An Introduction to Malliavin calculus and its applications

Transcription

1 An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart (Kansas University) May / 28

2 Stochastic Integral Representation W = {W t, t T } Brownian motion on the canonical probability space (Ω, F, P). For any t [, T ], F t is the σ-algebra generated by {W s, s t}. A stochastic process u = {u t, t [, T ]} is adapted if for any t [, T ], u t is F t -measurable L 2 a is the space of adapted process with E T u2 t dt <. Theorem (Itô s Representation) Let F L 2 (Ω). Then, there exists a unique process u L 2 a such that T F = E(F ) + u t dw t. David Nualart (Kansas University) May / 28

3 Clark-Ocone formula Let F D 1,2. Then, that is, u t = E(D t F F t ). T F = E(F) + E(D t F F t )dw t, Clark 7, Haussmann 78 : F = X t, where X is a diffusion process. Ocone 84 : F D 1,2. David Nualart (Kansas University) May / 28

4 Proof : For any v L 2 a we can write, using the duality relationship ( ) ( T ) T E F v t dw t = E(F δ(v)) = E D t Fv t dt = E[E(D t F F t )v t ]dt. If we assume that F = E(F) + T u tdw t, then by the Itô isometry ( E F T T v t dw t ) = T E(u t v t )dt. Comparing these two expressions we deduce that u t = E(D t F F t ) almost everywhere in Ω [, T ]. David Nualart (Kansas University) May / 28

5 Example 1 F = W 3 t. Then D sf = 3W 2 t 1 [,t](s) and Therefore, E(D s F F s ) = 3E[(W t W s + W s ) 2 F s ] = 3[t s + W 2 s ]. W 3 t = 3 Compare with Itô s formula [t s + W 2 s ]dw s. Wt 3 = 3 Ws 2 dw s + 3 W s ds. David Nualart (Kansas University) May / 28

6 Example 2 The local time of the Brownian motion {L x t, t [, T ], x R} is defined as the density of the occupation measure : for any Borel set C B(R). Formally, L x t 1 {Ws C}ds = C L x t dx, = δ x(w s )ds. Then, if p ε (x) = (2πε) 1 2 e x 2 /2ε, F ε = p ε (W s x)ds L2 (Ω) L x t. David Nualart (Kansas University) May / 28

7 We have D r F ε = p ε(w s x)d r W s ds = r p ε(w s x)ds, and E(D r F ε F r ) = = r r E ( p ε(w s W r + W r x) F r ) ds p ε+s r (W r x) ds. As a consequence, ( ) L x t = E(L x t ) + p s r (W r x)ds dw r. r David Nualart (Kansas University) May / 28

8 CLT for local time incremens {L x t Theorem, t, x R} local time of W. h 3 2 ( ) (L x+h t L x t ) 2 dx 4th R αt 8 3 η, as h, where α t = R (Lx t )2 dx and η is a N(, 1) random variable independent of W. Chen, Li, Marcus, Rosen 9 : Method of moments. Hu, Nualart 9 : Proof using Clark-Ocone formula. David Nualart (Kansas University) May / 28

9 Motivation Study of the Hamiltonian for the critical attractive random polymer in dimension one : H n = n i,j=1,i j 1 {Si =S j } 1 2 n i,j=1,i j 1 { Si S j =1}, where {S n, n =, 1, 2,...} is a simple random walk on the integers. Notice that H n = (ln x ln n+1 ) 2 x Z where l x n = n i=1 1 {S n=x} is the local time of the random walk S n. David Nualart (Kansas University) May / 28

10 Connection with self-intersection local times R In the same way, F t,h = (L x t ) 2 dx = = 2 (L x+h t R v = 2 ( 2 δ x (W s )ds) dx R v L x t ) 2 dx δ (W v W u )dudv. [δ (W v W u + h)δ (W v W u h) 2δ (W v W u )]dudv. David Nualart (Kansas University) May / 28

11 Integral representation of F t,h = R (Lx+h t L x t )2 dx We apply Clark-Ocone formula to F f,h. First we need to compute the derivative of F t,h. For u < r < v we have E (D r [δ (W v W u + h) + δ (W v W u h) 2δ (W v W u )] F r ) = E (δ (W v W u + h) + δ (W v W u h) 2δ (W v W u ) F r )) = p v r (W r W u + h) + p v r (W r W u h) 2p v r (W r W u ) = = 2 h h [p v r (W r W u + y) p v r (W r W u y)]dy [ p v r v where p ɛ = (2πɛ) 1 2 exp( x 2 2ɛ ). (W r W u + y) p v r v (W r W u y)]dy, David Nualart (Kansas University) May / 28

