book11-2homepage.fudan.edu.cn/jgying/files/2011/09/金融随机分析.pdf · Title: book11-2.dvi...

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1. 4_: 2C&qG#4iOz 4_>, bO 300 nq Leibniz Z?.

2. !4_: 3jI;AK4_, j!z4_, bO 1940 nj Ito Z?.

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3. Y>!4_: UY>=>si!4_, bO 1967 nBlack Scholes iOÆ.=,RZ?.

4. Li?: UY>=>siCn!4_, B>U O.jG#4.

5. L_: aV UY>=LH&Fw!4_3*sHF!4_, 'jjY>!4_ tU,'.

6. &' X: QI;>.K7 Tt-jT, L),R(?, 04, i?jQy, >Wj fU, 04j,, Xj>wUL, ,-W!4_UXpsI. 2?,04, ILTWT>jfQw, Dtj_, b53jzQ, $047XfVO2 [4], bLOW.??NAog 2011 n

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W0? :GA=. zb&&j$>!oyk5, OUI;>!4_j>H-j>H3*sH>1C&[, 3XMAfI;>, Ip, !#*4.zUÆE, ÆE , ,CF4., %v4., zi4., 4., <!&j 4..iF^

• !oy• I;A• kU&kI;• !#*U4.z• ÆE• 4.

1

Ux N~i 2

1. Y>LHU!4_Lig? ;LY>L !oy, <CEjje=T8i.

2. !oy!p": Y'Z;Ttoyp". C4,C, 6, 8tZ, e,. -J,8Mw!oy, 3, !oyYZ-n.

3. j;L!oy? j,UfeW), VUW)'L9, A. Einstein : K-C4. 'jj:LIL!oy. iOUW)eY j(D, k, ;L3j,k, l&o oyÆ3Y'Z;j;LXM1j$-a53I;:_-ak=\A-a, ,, XsW,,on?G1b2,, C\jRj(RnrA'?G1b2,, jg4Al,XMY'a2T. XM-ij-jL!oyW), inÆ&2\XMj-j2H&V,t-j, jj!oy.

4. #u: feLIL#uW)? ^j,UgNg#, \AN3/w&rV\UYo. qXMiO28O*-u-n, 3%N|W), XMf2n#ujA!#j, (i. XM-j;LT)`6jUC4JRj6.

5. I;>: I;>jf!oy, ,'. qj&*5j, I;>j, Yd', -jKUx!oy-jZ;W), -&ÆE,UI;9jaT6J~A. I;>LFZ2Cjng%B, jaRXM415

Ux N~i 37K (i, qiOU*-j|Y57.

6. !4_H: X:L!4_<!&jOBMw, V?G2u7f)r~$OBk:_, LU,B, 3jU!4_3*nY>1W?F.

7. Hj e? e j,U1C!&*, n!4_'jYZ-n.

e = limn→∞

(1 + 1/n)n.R<n ;j r > 0, tJnT,` , bH,n 1 [#J 1 + r [; tJgT , bH,n#J (1 + r/12)12; tJ,T , bH,nj (1+ r/365)365; ;_t&T , bH 1[,nj er [. t r [bA<,U\A#J 1, bHXM r jU\Ao;, o;jj ;u.

8. I;A: ">U!oy\, LjTt/*HL%A, 2L Ω. !, C%Aj R, (R ; C4%Aj 1, 2, 3, 4, 5, 6. I;j,U*, +n 0,1 ÆA, 2CXMjgGI;, 3XMF,U=~ F &dgG),HLgGW, ,UgGj%A,U4), %A[HL gG, !a,?yR[,y, ‘[w,U9’ jj,UgG, ?L)jL9/*. nI;\, gGW-jbH!&, qn!4_gGWj1C!&, 2CUL . gGF A &d, bH P(A) &d ‘gG A &VI;’.I;<!& j: : tgG A,B -3\&V, bHÆ,&VI;j)UI;, +P(A ∪B) = P(A) + P(B).

Ux N~i 4-\_, KY!&j.: : t A1, · · · , An, · · · ))-3\&V, bHP(

∞⋃

i=1

Ai) =∑

i

P(Ai). (1.1)J_;T2?,U : gG A &VI;U-&VI;ÆL 1, +P(Ac) = 1− P(A).FU=~0n, (Ω,F ,P) jHLj,UI;A, 3XMj3*">I;W)( 1,L,gGI;,.

9. I;&kI;: L,KL57, 'jjRg:ÆkL57, XM&'!p"j3"!A, C4, C, [y. m3 !oyj-j!A, W,Kw,, JU&8tZ. -j!A!oy">I;fLHr*57, Y1j*o, &k;;,3HL&kI; (s?, 57KI;HLkI;).L8: ‘BtZw 90 I;j 85%’, ‘W,KwkTI;j 20%’, XMY''Y' . qjtI;>j$>3!Aoy, bH?F*JQQv. # k, XM3,U-j!Aoy\!A?FI;t,'. n$> 8Bu\, ,U&8-j!A, qXM3nÆ k8t>U8!A; n$>e8i\n3DÆ# k8it>!A, qj%zwTtj4157kTt, T>-jFO">9,U&8. 8MCC?4Y5_ (L,S) (xLÆ, U8jHH, 3LHH

Ux N~i 5 T, t3'-xV,Y(j9f5H>3^vg"#dw-R8_.

10. 2: 95q,Us8, 3u 80 I;jkI;, #r*U83u 80 I;j&kI;, 8Mn$>ÆW)\?G*5)X)'. kRnn!\'-&'-)43!A-!A!oy, qj?GL.

11. !#*: !#*j%AKz. !8tZ, W,T*, C4, oKK7K\A. !#*jXMbCXwg#. !#*3|Jj*!oy, +j!. L!#**gjL57, tZ, L*\\j=~, C(R4'F 0, 1 &dJL!#*, gjIL57. !#*2CF6^ X , Y &d. 3,U!oy;Li*W)-3#L-3!#*, a,?y+ 10 y, L8i4U X , L8i#U Y , %jL-3!#*. !#*ÆALig, #U ,, Mw4jM,, t3 ig-j2, j, si .

