Dominated Convergence Theorem for stochastic integrals, see [28,Theorem 32, Chapter IV], we have the desired convergence.
We start this section by stating the assumptions on the functional f such that the Meyer-Tanaka formula will hold. Hypotheses 3.5. f h -time functional differentiable as in Definition 2.7;2. f and ∆ t F are Λ - φ -equicontinuous as in Definition 2.5;3. ∂ − y F ( Y t , y ) exists and is of bounded variation for ( t , y ) jointly and for y sepa-rately, for any Y t ∈ Λ .4. ( ∂ − h F ( X t , − h )) t ∈ [ , T ] is bounded in h ∈ [ ρ min , ρ max ] by an x -integrable process.13e are ready then to prove the main result of this paper. Theorem 3.6 (Functional Meyer-Tanaka Formula) . Suppose f satisfies Hypotheses 3.5and let x be a continuous semimartingale. Then, the functional Meyer-Tanaka formulaholds f ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) t ∆ − x f ( X s ) dx s (3.6) + (cid:90) R L x ( t , y ) d y ∂ − y F ( X t , y ) − (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) P − a.s.Proof. As we studied in Section 2, f n belongs to C , ∞ and by the functional Itˆo formula,Theorem 1.1, we find f n ( X t ) = f n ( X ) + (cid:90) t ∆ t f n ( X s ) ds + (cid:90) t ∆ x f n ( X s ) dx s + (cid:90) t ∆ xx f n ( X s ) d (cid:104) x (cid:105) s . Moreover, by what was shown in Proposition 3.4, the following convergences holdlim n → + ∞ f n ( Y t ) = f ( Y t ) , (3.7)lim n → + ∞ (cid:90) t ∆ t f n ( Y s ) ds = (cid:90) t ∆ t f ( Y s ) ds , (3.8)lim n → + ∞ (cid:90) t ∆ x f n ( X s ) dx s = (cid:90) t ∆ − x f ( X s ) dx s u.c.p. (3.9)for any Y t ∈ Λ . Let us now analyse the Itˆo term. Remember F is defined by Equation(2.9). If we denote the mollification of F with respect to the y variable by F n , we caneasily conclude, by Equation (2.10), F n ( Y yt − , h ) = F n ( Y t , y + h ) and then ∂ ( k ) h F n ( Y yt − , h ) = ∂ ( k ) y F n ( Y t , y + h ) . In particular, ∆ ( k ) x f n ( Y yt − ) = ∂ ( k ) h F n ( Y yt − , ) = ∂ ( k ) y F n ( Y t , y ) . So, by Equation (3.3),12 (cid:90) t ∆ xx f n ( X s ) d (cid:104) x (cid:105) s = (cid:90) R (cid:18) (cid:90) t ∂ yy F n ( X s , y ) d s L x ( s , y ) (cid:19) dy = (cid:90) R ∂ yy F n ( X t , y ) L x ( t , y ) dy − (cid:90) t (cid:90) R ∆ t ∂ yy F n ( X s , y ) L x ( s , y ) dyds , Hence, for g : R −→ R and q : R −→ R smooth and compactly supported, wehave, by the computations performed in Section 2.3, (cid:90) R ∂ yy F n ( Y t , y ) g ( y ) dy = − (cid:90) R g (cid:48) ( y ) ∂ y F n ( Y t , y ) dy (3.10) n → + ∞ −→ − (cid:90) R g (cid:48) ( y ) ∂ − y F ( Y t , y ) dy = (cid:90) R g ( y ) d y ∂ − y F ( Y t , y ) , (cid:90) t (cid:90) R ∆ t ∆ xx f n ( Y ys − ) q ( s , y ) dyds = − (cid:90) R ∂ y F n ( Y s , y ) ∂ y q ( s , y ) (cid:12)(cid:12)(cid:12) t dy (3.11) + (cid:90) t (cid:90) R ∂ y F n ( Y s , y ) ∂ ty q ( s , y ) dyds n → + ∞ −→ − (cid:90) R ∂ − y F ( Y s , y ) ∂ y q ( s , y ) (cid:12)(cid:12)(cid:12) t dy + (cid:90) t (cid:90) R ∂ − y F ( Y s , y ) ∂ ty q ( s , y ) dyds = (cid:90) t (cid:90) R q ( s , y ) d s , y ∂ − y F ( Y s , y ) , where the last equalities in (3.10) and (3.11) follow from item 3 of Hypotheses 3.5.Therefore, using well-known arguments along the lines of [11, Proof of Theorem 2.1],we can extend the formulas above for g ( y ) = L x ( t , y ) and q ( s , y ) = L x ( s , y ) , and finallyconclude lim n → + ∞ (cid:90) t ∆ xx f n ( X s ) d (cid:104) x (cid:105) s = (cid:90) R L x ( t , y ) d y ∂ − y F ( X t , y ) − (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) , as desired. Remark 3.7.
