White noise generalization of the Clark-Ocone formula under change of measure

Transcription

1 White noise generalization of the Clark-Ocone formula under change of measure Yeliz Yolcu Okur Supervisor: Prof. Bernt Øksendal Co-Advisor: Ass. Prof. Giulia Di Nunno Centre of Mathematics for Applications (CMA) University of Oslo, Norway Second General AMAMEF Conference and Banach Center Conference, Bedlewo, 3 April - 5 May 27

2 Outline 1. Introduction Chaos Expansions (Hida)Malliavin Derivative Skorohod Integral 5. Clark-Ocone Formula under Change of Measure Construction of G and G Clark-Ocone Theorem for L 2 (P) Clark-Ocone Theorem for L 2 (P) under Change of Measure 6.

3 Introduction Gaussian white noise theory was first introduced by Hida [Hida82]. Contribution of white noise theory to finance, F G L 2 (P), [AØPJ] F D 1,2 [Ocone84] F (ω) = E[F ] + T E[D t F F t ] Ẇ (t)dt. T F (ω) = E[F ] + E[D t F F t ]dw (t)

4 Introduction W (t) be a Brownian motion on the filtered white noise probability space (Ω, B, {F t } t, P) Let Q be the probability measure equivalent P such that Ŵ (t) is a Brownian motion with respect to Q, in virtue of the Girsanov theorem. Ŵ t be defined as dŵ (t) = u(t) + dw (t), where u(t) is an F t -measurable process satisfying certain conditions.

5 L 2 (P) = {F : Ω R s.t. F 2 L 2 (P) := Ω F 2 (ω)dp(ω) < }. F L 2 (F t, P), F = I n (f n ) n= F L2 (P)= n= n! f n L2 [,T ] n F D1,2 := n= nn! f n L2 [,T ] n D 1,2 L 2 (P)

6 F L 2 (P) T T F (ω) = E Q [F ]+ E Q [(D t F F D t u(s)dŵ (s)) F t ] dŵ (t). t [YO7]

7 Bochner-Minlos Theorem Construction of Brownian motion S = S(R) be the Schwartz space of rapidly decreasing smooth functions φ C (R) such that φ k,n = sup φ (j) (x)x n < for all j k, n N. x R The space S(R) equipped with the family of seminorms. k,n constitutes a Frechet space. S (R) is called the space of tempered distributions equipped with the weak-star topology. If ω S (R) and φ S(R) we set ω(φ) =< ω, φ > that denotes the action of the linear functional ω on the test function φ.

8 Bochner-Minlos Theorem Construction of Brownian motion Bochner-Minlos Theorem Let g : S(R) R be a given function. Then there exists a probability measure P on Ω := S (R) s.t. e i<ω,φ> dp(ω) = g(φ) ; φ S(R) (1) if and only if g()=1 S (R) g is positive definite g is continuous in Frechet topology.

9 Bochner-Minlos Theorem Construction of Brownian motion White noise analysis g(φ) = e 1 2 φ 2 L 2 (R) e i<ω,φ> dp(ω) = e 1 2 φ 2 L 2 (R) S (R) E[< ω, φ >] := E[< ω, φ > 2 ] := S (R) S (R) < ω, φ > dp(ω) = (2) < ω, φ > 2 dp(ω) = φ 2 L 2 (R) (3)

10 Bochner-Minlos Theorem Construction of Brownian motion Construction of Brownian motion Extend the definition of < ω, φ > from φ S(R) to φ L 2 (R) by using the properties above and the fact that S(R) is dense in L 2 (R) and L 2 (P) is complete. W (t, ω) = W (t) =< ω, χ [,t] (.) > where χ [,t] belongs to L 2 (R) for all t and defined as follows: { 1 if s [, t] or s [t, ](t < ) χ [,t] (s) = otherwise

11 Bochner-Minlos Theorem Construction of Brownian motion Construction of Brownian motion cont. W (t) is a gaussian process with mean o and variance t. By the Kolmogorov continuity theorem it has continuous version W (t) = W (t, ω)[loève78]. Note that since W (t, ω), ω Ω = S (R) is constructed in a special way ω can be regarded as a tempered distribution.