12. !#*4.: t3!#*!&& j4., 4.U6-j-ÆUl, j!#*+j 4.# , g # k, ,9F&d!#*4.# , pJ%4.9, 84.9, qK4.jsHgh? R Xj,U!#*, L) CF**o4.: 4.zUOz.

Ux N~i 6

13. 4.z: !#* X 4.z7LF (x) = P(X ≤ x), x ∈ R, (1.2)M j X -Eu x I;. z F j^)Kz, 3Wjt p*, XMjp*j

limx→−∞

F (x) = 0, limx→∞

F (x) = 1.m3P(X ∈ (a, b]) = F (b)− F (a), a, b ∈ R, a < b,j X +n a, b ÆAI;j F (b)− F (a), *o,4.x7. tXM|w,U4.z9y\, sHz4.# h?;L4.-O4.z>, 39y,4.O), (Æ9y,4.bM. '34. M4.zvJ.

14. Oz: Oz3Yk*o!#*4.. Oz7L4.zv, +t F (x) j!#* X 4.z,bHOzL f(x) = F ′(x). m3Oz7j1C ∫ +∞

−∞f(x)dx = 1.2COzSg1b*o X 4.# , 3YLk, g9|e,9,%k.

15. *5: -jL4.zLOz, LOz4.zHLj&.

16. -UCF4.:

Ux N~i 7

(1) %4., +)U!#*4., !CTt,C4\6j;To;

(2) %v4., !A n `J[I;L p !p"J[`X , bH

P(X = k) =

(n

k

)pk(1− p)n−k, (

nk

) j n U-3\*+ k U;.

(3) zi4.: a)A [a, b] K (zi) !+U2L X , bHX 4.zj

P(X ≤ x) =

0, x < a;

x−ab−a

a ≤ x ≤ b;

1, x > b,Ozjf(x) =

1b−a

, x ∈ (a, b);

0, x 6∈ (a, b).zij*onOzn (a, b) KjC. &*5j, a(a, b)!+,UGg#LnfVj>w, no^Vj>-w, qj3\ko^fV, # ZYCn, ^4K'jG#4?F.

(4) 4.: !#* X >a4.jOzj [0,∞)Kzf(x) =

αe−αx, x ≥ 0;

0, x < 0.

Ux N~i 8 α > 0 j7. 4.L/F P(X > x+ y|X > x) = P(X > y), x, y > 0.

17. ": "j1C!&Ip, !#* X, Y ", t9x, y ∈ R, L

P(X ≤ x, Y ≤ y) = P(X ≤ x)P(Y ≤ y), (1.3)'jj%4.zj `4.zK#. tMLOz, bH%Ozj `OzK#. k, "j X +U Y +Yi. Fz T, (1.3) LE[1(−∞,x](X)1(−∞,y](Y )] = E[1(−∞,x](X)] · E[1(−∞,y](Y )].3WtK n x, y $L^J"\kx19LV

(;) z f, g LE[f(X)g(Y )] = E[f(X)]E[g(Y )]. (1.4)2)U\ , nZ"!#*\LF.

18. !: R X1, · · · , Xn j" >a7L 1 4., JU%, awg2LX(1), · · · , X(n),HL41*. R X(0) = X0 = 0, 9 x1, · · · , xn, JH z

P(X(k) −X(k−1) > xk, 1 ≤ k ≤ n)

= n!P(Xk −Xk−1 > xk, 1 ≤ k ≤ n)

Ux N~i 9

= e−xn−2xn−1−···−nx1,;_ X(1), X(2)−X(1), · · · , X(n)−X(n−1) " 4'>a7 n, n−1, · · · , 1 4..

19. ÆEU,>: !#* X ÆE7LEX =

∫ +∞

−∞xf(x)dx, f j X Oz, nGB,

EX =∑

n

xnP(X = xn),3ÆE'3j4.zOz4.:ÆE. ÆE'HLz, *o!#*n 57kz. ÆEU^kKj, tl&Z;!#* X +, bHjEX j<~ , JXMRW. ,>7L

Var(X) = E[(X − EX)2],kKS,!#*OÆEz>, j!#*)L. ÆE,>j, Mj!#*<!&)U6&41*, 3,U8SK)U&, SQ*!. qjM'\\j)U&, -ja2!o4.,-\_L\j^v, ,U)L 100 989, L 999qAn 1000 [3k, 1 9qAn,B, bHzqA|Q, q^4# A/s(, 3|ÆE,>jCn &[, j-&u4|!.

20. nKRqA!4, Y(D^4# , jSOASbUqA.

Ux N~i 10

21. ÆE,> : ÆEjjr , R a, b jC,

E(aX + bY ) = aEX + bEY, (1.5)q(*5j."# (-y|wbH?4W, t3XM-np9-W. ,>-jr Var(aX + b) = a2Var(X).qD X , Y ", r

Var(X + Y ) = Var(X) + Var(Y ). (1.6)'jj0&Oj-H#,>. CÆE-##,>L0.

22. <Z;: 9^ a, XML- E(X − EX)2 ≤ E(X − a)2, (1.7)'jjÆEj!#*<^,UC, FZ; (n 57K) j<G.

23. !: R n <F7:[, M!-[ , Xn &d<F_. 5W<F!U~, F Ak &d k <FgG, 1Ak&dd , bH

Xn = 1A1+ · · ·+ 1An

.;T E[Xn] = nP(A1) = 1, ;LP(A1) = · · · = P(An) =

1

n.Æ

E[X2n] = E[Xn] +

∑

i 6=j

P(Ai ∩Aj) = 2,;_ Var(Xn) = 1.

Ux N~i 11

24. ÆE1\ : Bk, XMÆ&1!#*zÆE, EX2, R X j!#*, OzL f(x), y = g(x) j^WK&z, bH g(X) 'j!#*, tKR\ ÆE, XMÆ&T g(X) Oz, q^L,UCn\ , 3QF X O1, 1C!&.Xs1.1

E[g(X)] =

∫

R

g(x)f(x)dx, (1.8)#4 ∫R&da −∞ w +∞ #4.

25. 4.: 4.jI;>3*!4_<!&4., -jÆ,. -UV,3a-3x7zwU4., 's De

Moivre a&qN, ys Gauss a9;*^>N, A. Einstein a4_44jprN, 'W, 4.kFo+!& . 4.Ozk7f(x) =

1√2πσ

exp

(−(x− µ)2

2σ2

), x ∈ R, µ ∈ R, σ > 0 j)U7. t!#* X OzK,XMj X >a 4., L X ∼ N(µ, σ2).

