A proof of Ito's formula using a di Title formula. Author(s) Fujita, Takahiko; Kawanishi, Yasuhi. Studia scientiarum mathematicarum H Citation

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1 A proof of Io's formula using a di Tile formula Auhor(s) Fujia, Takahiko; Kawanishi, Yasuhi Sudia scieniarum mahemaicarum H Ciaion Issue 8-3 Dae Type Journal Aricle Tex Version auhor URL hp://hdl.handle.ne/186/15878 This is an auhor's version of an a Righoriginal publicaion is available a hp://dx.doi.org/1.1556/sscmah. Hiosubashi Universiy Reposiory

2 A proof of Iô s formula using a discree Iô s formula 1 Takahiko Fujia Graduae School of Commerce and Managemen, Hiosubashi Universiy, Naka -1, Kuniachi, Tokyo, , Japan, fujia@mah.hi-u.ac.jp Yasuhiro Kawanishi Graduae School of Commerce and Managemen, Hiosubashi Universiy, cd43@srv.cc.hi-u.ac.jp Absrac. In his paper, we will prove Iô s formula for Brownian moion in he case of f C (R), using a discree Iô s formula. AMS subjec classificaion : 6G5, 6H5, 6J65 1 Inroducion Le Z ; =, 1, } be a Z -valued symmeric random walk, ha is, Z =, Z = ξ 1 + ξ + + ξ where ξ 1,ξ, are independenly and idenically disribued wih P ξ 1 =1=P ξ 1 = 1 = 1/. We have he following. Lemma ( Discree Iô s Formula ) For any f : Z R and any nonnegaive ineger, i holds ha f(z +1 ) f(z )= f(z +1) f(z 1) (Z +1 Z ) + f(z +1) f(z )+f(z 1). (1) This is called a discree Iô s formula. I was discovered by he firs auhor. The proof is very easy. We only have o consider he difference beween he lef-hand side(henceforh, abbreviaed LHS) and he second erm of he righhand side(henceforh, abbreviaed RHS) of he above equaion. For he deails of his discree Iô s formula, see 1,,3. In he nex secion, we will give a proof of Iô s formula for Brownian moion in he case of f C (R) using he above discree Iô s formula. I seems naural ha Iô differenial formula can be approximaed by he discree Iô (Iô difference) formula. In he proof, i is imporan ha how we approximae Brownian moion by random walks. For he approximaion mehod, he reader is referred o Iô and Mckean 4 secion This paper appeared on Sudia Scieniarum Mahemaicarum Hungarica, Vol.45, No.1, (8),

3 The proof Le W s ; s } be a sandard Brownian moion and f be in C (R). In he sequel, we are going o prove he following saemen: P f(w ) f() = f (W s ) dw s + 1 f (W s ) ds for =1. () Le us begin he proof. As i is well-known o anyone who has proved Iô s formula, i is sufficien ha we show P f(w ) f() = f (W s ) dw s + 1 f (W s ) ds =1, (3) where f C (R) has a compac suppor and >. So we suppose ha f C (R) has a compac suppor and fix >. We will inroduce an approximaion o Brownian moion by random walks. For n =1,,, define ha τ n, :=, } τ n,i := inf s>τ n,i 1 ; W s W τn,i 1 = n for i =1,,. Then by he srong Markov propery of Brownian moion, τ n,i τ n,i 1 (i = 1,, ) are independenly and idenically disribued. And so are W τn,i W τn,i 1 (i =1,, ). In addiion o ha, i holds ha EW τn,1 =, Eτ n,1 =, (4) n Eτ n,1 = n Acually, W s ; s }, W s s; s } and W 4 s 6sW s +3s ; s } are maringales. And also, by Doob s opional sampling heorem, W s τn,1 ; s }, W s τn,1 (s τ n,1 ); s } and W 4 s τn,1 6(s τ n,1 )W s τn,1 +3(s τ n,1 ) ; s } are maringales. Thus, i follows ha EW s τn,1 =, Es τ n,1 = EW s τn,1, E(s τ n,1 ) = 1 3 EW s τ n,1 4 +E(s τ n,1 )W s τn,1. Then if we le s, we can obain (4). Furhermore, by he above-menioned facs, τ n,i i ; i =, 1,, } is a maringale. Thus, by (4) and he submaringale inequaliy, i holds ha n P sup τ n,i i 1i n >ε ε E(τ n, n ) n = ε. (5) 3 n

