A financial market with singular drift and no arbitrage
aa r X i v : . [ q -f i n . M F ] A ug A financial market with singular drift and no arbitrage
Nacira AGRAM and Bernt ØKSENDAL
21 August 2020
MSC(2010):
Keywords:
Jump diffusion, financial market with a local time drift term; arbitrage; opti-mal portfolio; delayed information; Donsker delta function; white noise calculus.
Abstract
We study a financial market where the risky asset is modelled by a geometric Itˆo-L´evy process, with a singular drift term. This can for example model a situation wherethe asset price is partially controlled by a company which intervenes when the priceis reaching a certain lower barrier. See e.g. Jarrow & Protter [JP] for an explanationand discussion of this model in the Brownian motion case. As already pointed out byKaratzas & Shreve [KS] (in the continuous setting), this allows for arbitrages in themarket. However, the situation in the case of jumps is not clear. Moreover, it is notclear what happens if there is a delay in the system.In this paper we consider a jump diffusion market model with a singular drift termmodelled as the local time of a given process, and with a delay θ > θ >
0. This implies that there is no arbitrage in the market in thatcase. However, when θ goes to 0, the value goes to infinity. This is in agreement withthe above result that is an arbitrage when there is no delay.Our model is also relevant for high frequency trading issues. This is because highfrequency trading often leads to intensive trading taking place on close to infinitesimal Department of Mathematics, Linnaeus University SE-351 95 V¨axj¨o, Sweden.Email: [email protected]. Department of Mathematics, University of Oslo, Box 1053 Blindern, NO-0316 Oslo, Norway.Email: [email protected].
This research was carried out with support of the Norwegian Research Council, within the researchproject Challenges in Stochastic Control, Information and Applications (STOCONINF), project number250768/F20. engths of time, which in the limit corresponds to trading on time sets of measure 0.This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al [LLLL] and the references therein. It is well-known that in the classical Black-Scholes market, there is no arbitrage. However,if we include a singular term in the drift of the risky asset, it was first proved by Karatzas &Shreve [KS] (Theorem B2, page 329), that arbitrages exist. Subsequently this type of markethas been studied by several authors, including Jarrow & Protter [JP]. They explain how asingular term in the drift can model a situation where the asset price is partially controlledby a large company which intervenes when the price is reaching a certain lower barrier, inorder to prevent it from going below that barrier. They also prove that arbitrages can occurin such situations.The purpose of our paper is to extend this study in two directions:First, we introduce jumps in the market. More precisely, we study a jump diffusion marketdriven by a Brownian motion B ( · ) and an independent compensated random measure e N ( · , · )with an added singular drift term, modelled by a local time of an underlying L´evy process Y ( · ). In view of the unstable financial markets we have seen in recent years, and in particularduring the economic crises in 2008 and the corona virus crisis this year, we think that jumpsare useful in an attempt to obtain more realistic financial market models.Introducing jumps in the stock price motion goes back to Cox & Ross [CR] and to Merton[M].Second, we assume that the trader only has access to a delayed information flow, representedby the filtration F t − θ , where θ > F t is the sigma-algebra generatedby both { B ( s ) } s ≤ t and { N ( s, · ) } s ≤ t . This extension is also motivated by the effort to get morerealistic market models. Indeed, in all real-life, markets there is delay in the information flowavailable, and traders are willing to pay to get the most recent price information. Especially,when trading with computers even fractions of seconds of delays are important.We represent the singular term by the local time of a given process and show that as longas θ > viable , in the sense that the value of the optimal portfolio problem with logarithmic utilityis finite. However, if the delay goes to 0, the value of the portfolio goes to infinity, at leastunder some additional assumptions.We emphasize that our paper deals with delayed information flow , not delay in the coefficientsin the model, as for example in the paper by Arriojas et al [AHMP1]. There are many paperson optimal stochastic control with delayed information flow, also by us. However, to the bestof our knowledge the current paper is the first to discuss the effect of delay in the informationflow on arbitrage opportunities in markets with a singular drift coefficient. We will show thatby applying techniques from white noise theory we can obtain explicit results. Specifically,our model is the following: 2uppose we have a financial market with the following two investment possibilities: • A risk free investment (e.g. a bond or a (safe) bank account), whose unit price S ( t )at time t is described by ( dS ( t ) = r ( t ) S ( t ) dt ; t ∈ [0 , T ] ,S (0) = 1 . (1.1) • A risky investment, whose unit price S ( t ) at time t is given by a linear stochasticdifferential equation (SDE) of the form ( dS ( t ) = S ( t − )[ µ ( t ) dt + α ( t ) dL t + σ ( t ) dB ( t ) + R R γ ( t, ζ ) e N ( dt, dζ )]; t ∈ [0 , T ] ,S (0) > , (1.2) where R = R \ { } . Here B ( · ) and e N = N ( dt, dζ ) − ν ( dζ ) dt is a standard Brownianmotion and an independent compensated Poisson random measure, respectively, defined ona complete filtered probability space (Ω , F , P ) equipped with the filtration F = {F t } t ≥ generated by the Brownian motion B ( · ) and N ( · ). The measure ν is the L´evy measure ofthe Poisson random measure N , and the singular term L t = L t ( y ) is represented as the localtime at a point y ∈ R of a given F -predictable process Y ( · ) of the form Y ( t ) = Z t φ ( s ) dB ( s ) + Z t Z R ψ ( s, ζ ) ˜ N ( ds, dζ ) , (1.3)for some real deterministic functions φ : [0 , T ] → R , ψ : [0 , T ] × R → R satisfying0 < Z Tt n φ ( t ) + Z R ψ ( t, ζ ) ν ( dζ ) o dt < ∞ a.s. for all t ∈ [0 , T ] . (1.4)The coefficients r ( t ) , µ ( t ) , α ( t ) , σ ( t ) > γ ( t, ζ ) > F -predictableprocesses, with σ ( t ) bounded away from 0.In this market we introduce a portfolio process u : [0 , T ] × Ω → R giving the fraction of thewealth invested in the risky asset at time t , and a consumption rate process c : [0 , T ] × Ω → R + giving the fraction of the wealth consumed at time t . We assume that at any time t both u ( t ) and c ( t ) are required to be adapted to a given possibly smaller filtration G = {G t } t ∈ [0 ,T ] with G t ⊆ F t for all t . For example, it could be a delayed information flow, with G t = F max(0 ,t − θ ) , t ≥ , for some delay θ > . (1.5)This case will be discussed in detail later.Let us denote by A G the set of all admissible consumption and portfolio processes. Wesay that c and u are admissible and write c, u ∈ A G if, in addition, u is self-financing and3 h R T ( u ( t ) + c ( t ) ) dt i < ∞ , where E denotes expectation with respect to P . Note that if c, u areadmissible, then the corresponding wealth process X ( t ) = X c,u ( t ) is described by the equation dX ( t ) = X ( t − )[(1 − u ( t )) r ( t ) + u ( t ) µ ( t ) − c ( t )] dt + u ( t ) α ( t ) dL t (1.6)+ u ( t ) σ ( t ) dB ( t ) + u ( t ) R R γ ( t, ζ ) e N ( dt, dζ )] . For simplicity, we put the initial value X (0) = 1 . The optimal consumption and portfolio problem we study is the following:
Problem 1.1
Let a > , b > be given constants. Find admissible c ∗ , u ∗ , such that J ( c ∗ , u ∗ ) = sup c,u J ( c, u ) , (1.7) where J ( c, u ) = E h Z T a ln( c ( t ) X ( t )) dt + b ln( X ( T )) i . (1.8)Our results are the following:Using methods from white noise calculus we find explicit expressions for the optimal consumptionrate c ∗ ( t ) and the optimal portfolio u ∗ ( t ). Then we show that the value is finite for all positivedelays in the information flow. In particular, this shows that there is no arbitrage in that case.This result appears to be new.We also show that, under additional assumptions, the value goes to infinity when the delay goesto 0. This shows in particular that also when there are jumps the value is infinite when there isno delay, in agreement with the arbitrage results of Karatzas & Shreve [KS] and Jarrow & Protter[JP] in the Brownian motion case. Remark 1.2
In our problem we are using the logarithmic utility function, both for the consumptionand for the terminal value. It is natural to ask if similar results can be obtained for other utilityfunctions. The method used in this paper is quite specific for the logarithmic utility and will notwork for other cases. This issue will be discussed in a broader context in a future research.
