Itô s formula Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Itô s formula Probability Theory 1 / 41
Outline 1 Itô s integral 2 Itô s formula 3 Extensions and consequences of Itô s formula Samy T. Itô s formula Probability Theory 2 / 41
Outline 1 Itô s integral 2 Itô s formula 3 Extensions and consequences of Itô s formula Samy T. Itô s formula Probability Theory 3 / 41
Introduction Aim: Differential calculus with respect to W, i.e formula of the form: Problem: δf (W ) st = d j=1 Ẇ nowhere defined! s xj f (W r ) dw j r = d j=1 Strategy: Define s u r dw r for piecewise constant processes Take limits invoking of increments of W Limit in L 2 (Ω) Remark: for notational sake Computations done for real valued processes Generalization to R d : add indices s xj f (W r ) Ẇ j r dr Samy T. Itô s formula Probability Theory 4 / 41
Elementary processes Definition 1. Let: (Ω, F, P) probability space. u = {u t ; t [, τ]} process defined on (Ω, F, P) Process u is called simple if it can be decomposed as: with u t = ξ 1 {} (t) + N 1 i=1 ξ i 1 (ti,t i+1 ](t), N 1, (t 1,..., t N ) partition of [, τ] with t 1 =, t N = τ ξ i F ti and ξ i c with c > Samy T. Itô s formula Probability Theory 5 / 41
Integral of a simple process Definition 2. Let u simple process. Let s, t [, τ] such that s t and: t m < s t m+1, and t n < t t n+1, m n We define J st (u dw ) = s u wdw w as: J st (u dw ) = ξ m δw stm+1 + n 1 i=m+1 ξ i δw ti t i+1 + ξ n δw tnt Remark: For simple processes Stochastic integral and Riemann integral coincide. Samy T. Itô s formula Probability Theory 6 / 41
Integral for simple processes: properties Proposition 3. Consider: u a simple process J st (u dw ) its stochastic integral. Then for α, β R: 1 J tt (u dw ) = 2 J st ((αu + βv) dw ) = αj st (u dw ) + βj st (v dw ) 3 E[J st (u dw ) F s ] =, i.e martingale property 4 E[(J st (u dw )) 2 ] = s E[u2 τ ] dτ 5 E[(J st (u dw )) 2 F s ] = s E[u2 τ F s ] dτ Samy T. Itô s formula Probability Theory 7 / 41
Proof of point 5 If i < j, then (independence of increments of B) ] E [ξ i δw ti ti+1 ξ j δw tjtj+1 F tj ] = ξ i ξ j δw ti t i+1 E [δw tjtj+1 F tj = ξ i ξ j δw ti t i+1 E [ ] δw tj t j+1 = Samy T. Itô s formula Probability Theory 8 / 41
Proof of point 5 (2) Thus: E[(J st (u dw )) 2 F s ] n 1 2 = E ξ m δw stm + ξ i δw ti t i+1 + ξ n δw tnt F s i=m+1 n 1 = E ξ m 2 δwst 2 m + ξi 2 δwt 2 i t i+1 + ξn 2 δwt 2 nt F s i=m+1 = E[ξ 2 m F s ](t m s) + = s E[u 2 r F s ] dr n 1 i=m+1 E[ξ 2 i F s ](t i+1 t i ) + E[ξ 2 n F s ](t t n ) Samy T. Itô s formula Probability Theory 9 / 41
Space L 2 a Definition 4. We denote by L 2 a([, τ]) the set of process u such that: 1 u square integrable 2 u right continuous 3 u t F t The norm on L 2 a is defined par: τ u 2 L E [ ] u 2 2 a s ds Remark: Condition u t F t and u right continuous Ensures predictable property for u See discrete time stochastic integral Samy T. Itô s formula Probability Theory 1 / 41
Density of simple processes in L 2 a Proposition 5. Let u L 2 a. There exists a sequence (u n ) n of simple processes such that: lim u n un L 2 a =. Samy T. Itô s formula Probability Theory 11 / 41