26. *r: 4.9w3,u, 3HL Bell curve,+*r. 9wiO x = µH, *r<QQj 1√2πσ

,;LU x $7)WR#7j 1, 3 σ e, *rjeQ#x,!#*je), (Æ σ eg, *rj~#, !#*je4G, + σ *o X ).

27. %0 4.: t µ = 0, σ2 = 1 \ 4.j%0 4., %0 4.9wjwz, *%0. %0 4.4.z

Ux N~i 12F Φ &d, +Φ(x) =

1√2π

∫ x

−∞e−

1

2t2dt, x ∈ R.UzILU_&d , 'jj#z-#4Y'FSz&d, 2CI;O2<EK%0 4.&.

28. f): R X1, · · · , Xn " >a%0 4., ' Y = X21 + · · ·+

X2n Oz.

29. ÆE,>: R X ∼ N(µ, σ2), bH EX = µ, Var(X) = σ2, 3)U757Ukj=. ^4K, t X j%0 4., bHO n,

E(X2n) = (2n− 1)!!.32u40#4q 'W. (u, KRm'K,Æ,%0 4., +tE(X2n) = (2n− 1)!!, E(X2n−1) = 0,bH X ∼ N(0, 1).

30. \k: n^41\, \k,UW4.z!#*j1CLF. g!LV,L,U>a [0, 1] zi4.!#*, HL!, 3^4KjO!, q'j_,. sHFVJ4.!#*h? F U &d!, t&,U4.zL F !#* X , b\ jX = F−1(U),

Ux N~i 13 F−1 j F mz,

F−1(x) := infy : F (y) ≥ x.\ 3Cn+, qj;Lmz-7jbH?4&d, 3Ff. 4.zmj-?4&d. LILYL,'? XN,U. G# j-,, PCn. XM<R F Ozj f . <RXM1b3\kOL g !#*, bnC c azL y, t g(y) 6= 0 \,

f(y)

g(y)≤ c,3jkRL

(a) V,U! U , mV,UOj g !* Y ;

(b) t U ≤ f(Y )/cg(Y ), bH X = Y , ;r!A,/.W X >aO f Z?4?f). XMUl,ksH\k>a%0 4.!* Z.

(a) 7 14.!* V ?4\k, 4.zj 1−e−x,mzj − log(1− y), ;_V,U! U , V = − log(1−U) >a7 1 4..

(b) -&E O f O. 3nzw Y mUC, R Y = V , (R Y = −V , bH Y Ozjg(x) =

1

2e−|x|, x ∈ R.%

f(x)

g(x)=

2√2π

ex−1

2x2

=2√e√

2πe−

1

2(x−1)2 ≤ 2.

Ux N~i 14

(c) mV,U! U , t U ≤ f(Y )/2g(Y ), bH Z = Y , ;roy(!AK/%.

31. f): W\kV!* X >aOz f .

32. f): t!#* X 4.z F &, W!#*zF (X) >a [0, 1] Kzi4..

W^? :GA II

=. zja 4.&zUm b, "XN,&zU!&Zr, 3XNI;><!&-U&q, Bernoulli g:&q, j!, !&=, -p, W 4.S, <=A.GÆEIp.iF^• &z• 4.mV • g:• Chebyshev - • &q• .GÆE

15

UX N~i II 16

1. &z: OzL f(x) !#* X &z7Lφ(t) = E[eitX ] =

∫ +∞

−∞eitxf(x)dx, t ∈ R, (2.1) i =

√−1. *57M W&zL4.zK,2.&z-?4, bC&FwAz#4 , qj&z1C!&, 'HL Fourier #, L)U!& :

(1) t X U Y ", bH X + Y &zj X &zU Y &zK#.

(2) t X U Y &zs, bHM4.s, 'jj&zK,x4.z.&zjwBG Y,', 3a2A4.z.

2. &zK, : jI;><fUt'j<<&Zr. !#*4.jJ,U9a7DzKÆE2,7FX(x) = P(X ≤ x) = E(1(−∞,x](X)),+Jd z

1(−∞,x](X) : x ∈ R2, nK"\1b(wu. 2?jjFNzeixX : x ∈ R,j/* (LV;) z g(X) : g ,U4), qjR)UPCnz K1b9a7Dw3(DT!#* X 4.& ,. ,UÆE ORz:J

UX N~i II 17"j3;T/*zJ". n # k, FUz3s+3zwÆTt. !, t X , Y &z,%, bHE(eixX) = E(eixY ) L x J", \3+Ld zz

FX(x) = E(1(−∞](X)) = E(1(−∞](Y )) = FY (x)L x J", ;_M34.. m!, R X, Y j!#*, E[Y eixX ] = 0 L x ∈ R J", bH

E[Y g(X)] = 0LLV (;) zJ",. ^4K<,KWB&djFUt, t3^4Kj Fourier #t.

3. 4.&z: R X ∼ N(µ, σ2), XMp11 X &z. XM-Al<R X j%0 4.. 6 ex Taylorw ex = 1 + x+ · · ·+ xn

n!+ · · · .UwA'J", ;_

eitX = 1 + itX + · · ·+ (itX)n

n!+ · · · .3) +ÆE, F 29 \ , M n j\ÆEL0, fL

φ(t) = E[eitX ]

=

∞∑

n=0

(it)2nEX2n

(2n)!

=

∞∑

n=0

(−1)nt2n

(2n)!!

UX N~i II 18

=

∞∑

n=0

(−1)nt2n

2nn!

= exp(−1

2t2)L,%0 4.&z, , 4.&zjCn,,;L (X − µ)σ−1 >a%0 4.. ;_

E[exp(itσ−1X)]e−itσ−1µ = E[exp(it(X − µ)σ−1)] = e−1

2t2 ,?# z = tσ−1 ;T, 4.&zj

φ(z) = exp(izµ − 1

2σ2z2), z ∈ R. (2.2)

4. f): R X ∼ N(µ, σ2), FÆKR,'1 E(etX), t ∈ R.

