The Wiener Itô Chaos Expansion

Transcription

1 1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in the sequel. This result which concerns the representation of square integrable random variables in terms of an infinite orthogonal sum was proved in its first version by Wiener in 1938 [227]. Later, in 1951, Itô [12] showed that the expansion could be expressed in terms of iterated Itô integrals in the Wiener space setting. Before we state the theorem we introduce some useful notation and give some auxiliary results. 1.1 Iterated Itô Integrals Let W = W (t) =W (ω, t), ω Ωt [,T](T > ), be a one-dimensional Wiener process, or equivalently Brownian motion, on the complete probability space (Ω,F,P) such that W () = P -a.s. For any t,letf t be the σ-algebra generated by W (s), s t,augmented by all the P -zero measure events. We denote the corresponding filtration by F = {F t, t [,T]}. (1.1) Note that this filtration is both left- and right-continuous, that is, { F t = lim F s := σ F s }, s t s<t respectively, F t = lim F u := F u. u t u>t See, for example, [129] or [27]. G.Di Nunno et al., Malliavin Calculus for Lévy Processes with Applications 7 to Finance, c Springer-Verlag Berlin Heidelberg 29

2 8 1 The Wiener Itô Chaos Expansion Definition 1.1. Arealfunctiong :[,T] n R is called symmetric if g(t σ1,...,t σn )=g(t 1,...,t n ) (1.2) for all permutations σ =(σ 1,..., σ n ) of (1, 2,...,n). Let L 2 ([,T] n ) be the standard space of square integrable Borel real functions on [,T] n such that g 2 L 2 ([,T ] n ) := g 2 (t 1,...,t n )dt 1 dt n <. (1.3) [,T ] n Let L 2 ([,T] n ) L 2 ([,T] n )bethespaceofsymmetric square integrable Borel real functions on [,T] n. Let us consider the set S n = {(t 1,...,t n ) [,T] n : t 1 t 2 t n T }. Note that this set S n occupies the fraction 1 n! of the whole n-dimensional box [,T] n. Therefore, if g L 2 ([,T] n )theng Sn L 2 (S n )and g 2 L 2 ([,T ] n ) = n! g 2 (t 1,...,t n )dt 1...dt n = n! g 2 L2 (S, (1.4) n) S n where L 2 (S denotes the norm induced by n) L2 ([,T] n )onl 2 (S n ), the space of the square integrable functions on S n. If f is a real function on [,T] n,thenitssymmetrization f is defined by f(t 1,...,t n )= 1 f(t σ1,...,t σn ), (1.5) n! where the sum is taken over all permutations σ of (1,...,n). Note that f = f if and only if f is symmetric. Example 1.2. The symmetrization of the function f(t 1,t 2 )=t t 2 sin t 1, (t 1,t 2 ) [,T] 2, is f(t 1,t 2 )= 1 [ t t t ] 2 sin t 1 + t 1 sin t 2, (t1,t 2 ) [,T] 2. Definition 1.3. Let f be a deterministic function defined on S n (n 1) such that f 2 L 2 (S := n) f 2 (t 1,...,t n )dt 1 dt n <. S n Then we can define the n-fold iterated Itô integralas J n (f) := T t n t 3 t 2 σ f(t 1,...,t n )dw (t 1 )dw (t 2 ) dw (t n 1 )dw (t n ). (1.6)