By the same arguments presented in [12], we could show that (cid:82) R L x ( t , y ) d y ∂ − y F ( X t , y ) is of bounded variation in t in [ , T ] . Therefore, ( f ( X t )) t ∈ [ , T ] is a semi-martingale. Remark 3.8.
Following the idea of [11, Theorem 2.3], we could consider the process x (cid:63) t = x t − a t , where ( a t ) t ≥ is a continuous process of finite variation. It is obvious that x (cid:63) is also a semimartingale. Denote the local time of x (cid:63) by L x − a . Therefore, the sameargument of [11, Theorem 2.3] applied to the computation we have just performedin (3.10) and (3.11) gives us the following version of the functional Meyer-Tanakaformula f ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) t ∆ − x f ( X s ) dx s (3.12) + (cid:90) R L x − a ( t , y ) d y ∂ − y F ( X t , y + a t ) − (cid:90) t (cid:90) R L x − a ( s , y ) d s , y ∂ − y F ( X s , y + a s ) . This version of the formula will be used in the running maximum example in Section4.2.
In this section we define the notion of convexity for functionals and then discuss someof its basic properties. The main interesting consequence is that some of the technical15ssumptions in Hypotheses 3.5 can be weakened.
Definition 4.1 (Convex Functionals) . We say f is a convex functional if F ( Y t , · ) is aconvex real function for any Y t ∈ Λ .Notice that, for f ∈ C , , convexity implies that ∆ xx f ( Y t ) ≥
0, for any Y t ∈ Λ . Remark 4.2.
Another possible definition for convexity of a functional would be f ( λ Y t + ( − λ ) Z t ) ≤ λ f ( Y t ) + ( − λ ) f ( Z t ) , (4.1)for all λ ∈ [ , ] and Y t , Z t ∈ Λ . Observe Y t and Z t must be in the same Λ t space. Thisclearly implies the previous definition of convexity because F ( Y t , λ h + ( − λ ) h ) = f ( λ Y h t + ( − λ ) Y h t ) . However, condition (4.1) is stronger than necessary for what follows.For a convex functional f , for any Y t ∈ Λ , F ( Y t , · ) is continuous, ∂ ± h F ( Y t , h ) existfor any h ∈ R and is non-decreasing in h . Moreover, ∂ ± h F ( Y t , h ) = ∆ ± x F ( Y t , h ) , wherethese one-sided functional derivatives are obviously defined as ∆ ± x f ( Y t ) = lim h → ± f ( Y ht ) − f ( Y t ) h . Proposition 4.3.
Assume f is convex. The following facts hold true:1. f n is convex. Moreover, ∆ x f n ( Y t ) increasingly converges to ∆ − x f ( Y t ) .2. If ( ∂ − h F ( X s , − h )) s ∈ [ , T ] is x-integrable for h = ρ min and h = ρ max , then lim n → + ∞ (cid:90) t ∆ x f n ( X s ) dx s = (cid:90) t ∆ − x f ( X s ) dx s u.c.p.Proof.