5. 4.: t X j!#*, logX >a 4., bHXM X >a 4.. ^4K, t Y >a 4., bH eY >a 4..

6. 4.mV : t X ∼ N(µ1, σ21), Y ∼ N(µ2, σ

22) X, Y", bH

X + Y ∼ N(µ1 + µ2, σ21 + σ2

2),U HLmV . &WGg#, F&z , Æ&W X + Y &zOexp

(it(µ1 + µ2)−

1

2(σ2

1 + σ22)t

2

)j3,, #3jk=|Tj X U Y &zK#, mJ&z , ;L X U Y ", X + Y &zjOM)U&zK#, %jW,mV .

UX N~i II 19

7. LqM 4.: RTou,M%M 4., qj&s"WBrown jLqM4., XMAf,kLqM 4.. Æ&,m _, ,U n RH, A HL, t910 (4*-/L0) n Mx* (.x*) x ∈ R

n LxtAx > 0,MKN t &dm,. t > HL ≥, bH A HL1C. m3,U1Cmjt \tj1<, +. 10. iOHm<!&TtjHm A &j^# bnMm Q az QtAQ jNm, Nr[j&/*. t A j1C\, &j1C; t Aj\, &j. ,U n M!x* X = (X1, · · · , Xn)j (1<) 4., tOz3Lk :

p(x) := C · exp(−1

2(x− a)tA−1(x− a)

), x ∈ R

n, (2.3) a ∈ Rn, A j n- Rm, A−1 j A m. m3x* a j X ÆEx*, + E[X ] = a, E X − a Oz?Gj

p(x+ a) = C · exp(−1

2xtA−1x

).

C jp*CazO#4O 1. \XM X >a 4. N(a, A). t A jnSm I, a = 0 \,

p(x) =

(1√2π

)n

e−|x|2

2 , |x| = xtx = x21+ · · ·+x2n, \ (X1, · · · , Xn) j" >a%0 4., XMj X >a n- M%0 4.. O

UX N~i II 20m A, bnK,m S az A = S2, \' S =√A.R X ∼ N(a, S2), Æ,M# k%0>r #

Y = S−1(X − a),bHJ [#4[\ Y OzLpY (y) = p(Sy + a)|S|

= C|S| · exp(−1

2ytSA−1Sy

)

= C|S| · exp(−1

2|y|2), |S| j S . , 'j[ Jacobi . . J_;T

C =1√

(2π)n|S|.x* a571b |, m A57jHh? ^4Kj X ,>m. J7 X ,>mj!#*m (X−a)(X−

a)t Vv'ÆEzwm, 2Lcov(X) = E[(X − a)(X − a)t].F# (X − a) = SY z

cov(X) = E[SY Y tS] = SE[Y Y t]S = S2 = A, E[Y Y t] j%0 4.,>m, m3jnSm.4s2.1 ,U n M!x* X j 4.t \tbn,U1<r # T x* a az T (X − a) j n M%0 4.!x*. ;_ 4.!x*1<r #-3>a 4..

UX N~i II 21

8. f): t X, Y %Ozj f(x, y), A j1<%R,,?#(U, V ) = (X, Y )A.' (U, V ) %Oz.

9. f): R X1, X2 " >a%0 4., (1)' Y = X2/X1 Oz. (2)2 X = (X1+X2)/2,W: X U (X1−X)2+(X2−X)2".

10. &z: !x* (X1, · · · , Xn) &z7Lφ(x1, · · · , xn) := E

(ei(x1X1+···+xnXn)

)= E

(eix

tX).XM 4.&z. sl'%0 4.!x*

X &z, \ X1, · · · , Xn "%0 4., ;_φ(x1, · · · , xn) = E

(eix1X1

)· · ·E

(eixnXn

)= e−

1

2|x|2.,, t X Ozj (2.3), bH √

A−1(X − a) j%0 , ;_E

(eix

t√A−1(X−a)

)= e−

1

2xtx,>#

y =√A−1x, x =

√Ay

E

(eiy

t(X−a))= e−

1

2ytAy, e−iyta KwM m y J x, zw

φ(x) = E

(eix

tX)= e−

1

2xtAx+ixta.

UX N~i II 22

11. g:: g:jI;><!&TtÆ,, j'Y9 (V) Bernoulli n 1713 nTWT, jtl!A,UJ[I;L p !p" `, bHJ[;I^I;. FV, !A,UJ[ (gG A) I;L p !p" n `, F n(A) &dJ[`, bHlimn

n(A)

n= pg:WT,B"I;,U,', ^4K, I;j,U!gG,U , V,n,Cn# <Rk, U -L8 . bH;jXMY[I;, , . qj n &O4g, 'jjXMRLt,U!p"3!A\,XM3LjY[wI;, I;3L57.

12. g:,: Y,, XMg:%.Xs2.1 RX1, X2, · · · , Xn, · · ·j" 4.s3!#*., bH

limn

X1 +X2 + · · ·+Xn

n= EX1. (2.4)" 4.s3 (, "34.) ^4Kj ‘"!A’ 5.

13. ?F: g:?Fo+, -L!=&Fwg:. T!, jj3Uh=TDE, #<=j!, nXMUJQj&$, 6n, <jB, !UBR1,U6*!(slot machine), ZlpToU9W MJ, t39

UX N~i II 23j!To, 3q8j,U!#*, =jg?G0 M,`Ki. bH<'j,`Kizwzq8, Fg:, B3~,U8&,Ug, 2k7z,3V3`, gI>- jj,`Kizq8,, U,'CnL, HL Monte Carlo ,'.

14. W: W-f, Æ&F Chebyshev - j9a,, 9!#*9 ε LP(|X| > ε) ≤ E[X2]

ε2. (2.5)4XMaCn- b.

(1) 1C!#*ÆE'j1C. t X ≥ 0. bH EX ≥ 0.

(2) t X ≥ Y , bH EX ≥ EY .

(3) O,UgG A, 7,U!#* 1A, n) A KO 1,P,O 0. bH E[1A] = P(A).

(4) sl?4|TX2 ≥ ε2 · 1|X|>ε,3

E[X2] ≥ ε2E[1|X|>ε] = ε2P(|X| > ε).