4 So if we apply Borel-Canelli lemma, we obain ha P lim sup τ n,i i n 1i n = =1. (6) n Then by (6) and he uniform coninuiy of a coninuous funcion defined on a compac inerval, i follows ha P lim sup W (τ n,i ) W (i n 1i n ) = =1. (7) n Here by he discree Iô formula and appropriae scaling, we obain ha f(w (τ n, n)) f() = n 1 i= + 1 f(w τn,i + n ) f(w τn,i n ) f(w τn,i + n 1 i= (W τn,i+1 W τn,i) n n ) f(w τ n,i )+f(w τn,i n )}. Firs, by (7), he LHS of (8) converges o f(w ) f() almos surely as n. Nex, we can show ha he second erm of he RHS of (8) converges o 1 f (W s ) ds a.s. as n. In fac, we have ha n 1 f(w τn,i + i= n ) f(w τ n,i )+f(w τn,i n )} n 1 f(w τn,i + + i= n 1 i= n 1 i= f (W s ) ds n ) f(w τ n,i )+f(w τn,i n )} f (W (i )) n n f (W (i )) n n f (W s ) ds. The second erm of he RHS of he above inequaliy converges o zero a.s. as n, because f (W s ) is Riemann inegrable on,. As for he firs erm of he RHS, when we pu i as A n and represen second order remainder erms (8) 3

5 of Taylor expansion in inegral forms, we have he following. A n n 1 + i= n 1 i= + sup Wτn,i + n W τn,i f (s)(w τn,i + Wτn,i n W τn,i f (s)(w τn,i n s) ds 1 f (W τn,i ) n n s) ds 1 f (W τn,i ) n f (u) f (v) ; u v sup W (τ n,i ) W (i } 1i n ). n Here i holds ha for x, y R, y f (s)(y s) ds 1 f (x)(y x) (y x) x u v y x sup f (u) f (v). So we obain ha } A n sup f (u) f (v) ; u v n + sup f (u) f (v) ; u v sup W (τ n,i ) W (i } 1i n ). n By he uniform coninuiy of f and (7), he RHS of he above inequaliy converges o zero a.s. as n. Therefore, he second erm of he RHS of (8) converges o 1 f (W s ) ds a.s. as n. Las, le us show ha he firs erm of he RHS of (8) converges o f (W s ) dw s in probabiliy as n. We define ha H n (s) := n 1 i= f(w τn,i + ) f(w n τn,i ) n 1 (τn,i,τ n,i+1(s). n Then he firs erm of he RHS of (8) can be wrien as τn, n H n (s) dw s. Le 4

6 ε, δ be sricly posiive. Firs, we have τn, n p n := P H n (s) dw s f (W s ) dw s >δ τn, n P (H n (s) f δ (W s )) dw s > τ n, n ε +P τ n, n >ε τn, n +P f δ (W s ) dw s > τ n, n ε τn, n +P f δ (W s ) dw s > ε τ n, n < r P sup (H n (s) f δ (W s )) dw s > r+ε +P τ n, n >ε r +P sup f δ (W s ) dw s > r+ε +P f δ (W s ) dw s > + P sup 4 ε εr r ε f δ (W s ) dw s >. 4 Here, r (H n(s) f (W s )) dw s ; r }, r f (W s ) dw s ; r } and r ε f (W s ) dw s ; r ε} are coninuous maringales and hese Iô inegrals have he Iô isomery because f is bounded. So by he submringale inequaliy, Chebyshev s inequaliy, Jensen s inequaliy and Iô inegral s isomery, i holds ha p n +ε δ E (H n (s) f (W s )) dw s + ε E τ n,n + +ε δ E f (W s ) dw s + 8 δ E f (W s ) dw s ε +ε } 1/ E (H n (s) f (W s )) ds + δ ε E(τ n, n ) } 1/ + +ε } E f (W s ) 1/ 8 1/ ds + E f (W s ) ds}. δ δ Furhemore, leing M be he maximum of f, we have he following. p n δ E +ε ε } 1/ (H n (s) f (W s )) ds + ε E(τ n, n ) } 1/ + 1 δ Mε1/. (9) As for he firs erm of he RHS of (9), we have he following upper bound by he mean value heorem(wih θ = θ(w τn,i ) and θ < 1), Hölder s inequaliy 5