1. Indeed, F n ( Y t , λ h + ( − λ ) h ) = (cid:90) R F ( Y t , ( λ h + ( − λ ) h ) − y ) ρ n ( y ) dy = (cid:90) R F ( Y t , ( λ ( h − y ) + ( − λ )( h − y )) ρ n ( y ) dy ≤ λ F n ( Y t , h ) + ( − λ ) F n ( Y t , h ) . Hence, since f n is smooth, ∆ xx f n ( Y t ) ≥
0. The second affirmation follows from: ∆ x f n ( Y t ) = (cid:90) ρ max ρ min ρ ( ξ ) ∂ − h F (cid:18) Y t , − ξ n (cid:19) d ξ , (4.2)and it is easy to see the desired result using the fact that ∂ − h F ( Y t , h ) is non-decreasingin h , because of the convexity of f . 16. Since ∂ − h F ( Y t , h ) is non-decreasing in h , by Equation (4.2), ∂ − h F ( Y s , − ρ max ) ≤ ∂ − h F (cid:16) Y s , − ρ max n (cid:17) ≤ ∂ − h F (cid:18) Y s , − ξ n (cid:19) , ∂ − h F (cid:18) Y s , − ξ n (cid:19) ≤ ∂ − h F (cid:16) Y s , − ρ min n (cid:17) ≤ ∂ − h F ( Y s , − ρ min ) , where we are using the fact that ρ min < ρ max >
0; see Remark 3.3. Hence | ∆ x f n ( Y s ) | ≤ | ∂ − h F ( Y s , − ρ min ) | + | ∂ − h F ( Y s , − ρ max ) | , and the convergence follows as in Proposition 3.4Therefore, we might then consider the following class of convex functionals, wherewe have weakened conditions 3 and 4 of Hypotheses 3.5: Hypotheses 4.4. f h -time functional differentiable as in Definition 2.7;2. f and ∆ t F are Λ - φ -equicontinuous as in Definition 2.5;3. ∂ − y F ( Y s , y ) is of bounded variation for ( s , y ) jointly, for any Y ∈ Λ ;4. ( ∂ − h F ( X s , − h )) s ∈ [ , T ] is x -integrable for h = ρ min and h = ρ max ;5. f is convex; Remark 4.5.
It is straightforward to notice that if f satisfies Hypotheses 4.4, then f also satisfies Hypotheses 3.5. So, the functional Meyer-Tanaka formula, Theorem 3.6,holds for f .Similarly as in [28], we may analyze the limit of f n ( Y t ) without identifying the limitof the Itˆo term. Theorem 4.6.
Let f be a functional satisfying Hypotheses 4.4 and x a continuoussemimartingale. Thenf ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) t ∆ − x f ( X s ) dx s + A ft P − a.s. , (4.3) where A ft is a continuous and increasing process. roof. As we have seen in the proof of Theorem 3.6, f n ( X t ) = f n ( X ) + (cid:90) t ∆ t f n ( X s ) ds + (cid:90) t ∆ x f n ( X s ) dx s + (cid:90) t ∆ xx f n ( X s ) d (cid:104) x (cid:105) s . Consider now the continuous process A nt = (cid:90) t ∆ xx f n ( X s ) d (cid:104) x (cid:105) s . This process is increasing because f n is convex, which means ∆ xx f n ≥
0. Hence, byEquations (3.7)–(3.8), A nt converges u.c.p. to a continuous increasing process A ft thatsatisfies Equation (4.3). Remark 4.7.
As in the classical case, Equation (4.3) shows that the convex functionalof a continuous semimartingale is also a continuous semimartingale.
The running maximum (or more precisely, supremum) is defined as m ( Y t ) = sup ≤ s ≤ t y s , (4.4)for any Y t ∈ Λ .Let us first verify that m is Λ -continuous. Notice m ( Y t ) = m ( Y t , r ) , for any Y t ∈ Λ and r ≥
0. Hence, if we fix Y t , Z s ∈ Λ with s ≥ t , we find | m ( Y t ) − m ( Z s ) | = | m ( Y t , s − t ) − m ( Z s ) | = (cid:12)(cid:12)(cid:12)(cid:12) sup ≤ u ≤ s Y t , s − t ( u ) − sup ≤ u ≤ s Z s ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup ≤ u ≤ s | Y t , s − t ( u ) − Z s ( u ) | ≤ d Λ ( Y t , Z s ) . Therefore, the running maximum is (Lipschitz) Λ -continuous. Moreover, one couldalso verify that ∆ t m ( Y t ) =