(5) 2Yn =

X1 +X2 + · · ·+Xn

n− EX1 =

1

n

n∑

k=1

(Xk − EXk).F Yn jA Chebyshev - , XMÆ1 EY 2n . ;L Xn"34. ,

EY 2n = n−2Var(X1 + · · ·+Xn) = n−1Var(X1),

UX N~i II 24;_9 ε > 0,

limn

P(|Yn| > ε) ≤ limnn−1ε−2Var(X1) = 0.

15. q'57: K7P*!#*. Xn q'LF -3x7, ,j-ZZq', jj-L% ω, Xn(ω) −→X(ω); %jz,q', +

E[(Xn −X)2] −→ 0;Fj-I;q', jjNN, 9 ε > 0,

P(|Xn −X| > ε) −→ 0.KR^4K1bW,z,q';T-I;q'. -fW-ZZq''kx1-I;q'. qj-ZZq'z,q'ILHskxig.

16. Borel-Cantelli =: t-I;q'\, '3;T-ZZq'. UTtRFw.4s2.2 t9 ε > 0 L∑

n

P(|Xn −X| > ε) <∞,r Xn -ZZq'O X .

17. f): tbn(O0. εn az∑

n

P(|Xn −X| > εn) <∞,bH Xn -ZZq'O X .

UX N~i II 25

18. &q: XMj<R Xn j"34., ÆEj µ, ,>j σ2. t Xi − µ |J,`;*^>, bH ∑ni=1(Xi − µ)j n `;*^>7, mV3 n jz^>, g:z^>^ 0. onXMb17^>4., m3V3 n ,, ;L7^>,>j nσ2, bH

Yn :=

∑ni=1(Xi − µ)√

nσÆEO 0 #,>O 1, j,U%0!#*.Xs2.2 (&q) 9 x ∈ R,

limn

P(Yn ≤ x) = Φ(x), (2.6) Φ j%0 4.4.z.

De Moivre<lWT%v4.&q, LaplaceH,W, &q-H, KR j Lindeberg-Levy&q. &q)6W, 4.nI;>S. WPf, Æ&Y Zr, 37I;O2, [3].

19. \k: &qW,XM,UCn,'^Æ\k 4.!#*. + Ui : 1 ≤ i ≤ n j n U!, 7Xi = 1Ui<1/2 − 1Ui>1/2,bH Xi "34., ÆE 0, ,> 1, t n g\

X =1√n

n∑

i=1

Xi^Æ%0 4.. n1!eegZ,, +,!?G-jgW).

UX N~i II 26

20. .GI;U.GÆE: ngG A &V.Gk, gG B &VI;LP(B|A) = P(B ∩ A)

P(A).!#* X ngG A &V.GkÆEL

E(X|A) = E(X ;A)

P(A), E(X ;A) = E(X1A). )k, 1 8L)Uv4, n1 ML,Ut$.Gk, )Ujt$I;j 1/3.</s91znqAj 50000, Kw91znqAj

80000, 9+,Us891qA2L X , bH EX = 50000, t x_8jKw8, bH191qAj.GÆE E(X ;A) =

80000.

21. /I;\ ;o: "# , tn,U!p", gG ΩnL \L,U&V, bHP(A) =

∑

n

P(A|Ωn)P(Ωn).3%,

E[X ] =∑

n

E(X|Ωn)P(Ωn). (2.7)

22. !: C(R4'2L 1, 0, &,UCzw,U 01.,XM x, m 1 ToÆ\A T1 j-4., ÆEj E[T1] = 2,bHm 11 To\A T11 h? -j-4.. L,, XM<RCRI;j p, bHJKR\ E[T11] = E[T11|0]q + E[T11|1]p,

UX N~i II 27

E[T11|0] = 1 + E[T11],

E[T11|1] = E[T11|10]q + E[T11|11]p = (E[T11] + 2)q + 2pzw,LE[T11] = (1 + E[T11])q + (E[T11] + 2)pq + 2p2,;_

E[T11] =q + 2pq + 2p2

1− q − pq=

1 + p

p2.

23. f): C(R4'2L 1, 0, &,UCzw,U 01 ., 'nU., (1) 11 s`Toz\A, (2) 01 s`Toz\A, (3) 101 s`Toz\A.

W? j)=..GÆEj!4_>< ,UIp,z7(WT,.GÆEV Æ, K,nr*# ksH1.GÆE, <!&j.T,.GÆE .!& , MnR>bCFw, <a!K =T,BIp.iF^

• .GÆE• <GZ;• .GÆEÆ• .GÆE1• .GÆE • !KUB

28

Uf p^lv r 29

1. ,`: .GÆE7 jJ%,39A. Kolmogorov H".

2. .GÆE: nZ;\, j!&, x, , Z;jY02,, 3ÆEj #. bHHj h? XM j%A,U4, 8j,U%A, etjj,U4, s(j,U4, UL34. l x%nU4jj xY ,. !ljs941l, xV 416, onWl,Us8, yn?R, &l1/nqA, l/,Y , bl3/szqA1;; on/,, l xjUt, l53j3/st zqA1;; jKw8, bHlj3Kwt zqA1;. 'jjÆEj1 #H#, .GÆEj1 kj?T<8vq. m!fy\, XpTy# , JU8n, 3?T<8Tyq. aUN, .GÆEjYZ-n.

3. iO!#*.GÆE: XMRFwg # j3!#*!x*?L _.GÆE, ,U!!#*^4KWT%A4, 8qA?L!#*WT,8XpqA4, 8(m?L!#*WT,(m4. iO!#* X ,UZ;^4Kj!#* X ,Uz φ(X),2,U!#* Y n1 X .GkÆEjnL1 XZ; Y <^,UZ; φ(X), sjz,s, +E[(Y − φ(X))2] = min

gE[(Y − g(X))2]. (3.1)

4. !#*?L : I;>!4_<!&Ipj;

Uf p^lv r 30 . nKT7; Æ& , qXMonCn; ULz. R X j,U!#* U!#*;J!x*, Y j!#*, Y J X x Y j X zJL Y iO X ;, +X33.1 tbn,Uz g az Y = g(X), bH Y iO X;, 2L Y ∈ σ(X).)UC44'L X1, X2, X j, bH XiO X1, X2 ;, X ∈ σ(X1, X2). ,UC4L X , X1 =

1X=6, X2 = 1X=2,4,6, bHM)iO-;.