As we have mentioned above, we will use white noise calculus to find explicit expressions for theoptimal consumption and the optimal portfolio. Specifically, we will define the local time in theterms of the Donsker delta function which is an element of the Hida space of stochastic distributions( S ) ∗ . A brief introduction to white noise calculus is given in the Appendix. For more informationon the underlying white noise theory we refer to Hida et al [HKPS], Oliveira [O], Holden et al [HOUZ] and Di Nunno et al [DOP] and Agram & Øksendal [AO]. We now define the Donsker delta function and give some of its properties. It will play a crucialrole in our computations. efinition 2.1 Let Y : Ω → R be a random variable which also belongs to the Hida space ( S ) ∗ ofstochastic distributions. Then a continuous functional δ Y ( · ) : R → ( S ) ∗ (2.1) is called a Donsker delta function of Y if it has the property that Z R g ( y ) δ Y ( y ) dy = g ( Y ) , a.s. (2.2) for all (measurable) g : R → R , such that the integral converges. Explicit formulas for the Donsker delta function are known in many cases. For the Gaussian case,see Section 3.2. For details and more general cases, see e.g. Aase et al [AaU].In particular, for our process Y described by the diffusion (1.3), it is well known (see e.g. [MOP],[DiO], [DOP]) that the Donsker delta functional exists in ( S ) ∗ and is given by δ Y ( t ) ( y ) = 12 π R R exp ⋄ (cid:2)R t R R ( e ixψ ( s,ζ ) −
1) ˜ N ( ds, dζ ) + R t ixφ ( s ) dB ( s )+ R t { R R ( e ixψ ( s,ζ ) − − ixψ ( s, ζ )) ν ( dζ ) − x φ ( s ) } ds − ixy (cid:3) dx, (2.3)where exp ⋄ denotes the Wick exponential.Moreover, if 0 ≤ s ≤ t, we can compute the conditional expectation E [ δ Y ( t ) ( y ) |F s ] (2.4)= 12 π R R exp (cid:2) R s R R ixψ ( r, ζ ) ˜ N ( dr, dζ ) + R s ixφ ( r ) dB ( r )+ R ts R R ( e ixψ ( r,ζ ) − − ixψ ( r, ζ )) ν ( dζ ) dr − R ts x φ ( r ) dr − ixy (cid:3) dx. Note that if we put s = 0 in (2.4), we get E [ δ Y ( t ) ( y )] = π R R exp (cid:16) − x R t φ ( r ) dr + R t R R ( e ixψ ( r,ζ ) − − ixψ ( r, ζ )) ν ( dζ ) dr − ixy (cid:17) dx < ∞ . Putting ν = 0 in (2.4), yields12 π Z R exp h Z s ixφ ( r ) dB ( r ) − R ts x φ ( r ) dr − ixy i dx = (cid:16) π R ts φ ( r ) dr (cid:17) − exp (cid:16) − (cid:0)R s φ ( r ) dB ( r ) − y (cid:1) R ts φ ( r ) dr (cid:17) , (2.5)where we have used, in general, for a > , b ∈ R , that Z R e − ax − bx dx = r πa e b a . (2.6) n particular, applying the above to the random variable Y ( t ) := B y ( t ) for some t ∈ (0 , T ]where B y is Brownian motion starting at y , we get for all 0 ≤ s < t , E [ δ B ( t ) ( y ) |F s ] = (2 π ( t − s )) − exp h − ( B ( s ) − y ) t − s ) i . (2.7)We will also need the following estimate: Lemma 2.2
Assume that ≤ s ≤ t ≤ T . Then E [ δ Y ( t ) ( y ) |F s ] ≤ (cid:16) π R ts n φ ( r ) + R R ψ ( r, ζ ) ν ( dζ ) o dr (cid:17) − . (2.8)Proof. From (2.4) we get, with i = √− | E [ δ Y ( t ) ( y ) |F s ] | ≤ π R R exp hR ts R R Re ( e ixψ ( r,ζ ) − − ixψ ( r, ζ )) ν ( dζ ) dr − R ts x φ ( r ) dr i dx ≤ π R R exp hR ts R R − x ψ ( r, ζ ) ν ( dζ ) dr − R ts x φ ( r ) dr i dx = π R R exp h − x R ts n φ ( r ) + R R ψ ( r, ζ )) ν ( dζ ) o dr i dx = (cid:16) π (cid:16)R ts n φ ( r ) + R R ψ ( r, ζ ) ν ( dζ ) o dr (cid:17) (cid:17) − . (cid:3) In this subsection we define the local time of Y ( · ) at y and we give a representation of it in termsof the Donsker delta function. Definition 2.3
The local time L t ( y ) of Y ( · ) at the point y and at time t is defined by L t ( y ) = lim ǫ → ǫ λ ( { s ∈ [0 , t ]; Y ( s ) ∈ ( y − ǫ, y + ǫ ) } ) , where λ denotes Lebesgue measure on R and the limit is in L ( λ × P ) . Remark 2.4