0. Define now the subset of Λ where the supremum is at-tained at the last value: S = { Y t ∈ Λ ; m ( Y t ) = y t } . For paths in S , the space functional derivative is not defined: the right derivative is1 and the left derivative is 0. For paths outside S , the space functional derivative iswell-defined and it is 0: ∆ x m ( Y t ) =
0, for Y t / ∈ S .We show below that the running maximum is Λ -equicontinuous according to Defi-nition 2.5. One can easily see that M ( Y t , ξ ) = m ( Y ξ t ) = max (cid:8) m ( Y t ) , y t + ξ + (cid:9) . Y t , Z s ∈ Λ , | M ( Y t , ξ ) − M ( Z s , ξ ) | ≤ d Λ ( Y ξ t , Z ξ s ) = d Λ ( Y t , Z s ) . Since the bound above is independent of ξ , the running maximum is Λ -equicontinuous.Besides, we notice that m (( Y t , δ t ) ξ ) = max (cid:8) m ( Y t ) , y t + ξ + (cid:9) = m (( Y ξ t ) t , δ t ) , and therefore, m satisfies Definition 2.7. Furthermore, this shows that the runningmaximum is locally weakly path-dependent, i.e. the Lie bracket is zero (in the limitcharacterization), see Remark 2.9.Additionally, one can easily prove that the running maximum m ( Y t ) is a (non-smooth) convex functional. It is actually convex in the stronger sense of (4.1). Ad-ditionally, ∂ − h M ( Y t , h ) = ∆ − x m ( Y ht ) = ∆ t m ( Y t ) = ∆ − x m ( Y t ) = , ∀ Y t ∈ Λ . Notice now m ( Y t − ) = sup ≤ s < t y u (time t is not allowed in the supremum) and notice that M ( Y t , y + h ) = m ( Y y + ht − ) = max { y + h , m ( Y t − ) } . Hence, we can compute ∂ − y M ( Y t , y ) = { y > m ( Y t − ) } ⇒ d y ∂ − y M ( Y t , y ) = δ m ( Y t − ) ( dy ) , where δ c is the Dirac mass concentrated at c ∈ R . We then face a problem, because d t , y ∂ − y M ( Y t , y ) is not easily computed. However, we notice that ∂ − y M ( Y t , y + m ( Y t − )) = { y > } ⇒ d y ∂ − y M ( Y t , y + m ( Y t − )) = δ ( dy ) and d t , y ∂ − y M ( Y t , y + m ( Y t − )) = . Hence, we have seen that m satisfies Hypotheses 4.4. We will then apply formula (3.12)with a t = m ( X t ) = m ( X t − ) , which is clearly a continuous process of finite variation,since ( x t ) t ≥ is a continuous semimartingale. These equalities hold because the process x is continuous. Therefore, by Equation (3.12), we finally find the pathwise version ofthe important formula of L´evy: m t = max ≤ s ≤ t x s = x + L x − m ( t , ) , L x − m is the local time of the process ( x t − m t ) t ∈ [ , T ] .Furthermore, the same analysis could be performed for the running minimum. In-deed, m ( Y t ) = inf ≤ s ≤ t y s = − m ( − Y t ) , (4.5)where − Y t ( u ) = − y u , for all u ≤ t . Therefore, m satisfies Hypotheses 4.4 as well and M ( Y t , y ) = − M ( − Y t , − y ) ⇒ ∂ − y M ( Y t , y ) = { y < m ( Y t − ) } , where m ( Y t − ) = inf ≤ s < t y u . Then, m t = min ≤ s ≤ t x s = x − L x − m ( t , ) , In the articles [24, 25], the authors studied the problem of complete characterization oflocal martingales that are functions of the current state of a continuous local martingaleand its running maximum. In this section, we will show how the functional Itˆo calculusframework can be used to study this problem.
Theorem 4.8.
Let ( x t ) t ≥ be a continuous local martingale and consider a functionalf satisfying Hypotheses 3.5. Then ( f ( X t )) t ≥ is a local martingale if and only if (cid:90) t ∆ t f ( X s ) ds + (cid:90) R L x ( t , y ) d y ∂ − y F ( X t , y ) (4.6) = (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) , ∀ t ≥ , P − a.s.hen, if ∆ t f ( X s ) = , ∀ s ≥ , P -a.s. and if ∂ y ∂ − y F ( X s , y ) exists, Equation (4.6) isequivalent to (cid:90) R (cid:90) t ∂ y ∂ − y F ( X s , y ) d s L x ( s , y ) dy = . (4.7) Proof.