Concrete approximation Generic partition: We consider π = {s,..., s n } with = s < < s n = t π = max{ s j+1 s j ; j n 1} Proposition 6. Let: u L 2 a such that u t M < for all t τ a.s Assume u is continuous (almost surely) {π m ; m 1} sequence of partitions of [, τ] such that lim m π m = u m s j π m u s j 1 [sj,s j+1 ) Then we have: lim m u um L 2 a = Samy T. Itô s formula Probability Theory 12 / 41
Proof Expression for u m : u m t = s j π m u sj 1 [sj,s j+1 )(t) Properties of u m : 1 Almost surely: lim m u m t = u t for all t [, τ] 2 u m t Z t with Z t M 3 Z L 2 (Ω [, τ]) Convergence of u m : by dominated convergence, τ lim E [ u t u m t m 2] = Samy T. Itô s formula Probability Theory 13 / 41
Stochastic integral: extended definition Proposition 7. Let: u L 2 a (u n ) n sequence of simple processes such that lim n u u n L 2 a = Then for s < t: The sequence (J st (u n dw )) n converges in L 2 (Ω) Its limit does not depend on the sequence (u n ) n Samy T. Itô s formula Probability Theory 14 / 41
Proof Cauchy sequence u n : We know that: u n u m L 2 a = s E [ (u n w u m w ) 2] dw n. Cauchy sequence J st (u n ): According to property: we have E [ (J st ((u n u m ) dw ) 2] = is a Cauchy sequence in L 2 (Ω). (Z n ) n (J st (u n )) n s E [ (u n w u m w ) 2] dw Samy T. Itô s formula Probability Theory 15 / 41
Stochastic integral: extended definition (2) Definition 8. Let u L 2 a. The stochastic integral of u with respect to W is the process J (u dw ) such that for all s < t, J st (u dw ) = L 2 (Ω) lim n J st (u n dw ), where (u n ) n is an arbitrary sequence of simple processes converging to u in L 2 a. Samy T. Itô s formula Probability Theory 16 / 41
Integral of a L 2 a process: properties Proposition 9. Let u a process of L 2 a, J st (u dw ) its stochastic integral. Then for α, β R: 1 J tt (u dw ) = 2 J st ((αu + βv) dw ) = αj st (u dw ) + βj st (v dw ) 3 E[J st (u dw ) F s ] =, i.e martingale property 4 E[(J st (u dw )) 2 ] = s E[u2 w] dw 5 E[(J st (u dw )) 2 F s ] = s E[u2 w F s ] dw Remark: For this construction, crucial use of: Independence of increments for W L 2 convergence in probabilistic sense Samy T. Itô s formula Probability Theory 17 / 41
Wiener integral Proposition 1. Let: W a 1-dimensional Wiener process, and τ >. h 1, h 2 deterministic functions in L 2 ([, τ]). Then the following stochastic integrals are well defined: W (h 1 ) = In addition: τ h 1 r dw r, W (h 2 ) = τ 1 (W (h 1 ), W (h 2 )) centered Gaussian vector. 2 Covariance of W (h 1 ), W (h 2 ): E [ W (h 1 ) W (h 2 ) ] = τ h 1 r h 2 r dr. h 2 r dw r. Samy T. Itô s formula Probability Theory 18 / 41
Proof Approximation: For m 1 we set s j = s m j = jτ m and h 1,m = m 1 j= h 1 s j 1 [sj,s j+1 ), h 2,m = m 1 j= h 2 s j 1 [sj,s j+1 ). Then: W (h 1,m ) = m 1 j= h 1 s j δw sj s j+1, W (h 2,m ) = Approximation properties: we have m 1 j= h 2 s j δw sj s j+1. (W (h 1,m ), W (h 2,m )) Gaussian vect. since W Gaussian process. E[W (h 1,m ) W (h 2,m )] = τ h1,m r hr 2,m dr. Then limiting procedure for Gaussian vectors. Samy T. Itô s formula Probability Theory 19 / 41
Outline 1 Itô s integral 2 Itô s formula 3 Extensions and consequences of Itô s formula Samy T. Itô s formula Probability Theory 2 / 41
Functional set C 1,2 Definition 11. Let τ > f : [, τ] R d R We say that f C 1,2 b,τ if: 1 t f (t, x) is C 1 ([, τ]) for all x R d 2 x f (t, x) is C 2 (R d ) for all t [, τ] 3 f and its derivatives are bounded Samy T. Itô s formula Probability Theory 21 / 41