Note that this definition differs from the definition in Protter [P] Corollary 3, page230, in two ways:(i) We are using Lebesgue measure dλ ( s ) = ds as integrator, not d [ Y, Y ] s (ii) Protter [P] is defining left-sided and right-side local times. Our local time corresponds to theaverage of the two.If the process Y is Brownian motion both definitions coincide with the standard one. We chooseour definition because it is convenient for our purpose. There is a close connection between local time and the Donsker delta function of Y ( t ), given bythe following result. heorem 2.5 The local time L t ( y ) of Y at the point y and the time t is given by the following S ∗ -valued integral L t ( y ) = Z t δ Y ( s ) ( y ) ds, (2.9) where the integral converges in ( S ) ∗ . Proof. In the following we let χ F denote the indicator function of the Borel set F , i.e. χ F ( x ) = ( x ∈ F, x / ∈ F. (2.10)By definition of the local time and the Donsker delta function, we have L t ( y ) = lim ǫ → Z t ǫ χ ( y − ǫ,y + ǫ ) ( Y ( s )) ds = lim ǫ → R t (cid:16) R R ǫ χ ( y − ǫ,y + ǫ ) ( x ) δ Y ( s ) ( x ) dx (cid:17) ds = lim ǫ → Z R ǫ χ ( y − ǫ,y + ǫ ) ( x ) (cid:16) R t δ Y ( s ) ( x ) ds (cid:17) dx = R t δ Y ( s ) ( y ) ds, because the function y δ Y ( s ) ( y ) is continuous in ( S ) ∗ . (cid:3) We now return to the model in the Introduction. Thus we consider the optimal portfolio andconsumption problem (1.7)-(1.8) of an agent in the financial market (1.1) & (1.2). The agent hasaccess to a partial information flow G = {G t } t ≥ where G t ⊆ F t for all t . It is known that if G = F ,i.e. G t = F t for all t , and if there are no jumps ( N = ν = 0), then the market is complete and itallows an arbitrage. See Karatzas & Shreve [KS] and Jarrow & Protter [JP]. It is clear that ourmarket with jumps is not complete, even if G = F . However, we will show that if G t = F t − θ for somedelay θ >
0, then the market is viable (i.e. the optimal consumption and portfolio problem hasa finite value) and it has no arbitrage. Moreover, we will find explicitly the optimal consumptionand portfolio rates. If the delay goes to 0, we show that the value goes to infinity, in agreementwith the existence of arbitrage in the no-delay case.First we need the following auxiliary result.
Lemma 3.1
Suppose that E [ δ Y ( t ) ( y ) |G t ] ∈ L ( P ) and that µ ( t ) − r ( t ) + α ( t ) E [ δ Y ( t ) ( y ) |G t ] > . Then there exists a unique solution u ( t ) = u ∗ ( t ) > of the equation ( a + b ) σ ( t ) u ∗ ( t ) + [ a ( T − t ) + b ] Z R u ∗ ( t ) γ ( t, ζ )1 + u ∗ ( t ) γ ( t, ζ ) ν ( dζ )= ( a ( T − t ) + b )[ µ ( t ) − r ( t ) + α ( t ) E [ δ Y ( t ) ( y ) |G t ]] . roof. Define F ( u ) = a u + a Z R uγ ( t, ζ )1 + uγ ( t, ζ ) ν ( dζ ) , u ≥ , where a = ( a + b ) σ ( t ) , a = a ( T − t ) + b. Then F ′ ( u ) = a + a Z R γ ( t, ζ )(1 + γ ( t, ζ )) ν ( dζ ) > , and F (0) = 0 , lim u →∞ F ( u ) = ∞ . Therefore, for all a > u > F ( u ) = a . (cid:3) We can now proceed to our first main result:
Theorem 3.2 (Optimal consumption and portfolio)
Assume that α and γ > are G -adaptedand that E [ δ Y ( t ) ( y ) |G t ] ∈ L ( λ × P ) and E [ µ ( t ) − r ( t ) |G t ] + α ( t ) E [ δ Y ( t ) ( y ) |G t ] > , for all t ∈ [0 , T ] . Then the optimal consumption rate is c ∗ ( t ) = c ∗ ( t ) = ab + a ( T − t ) , and the optimal portfolio is given as the unique solution u ∗ ( t ) > of the equation ( a + b ) E [ σ ( t ) |G t ] u ∗ ( t ) + ( a ( T − t ) + b ) Z R u ∗ ( t ) γ ( t, ζ )1 + u ∗ ( t ) γ ( t, ζ ) ν ( dζ )= ( a ( T − t ) + b ) (cid:16) E [ µ ( t ) − r ( t ) |G t ] + α ( t ) E [ δ Y ( t ) ( y ) |G t ] (cid:17) . In particular, if there are no jumps ( N = ν = 0 ), the optimal portfolio will be u ∗ ( t ) = ( a ( T − t ) + b ) (cid:16) E [ µ ( t ) − r ( t ) |G t ] + α ( t ) E [ δ Y ( t ) ( y ) |G t ] (cid:17) ( a + b ) E [ σ ( t ) |G t ] . Proof. By the Itˆo formula for semimartingales, see e.g. Protter [P], we get that the solution of(4.2) is X ( t ) = exp (cid:16) Z t u ( s ) σ ( s ) dB ( s ) + Z t (cid:26) r ( s ) + [ µ ( s ) − r ( s )] u ( s ) − c ( s ) − σ ( s ) u ( s ) (cid:27) ds + Z t u ( s ) α ( s ) dL s + Z t Z R { ln(1 + u ( s ) γ ( s, ζ )) − u ( s ) γ ( s, ζ ) } ν ( dζ ) ds + Z t Z R ln { u ( s ) γ ( s, ζ ) } e N ( ds, dζ ) (cid:17) . ince σ and γ are bounded and u ∈ A G the stochastic integrals in the exponent have expectation0. Therefore we get E | ln( X ( t ))] = E h Z t { r ( s ) + [ µ ( s ) − r ( s )] u ( s ) − c ( s ) − σ ( s ) u ( s ) } ds + Z t u ( s ) α ( s ) dL s + Z t Z R { ln(1 + u ( s ) γ ( s, ζ )) − u ( s ) γ ( s, ζ ) } ν ( dζ ) ds i . (3.1)Formulas (4.2) and (1.8) and the Itˆo formula, lead to J ( c, u ) = E h Z T a ln( c ( t ) X ( t )) dt + b ln( X ( T )) i = E h Z T n a ln( c ( t )) + a ln( X ( t )) + b (cid:16) r ( t ) + [ µ ( t ) − r ( t )] u ( t ) − c ( t ) − σ ( t ) u ( t ) (cid:17)o dt + b Z T u ( t ) α ( t ) dL t + b Z T Z R { ln(1 + u ( t ) γ ( t, ζ )) − u ( t ) γ ( t, ζ ) } ν ( dζ ) dt i . Substituting (3.1) in the above, gives J ( c, u ) = E h Z T n a ln( c ( t )) + a (cid:16) Z t { r ( s ) + [ µ ( s ) − r ( s )] u ( s ) − c ( s ) − σ ( s ) u ( s ) } ds + Z t u ( s ) α ( s ) dL s + Z t Z R { ln(1 + u ( s ) γ ( s, ζ )) − π ( s ) γ ( s, ζ ) } ν ( dζ ) ds (cid:17) + b (cid:16) Z T n r ( t ) + [ µ ( t ) − r ( t )] u ( t ) − c ( t ) − σ ( t ) u ( t ) (cid:17)o dt + Z T u ( t ) α ( t ) dL t + Z T Z R { ln(1 + u ( t ) γ ( t, ζ )) − u ( t ) γ ( t, ζ ) } ν ( dζ ) dt (cid:17)i . Note that in general, we have, by the Fubini theorem, Z T (cid:16) Z t h ( s ) ds (cid:17) dt = Z T (cid:16) Z Ts h ( s ) dt (cid:17) ds = Z T ( T − s ) h ( s ) ds = Z T ( T − t ) h ( t ) dt, and Z T (cid:16) Z t h ( s ) dL s (cid:17) dt = Z T (cid:16) Z Ts h ( s ) dt (cid:17) dL s = Z T ( T − s ) h ( s ) dL s = Z T ( T − t ) h ( t ) dL t . Therefore, using that dL t = dL t ( y ) = δ Y ( t ) ( y ) dt, e get from the above that J ( c, u ) = E h Z T E hn a (cid:16) ln( c ( t )) + ( T − t ) { r ( t ) + [ µ ( t ) − r ( t )] u ( t ) − c ( t ) − σ ( t ) u ( t ) (3.2)+ ( T − t ) u ( t ) α ( t ) δ Y ( t ) ( y )+ ( T − t ) Z R { ln(1 + u ( t ) γ ( t, ζ )) − u ( t ) γ ( t, ζ ) } ν ( dζ ) (cid:17) dt + b (cid:16) r ( t ) + [ µ ( t ) − r ( t )] u ( t ) − c ( t ) − σ ( t ) u ( t ) } + u ( t ) α ( t ) δ Y ( t ) (0)+ Z R { ln(1 + u ( t ) γ ( t, ζ )) − u ( t ) γ ( t, ζ ) } ν ( dζ ) (cid:17)o(cid:12)(cid:12)(cid:12) G t i dt i . Using that c, u, α and γ are G -adapted, we obtain J ( c, u ) = E h Z T n a (cid:16) ln( c ( t )) + ( T − t ) { E [ r ( t ) |G t ] + E [ µ ( t ) − r ( t ) |G t ] u ( t ) − c ( t ) − E [ σ ( t ) |G t ] u ( t ) } + ( T − t ) u ( t ) α ( t ) E [ δ Y ( t ) ( y ) |G t ]+ ( T − t ) Z R { ln(1 + u ( t ) γ ( t, ζ )) − u ( t ) γ ( t, ζ ) } ν ( dζ ) (cid:17) + b (cid:16) E [ r ( t ) |G t ] + E [ µ ( t ) − r ( t ) |G t ] u ( t ) − c ( t ) − E [ σ ( t ) |G t ] u ( t ) + u ( t ) α ( t ) E [ δ Y ( t ) ( y ) |G t ]+ Z R { ln(1 + u ( t ) γ ( t, ζ )) − u ( t ) γ ( t, ζ ) } ν ( dζ ) (cid:17)o dt i . (3.3)This we can maximise pointwise over all possible values c, u ∈ A G by maximising for each t theintegrand. Then we get the optimal consumption rate c ∗ ( t ) = ab + a ( T − t ) , and the optimal portfolio is given as the unique solution u ∗ ( t ) > a + b ) E [ σ ( t ) |G t ] u ∗ ( t ) + [ a ( T − t ) + b ] R R u ∗ ( t ) γ ( t,ζ )1+ u ∗ ( t ) γ ( t,ζ ) ν ( dζ )= ( a ( T − t ) + b ) h E [ µ ( t ) − r ( t ) |G t ] + α ( t ) E [ δ Y ( t ) ( y ) |G t ] i . In particular, if there are no jumps ( N = ν = 0), we get u ∗ ( t ) = ( a ( T − t ) + b ) h E [ µ ( t ) − r ( t ) |G t ] + α ( t ) E [ δ Y ( t ) ( y ) |G t ] i ( a + b ) E [ σ ( t ) |G t ] . (cid:3) G t = F t − θ , t ≥ From now on we restrict ourselves to the subfiltration G t = F t − θ , t ≥ θ >
0, where we put F t − θ = F for t ≤ θ . In this case we can compute the optimal portfolio andthe optimal consumption explicitly. By (2.4) we have the following result: emma 3.3 Assume that α and γ > are G θ -adapted, where G θ = {F t − θ } t ≥ . For t ≥ θ we have E [ δ Y ( t ) ( y ) |F t − θ ] = π R R exp (cid:2) R t − θ R R ixψ ( r, ζ ) e N ( dr, dζ ) + R t − θ ixφ ( r ) dB ( r )+ R tt − θ R R ( e ixψ ( r,ζ ) − − ixψ ( r, ζ )) ν ( dζ ) dr − R tt − θ x φ ( r ) dr − ixy (cid:3) dx. (3.4) In particular, if ψ = 0 and φ = 1 , we get Y = B and (see also (2.7) ) E [ δ B ( t ) ( y ) |F t − θ ] = (2 πθ ) − exp h − ( B ( t − θ ) − y ) θ i . (3.5)Then by Theorem 3.2, we get Theorem 3.4