5. ;&dUI;: R!#* X + 1,2,3, bHXM Y iO X;t \tbn a1, a2, a3 ∈ R,

Y = a11X=1 + a21X=2 + a31X=3.j Y ;&d, UI;;|Yig. qD P(X = 3) = 0,bHXM3 Y iO X ;t \t a1, a2 ∈ R, n-ZZ57kY = a11X=1 + a21X=2.j Y -ZZ57k;&d, 2C;&dCn,U;gLig. 3KR7 Y = g(X) &'-ZZJ"j3,, !4_B&dW)jj &d.

6. GW)U: Y iO X .GÆE φ(X), 2L E[Y |X ], jGW)E[(Y − φ(X))2] = min

gE[(Y − g(X))2]U, φ, g j^z. KRU<GW)LILK,U?sH'U? kR3(XM.

Uf p^lv r 31Xs3.1 φ(X) jUt \tF99 X z h(X) LE[(Y − φ(X))h(X)] = 0. (3.2)

7. W: lRKR.GF9, bHLE[(Y − g(X))2] = E[(Y − φ(X) + φ(X)− g(X))2]

= E[(Y − φ(X))2] + E[(φ(X)− g(X))2],;_ φ(X) jGW)U, # K,. (u, t φ(X) jGW)U, bH9^ λ,

E[(Y − φ(X))2] ≤ E[(Y − φ(X) + λh(X))2],=O2λE[(Y − φ(X))h(X)] + λ2E[h(X)2] ≥ 0,bH λ gj 0, f#L (3.2) J".

8. 7: φ(X) j Y iO X .GÆEt \t9 hF9 (3.2). U7n^4aF\1C,". n$> Y iO!#*.GÆE\, ,U/~ .Gj# , +E|Y | <∞.XM-C(w\;jXM7j<R# .GF9.

9. f): R X, Y j)U!#*, t9 x ∈ R, LE(eixX |Y

)= E

(eixX

),?F 3.1 W X U Y ".

Uf p^lv r 32

10. Uz y = φ(x) -Æ&9z h F9 (3.2), ^4KÆ&,&yzF9j3,, +9^ x,

E[φ(X);X ≤ x] = E(Y ;X ≤ x). (3.3)UCWPQy, W=.

11. .GÆE1: tXM x (X, Y ) %Oz f(x, y) 3* X Oz fX(x), bH (3.3) 3L∫ x

−∞φ(t)fX(t)dt =

∫ +∞

∞dy

∫ x

−∞yf(t, y)dt.) x 'vz

φ(x)fX(x) =

∫ +∞

−∞yf(x, y)dy,a#z

φ(x) =

∫ +∞

−∞yf(x, y)

fX(x)dy. (3.4);Lh x L ∫

f(x, y)dy = fX(x), 3 f(x,y)fX(x)

iO y jOz, HLW X = x \ Y , L\J Y |X=x, .GOz. #φ(X) jW X .Gk Y ÆE, 2L E(Y |X), φ(x) jW X = x .Gk Y ÆE, 2L E(Y |X = x), JK 3|T, OW X = x \ Y .GOÆE.

12. onXM x.GÆEk573*sH1.GÆE,:

φ(X) = E(Y |X), φ(x) = E(Y |X = x).k|φ(x) =

E(Y ;X = x)

P(X = x),

Uf p^lv r 33R.GzIps=. m3t X, Y ", bH X Z; Y IL(, IL j,%, ;_ E(Y |X) = EY . t Y Sjj X z, 'jj X A/2, Y , bHY SjKR<GW)U, + E(Y |X) = Y .

13. t X G\, E(Y |X) jG!#*, n X = x \+j E(Y |X = x), bHP(E(Y |X) = E(Y |X = x)) = P(X = x).;_

E[E(Y |X)] =∑

x

E(Y |X = x)P(X = x) = EY.W, ÆEjs?. n!, <RlLp, xVWkUzqA, qjl- xU8jaUW, bHbHljjF/szqAZ;. U\ RXM/szqA3XpVWzqA1, jVWzqA:.z. ÆOXM|e/I;\ : ,UgGI;j-34# kI;:.z. Y,, t; x 2 , ;Z;u!#*m42Z;U,bj42Z;j,%.

14. .GA/ : nR1.GÆEB- , Y!&jaF.GÆE UxW). nkR , XM-. X tyJ!x*, jLq U!#*), \KRb1\ ;3L. 2?kR W'LQy, qjkKUj-f.Xs3.2 .GÆE :

(a) E(1|X) = 1;

Uf p^lv r 34

(b) E(a1Y1 + a2Y2|X) = a1E(Y1|X) + a2E(Y2|X);

(c) t Y1 ≤ Y2 bH E(Y1|X) ≤ E(Y2|X);

(d) |E(Y |X)| ≤ E(|Y ||X);

(e) E[E(Y |X)] = E[Y ];

(f) t X1 J X2 2, bH E[E(Y |X2)|X1] = E(Y |X1);

(g) t Y J X x, bH E(Y |X) = Y ;

(h) t H J X 2, bHE(H · Y |X) = H · E(Y |X).j-#b-W|K-m3 `, qjj%, m3 `'jÆ&W, &sÆ, ljjaW. 3.1 jXMW &&Zr. 4XMW<,U . R φ(X) = E(Y |X), h(X) = H , ψ(X) = h(X)φ(X), XM&W E(H · Y |X) = ψ(X), 7, jjW9 g L

E[(HY − ψ(X))g(X)] = 0.#= O E[(Y − φ(X))h(X)g(X)], ?F7 3.2 +;TO0.

15. W: I;>iO!#* - , ,jn-357kJ", 'jj-J"I;O 0. &*5-jgGI; 0 gGG>'.

16. !: R (X, Y ) %Ojf(x, y) =

1

2π√1− r2

exp

(−x

2 − 2rxy + y2

2(1− r2)

),

Uf p^lv r 36uL, e, *. !#*) X1, · · · , Xn 3|Jj\ n , 3HL,U 5, 1\A5h, ee , XM-XMz:=. L\, XMFσX1, · · · , Xn &dx , F X ∈ σX1, · · · , Xn &d!#* X J X1, · · · , Xn x.