1-d Itô s formula Theorem 12. Let: W real-valued Brownian motion f C 1,2 b,τ Then f (t, W t ) can be decomposed as: f (t, W t ) = f (, ) + + Remark: Proof for the 1-d formula only r f (r, W r ) dr f (r, W r ) dw r + 1 2 f (r, W r ) dr. Samy T. Itô s formula Probability Theory 22 / 41
Multidimensional Itô s formula Theorem 13. Let: W Brownian motion, d-dimensional f C 1,2 b,τ Laplace operator on R d, i.e = d j=1 2 x j x j Then f (t, W t ) can be decomposed as: f (t, W t ) = f (, ) + + d j=1 r f (r, W r ) dr xj f (r, W r ) dw j r + 1 2 f (r, W r ) dr Samy T. Itô s formula Probability Theory 23 / 41
Itô s formula for physicists Simplification: We consider f : R R only. Intuition: Taylor expansion of f (W ) up to o(t). dw t can be identified with dt. Heuristic computation: We get df (W t ) = f (W t ) dw t + 1 2 f (W t ) (dw t ) 2 + o ( (dw t ) 2) = f (W t ) dw t + 1 2 f (W t ) dt + o(dt). Itô s formula is then obtained by integration. Samy T. Itô s formula Probability Theory 24 / 41
Proof in the real case Simplification: We consider f : R R only Generic partition: We consider π = {t,..., t n } with = t < < t n = t π = max{ t j+1 t j ; j n 1} Taylor s formula: we have δf (W ) t = = n 1 j= n 1 j= δf (W ) tj t j+1 f (W tj ) δw tj t j+1 + 1 2 f (ξ j ) ( δw tj t j+1 ) 2, where ξ j [W tj, W tj+1 ]. Samy T. Itô s formula Probability Theory 25 / 41
Proof in the real case (2) Notation: We set n 1 Jt 1,π n 1 = f (W tj ) δw tj t j+1, Jt 2,π = f (ξ j ) ( ) 2 δw tj t j+1 j= j= Aim: Find a sequence (π m ) m 1 such that a.s lim m π m = lim m Jt 1,m lim m J 2,m t = f (W s ) dw s with Jt 1,m = Jt 1,πm = f (W s ) ds with Jt 2,m = J 2,πm t Samy T. Itô s formula Probability Theory 26 / 41
Proof in the real case (3) Analysis of the term Jt 1,m : we have Jt 1,m = us m dw s, with u m u tj 1 [tj,t j+1 ) t j π m Since u f (W ) L 2 a, continuous and bounded according to Proposition 6 we get: L 2 (Ω) lim J 1,m m t = f (W s ) dw s a.s convergence: For a subsequence π m π mk we have a.s lim J 1,m m t = f (W s ) dw s. Samy T. Itô s formula Probability Theory 27 / 41
Proof in the real case (4) Analysis of the term Jt 2,m, strategy: We set n 1 Jt 3,π = f (W tj ) ( ) n 1 2 δw tj t j+1, J 4,π t = f (W tj ) (t j+1 t j ) j= We will show that a.s, for a subsequence π m with π m : This will end the proof. j= lim J 2,πm m t Jt 3,πm = (1) lim J 3,πm m t Jt 4,πm = (2) lim m J t 4,πm f (W s ) ds = (3) Samy T. Itô s formula Probability Theory 28 / 41
Proof in the real case (5) Recall: we have n 1 Jt 4,πm = f (W tj ) (t j+1 t j ) j= Thus J 4,πm t Riemann sum for f (W ) Proof relation (3): Since s f (W s ) continuous, we have lim J 4,πm m t = f (W s ) ds. Samy T. Itô s formula Probability Theory 29 / 41
Proof in the real case (6) Proof relation (1): we have J 3,π t J 2,π t = n 1 j= ( f (W tj ) f (ξ j ) ) ( δw tj t j+1) 2 Additional Hyp. for sake of simplicity: f C 3 b. Then for < γ < 1 2 we have: ξ j [W tj, W tj+1 ] δw tj t j+1 c W,γ t j+1 t j γ f (W tj ) f (ξ j ) c W,γ,f π m γ Thus applying Theorem 27, Brownian Chapter: n 1 Jt 3,πm Jt 2,πm c W,γ π m γ ( 2 δwtj t j+1) j= c W,γ π m γ V 2,t(W ) = c W,γ t π m γ m Samy T. Itô s formula Probability Theory 3 / 41