7 and(4): +ε E (H n (s) f (W s )) ds +ε = E = ( n 1 i= ) f (W τn,i + θ n )1 (τ n,i,τ n,i+1(s) f (W s ) ds n 1 +ε E (f (W τn,i + θ i= n ) f (W s )) 1 (τn,i,τ n,i+1(s) ds +ε +E f (W s ) 1 (τn, n, )(s) ds n 1 i= ( ) (τn,i+1 E sup f (W τn,i + θ τ n,isτ n,i+1 n ) f (W s ) τ n,i ) +M E τ n, n ε n 1 i= ( E ) 4 sup f (W τn,i + θ τ n,isτ n,i+1 n ) f (W s ) E(τ n,i+1 τ n,i ) } 1/ +M E(τ n, n ) } 1/ + M ε 5 ( 3 E sup f (u) f (v) ; u v sup i n 1 τ n,isτ n,i+1 }) 4 } 1/ W τn,i + θ n W s +M E(τ n, n ) } 1/ + M ε. (1) Here by (6), i holds ha wih probabiliy one, sup (s τ n,i) = sup (τ n,i+1 τ n,i ) i n 1 i n 1 τ n,isτ n,i+1 sup 1i n τ n,i (ω) i n + n (n ). From his fac, he uniform coninuiy of a coninuous funcion defined on a compac inerval and he uniform coninuiy of f, i holds ha wih probabiliy 6

8 one, } lim sup f (u) f (v) ; u v sup W τn,i + θ n i n 1 n W s =. τ n,isτ n,i+1 So, by dominaed convergence heorem, he firs erm of he las RHS of (1) converges o zero as n. In addiion, by (5), he second erm of he las RHS of (1) and he second erm of he RHS of (9) converges o zero as n. Therefore, we have ha lim sup p n 1 n δ Mε1/ (ε ). This means ha he firs erm of he RHS of (8) converges o in probabiliy as n. So we obain (3). (Q.E.D.) Remark f (W s ) dw s In 5, Szabados obained anoher ype of discree Iô formula as he following. 1 g(z )= f(z i )(Z i+1 Z i )+ 1 i= j=1 1 i= f(z i+1 ) f(z i ) Z i+1 Z i, (11) where g is defined as k 1 1 g(k) = sgn(k) f() + f(j sgn(k)) + 1 } f(k). Furhermore, for f C 1 (R), he derived a new represenaion of f(w s) dw s, using Iô s formula, his discree Iô s formula and he same approximaion mehod of Brownian moion by random walks. Even if we use his discree Iô s formula insead of (1), we can prove Iô s formula. Therefore (1) and (11) are no differen in he limi case. Bu in he discree case, hey are differen in ha hough (1) gives Doob-Meyer decomposiion, (11) does no generally do so. We can prove Iô s formula for f C 1, (R + R), using discree Iô s formula in he explicily ime-dependen case: g( +1,Z +1 ) g(, Z )= g( +1,Z +1) g( +1,Z 1) (Z +1 Z ) + g( +1,Z +1) g( +1,Z )+g( +1,Z 1) + g( +1,Z ) g(, Z ), 7

9 where g : Z + Z R and is a nonnegaive ineger. In fac, we have he following from his discree Iô s formula and appropriae scaling. f(, W (τ n, n)) f(, ) = n 1 i= i= f((i +1) n,w τn,i + n ) f((i +1) n,w τn,i n ) n (W τn,i+1 W τn,i ) + 1 n 1 f((i +1) n,w τ n,i + ) f((i +1) n n,w τ n,i ) i= +f((i +1) n,w τ n,i n )} n 1 + f((i +1) n,w τ n,i ) f(i n,w τ n,i )}, where f C 1, (R + R) has a compac suppor and > is fixed. Here we only have o consider he limi of each erms when we le n. References 1 Fujia, T. :A random walk analogue of Lévy s heorem, in his volume. Fujia, T. :Maringale mehods in pricing derivaives (in Japanese), The Hiosubashi Review, Vol. 15, No.1, (1), Fujia, T. () Sochasic Calculus for Finance (in Japanese), Kodansha. 4 Iô, K. and H. McKean (1974) Diffusion Processes and heir Sample Pahs, Springer. 5 Szabados, T. (1989) A discree Iô formula Limi heorems in probabiliy and saisics (Pécs,1989), Colloquia Mahemaica Socieais János Bolyai 57, Norh-Holland, Amserdam, (199), Szabados, T. (1996) An elemenary inroducion o he Wiener process and sochasic inegrals, Sudia Scieniarum Mahemaicarum Hungarica 31 (1996),