20. !K: a<Cn*b, ;2)8*, J`zA,[, ;AI;L p, zI;j q = 1 − p, F&d, R ξn&d"34.!.,

P(ξn = 1) = p, P(ξn = −1) = q, p+ q = 1,3 Sn(a) = a+∑n

i=1 ξi, bH Sn(a)HLa aT&!K (tzAI;,%\), 1H!K (tzAI;-3\). ta = 0, XMQ Sn. U!KjU!4_,/.

21. s`q\A: lF1CS,'. ;2)U:3\CJ`,[zA*, S0 = a j;i2, Sn j n l;Xk, 3T = minn ≥ 1 : Sn = aHLjs`q\A, jT!\A, t n ≥ 1 : Sn = aj)\, 7 T LY%. XM1 T 4.(W P(T <∞) =

1, qj E[T ] = ∞, 'jjnLq\Aiqqzq\AjY%. XMF1CS,'>, -.R a = 0, sln XYR|S0, S1, · · · , Sn, · · · ^4Kj,.Trx (n, Sn) : n ≥ 0, 30g,e=T8i, &jKk,T. m3 T jjw, ;L S2n−1 6= 0,

Uf p^lv r 37fP(T = 2) = 1/2;

P(T = 2n) = P(S2 6= 0, · · · , S2n−1 6= 0, S2n = 0)

= 2P(S2 > 0, · · · , S2n−2 > 0, S2n = 0)sHUI;? 4XM=A(O\, jJNg Maxwell Kelvin sl?F.Xs3.3 R a, b > 0, bHa (m, a) w (n, b) w x- $Trx7Oa (m, a) w (n,−b) Trx7.Æ& S2 6= 0, · · · , S2n−1 6= 0, S2n = 0 Trx, 'jja (0, 0)w (2n, 0)A-w x-$Trx7, (uXMAwu x- $Trx, Oa (1, 1) w (2n− 1, 1)wu x-$Trx, J&O\O (1, 1)w (2n−1,−1)Trx, ;_

P(S2 > 0, · · · , S2n−2 = 0, S2n = 0) =1

22n(N2n,0 −N2n−2,−2), Nn,a ja (0, 0) w (n, a) Trx7, -f1

Nn,a =

(n

(n+ a)/2

).jAz

P(S2 > 0, · · · , S2n−2 > 0, S2n = 0)]

=1

22n

[(2n

n

)−(2n− 2

n− 2

)]

=1

2n− 1

1

22n+1

(2n

n

),

Uf p^lv r 38;_L T 4.:P(T = 2n) =

1

2n− 1

1

22n

(2n

n

)

=

(2n

2n− 1− 1

)1

22n

(2n

n

)

=1

22n−2

(2n− 2

n− 1

)− 1

22n

(2n

n

),a#;T P(T <∞) = 1. ?F Stirling \

n! ∼√2πnn+ 1

2 e−nzE[T ] ∼

∑

n≥1

1√n= ∞.

22. f): ' P(S2 > 0, · · · , S2n > 0).

23. f): R Tk js`d k \A, F&O\ T1 4.:.

24. .*^z: ^zjf4.:, 1CL,'. . an : n ≥ 0 ^z (generating function) 7LG(z) := a0 + a1z + · · ·+ anz

n + · · · , z ∈ R.M jP,, Lq'Æd, tq'Æd, bHP,3(uK,x., \^z3L57, z 7WjM q')Wi. '3:LM j= P,w, -U|e Taylorw1

1− z= 1 + z + z2 + · · ·+ zn + · · · ;

−log(1− z) = z +z2

2+ · · ·+ zn

n+ · · · ;

Uf p^lv r 39

ez = 1 + z +z2

2+ · · ·+ zn

n!+ · · · ;

· · · · · · · · · .t bn ^zj H(z), bHG(z)H(z) = a0b0 + (a0b1 + a1b0)z

+ · · ·+ (a0bn + a1bn−1 + · · ·+ anb0)zn + · · · ,;_)U.u#^zO^zK#, j^z<!& . ^z,'=I;>, 3F$>+1C!#* ξ 4.:

G(z) = P(ξ = 0) + zP(ξ = 1) + · · ·+ znP(ξ = n) + · · · .bH^zx4.:.

25. !KBm : j90T&!K Sn, Tk &ds`d k \A, Sn d k 34L)/, (1) d k − 1; (2) ak − 1 d k. td k − 1 !\A, j,Ua k − 1 T&!K, UÆL", ;_a k − 1 d k, jstOa 0 d 1, W Tk O Tk−1 :K,UUÆ" U T1 34.*. j!KBm , |k, W-f, L,35.pp. F^z&djj

E[zTk ] = E[zTk−1 ]E[zT1 ] =(E[zT1 ]

)k.,, on4!K8,/, | S1, ) j, tO 1, bH T1 = 1, tO −1, bH&a −1 T&d 1, stOa

0 T&d 2, J_;TE[zT1 ] =

1

2E[zT1 |S1 = 1] +

1

2E[zT1 |S1 = −1]

Uf p^lv r 40

=1

2(z + E[z1+T2 ]

=1

2z[1 +

(E[zT1 ]

)2],UU,L

E[zT1 ] =1−

√1− z2

z, z ∈ [0, 1]. (3.5)

P(T1 <∞) = limz↑1

E[zT1 ] = 1,Æ, mF/I;\ E[zT0 ] =

1

2(E[zT0 |S1 = 1] + E[zT0 |S1 = −1])

= zE[zT1 ] = 1−√1− z2.M w

1−√1− z2 =

1

2z2 +

1

2!22z4 + · · ·+ (2n− 3)!!

n!2nz2n + · · · ,+zwR,%e

P(T0 = 2n) =1

2n− 1

(2n

n

)1

22n.

26. f): R ξ ^zj G(z), W:

P(ξ <∞) = limz↑1

G(z).

27. f): O,U-H!K Sn, F^z,'s\Tk 4.:.