3 1.1 Iterated Itô Integrals 9 Note that at each iteration i =1,..., n the corresponding Itô integralwith respect to dw (t i ) is well-defined, being the integrand t i t 2 f(t 1,..., t n ) dw (t 1 )...dw (t i 1 ), t i [,t i+1 ], a stochastic process that is F-adapted and square integrable with respect to dp dt i. Thus, (1.6) is well-defined. Thanks to the construction of the Itô integralwehavethatj n (f) belongs to L 2 (P ), that is, the space of square integrable random variables. We denote the norm of X L 2 (P )by X L2 (P ) := ( E [ X 2] ) 1/2 = ( Ω X 2 (ω)p (dω)) 1/2. Applying the Itô isometry iteratively, if g L 2 (S m )andh L 2 (S n ), with m<n,wecanseethat [ ] [( T E J m(g)j n(h) = E ( T s m t 2 s m s 2 ) g(s 1,...,s m)dw (s 1 ) dw (s m) )] h(t 1,...,t n m,s 1,...,s m)dw (t 1 ) dw (t n m )dw (s 1 ) dw (s m) T [( sm s 2 ) = E g(s 1,...,s m 1,s m)dw (s 1 ) dw (s m 1 ) ( sm t 2 )] h(t 1,...,s m 1,s m)dw (t 1 ) dw (s m 1 ) ds m =... T = s m s 2 g(s 1,s 2,...,s m)e [ s 1 t 2 ] dw (t 1 ) dw (t n m ) ds 1 ds m = h(t 1,...,t n m,s 1,...,s m) (1.7) because the expected value of an Itô integral is zero. On the other hand, if both g and h belong to L 2 (S n ), then E [ J n (g)j n (h) ] T = = E s n T [ sn s 2 s 2 s 2 We summarize these results as follows. g(s 1,...,s n )dw (s 1 ) dw (s n 1 ) ] h(s 1,...,s n )dw (s 1 ) dw (s n 1 ) ds n =... (1.8) g(s 1,...,s n )h(s 1,...,s n )ds 1 ds n =(g, h) L 2 (S n)

4 1 1 The Wiener Itô Chaos Expansion Proposition 1.4. The following relations hold true: {,n m E[J m (g)j n (h)] = (m, n =1, 2,...), (1.9) (g, h) L 2 (S n),n= m where (g, h) L2 (S n) := g(t 1,...,t n )h(t 1,...,t n )dt 1 dt n is the inner product of L 2 (S n ). In particular, we have S n J n (h) L2 (P ) = h L2 (S n). (1.1) Remark 1.5. Note that (1.9) also holds for n = or m = if we define J (g) =g,wheng is a constant, and (g, h) L 2 (S ) = gh,wheng, h are constants. Remark 1.6. It is straightforward to see that the n-fold iterated Itô integral L 2 (S n ) f = J n (f) L 2 (P ) is a linear operator, that is, J n (af + bg) =aj n (f)+bj n (g), for f,g L 2 (S n ) and a, b R. Definition 1.7. If g L 2 ([,T] n ) we define I n (g) := g(t 1,...,t n )dw (t 1 )...dw(t n ):=n!j n (g). (1.11) [,T ] n We also call n-fold iterated Itô integralsthei n (g) here above. Note that from (1.9) and (1.11) we have I n (g) 2 L 2 (P ) = E[I2 n(g)] = E[(n!) 2 J 2 n(g)] =(n!) 2 g 2 L 2 (S n) = n! g 2 L 2 ([,T ] n ) (1.12) for all g L 2 ([,T] n ). Moreover, if g L 2 ([,T] m )andh L 2 ([,T] n ), we have {,n m E[I m (g)i n (h)] = (m, n =1, 2,...), (g, h) L2 ([,T ] n ),n= m with (g, h) L 2 ([,T ] n ) = n!(g, h) L 2 (Sn). There is a useful formula due to Itô [12] for the computation of the iterated Itô integral. This formula relies on the relationship between Hermite polynomials and the Gaussian distribution density. Recall that the Hermite polynomials h n (x), x R, n =, 1, 2,... are defined by d n h n (x) =( 1) n e 1 2 x2 1 dx n (e 2 x2 ), n =, 1, 2,..., (1.13)