12 Hermite Polynomials Hermite Functions Hermite polynomials constitute an orthogonal basis for L 2 (R, µ(dx)) if µ = 1 2π e x2 2. d n h n (x) = ( 1) n e 1 2 x2 dx n (e 1 2 x2 ), x R, n =, 1, 2,... h n+1 (x) = xh n (x) nh n 1 (x), n 1 h n(x) = nh n 1 (x), n 1

13 Hermite Polynomials Hermite Functions For the tensor power of g L 2 ([, T ]) n! T tn where [Itô51] t2 g(t 1 ) g(t n ) dw (t 1 ) dw (t n ) = g n θ h n ( g ). θ =< ω, g >= T g = g L 2 ([,T ]) g(t)dw (t)

14 Hermite Polynomials Hermite Functions J n (g) := T tn t2 g(t 1, t 2,, t n ) dw (t 1 ) dw (t n ) n!j n (g) = I n (g) [DØP] I n (g) = T T T g(t 1, t 2, t 3,, t n )dw (t 1 ) dw (t n ).

15 Hermite Polynomials Hermite Functions Hermite Functions e k (x) = π 1/4 ((k 1)!) 1 2 e 1 2 x2 h k 1 ( 2x), k = 1, 2, 3,... (4) {e k } k 1 constitutes an orthonormal basis for L 2 (R) and e k S(R) for all k.

16 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Wiener-Itô Chaos Expansion I Theorem Let F (ω) be an F T -measurable random variable in L 2 (P) F = I n (f n ) n= where f n ˆL 2 ([, T ] n ) and the convergence is in L 2 (P). [DØP] F 2 L 2 (P) = E[F 2 ] = n! f n 2 L 2 ([,T ] n ) n=

17 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Wiener-Itô Chaos Expansion II Define θ k (ω) :=< ω, e k >= R e k (x)dw (x), ω Ω (5) Let I be the set of all finite multi-indices α = (α 1, α 2,..., α m ), where α m N = N {}, m=1,2,... and α = m i=1 α i. Define H α := m h αj (θ j (ω)), ω Ω. (6) j=1 The family {H α } α I is an orthogonal sequence that constitutes basis for the Hilbert space L 2 (P).

18 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Wiener-Itô Chaos Expansion II Theorem For all F L 2 (P) there exists unique constants c α R such that F = α I c α H α. (7) Moreover, there exists the following equality F 2 L 2 (P) = α I α! c 2 α, where α! = α 1!α 2!... α m! for α = (α 1, α 2,..., α m ). [DØP]

19 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Relation between Chaos Expansions where f n = α I: α =n e ˆ α = e α 1 1 and α = m i=1 α i then F = c α e ˆ α, n =, 1, 2,..., ˆ e α 2 2 I n (f n ) = n= ˆ... ˆ e αm m α I, α =n c α H α

20 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Hida Stochastic Test Function Space (2N) α = m i=1 (2i)α i, for α = (α 1,..., α m ) I. Definition f = α I a αh α L 2 (P) belongs to the Hida test function Hilbert space (S) k for k {1, 2,...} if f 2 k := α I α! a 2 α (2N) αk < Hida space of stochastic test functions (S) as (S) = (S) k, (projective topology) (8) k=1

21 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Hida Stochastic Distribution Space Let (S) q, q = 1, 2,..., be the set of all expansions F = α I b αh α such that F 2 q:= α I b 2 α α! (2N) αq <, Hida space of stochastic distributions (S) as (S) = (S) q, (inductive topology) (9) q=1

22 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Wick Product (S) is the dual of (S), the action of F = α I b αh α (S) on f = α I a αh α L 2 (P) (S) < F, f >= F (f ) = α I a α b α α! (S) (S) k L 2 (P) (S) q (S), for all k, q. X = α a α H α (S), Y = β b β H β (S) X Y := α, β a α b β H α+β

23 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral The (Hida)Malliavin Derivative (i) Let F L 2 (P) be random variable and let γ L 2 (R) Ω. Then the directional (or Gateaux) derivative of F in the direction of γ (ii) F (ω + εγ) F (ω) D γ F (ω) = lim ε ε if the limit exists in (S). D γ F (ω) = R ψ(t)γ(t)dt for all γ L 2 (R), then we say that F is (Hida) Malliavin differentible D t F = ψ(t) t R. (1)

24 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Chain Rule Theorem Suppose F L 2 (P) is (Hida) Malliavin differentiable. Let Φ C 1 (R) and assume that D t F L 2 (P) and Φ (F )D t F L 2 (λ P) where λ is the Lebesgue measure on R. Then Φ(F ) is Malliavin differentiable and D t (Φ(F )) = Φ (F )D t F (11) [DØP]

25 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Results of Chain Rule Lemma Let F = I n (f n ) L 2 (P) D t (F 1 F 2 ) = F 1 D t F 2 + F 2 D t F 1 D t (h n (< ω, f >)) = nh n 1 (< ω, f >)f (t). D t F = ni n 1 (f n (., t)), (12) where I n 1 (f n (., t)) stands for the (n-1)-iterated Itô integral which is defined with respect to the n 1 first variables t 1,..., t n 1 of f n (t 1, t 2,..., t n 1, t).