12 Integrating in u and v yields where u t,h (r) = F t,h = E(F t,h ) + 4 u t,h (r)dw r + 4 Ψ h (r)dw r, r h (p t r (W r W u + y) p t r (W r W u y))dydu and Ψ h (r) = r (1 [,h] (W r W u ) 1 [,h] (W u W r ))du. David Nualart (Kansas University) May / 28

13 Limit results E(F t,h ) = 4th + o(h 3 2 ). ( ) t E u t,h(r) 2 dr Ch 4. The martingale satisfies M h t = h 3 2 Ψ h (r)dw r M h t β( 4 3 α t), (1) where β is a Brownian motion independent of W and α t = R (Lx t )2 dx. David Nualart (Kansas University) May / 28

14 Sketch of the proof of (1) : The convergence holds if for any t, and M h t β( 4 3 α t) M h t = h 3 Ψ h (r) 2 dr 4 3 α t, in L 2 (Ω) (2) sup M h, W s = sup h 3 2 s t s t s Ψ h (r)dr, in L2 (Ω). (3) In fact, (2) and (3) imply that (W, β h ) converges in law to (W, β), where β is a Brownian motion independent of W, and β h is such that Mt h = β h ( M h t ) (asymptotic version of Ray-Knight s theorem). David Nualart (Kansas University) May / 28

15 Proof of (2) : Ψ h (r) = = R h ( 1[,h] (W r x) 1 [,h] (x W r ) ) L r (x)dx ( Wr y Lr L Wr +y r ) dy. Tanaka s formula for the Brownian motion {W r W s, s r} yields Wr y L Wr +y r Lr = 2y 2(W r y) + + 2(W r + y) + +2 r 1 {y> Wr W s }dŵs, where dŵs denote the backward stochastic Itô integral. Then, Ψ h (r) = h r h [ (Wr + y) + (W r y) +] dy (h W r W s ) + dŵs. David Nualart (Kansas University) May / 28

16 It suffices to show that, in L 2, ( r ) 2 4h 3 (h W r W s ) + dŵs dr 4 3 α t. By Itô s formula we can write ( r 2 r ( r ) (h W r W s ) dŵs) + = 2 (h W r W u ) + d B u s r (h W r W s ) + [ dŵs + (h Wr W s ) +] 2 ds = I 1 (r, h) + I 2 (r, h). The term I 1 (r, h) does not contribute to the above limit David Nualart (Kansas University) May / 28

17 We need to show that h 3 This follows from α t = r R r [ (h Wr W s ) +] 2 dsdr L 2 (Ω) 1 3 α t. (L x t ) 2 dx = 2 L x r L x dr dx = 2 R [ (h Wr W s ) +] 2 dsdr = L Wr r dr, [ (h Wr x ) +] 2 L x r dx, and R R [(h x ) + ] 2 h 3 dx = 2 3. David Nualart (Kansas University) May / 28

18 Derivative of the Brownian self-intersection local time Define γ t = d dx Proposition s γ t = lim L 2 (Ω) p ɛ ɛ(w s W u )duds. ( s δ x(w s W u )duds) x=. ( r ) γ t = 2 p t r (W r W u )du L Wr r dw r. David Nualart (Kansas University) May / 28

19 Proof : Use Clark-Ocone formula. Set γt ɛ = s p ɛ(w s W u )duds. Then s r D r γt ɛ = p ɛ (W s W u )1 [u,s] (r)duds = p ɛ (W s W u )duds r and E(D r γ ɛ t F r ) = = 2 = 2 r r r r r p ɛ+s r (W r W u )duds p ɛ+s r (W r W u )duds u (p ɛ+t r (W r W u ) p ɛ (W r W u ))du. As ɛ tends to zero, this converges in L 2 ([, t] Ω) to r p t r (W r W u )du L Wr r. David Nualart (Kansas University) May / 28

20 The process γ t is a Dirichlet process (it has zero quadratic variation and infinite total variation). Theorem (Rogers-Walsh 94) γ has a 4 3 -variation in L2 given by ( γ 4 3,t = K L Wr r ) 2 3 dr Proof was based on Gebelein s inequality for Gaussian random variables. Alternative proof by Hu-Nualart-Song 12 using the integral representation and ideas from fractional martingales. David Nualart (Kansas University) May / 28