5 Thus, the first Hermite polynomials are 1.2 The Wiener Itô Chaos Expansion 11 h (x) =1,h 1 (x) =x, h 2 (x) =x 2 1, h 3 (x) =x 3 3x, h 4 (x) =x 4 6x 2 +3,h 5 (x) =x 5 1x 3 +15x,.... We also recall that the family of Hermite polynomials constitute an orthogonal basis for L 2 (R,μ(dx)) if μ(dx) = 1 2π e x2 2 dx (see, e.g., [215]). Proposition 1.8. If ξ 1,ξ 2,... are orthonormal functions in L 2 ([,T]), we have that ( I n ξ α 1 ) m ( T 1 ˆ ˆ ξ αm m = h αk ξ k (t)w (t) ), (1.14) k=1 with α 1 ++α m = n. Here denotes the tensor power and α k {, 1, 2,...} for all k. See [12]. In general, the tensor product f g oftwo functions f,g is defined by (f g)(x 1,x 2 )=f(x 1 )g(x 2 ) and the symmetrized tensor product f ˆ g is the symmetrization of f g. In particular, from (1.14), we have T n! t n t 2 ( θ ) g(t 1 )g(t 2 ) g(t n )dw (t 1 ) dw (t n )= g n h n, (1.15) g for the tensor power of g L 2 ([,T]). Here above we have used g = T g L 2 ([,T ]) and θ = g(t)dw (t). Example 1.9. Let g 1andn =3,thenweget T 6 t 3 t 2 1 dw (t 1 )dw (t 2 )dw (t 3 )=T 3/2 h 3 ( W (T ) T 1/2 ) = W 3 (T ) 3T W(T ). 1.2 The Wiener Itô Chaos Expansion Theorem 1.1. The Wiener Itô chaos expansion. Let ξ be an F T - measurable random variable in L 2 (P ). Then there exists a unique sequence {f n } n= of functions f n L 2 ([,T] n ) such that ξ = I n (f n ), (1.16) n=

6 12 1 The Wiener Itô Chaos Expansion where the convergence is in L 2 (P ). Moreover, we have the isometry ξ 2 L 2 (P ) = n! f n 2 L 2 ([,T ] n ). (1.17) n= Proof By the Itô representation theorem there exists an F-adapted process ϕ 1 (s 1 ), s 1 T, such that and Define [ T E T ξ = E[ξ]+ ϕ 2 1(s 1 )ds 1 ] E [ ξ 2] (1.18) ϕ 1 (s 1 )dw (s 1 ). (1.19) g = E[ξ]. For almost all s 1 T we can apply the Itô representation theorem to ϕ 1 (s 1 ) to conclude that there exists an F-adapted process ϕ 2 (s 2,s 1 ), s 2 s 1, such that [ s1 ] E ϕ 2 2 (s 2,s 1 )ds 2 E[ϕ 2 1 (s 1)] < (1.2) and ϕ 1 (s 1 )=E[ϕ 1 (s 1 )] + Substituting (1.21) in (1.19) we get where T ξ = g + T g 1 (s 1 )dw (s 1 )+ s 1 s 1 g 1 (s 1 )=E[ϕ 1 (s 1 )]. Note that by (1.18), (1.2), and the Itô isometrywehave [( T E s 1 ) 2 ] T ϕ 2 (s 2,s 1 )dw (s 2 )dw (s 1 ) = ϕ 2 (s 2,s 1 )dw (s 2 ). (1.21) ϕ 2 (s 2,s 1 )dw (s 2 )dw (s 1 ), (1.22) s 1 E[ϕ 2 2 (s 2,s 1 )]ds 2 ds 1 E[ξ 2 ]. Similarly, for almost all s 2 s 1 T, we apply the Itô representation theorem to ϕ 2 (s 2,s 1 )andwegetanf-adapted process ϕ 3 (s 3,s 2,s 1 ), s 3 s 2, such that

7 and E [ s2 1.2 The Wiener Itô Chaos Expansion 13 ϕ 2 3(s 3,s 2,s 1 )ds 3 ] E[ϕ 2 2(s 2,s 1 )] < (1.23) ϕ 2 (s 2,s 1 )=E[ϕ 2 (s 2,s 1 )] + Substituting (1.24) in (1.22) we get where T ξ = g + + T s 1 s 2 s 2 T g 1 (s 1 )dw (s 1 )+ ϕ 3 (s 3,s 2,s 1 )dw (s 3 ). (1.24) s 1 g 2 (s 2,s 1 )dw (s 2 )dw (s 1 ) ϕ 3 (s 3,s 2,s 1 )dw (s 3 )dw (s 2 )dw (s 1 ), g 2 (s 2,s 1 )=E[ϕ 2 (s 2,s 1 )], s 2 s 1 T. By (1.18), (1.2), (1.23), and the Itô isometrywehave [( T E s 1 s 2 ) 2 ] ϕ 3 (s 3,s 2,s 1 )dw (s 3 )dw (s 2 )dw (s 1 ) E [ ξ 2]. By iterating this procedure we obtain after n steps a process ϕ n+1 (t 1,t 2,..., t n+1 ), t 1 t 2 t n+1 T, and n + 1 deterministic functions g,g 1,...,g n,withg constant and g k defined on S k for 1 k n, such that where ξ = n J k (g k )+ k= S n+1 ϕ n+1 dw (n+1), ϕ n+1 dw (n+1) := S n+1 T t n+1 t 2 ϕ n+1 (t 1,...,t n+1 )dw (t 1 ) dw (t n+1 ) is the (n + 1)-fold iterated integral of ϕ n+1.moreover, [( E S n+1 ϕ n+1 dw (n+1) ) 2 ] E [ ξ 2].