Suppose G t = F t − θ with θ > . Then the optimal consumption rate is given by c ∗ ( t ) = ab + a ( T − t ) , and the optimal portfolio is given as the unique solution u ∗ ( t ) > of the equation ( a + b ) E [ σ ( t ) |F t − θ ] u ∗ ( t ) + ( a ( T − t ) + b ) Z R u ∗ ( t ) γ ( t, ζ )1 + u ∗ ( t ) γ ( t, ζ ) ν ( dζ )= ( a ( T − t ) + b ) (cid:0) E [ µ ( t ) − r ( t ) |F t − θ ] + α ( t ) E [ δ Y ( t ) ( y ) |F t − θ ] (cid:1) . In particular, sup c,u J ( c, u ) = J ( c ∗ , u ∗ ) < ∞ , and there is no arbitrage in the market. In this section we concentrate on the delay case and with optimal portfolio only, i.e. withoutconsumption. Thus we are only considering utility from terminal wealth, and we put a = 0 and b = 1 in Theorem 3.2. Moreover, we assume that φ = 1 and ψ = 0, i.e. that Y ( t ) = B ( t ); t ∈ [0 , T ] . (4.1)Also, to simplify the calculations we assume that r = 0 and µ ( t ) = µ > , α ( t ) = α > , σ ( t ) = σ > γ ( t, ζ ) = γ ( ζ ) is deterministic and does not depend on t . Then the wealthequation will take the form dX ( t ) = X ( t )[ u ( t ) µdt + u ( t ) αdL t + u ( t ) σdB ( t ) + u ( t ) R R γ ( ζ ) e N ( dt, dζ )]; t ∈ [0 , T ] , X (0) = 1 , (4.2)where the singular term L t = L t ( y ) is represented as the local time at a point y ∈ R of B ( · ). Theperformance functional becomes J ( u ) = E [ln X ( u ) ( T )]; u ∈ A θ , here A θ denotes the set of all F t − θ -predictable control processes. This now gets the form J ( u ) = E h Z T n µu ( t ) − σ u ( t ) + u ( t ) α E [ δ Y ( t ) ( y ) |F t − θ ]+ Z R { ln(1 + u ( t ) γ ( ζ )) − u ( t ) γ ( ζ ) } ν ( dζ ) o dt i . (4.3)Our second main result is the following: Theorem 4.1
Suppose in addition to the above that Z R γ ( ζ ) ν ( dζ ) < σ . Then lim θ → + sup u ∈A θ J ( u ) = ∞ . In particular, if there is no delay ( θ = 0 ) the value of the optimal portfolio problem is infinite. Proof. For given θ > u θ ( t ) = µ + αRσ , where we for simplicity put R = R θ = E [ δ B ( t ) ( y ) |F t − θ ] . Then we see that J ( u θ ) ≥ E [( µ + αR ) ] (cid:16) − R R γ ( ζ ) ν ( dζ ) σ (cid:17) =: C E [( µ + αR ) ] ≥ C + C E [ R ] , since, by (2.8), E [ R ] < ∞ . Here C , C , C are positive constants.It remains to prove that E [ R θ ] → ∞ when θ → + . To this end, note that by (2.5) we have E [ R θ ] = E [( δ B ( t ) ( y ) |F t − θ ) ] = (2 πθ ) − E h exp (cid:16) − B ( t − θ ) − y ) θ (cid:17)i . (4.4)By formula 1.9.3(1) p.168 in [BS] we have, with κ > E [exp( − κ ( B ( t − θ ) − y ) )] = 11 + 2 κ ( t − θ ) exp (cid:16) − κy ( t − θ )1 + 2 κ ( t − θ ) (cid:17) . (4.5)Applying this to κ = θ we get E [ R θ ] = 12 π √ θ √ t − θ exp (cid:16) − y t − θ (cid:17) → ∞ , (4.6)when θ → (cid:3) The Brownian motion case
In the case when Y ( t ) = B ( t ) the computations above can be made more explicit. We now illustratethis, assuming for simplicity that y = 0. Then by Theorem 3.4 the optimal portfolio b u ( t ) is givenby b u ( t ) = µ + α Λ( t ) σ , where Λ( t ) = E [ δ B ( t ) (0) |F t − θ ] = (2 πθ ) − exp h − B ( t − θ ) θ i , t ≥ θ, Λ( t ) = 1 √ πθ , ≤ t ≤ θ. (5.1)By (4.3) and (5.1) we see, after some algebraic operations, that the corresponding performance b J θ = J (0 , b π ) is b J θ = E h Z T (cid:16) µ b π ( t ) − σ b π ( t ) + b π ( t ) α Λ( t ) (cid:17) dt i = E h Z T ( µ + α Λ( t )) σ dt i = A + A + A , where A = µ σ T, A = µασ E h Z T Λ( t ) dt i , A = α σ E h Z T Λ ( t ) dt i . Using the density of B ( s ), we get E h exp (cid:16) − B ( s )2 θ (cid:17)i = Z R exp (cid:18) − y θ (cid:19) √ πs exp (cid:18) − y s (cid:19) dy = 1 √ πs Z R exp (cid:18) − y ( 1 θ + 1 s ) (cid:19) dy. (5.2)In general we have, for a > Z R exp( − ay ) dy = r πa , we conclude, by putting s = t − θ in (5.2) , that A = θ √ πθ + Z Tθ µασ √ πθ r θt dt = r θ π + 2 µα ( √ T − √ θ ) σ √ π . Finally we use similar calculations to compute A = α σ (2 πθ ) − (cid:18) θ + Z Tθ Ψ( t ) dt (cid:19) , where, putting t − θ = s , ψ ( t ) = E (cid:20) exp (cid:18) − B ( s ) θ (cid:19)(cid:21) = Z R e − y θ √ πs e − y s dy = 1 √ πs Z R exp (cid:18) − y ( 1 θ + 12 s ) (cid:19) dy = 1 √ πs r π θ + s = 1 q sθ + 1 . his gives A = α σ (cid:18) π + Z T − θ π √ θ √ s + θ ds (cid:19) = α πσ √ T − θ − √ θ √ θ ! . We have proved the following:
Theorem 5.1
The optimal performance with a given delay θ > is given by b J θ = µ σ T + q θ π + µα ( √ T −√ θ ) σ √ π + α πσ (1 + √ T − θ −√ θ √ θ ) . In particular, b J θ → ∞ when θ → . Corollary 5.2 (i) For all information delays θ > the value of the optimal portfolio problem isfinite.(ii) When there is no information delay, i.e. when θ = 0 , the value is infinite. In this section we give a brief survey of the underlying theory of white noise analysis used in thispaper. For more details see e.g. Di Nunno et al [DOP] and Holden et al [HOUZ] and the referencestherein.