Proof in the real case (7) Notation: We set δt j (t j+1 t j ), Z j ( δw tj t j+1 ) 2 δtj Proof relation (2), strategy: we have J 4,πm t We proceed as follows: 1 We show lim m E[ J 4,πm t n 1 Jt 3,πm = j= f (W tj ) Z j. J 3,πm t 2 ] =. 2 For a subsequence π m π mk we deduce: a.s lim J 4,πm m t J 3,πm t =. Samy T. Itô s formula Probability Theory 31 / 41
Proof in the real case (8) Decomposition of J 4,πm t we have J 4,πm t with J 3,πm t K m,1 t = K m,2 E [ J 4,π m t t = 2 Jt 3,πm : = n 1 j= f (W tj ) Z j. Thus: n 1 j= J 3,πm t 2] = Kt m,1 + Kt m,2 [ (f ) ] E 2 (W tj) Z j j<k n 1 E [ f (W tj ) Z j f (W tk ) Z k ] Lemma: Let X N (, σ 2 ). Then: [ (X E 2 σ 2) 2 ] = 2σ 4 Samy T. Itô s formula Probability Theory 32 / 41
Proof in the real case (9) Convergence of K m,1 t : We have δw tj t j+1 N (, δt j ) and δw tj t j+1 F tj. Therefore: K m,1 t = = n 1 j= n 1 j= [ (f ) ] E 2 (W tj) Z j E n 1 = 2 j= c f π m [ (f (W ) 2 ] tj) E [ ] Zj 2 [ (f E (W ) 2 ] tj) (δt j ) 2 n 1 j= δt j = c f t π m m. Samy T. Itô s formula Probability Theory 33 / 41
Proof in the real case (1) Proof relation (2): we have seen lim K m,1 m t =. In the same way, one can check that: Thus: K m,2 t =. 4,πm lim E[ J m t Jt 3,πm 2 ] =. This finishes the proof of (2) and of Itô s formula. Samy T. Itô s formula Probability Theory 34 / 41
Outline 1 Itô s integral 2 Itô s formula 3 Extensions and consequences of Itô s formula Samy T. Itô s formula Probability Theory 35 / 41
Itô processes Definition 14. Let: X : [, τ] R n process of L 2 a a R n initial condition {b j ; j = 1,..., n} real bounded and adapted process {σ jk ; j = 1,..., n, k = 1,... d} process of L 2 a We say that X is an Itô process if it can be decomposed as: for j = 1,..., n. X j t = a j + b j s ds + σ jk s dw k s, Remark: An Itô process is a particular case of semi-martingale. Samy T. Itô s formula Probability Theory 36 / 41
Itô s formula for Itô processes Theorem 15. Let: X Itô process, defined by a, b, σ. f C 1,2 b,τ. Then f (t, X t ) can be decomposed as: f (t, X t ) = n f (, a) + r f (r, X r ) dr + xj f (r, X r ) br j dr j=1 n d + xj f (r, X r ) σr jk dwr k j=1 k=1 + 1 n d x 2 2 j1 x j2 f (r, X r ) σ j 1k r σ j 2k r dr. j 1,j 2 =1 k=1 Samy T. Itô s formula Probability Theory 37 / 41
Infinitesimal generator of Brownian motion Proposition 16. Let: Then: W a F s -Wiener process in R d. s R +. f C 2 b. E [δf (W ) s,s+h F s ] lim h h = 1 2 f (W s). Samy T. Itô s formula Probability Theory 38 / 41
Proof Recasting Itô s formula: Let M t d j=1 Then Itô s formula can be written as: δf (W ) st = δm st + 1 2 xj f (W r ) dw j r. s f (W r ) dr. Conditional expectation: According to Proposition 9, M is a martingale. Thus: E [δf (W ) s,s+h F s ] = 1 2 s+h s E [ f (W r ) F s ] dr. Samy T. Itô s formula Probability Theory 39 / 41
Proof (2) Limiting procedure: Applying dominated convergence for conditional expectation we get: r E [ f (W r ) F s ] continuous. Thus: lim h and 1 2h s+h s E [ f (W r ) F s ] dr = 1 2 E [ f (W s) F s ] = 1 2 f (W s), E [δf (W ) s,s+h F s ] lim h h = 1 2 f (W s). Samy T. Itô s formula Probability Theory 4 / 41
Extension for Itô processes Theorem 17. Let: Then: X Itô process, defined by a, b, σ. f C 2 b. E [δf (X) s,s+h F s ] lim h h n = xj f (X s ) bs j + 1 j=1 2 n d j 1,j 2 =1 k=1 2 x j1 x j2 f (X s ) σ j 1k s σ j 2k s. Samy T. Itô s formula Probability Theory 41 / 41