8 14 1 The Wiener Itô Chaos Expansion In particular, the family ψ n+1 := S n+1 ϕ n+1 dw (n+1), n =1, 2,... is bounded in L 2 (P ) and, from the Itô isometry, for k n, f k L 2 ([,T] k ). Hence we have In particular, ξ 2 L 2 (P ) = (ψ n+1,j k (f k )) L 2 (P ) = (1.25) n k= J k (g k ) 2 L 2 (P ) + ψ n+1 2 L 2 (P ). n J k (g k ) 2 L 2 (P ) <, n =1, 2,... k= and therefore J k (g k )isconvergentinl 2 (P ). Hence k= lim n ψ n+1 =: ψ exists in L 2 (P ). But by (1.25) we have (J k (f k ),ψ) L2 (P ) = for all k and for all f k L 2 ([,T] k ). In particular, by (1.15) this implies that [ ( θ ) ] E h k ψ = g for all g L 2 ([,T]) and for all k, where θ = T g(t)dw (t). But then, from the definition of the Hermite polynomials, for all k, which again implies that Since the family E[exp θ ψ] = {exp θ : E[θ k ψ] = k= 1 k! E[θk ψ] =. g L 2 ([,T])} is total in L 2 (P ) (see [179, Lemma 4.3.2]), we conclude that ψ =. Hence, we conclude

9 1.2 The Wiener Itô Chaos Expansion 15 ξ = J k (g k ) (1.26) k= and ξ 2 L 2 (P ) = J k (g k ) 2 L 2 (P ). (1.27) k= Finally, to obtain (1.16) (1.17) we proceed as follows. The function g n is defined only on S n, but we can extend g n to [,T] n by putting g n (t 1,...,t n )=, (t 1,...,t n ) [,T] n \ S n. Now define f n := g n to be the symmetrization of g n - cf. (1.5). Then I n (f n )=n!j n (f n )=n!j n ( g n )=J n (g n ) and (1.16) and (1.17) follow from (1.26) and (1.27), respectively. Example What is the Wiener Itô expansion of ξ = W 2 (T )? From (1.15) we get T t 2 ( W (T ) ) 2 1 dw (t 1 )dw (t 2 )=Th 2 = W 2 (T ) T, T 1/2 and therefore ξ = W 2 (T )=T + I 2 (1). Example Note that for a fixed t (,T)wehave T t2 T χ {t1 <t<t 2 } (t1,t2)dw (t1)dw (t2)= W (t)dw (t 2)=W (t) ( W (T ) W (t) ). Hence, if we put t we can see that where ξ = W (t)(w (T ) W (t)), g(t 1,t 2 )=χ {t1<t<t 2} ξ = J 2 (g) =2J 2 ( g )=I 2 (f 2 ), f 2 (t 1,t 2 )= g(t 1,t 2 )= 1 2 ( χ{t1<t<t 2} + χ {t2<t<t 1}). Here and in the sequel we denote the indicator function by { 1, x A, χ = χ A (x) =χ {x A} :=, x / A.