26 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Lemma (i) Let G (S). Then D t G (S) for a.a. t R. (ii) Suppose G, G n (S) for all n N and G n G in (S). Then there exist a subsequence {G nk } k 1 such that D t G nk D t G in (S), for almost all t >. [YO7]

27 Wiener-Itô Chaos expansions (Hida)Malliavin Derivative Skorohod Integral Skorohod Integral Let u(t) L 2 (P), F T measurable stochastic process. u(t) = I n (f n (., t)) δ(u) := T n= u(t)δw (t) = I n+1 ( f n ), (13) n= where f n are the symmetric functions derived from f n (., t) for n = 1, 2,.... Note that E[δ(u) 2 ] = (n + 1)! f n 2 L 2 ([,T ] n+1 ) <. n=

28 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Construction of spaces G and G (i) Let λ R. G λ consists of all formal expansions F = I n (f n ) n= (ii) Define F 2 G λ := n! f n 2 L 2 (R n ) e2λn <. n= G is the dual of G. G = λ R G λ G = λ R G λ

29 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Lemma (i) Suppose F G. Then D t F G for a.a. t R. (ii) Suppose F, F n G for all n N and F n F in G. Then there exists a subsequence {F nk } k 1 such that D t F nk D t F in G, (14) for almost all t >. [AØPJ]

30 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) (S) G L 2 (P) G (S) Theorem Let F = n= I n(f n ) L 2 (P). Then [YO7] D t F = ni n 1 (f n (., t)). (15) n=1

31 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Sketch of the proof Let F L 2 (P). Define F m = n= I n(f n ) F m F in L 2 (P) implies F m F in G. By Lemma, there exists a subsequence F mk D t F mk D t F in G, i.e., such that m k ni n 1 (f n (., t)) D t F n= in G.

32 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Sketch of the proof We want to prove that m k ni n 1 (f n (., t)) n= n= ni n 1 (f n (., t)) ni n 1 (f n(., t)) 2 G q 1 = (n 1)!n 2 f n(., t) 2 L 2 (R n 1 ) e 2q(n 1). m k+1 m k+1 Take integral of both sides ni n 1 (f n(., t)) 2 G q dt K F L 2 (P) < R m k+1

33 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Conditional Expectation in G F = n= I n(f n ) G. E[F F t ] = I n (f n χ [,t] n). n= Note that this coincides with the usual expectation if F L 2 (P). E[F F t ] Gλ G Gλ, for all λ R then E[F F t ] G (16)

34 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Fundamental Theorem of Calculus Theorem Let u(s, ω) L 2 (P) be a stochastic process satisfying the following conditions: (i) E[ T u2 (s)ds] < (ii) s D t u(s) is Skorohod integrable in (S), for all t [, T ] (iii) E[ T ( T D t(u(s))δw (s)) 2 dt] < then δ(u) = T u(s)δw (s) L2 (P) and T T D t ( u(s)δw (s)) = D t u(s) δw (s) + u(t). (17)

35 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Corollary Let u = u(t) L 2 (P), t [, T ], be an F-adapted stochastic process. Then (i) D s u(t) is F adapted for all s, (ii) D s u(t) = for s > t.

36 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Theorem (The Clark-Ocone theorem for L 2 (P)) Let λ denote the Lebesgue measure on R. Suppose F (ω) L 2 (P) be F T -measurable. Then (t, ω) E[D t F F t ](ω) L 2 (λ P) and T F (ω) = E[F ] + E[D t F F t ]dw (t) (18) The proof can be found in Aase et al [AØPJ].

37 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Girsanov Theorem Theorem Let W (t) be a Brownian motion on the white noise probability space (Ω, F, P). Let (u(t)) t T be an adapted measurable process satisfying T u2 (s)ds < a.s. and such that Z(T ) defined by T Z(T ) = exp{ u(s)dw (s) 1 T u 2 (s)ds} (19) 2 is a martingale. Then under the white noise probability measure Q with density Z(T ) relative to P, the process Ŵ (t) defined by Ŵ (t) = W (t) + t u(s)ds is a Brownian motion under measure Q.