Definition 6.1
Let S ( R ) be the Schwartz space consisting of all real-valued rapidly decreasingfunctions f on R , i.e., lim | x |→∞ | x n f ( k ) ( x ) | = 0 , ∀ n, k ≥ . (6.1) Example 6.2
For instance C ∞ functions with compact support, f ( x ) = e − x , f ( x ) = e − x , ... areall functions in S ( R ) . For any n, k ≥ , define a norm k . k n,k on S ( R ) by k f k n,k = sup x ∈ R | x n f ( k ) ( x ) | . (6.2)Then the Schwartz space S ( R ), equipped with the topology defined by the family of seminorms {k . k n,k , n, k ≥ } is a Fr´echet space.Let S ′ ( R ) be the dual space of S ( R ). S ′ ( R ) is called the space of tempered distributions. Let B denote the family of all Borel subsets of S ( R ) equipped with the weak topology.From now on we will use the notation h a, b i that means a acting on b . Theorem 6.3 (Minlos)
Let E be a Fr´echet space with dual space E ∗ . A complex-valued function φ on E is the characteristic functional of a probability measure ν on E ∗ ,i.e., φ ( y ) = Z E ∗ e i h x,y i dν ( x ) , y ∈ E , (6.3) if and only if it satisfies the following conditions: . φ (0) = 1 ,2. φ is positive definite, i.e. n X j,k =1 z j ¯ z k φ ( a j − a k ) ≥ for all z j , z k ∈ C , a j , a k ∈ E , φ is continuous. Remark 6.4
The measure ν is uniquely determined by φ . Observe that φ (0) = ν ( E ∗ ) . Thus whencondition above is not assumed, then we can only conclude that ν is a finite measure. Let φ be the function on S ( R ) given by φ ( ξ ) = exp( − | ξ | ) , ξ ∈ S ( R ) , where | · | is the L ( R ) norm.Then it is easy to check that conditions 1-3 above are satisfied.Therefore, by the Minlos theorem there exists a unique probability measure P on S ′ ( R ) such thatexp (cid:18) − | ξ | (cid:19) = Z S ′ ( R ) e i h ω,ξ i dP ( ω ) , ξ ∈ S ( R ) . (6.4) Definition 6.5
The measure P is called the standard Gaussian measure on S ′ ( R ) . The probabilityspace ( S ′ ( R ) , B , P ) is called the white noise probability space. In the following we will use thenotation Ω = S ′ ( R ) and the elements of Ω are denoted by ω . The expectation with respect to P isdenoted by E [ · ] , Note that from (6.4) it follows that E [ h ω, ξ i ] = 0 for all ξ ∈ S ( R ) and, (6.5) E [ h ω, ξ i ] = | ξ | for all ξ ∈ S ( R ) (The Ito isometry). (6.6)Using the Ito isometry we see that we can extend the definition of h ω, ξ i from ξ ∈ S ( R ) to all φ ∈ L ( R ) as follows: h ω, φ i = lim n →∞ h ω, ξ n i (limit in L ( P )) , for any sequence ξ n ∈ S ( R ) converging to φ in L ( R ) . Thus for each t we can define B ( t, · ) ∈ L ( P ) by B ( t, ω ) = (cid:10) ω, χ [0 ,t ] ( · (cid:11) ) , t ≥ , ω ∈ Ω . Then the process { B ( t, ω ) } t ≥ ,ω ∈ Ω has stationary independent increments of mean 0 (by (6.5)), andthe variance of B ( t ) is t (by (6.6)). Moreover, by the Kolmogorov continuity theorem the processhas a continuous version. This version is a Brownian motion. This is the Brownian motion we workwith in this paper. .1.2 The Wiener-Itˆo chaos expansion Let the Hermite polynomials h n ( x ) be defined by h n ( x ) = ( − n e x d n dx n ( e − x ) , n = 0 , , , ... The first Hermite polynomials are h ( x ) = 1 , h ( x ) = x, h ( x ) = x − , h ( x ) = x − x, ... Let e k be the k th Hermite function defined by e k ( x ) := π − (( k − − e − x h k − ( √ x ) , k = 1 , , ... (6.7)Then { e k } k ≥ constitutes an orthonormal basis for L ( R ) and e k ∈ S ( R ) for all k . Define θ k ( ω ) := h ω, e k i = Z R e k ( x ) dB ( x, ω ) , ω ∈ Ω . (6.8)Let J denote the set of all finite multi-indices α = ( α , α , ..., α m ) , m = 1 , , ..., of non-negativeintegers α i . If α = ( α , ..., α m ) ∈ J , α = 0 , we put H α ( ω ) := m Y j =1 h α j ( θ j ( ω )) , ω ∈ Ω . (6.9)By a result of Itˆo we have that I m ( e b ⊗ α ) = Y j =1 h α j ( θ j ) = H α , (6.10)where I m denotes the m-iterated Itˆo integral, defined below.We set H := 1. Here and in the sequel the functions e , e , ... are defined in (6.7) and ⊗ and b ⊗ denote the tensor product and the symmetrized tensor product, respectively.The family { H α } α ∈J is an orthogonal basis for the Hilbert space L ( P ). In fact, we have thefollowing result. Theorem 6.6 (The Wiener-Itˆo chaos expansion theorem (I))