10 16 1 The Wiener Itô Chaos Expansion 1.3 Exercises Problem 1.1. (*) Let h n (x), n =, 1, 2,..., be the Hermite polynomials defined in (1.13). (a) Prove that exp { tx t2 } t n = 2 n! h n(x). [Hint. Write exp{tx t2 2 } =exp{ 1 2 x2 } exp{ 1 2 (x t)2 } and apply the Taylor formula on the last factor.] (b) Show that if λ>then n= exp { tx t2 λ} t n λ n 2 ( x ) = h n λ. 2 n! n= (c) Let g L 2 ([,T]). Put Show that { T exp θ = T g(s)dw (s). g(s)dw (s) 1 2 g 2} = n= g n ( θ ) h n, n! g where g = g L2 ([,T ]). (d) Let t [,T]. Show that exp{w (t) 1 2 t} = n= tn/2 n! h n ( W (t) t ). Problem 1.2. Let ξ and ζ be F T -measurable random variables in L 2 (P )with Wiener Itô chaos expansions ξ = n= I n(f n )andζ = n= I n(g n ), respectively. Prove that the chaos expansion of the sum ξ +ζ = n= I n(h n )issuch that h n = f n + g n for all n =1, 2,... Problem 1.3. (*) Find the Wiener Itô chaos expansion of the following random variables: (a) ξ = W (t), where t [,T]isfixed, T (b) ξ = g(s)dw (s), where g L 2 ([,T]), (c) ξ = W 2 (t), where t [,T]isfixed, (d) ξ =exp{ T g(s)dw (s)}, whereg L 2 ([,T]) [Hint. Use (1.15).], (e) ξ = T g(s)w (s)ds, whereg L2 ([,T]).

11 1.3 Exercises 17 Problem 1.4. (*) The Itô representation theorem states that if F L 2 (P ) is F T -measurable, then there exists a unique F-adapted process ϕ = ϕ(t), t T, such that T F = E[F ]+ ϕ(t)dw (t). This result only provides the existence of the integrand ϕ, but from the point 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[D t F F t ], t T, where D t 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, by using the Itô formula. For example, find ϕ when (a) F = W 2 (T ) (b) F =exp{w (T )} (c) F = T W (t)dt (d) F = W 3 (T ) (e) F =cosw (T )[Hint. Check that N(t) :=e 1 2 t cos W (t), t [,T], is a martingale.] Problem 1.5. (*) This exercise is based on [18]. Suppose the function F of Problem 1.4 has the form F = f(x(t )), where X = X(t), t [,T], is an Itô diffusion given by dx(t) =b(x(t))dt + σ(x(t))dw (t); X() = x R. Here b : R R and σ : R R are given Lipschitz continuous functions of at most linear growth, so there exists a unique strong solution X(t) =X x (t), t [,T]. Then there is a useful formula for the process ϕ in the Itôrepresentation theorem. This formula is achieved as follows. If g is a real function such that then we define E[ g(x x (t)) ] <, u(t, x) :=P t g(x) :=E[g(X x (t))], t [,T], x R. Suppose that there exists δ> such that σ(x) δ for all x R. (1.28)

12 18 1 The Wiener Itô Chaos Expansion Then u(t, x) C 1,2 (R + R) and u t = b(x) u x σ2 (x) 2 u x 2 (this is the Kolmogorov backward equation, see, for example, [74, Volume 1, Theorem 5.11, p. 162 and Volume 2, Theorem 13.18, p. 53], [177, Theorem 8.1] for details on this issue). (a) Use the Itô formula for the process to show that Y (t) =g(t, X(t)), t [,T], with g(t, x) =P T t f(x) f(x(t )) = P T f(x)+ T [ σ(ξ) ] ξ P T tf(ξ) dw (t), (1.29) ξ=x(t) for all f C 2 (R). In other words, with the notation of Problem 1.4, we have shown that if F = f(x(t )), then [ E[F ]=P T f(x) and ϕ(t) = σ(ξ) ] ξ P T tf(ξ). (1.3) ξ=x(t) (b) Use (1.3) to compute E[F ] and find ϕ in the Itô representationofthe following random variables: (b.1) F = W 2 (T ) (b.2) F = W 3 (T ) (b.3) F = X(T ), where X(t), t [,T], is the geometric Brownian motion, that is, dx(t) = ρx(t)dt + αx(t)dw (t); X() = x R (ρ, α constants). (c) Extend formula (1.3) to the case when X(t) R n,t [,T], and f : R n R. In this case, condition (1.28) must be replaced by the uniform ellipticity condition η T σ T (x)σ(x)η δ η 2 for all x R n,η R n, (1.31) where σ T (x) denotes the transposed of the m n-matrix σ(x).