By the functional Meyer-Tanaka formula, Equation (3.6), we find f ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) t ∆ − x f ( X s ) dx s + (cid:90) R L x ( t , y ) d y ∂ − y F ( X t , y ) − (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) . ( f ( X t )) t ≥ is a local martingale if and only if (cid:90) t ∆ t f ( X s ) ds + (cid:90) R L x ( t , y ) d y ∂ − y F ( X t , y ) = (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) . Furthermore, Equation (4.7) follows from (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) = (cid:90) R (cid:90) t L x ( s , y ) d s ∂ y ∂ − y F ( X s , y ) dy = (cid:90) R (cid:18) L x ( t , y ) ∂ y ∂ − y F ( X t , y ) − (cid:90) t ∂ y ∂ − y F ( X s , y ) d s L x ( s , y ) (cid:19) dy . Remark 4.9.
Let ( x t ) t ≥ be a continuous local martingale. Denote C = (cid:91) t > supp ( m t ) ⊂ R , where supp ( z ) is the support of the random variable z . By the Dambis-Dubins-SchwarzTheorem [19, Theorem 4.6, Chapter 3], x t = b (cid:104) x (cid:105) t , where b is a Brownian motion, whichimplies that C = R + for any continuous local martingale. This will be useful in theproof of the next theorem. Theorem 4.10.
Let ( x t ) t ≥ be a continuous local martingale with x = and considerH : R −→ R in C ( R ) . Then ( H ( x t , m t )) t ≥ is a right-continuous local martingale inthe natural filtration of x if and only if there exists ψ : R −→ R in C ( R ) such thatH ( x , x ) = (cid:90) x ψ ( s ) ds − ψ ( x )( x − x ) + H ( , ) , ∀ ( x , x ) ∈ R . (4.8) Proof.
We start by defining the functional f ( Y t ) = H ( y t , m ( Y t )) .Since m ( Y t , δ t ) = m ( Y t ) and ∆ − x m ( Y t ) =
0, we easily conclude that ∆ t f ( Y t ) = ∆ − x f ( Y t ) = ∂ H ( y t , m ( Y t )) , where ∂ i denotes the derivative with respect to i th variableof H , i = ,
2. Smoothness of H implies that f satisfies Hypotheses 3.5.To ease the burden of notation, notice that m ( X t − ) = m ( X t ) = m t , since x is con-tinuous almost surely. By Theorem 4.8, ( f ( X t )) t ≥ is a local martingale if and onlyif (cid:90) R (cid:90) t ∂ y ∂ − y F ( X t , y + m t ) d s L x ( s , y ) dy = , ∀ t ≥ , P − a.s. (4.9)By a mollification argument, we may assume for the moment that H ∈ C ( R ) and thenwe are able to directly compute ∂ y ∂ − y F ( X t , y + m t ) .Note that F ( X t , y + m t ) = H ( y + m t , max { y + m t , m t } ) . Hence, since max { y + m t , m t } = m t + y + , we find ∂ − y F ( X t , y + m t ) = ∂ H ( y + m t , m t + y + )+ ∂ H ( y + m t , m t + y + ) { y > } , ∂ y ∂ − y F ( X t , y + m t ) = ∂ H ( y + m t , m t + y + )+ { y > } ( ∂ + ∂ ) H ( y + m t , m t + y + )+ ∂ H ( y + m t , m t + y + ) δ . Therefore, (cid:90) R (cid:90) t ∂ y ∂ − y F ( X t , y + m t ) d s L x − m ( s , y ) dy = (cid:90) R (cid:90) t ∂ H ( y + m t , m t + y + ) d s L x − m ( s , y ) dy + (cid:90) + ∞ (cid:90) t ( ∂ + ∂ ) H ( y + m t , m t + y ) d s L x − m ( s , y ) dy + (cid:90) t ∂ H ( m t , m t ) d s L x − m ( s , )= (cid:90) − ∞ (cid:90) t ∂ H ( y + m t , m t + y + ) d s L x − m ( s , y ) dy + (cid:90) t ∂ H ( m t , m t ) d s L x − m ( s , ) since L x − m ( t , y ) =
0, for y >
0. Then, by Equation (4.9) and Remark 4.9 and since themeasures d s L x − m ( s , y ) have disjoint supports for different y <