The family { H α } α ∈ J con-stitutes an orthogonal basis of L ( P ) . More precisely, for all F T -measurable X ∈ L ( P ) there exist(uniquely determined) numbers c α ∈ R , such that X = X α ∈J c α H α ∈ L ( P ) . (6.11) Moreover, we have k X k L ( P ) = X α ∈J α ! c α . (6.12) et us compare the above Theorem to the equivalent formulation of this theorem in terms ofiterated Itˆo integrals. In fact, if ψ ( t , t , ..., t n ) is a real deterministic symmetric function in its n variables t , ..., t n and ψ ∈ L ( R n ) , that is, k ψ k L ( R n ) := Z R n | ψ ( t , t , ..., t n ) | dt dt ...dt n then its n -iterated Itˆo integral is defined by I n ( ψ ) := Z R n ψdB ⊗ n = n ! Z ∞−∞ Z t n −∞ Z t n − −∞ ... Z t −∞ ψ ( t , t , ..., t n ) dB ( t ) dB ( t ) ...dB ( t n ) , where the integral on the right-hand side consists of n -iterated Itˆo integrals.Note that the integrand at each step is adapted to the filtration F . Applying the Itˆo isometry n times we see that E (cid:2) (cid:18)Z R n ψdB ⊗ n (cid:19) (cid:3) = n ! k ψ k L ( R n ) . (6.13)For n = 0 we adopt the convention that I ( ψ ) := Z R ψdB ⊗ = ψ = k ψ k L ( R ) , for ψ constant. Let e L ( R n ) denote the set of symmetric real functions on R n , which are squareintegrable with respect to Lebesgue measure.Then we have the following result: Theorem 6.7 (The Wiener Itˆo chaos expansion theorem (II))
For all F t - measurable X ∈ L ( P ) there exist (uniquely determined) deterministic functions f n ∈ e L ( R n ) such that X = ∞ X n =0 Z R n f n dB ⊗ n = ∞ X n =0 I n ( f n ) ∈ L ( P ) . (6.14) Moreover, we have the isometry k X k L ( P ) = ∞ X n =0 n ! k f n k L ( R n ) . (6.15)The connection between these two expansions in Theorem (6.6) and Theorem (6.7) is given by f n = X α ∈J , | α | = n c α e ⊗ α b ⊗ e ⊗ α b ⊗ ... b ⊗ e ⊗ α m m , n = 0 , , , ... where | α | = α + α ... + α m for α = ( α , ..., α m ) ∈ J , m = 1 , , ... Recall that the functions e , e , ... are defined in (6.7) and ⊗ and b ⊗ denote the tensor productand the symmetrized tensor product, respectively.Note that since H α = I m ( e b ⊗ α ) , for α ∈ J , | α | = m, we get that m ! k e b ⊗ α k L ( R m ) = α ! , (6.16)by combining (6.12) and (6.15) for X = X α . .1.3 Stochastic distribution spaces Analogous to the test functions S ( R ) and the tempered distributions S ′ ( R ) on the real line R , thereis a useful space of stochastic test functions ( S ) and a space of stochastic distributions ( S ) ∗ on thewhite noise probability space. We now explain this in detail:In the following we use the notation (2 N ) α = m Y j =1 (2 j ) α j , (6.17)if α = ( α , α , ... ).We define ε ( k ) = (0 , , ..., , ... ) , with 1 on the k th place. Thus we see that (2 N ) ε ( k ) = 2 k. The Kondratiev Spaces ( S ) , ( S ) − and the Hida Spaces ( S ) and ( S ) ∗ . Definition 6.8
Let ρ be a constant in [0 , . • Let k ∈ R . We say that f = P α ∈J a α H α ∈ L ( P ) belongs to the Kondratiev test functionHilbert space ( S ) k,ρ if k f k k,ρ := X α ∈J a α ( α !) ρ (2 N ) αk < ∞ . (6.18) • We define the Kondratiev test function space ( S ) ρ as the space( S ) ρ = \ k ∈ R ( S ) k,ρ equipped with the projective topology, that is, f n → f, n → ∞ , in ( S ) ρ if and only if k f n − f k k,ρ → , n → ∞ , for all k . • Let q ∈ R . We say that the formal sum F = P α ∈J b α H α belongs to the Kondratiev stochasticdistribution space ( S ) − q, − ρ if k f k − q, − ρ := X α ∈J b α ( α !) − ρ (2 N ) − αq < ∞ . (6.19)We define the Kondratiev distribution space ( S ) − ρ by( S ) − ρ = [ q ∈ R ( S ) − q, − ρ equipped with the inductive topology, that is, F n → F, n → ∞ , in ( S ) − ρ if and only if thereexists q such that k F n − F k − q, − ρ → , n → ∞ . If ρ = 0 we write ( S ) = ( S ) and ( S ) − = ( S ) ∗ . These spaces are called the
Hida test function space and the Hida distribution space , respec-tively. • If F = P α ∈J b α H α in ( S ) − , we define the generalized expectation E [ F ] of F by E [ F ] = b . (6.20)(Note that if F ∈ L ( P ), then the generalized expectation coincides with the usual expecta-tion, since E [ H α ] = 0 for all α = 0).Note that ( S ) − is the dual of ( S ) and ( S ) ∗ is the dual of ( S ). The action of F = P α ∈J b α H α ∈ ( S ) − on f = P α ∈J a α H α ∈ ( S ) is given by h F, f i = X α α ! a α b α . We have the inclusion ( S ) ⊂ ( S ) ⊂ L ( P ) ⊂ ( S ) ∗ ⊂ ( S ) − . Example 6.9
Since B ( t ) = (cid:10) ω, χ [0 ,t ] (cid:11) = ∞ X k =1 ( e k , χ [0 ,t ] ) h ω, e k i = ∞ X k =1 (cid:18)Z t e k ( s ) ds (cid:19) H ε ( k ) , we see that white noise • B ( t ) defined by • B ( t ) = ddt B ( t ) = ∞ X k =1 e k ( t ) H ε ( k ) , exists in ( S ) ∗ . In addition to a canonical vector space structure, the spaces ( S ) and ( S ) ∗ also have a naturalmultiplication given by the Wick product: Definition 6.10
Let X = P α ∈J a α H α and Y = P β ∈J b β H β be two elements of ( S ) ∗ . Then wedefine the Wick product of X and Y by X ⋄ Y = X α,β ∈J a α b β H α + β = X γ ∈J X α + β = γ a α b β H γ . xample 6.11 We have B ( t ) ⋄ B ( t ) = B ( t ) − t, and more generally (cid:18)Z R φ ( s ) dB ( s ) (cid:19) ⋄ (cid:18)Z R ψ ( s ) dB ( s ) (cid:19) = (cid:18)Z R φ ( s ) dB ( s ) (cid:19) . (cid:18)Z R ψ ( s ) dB ( s ) (cid:19) − Z R φ ( s ) ψ ( s ) ds, for all φ, ψ ∈ L ( R ) . Some basic properties of the Wick product.
We list some properties of the Wick product:1.