38 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Lemma Let F L 2 (P) be F T -measurable, Q and Z(T ) is defined as in Girsanov theorem. Then D t (Z(T )F ) = Z(T ){D t F F [u(t) + [YO7] T t D t u(s)dŵ (s)]} (2)

39 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Main Result Theorem Suppose F L 2 (P) is F T -measurable E Q [ F T T E Q [ E Q [ F ] < D t F 2 dt] < T T ( D t u(s)dw (s) + u(s)d t u(s)ds) 2 dt] < F (ω) = E Q [F ]+ T E Q [(D t F F T t D t u(s)dŵ (s)) F t]dŵ (t). [YO7]

40 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Sketch of the proof Y (t) = E Q [F F t ] t λ(t) = Z 1 (t) = exp{ u(s)dw (s) + 1 t u 2 (s)ds}. (21) 2 By Itô Formula, dλ(t) = λ(t)u(s)dŵ (s). By Bayes formula (Karatzas and Shreve Lemma [KS87]) and the settlement of Y (t), Y (t) = E[Z(T )F F t ]Z 1 = λ(t)e[z(t )F F t ].

41 Construction of G and G The Clark-Ocone theorem for L 2 (P) The Clark-Ocone formula under change of measure for L 2 (P) Sketch of the proof By Clark-Ocone formula, t E[Z(T )F F t ] = E[Z(T )F ]+ E[D s (Z(T )F ) F s ]dw (s) := U(t) dy (t) = {E Q [D t F F t ] E Q [Fu(t) F t ] E Q [F dŵ (s) F t ] + u(t)e Q [F F t ]}dŵ (t). T t D t u(s)

42 Market (i) Risk free asset (Bond) ds (t) = ρ(t)s (t)dt, S () = 1 (ii) Risky asset (Stock) ds 1 (t) = µ(t)s 1 (t)dt + σ(t)s 1 (t)dw (t) S 1 () > (iii) Contingent Claim (Digital Option) F = χ [K, ) (W (T ))

43 dv θ = θ (t)ds (t) + θ 1 ds 1 (t). µ(t) ρ(t) = σ(t)u(t). (22) dv θ (t) = ρ(t)v θ (t)dt + σ(t)θ 1 (t)s 1 (t)dŵ (t). Discounted value of the portfolio T U θ (T ) = V θ () + e t ρ(s)ds σ(t)θ 1 (t)s 1 (t)dŵ (t).

44 Apply Clark-Ocone formula under change of measure for U θ (T ), T U θ (T ) = E Q [U θ (T )] + E Q [(D t U θ (T ) T U θ (T ) D t u(s)dŵ (s)) F t ]dŵ (t). t

45 t θ 1 (t) = e ρ(s)ds σ 1 (t)s1 1 (t)e Q[(D t (e T ρ(s)ds F ) e T ρ(s)ds F T t D t u(s)dŵ (s)) F t ].

46 Choose ρ constant and µ, σ deterministic functions θ 1 (t) = e ρ(t t) σ 1 (t)s 1 1 (t)e Q[D t F F t ] F = χ [K, ) (W (T )) ( Aase et al. [AØU1])

47 Donsker Delta Function Definition Let Y : Ω R be a random variable which belongs to G. Then the continuous function δ Y (.) : R G is called Donsker delta function of Y if it has the following property, g(y)δ Y (y)dy = g(y ) a.s. R for all measurable function g : R R such that the integral converges.

48 Theorem Let φ : [, T ] R and α : [, T ] R be deterministic functions such that φ L 2 ([,T ]) and α L 2 ([,T ]) <. Define Y (t) = Y (t, ω) = t Let g : R R be bounded. Then t φ(s)dŵ (s) + φ(s)α(s)ds. T f (Y (T )) = V + Ψ(t, ω) (α(t) + Ŵ t )dt, (23)

49 where V = R g(y) 2π φ L 2 ([,T ]) y 2 exp[ ]dy, 2 φ L 2 ([,T ]) g(y) Ψ(t, ω) = φ(t) exp (y Y (t)) 2 [ ] y Y (t) dy R 2π φ L 2 ([,T ]) 2 φ L 2 ([,T ]) φ L 2 ([,T ])

50 Take g(y) = χ [K, ) (y) and Y (T ) = W (T ) Then φ(t) = 1, α(t) = u(t) where u(t) is defined in equation (22) Implies Ψ(t, ω) = ( 2π) 1/2 T 1/2 exp [ K (y W (t)) 2 2T ] y W (t) dy T

51 By the Clark-Ocone formula, T χ [K, ) (W (T )) = E Q [χ [K, ) (W (T ))]+ E Q [D t {χ [K, ) (W (T ))} F t ] dŵ (t) }{{} = (2πT ) 1/2 exp [ K (y W (t)) 2 2T ] y W (t) dy. T

52 By Aase et al, Lemma 3.8 [AØU1] = (2π) 1/2 (T t) 1/2 exp[ K (y W (t))2 2(T t) ] y W (t) dy. T t θ 1 (t) = e ρ(t t) (2π(T t)) 1/2 σ 1 (t)s 1 1 (t) K exp[ (y W (t))2 2(T t) ] y W (t) dy T t

53 Coordinates

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