21 CLT for the third integrated moment Theorem h 2 (L x+h R t L x t ) 3 dx 8 ( 3 (L x t ) 3 dx R ) 1 2 η, as h tends to zero, where η is a N(, 1) random variable independent of W Hu and Nualart 1 : Proof using Clark-Ocone formula applied to F t,h = h 2 R (Lx+h t L x t )3 dx. Rosen 11 : Proof using method of moments. David Nualart (Kansas University) May / 28

22 Sketch of the proof : We have F t,h = Φ r dw r, where Φ r = 4 Φ (1) r = 6 R Φ (2) r = 6 i=1 Φ(i) r ( ) 2 Lr z+h L z r 1[,h] (W r z)dz R h Φ (3) r = 12h r 2π ( L z+h h h, and r L z r ) 2 pt r (W r z y)dydz h 2 t r p t r h 2 (W r W s + y) z z 3 2 (1 e z 2 )dzdyds Φ (4) r = 12h r 1 [ h,h] (W r W s )ds 2π h 2 t r z 3 2 (1 e z 2 )dz David Nualart (Kansas University) May / 28

23 Limit results : [Φ(3) r + Φ (4) r ]dw r converges to 12γ t Applying Tanaka s formula to the time reversed Brownian motion {W r W s, s r}, we decompose (up to negligeable terms) [Φ(1) r + Φ (2) r ]dw r into two terms : One term converges to 12γ t A martingale term of the form ( t ( h ( r ) r 48h 2 1 [ h,] (W r W s x)dŵs 1 [ h,] (W r W σ x)dŵσ σ ) dx ) dw r, to which we can apply martingale techniques. David Nualart (Kansas University) May / 28

24 Applications in finance Black-Scholes model, under the risk-neutral probability ds t = rs t dt + σs t dw t, t T. Any payoff H, F T -measurable, E(H 2 ) <, can be replicated (complete market). That is, there exists a self-financing portfolio which value X t satisfies X T = H. The value X t satisfies dx t = t ds t + r(x t t S t )dt. if t is the amount of shares in the portfolio. Applying the Itô s representation theorem to e rt H yields e rt H = E(e rt H) + T u t dw t, and to have X T = H it suffices to take X = E(e rt rt ut H) and t = e σs t David Nualart (Kansas University) May / 28

25 Proposition u t = e rt E(D t H F t ) and the hedging portfolio of a derivative security with payoff H is given by t = e r(t t) σs t E(D t H F t ). In the case H = Φ(S T ), we have D t H = Φ (S T )D t S T = Φ (S T )σs T, and t = e r(t t) S t E(Φ (S T )S T F t ). By the Markov property, the price of the security is of the form v(t, S t ), where v(s, x) = e r(t t) E(Φ(x S T S t )). In that case, t = v x (t, S t ). David Nualart (Kansas University) May / 28

26 Applying the integration-by-parts formula of Malliavin calculus, we obtain e r(t t) t = S t (T t) E(Φ(S T )(W T W t ) F t ). This expression is well suited for Monte Carlo simulations (see Fournié, Lasry, Lebuchoux, Lions and Touzi 99, Kohatsu-Higa and Montero 3). David Nualart (Kansas University) May / 28

27 Generalization of Clark-Ocone formula : Suppose that W t = W t + θ s ds, where θ is adapted and T θ2 t dt < a.s. Suppose E(Z T ) = 1, where ( Z T = exp T θ s dw s 1 2 T θ 2 sds Let dq dp = Z T. Under Q, W is a Brownian motion (Girsanov theorem). In general F W T F W T, with a strict inclusion. ). David Nualart (Kansas University) May / 28

28 Can we represent a random variable FT W -measurable as a stochastic integral with respect to W using Clark-Ocone formula? Theorem Suppose F D 1,2, θ D 1,2 (L 2 ([, T ])), and E(ZT 2F 2 ) + E(ZT 2 DF 2 H ) < ( E ZT 2F ( 2 T θ t + T D t t θ s dw s + ) ) T 2 θ t s D t θ s ds dt <. Then T ) T F = E Q (F ) + E Q (D t F + F D t θ s d W s F t d W t. t Karatzas, Ocone 91 : Multidimensional case. Applications to hedging in a generalized Black-Sholes model. David Nualart (Kansas University) May / 28