X, Y ∈ ( S ) ⇒ X ⋄ Y ∈ ( S ) .2. X, Y ∈ ( S ) − ⇒ X ⋄ Y ∈ ( S ) − .3. X, Y ∈ ( S ) ⇒ X ⋄ Y ∈ ( S ).4. X ⋄ Y = Y ⋄ X .5. X ⋄ ( Y ⋄ Z ) = ( X ⋄ Y ) ⋄ Z .6. X ⋄ ( Y + Z ) = X ⋄ Y + X ⋄ Z .7. I n ( f n ) ⋄ I m ( g m ) = I n + m ( f n b ⊗ g m ) . In view of the properties (1) and (4) we can define the Wick powers X ⋄ n ( n = 1 , , ... ) of X ∈ ( S ) − as X ⋄ n := X ⋄ X ⋄ ... ⋄ X (n times) . We put X ⋄ := 1. Similarly, we define the Wick exponential exp ⋄ X of X ∈ ( S ) − byexp ⋄ X := ∞ X n =0 n ! X ⋄ n ∈ ( S ) − . In view of the aforementioned properties, we have that( X + Y ) ⋄ = X ⋄ + 2 X ⋄ Y + Y ⋄ , and also exp ⋄ ( X + Y ) = exp ⋄ X ⋄ exp ⋄ Y, for X, Y ∈ S − . Let E [ X ] denote the generalized expectation of an element X ∈ ( S ). It coincides with the standardexpectation if X ∈ L ( P ). Then we see that E [ X ⋄ Y ] = E [ X ] E [ Y ] , for X, Y ∈ ( S ) − . Note that independence is not required for this identity to hold. By induction,it follows that E [exp ⋄ X ] = exp E [ X ] , for X ∈ ( S ) − . .1.5 Wick product, white noise and Itˆo integration One of the spectacular results in white noise theory is the following, which combines Wick product,white noise and Itˆo integration:
Theorem 6.12
Let ϕ ( t ) ∈ L ([0 , T ] × Ω) be F -adapted. Then the integral R T ϕ ( t ) ⋄ • B ( t ) dt existsin ( S ) ∗ and Z T ϕ ( t ) dB ( t ) = Z T ϕ ( t ) ⋄ • B ( t ) dt. (6.21) Remark 6.13
Heuristically, we can see that we obtain this result by using that • B ( t ) = ddt B ( t ) . Ifwe work in ( S ) ∗ this argument can be made rigorous. The construction we did above for Brownian motion can be modified to apply to other processes.For example, we obtain a white noise theory for L´evy processes if we proceed as follows (see [DOP]for details):
Definition 6.14
Let ν be a measure on R such that Z R ζ ν ( dζ ) < ∞ . (6.22) Define h ( ϕ ) = exp( Z R Ψ( ϕ ( x )) dx ); ϕ ∈ ( S ) , (6.23) where Ψ( w ) = Z R ( e iwζ − − iwζ ) ν ( dζ ); w ∈ R , i = √− . (6.24) Then h satisfies the conditions (i) - (iii) of the Minlos theorem 6.3. Therefore there exists aprobability measure Q on Ω = S ′ ( R ) such that E Q [ e i h ω,ϕ i ] := Z Ω e i h ω,ϕ i dQ ( ω ) = h ( ϕ ); ϕ ∈ ( S ) . (6.25) The triple (Ω , F , Q ) is called the (pure jump) L´evy white noise probability space. One can now easily verify the following • E Q [ h· , ϕ i ] = 0; ϕ ∈ ( S ) • E Q [ h· , ϕ i ] = K R R ϕ ( y ) dy ; ϕ ∈ ( S ), where K = R R ζ ν ( dζ ) . As we did for the Brownian motion, we use an approximation argument to define e η ( t ) = e η ( t, ω ) = (cid:10) ω, χ [0 ,t ] (cid:11) ; a.a. ( t, ω ) ∈ [0 , ∞ ) × Ω . (6.26)Then the following holds: Theorem 6.15
The stochastic process e η ( t ) has a c`adl`ag version. This version η ( t ) is a pure jumpL´evy process with L´evy measure ν . .2.2 Chaos expansion From now on we assume that the L´evy measure ν satisfies the following condition:For all ε > λ > Z R \ ( − ε,ε ) exp( λ | ζ | ) ν ( dζ ) < ∞ (6.27)This condition implies that the polynomials are dense in L ( ρ ), where ρ ( dζ ) = ζ ν ( dζ ) (6.28)Now let { l m } m ≥ = { , l , l , ... } be the orthogonolization of (cid:8) , ζ, ζ , ... (cid:9) with respect to the innerproduct of L ( ρ ).Define p j ( ζ ) := k l j − k − L ( ρ ) ζ j − ( ζ ); j = 1 , , ... (6.29)and m := (cid:18)Z R ζ ν ( dζ ) (cid:19) = k l k L ( ρ ) = k k L ( ρ ) . (6.30)In particular, p ( ζ ) = m − ζ or ζ = m p ( ζ ) . (6.31)Then { p j ( ζ ) } j ≥ is an orthonormal basis for L ( ν ).Define the bijection κ : N × N −→ N by κ ( i, j ) = j + ( i + j − i + j − / . (6.32)(1) (2) (4) ( i ) • −→ • • · · · • −→ (3) ւ (5) ւ• • (6) ւ• ...( j ) •↓ Let { e i ( t ) } i ≥ be the Hermite functions. Define δ κ ( i,j ) ( t, ζ ) = e i ( t ) p j ( ζ ) . (6.33)If α ∈ J with Index ( α ) = j and | α | = m , we define δ ⊗ α by δ ⊗ α ( t , ζ , ..., t m , ζ m ) (6.34)= δ ⊗ α ⊗ ... ⊗ δ ⊗ α j j ( t , ζ , ..., t m , ζ m )= δ ( t , ζ ) · ... · δ ( t α , ζ α ) | {z } α factors · ... · δ j ( t m − α j +1 , ζ m − α j +1 ) · ... · δ j ( t m , ζ m ) | {z } α j factors . e set δ ⊗ i = 1 . Finally we let δ ˆ ⊗ α denote the symmetrized tensor product of the δ k ′ s : δ ˆ ⊗ α ( t , ζ , ..., t m , ζ m ) = δ ˆ ⊗ α ⊗ ... ⊗ δ ˆ ⊗ α j j ( t , ζ , ..., t m , z m ) . (6.35)For α ∈ J define K α := I | α | (cid:16) δ ˆ ⊗ α (cid:17) . (6.36) Theorem 6.16 Chaos expansion.
Any F ∈ L ( P ) has a unique expansion of the form F = X α ∈J c α K α . (6.37) with c α ∈ R . Moreover, k F k L ( P ) = X α ∈J α ! c α . (6.38) (i) Let ( S ) consist of all ϕ = P α ∈J a α K α ∈ L ( P ) such that k ϕ k k := X α ∈J a α α !(2 N ) kα < ∞ for all k ∈ N , (6.39)equipped with the projective topology, where(2 N ) kα = Y j ≥ (2 j ) kα j , (6.40)if α = ( α , α , , ... ) ∈ J .(ii) Let ( S ) ∗ consist of all expansions F = P α ∈J b α K α such that k F k − q := X α ∈J b α α !(2 N ) − qα < ∞ for some q ∈ N . (6.41)endowed with the inductive topology. The space ( S ) ∗ is the dual of ( S ) . If F = P α ∈J b α K α ∈ ( S ) ∗ and ϕ = P α ∈J a α K α ∈ ( S ) , then the action of F on ϕ is h F, ϕ i = X α ∈J a α b α α ! . (6.42)(iii) If F = P α ∈J a α K α ∈ ( S ) ∗ , we define the generalized expectation E [ F ] of F by E [ F ] = a . Note that E [ K α ] = 0 for all α = 0. Therefore the generalized expectation coincides with theusual expectation if F ∈ L ( P ). e can now define the white noise • η ( t ) of the L´evy process η ( t ) = Z t Z R ζ e N ( dt, dζ ) . and the white noise • e N ( t, ζ ) of e N ( dt, dζ ) as follows. • e N ( t, ζ ) = e N ( dt, dζ ) dt × ν ( dζ ) (Radon-Nikodym derivative). (6.43)Also note that • η is related to • e N by • η ( t ) = Z R • e N ( t, ζ ) ζν ( dζ ) . (6.44) We now proceed as in the Brownian motion case and use the chaos expansion in terms of { K α } α ∈J to define the (L´evy-) Wick product. Definition 6.17
Let F = P α ∈J a α K α and G = P β ∈J b β K β be two elements of ( S ) ∗ . Then wedefine the