0, we must have, for all ( x , x ) ∈ R × R + , ∂ H ( x , x ) = ∂ H ( x , x ) = . (4.10)These equations can be solved analytically. The first equation above implies thereexists ψ , ϕ ∈ C ( R ) such that H ( x , x ) = ψ ( x ) x + ϕ ( x ) . Then, by the first equation in (4.10) we find that ψ (cid:48) ( x ) x + ϕ (cid:48) ( x ) = , which means ϕ ( x ) = ϕ ( ) − (cid:90) x ψ (cid:48) ( s ) sds = ϕ ( ) − ψ ( x ) x + (cid:90) x ψ ( s ) ds . Moreover, notice that ϕ ( ) = H ( , ) . Therefore, a function H ∈ C ( R ) is such that ( H ( x t , m t )) t ≥ is a local martingale if and only if there exists ψ ∈ C ( R ) such that H ( x , x ) = (cid:90) x ψ ( s ) ds − ψ ( x )( x − x ) + H ( , ) , ∀ ( x , x ) ∈ R . emark 4.11. It is proved in [24] that ψ ( ξ ) = d (cid:104) x , H ( x , m ) (cid:105) t d (cid:104) x (cid:105) t (cid:12)(cid:12)(cid:12)(cid:12) t = T ξ , where T ξ = inf { t ; x t = ξ } . Within the functional framework, it easy to see that ψ ( m ( Y t )) = ∆ − x H ( y t , m ( Y t )) . (4.11)This formula could be evaluated pathwise to find ψ ( ξ ) . The functional Meyer-Tanaka formula, Theorem 3.6, could provide interesting resultseven when applied to smooth functionals. As an illustrative example, let us considerthe quadratic variation functional QV, see [23] for the proper pathwise definition anddiscussion on its smoothness. It is straightforward and intuitive that ∆ t QV = ( Y yt − ) = QV ( Y t − ) + ( y − y t − ) . Therefore, the functional Meyer-Tanaka formulagives us the well-known formula (cid:104) x (cid:105) t = (cid:90) R L x ( t , y ) dy , for any continuous semimartingale x . Definition 4.12.
A functional f : Λ −→ R is called increasing if f ( Y t ) ≥ f ( Y s ) , for all Y t ∈ Λ and s ≤ t , where Y s is the restriction of Y t to [ , s ] .Consider now an increasing functional f in C , with ∆ x f ∈ C , . Then, we findthat ∆ t f ≥ ( f ( Y t )) t ∈ [ , T ] is of finite variation for any Y T Λ T . Hence,if ( w t ) t ∈ [ , T ] is a Brownian motion in ( Ω , F , P ) , by the Functional Itˆo Formula, f ( W t ) = f ( W ) + (cid:90) t ∆ t f ( W u ) du + (cid:90) t ∆ x f ( W u ) dw u + (cid:90) t ∆ xx f ( W u ) du . Now, since the increasing process ( f ( W t )) t ≥ is of finite variation, by the uniquenessof the semimartingale decomposition, we conclude that ∆ x f ( W u ) =
0, for u ∈ [ , T ] .Since ∆ x f ∈ C , and the support of Brownian paths is the set of continuous paths, wehave ∆ xx f ( Y t ) = Y t , see [16, Theorem 2.2]. Therefore, f ( W t ) = f ( W ) + (cid:90) t ∆ t f ( W u ) du . Furthermore, by the Λ -continuity of the functionals involved in the equality above, weconclude that f ( Y t ) = f ( Y ) + (cid:90) t ∆ t f ( Y u ) du , Y t . What happens if the functional f is not smooth, but satisfiesHypotheses 3.5? In this case, for any local martingale x , f ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) t ∆ − x f ( X s ) dx s + (cid:90) R L x ( t , y ) d y ∂ − y F ( X t , y ) − (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) , and, for the same reason, the stochastic integral term vanishes and we conclude that f ( X t ) = f ( X ) + (cid:90) t ∆ t f ( X s ) ds + (cid:90) R L x ( t , y ) d y ∂ − y F ( X t , y ) − (cid:90) t (cid:90) R L x ( s , y ) d s , y ∂ − y F ( X s , y ) . Acknowledgements
Firstly, I express my gratitude to B. Dupire for proposing such interesting problemand for the helpful discussions. I am thankful to J.-P. Fouque and T. Ichiba for all theinsightful comments. Part of the research was carried out in part during the summerinternship of 2013 supervised by B. Dupire at Bloomberg LP.
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