28. f): nKRs`q\AW), 3 u2n := P(S2n = 0) 1 ,t S2n = 0, bH T ≤ 2n, ;_u2n =

n∑

k=1

P(S2n = 0|T = 2k)P(T = 2k)

Uf p^lv r 41

=

n∑

k=1

P(S2n−2k = 0)P(T = 2k)

=n∑

k=1

u2n−2kP(T = 2k), FUig;T P(T = 2k)?

W? C6=. B>j!4_UY><!&>, z&&K, DoobB , jj,U-.iOB!#4;3jB, UjU!4_ [, 3XMK0\Ip, J_=T L,W), 1,'jI;>b.iF^

• B• !#4• B • Doob 4U• 0\• t:=• znW)

42

Um zqh 43

1. ,`: BfjJ P.Levy n 1937 nb, &&Z?jJ.L.Doob n 1940 nAJ, g47Tn 1953 n+'.

2. aCn!Kb: R Sn &d,U\* n l;L2, j,U!K. ;L n lzATt Sn − Sn−1 U n l", ;_LE(Sn − Sn−1|S1, · · · , Sn−1) = E(Sn −Xn−1) = 0.t, 1 RTtZ;k,lTtIL9(, .GÆEj 0. '3J

E(Sn|S1, · · · , Sn−1) = Sn−1.UIpQyTjzwB7.

3. B7: R Xn j!., t9 n LE(Xn|X1, · · · , Xn−1) = Xn−1, (4.1)bHXM Xn jB. aKR4_3|T, Bjj\5, 'jjlF3 -jzwlL z. ÆEO 0"34.!#*04jB. tKR~J ≥, bHjee, HLkB, tJ ≤, bHee>, HLKB.KBkBj-\. ,, l-B*, BjkB, ljKB; 8M7jsenBeUjkB, bjkBh?'g53jKB.

4. B: Xn j!., t9 n LE(Xn|X1, · · · , Xn−1) = Xn−1,bHXM Xn jB. Bj\KijY^.

Um zqh 44

5. ÆE-# : m3E[Xn] = E[Xn−1] = E[X0],BÆEU n Yi, HLÆE-# , jB,U&.G.

6. "34.!#*#: R Y1, Y2, · · · , Yn, · · · j"34.!g., E[Yn] = 1. 3Xn = Y1Y2 · · ·Yn, n ≥ 1,bH Xn jB.

7. f): R ξn "34. P(ξn = 1) = p and P(ξ = −1) = q,

p+ q = 1. '^ a > 0 az Xn := aξ1+···+ξn jB.OUW), & E[aξ1 ] = ap+ a−1q = 1 j3,, 'jj%`,L a2p− a+ q = 0, U,LL)UU, ,Uj 12?,UU3F9XM&',.

8. ,UC_: !. Xn &d;n,Ul n \L,

Xn −Xn−1 &d n lz, K,U!#* Hn−1, Hn−1(Xn −Xn−1) j n lz,U, 3UlL2?,U82n;SK*. bH

Yn = Y0 +H0(X1 −X0) + · · ·+Hn−1(Xn −Xn−1) (4.2)j2nB*w n lLzA. C_, t;7:j\*, bH-l2sH+:= Hn, 'j,B\*, :=-jWi9Gi.

Um zqh 45

9. B : Doob VCk=wUC_( nK&dT, jHh? j298,%, -jn nlbZ n lTt, K, jjHn−1 ∈ σX1, · · · , Xn−1,'jj, j n − 1 l# 2:= Hn−1. (n O2K Hn−1 F Hn j+, bH Hn &'iO X1, · · · , Xn−1 ;, 2CHL- .)Xs4.1 (Doob) nU<Rk, Yn 'jB.g^K, Xp.GÆE ,

E(Yn − Yn−1|X1, · · · , Xn−1)

= E(Hn−1(Xn −Xn−1)|X1, · · · , Xn−1)

= Hn−1E(Xn −Xn−1|X1, · · · , Xn−1)

= 0.# Y1, · · · , Yn−1jJ X1, · · · , Xn−12, MO, ;_mF.GÆE ;TE(Yn − Yn−1|Y1, · · · , Yn−1) = 0.

10. !#4: KR!. Yn 'HLj Hn iO Xn !#4. B j!4_< , 3!4_>jjaUb. T, B jÆ&.G,&'JU Hn (iO%) jLV.

11. iOB5B: R Xn jB, bHFn = σX1, · · · , Xn

Um zqh 46HL,UB5. !. Yn HLj,UiOB5 Fn B, t9 n, Yn jiO Fn ; (\XM Yn iO5Fn k?) Yn iOB5 Fn jB, +9 n

E(Yn|Fn−1) = Yn−1.JB Hn iOB Xn !#4jiOB5B.

12. f): R Xn jB, Fn jB5.

(a) t Yn : 0 ≤ n ≤ N jiOB5B, bH Yn = E(YN |Fn),

n ≤ N .

(b) t Y ∈ FN , 7 Yn = E(Y |Fn), bH Yn : 1 ≤ n ≤ N jiOB5B.

(c) t Yn : 0 ≤ n ≤ N U Zn : 0 ≤ n ≤ N jiOB5B YN = ZN , bHL 0 ≤ n < N L Yn = Zn.

13. Doob 4U: R Xn jB, bH X2n jkB, ;L

E(X2n −X2

n−1|Fn−1) = E[(Xn −Xn−1)2|Fn−1] ≥ 0.kR Doob 4UnG\AILHgF, q&\An7!#4\L-+j!& .Xs4.2 (Doob 4U) bnK,tuL Zn az Z0 = 0,

Zn ∈ Fn−1 X2n − Zn jB, + X2

n 4ULB- (Zn ∈ Fn−1)tuLX2

n = (X2n − Zn) + Zn.^95kBJ". WCn, q 7 Z0 := 0

(L,K, ),

Zn := Zn−1 + E[X2n −X2

n−1|Fn−1],

Um zqh 473" Zn F9&'j3,, K, 4?f).

14. f): W Doob 4UK, .

15. B&dW): R Xn jB, bHiOB5Bj , B ,U(W)jj, iOB5Bj-j,jU!#4?'jjO9iOB5B Yn, j;bnk? Hn azYn jj Hn iO Xn !#4? +

Yn − Yn−1 = Hn−1(Xn −Xn−1).B&dW)jY><!&W), <ojJ Ito (T(nBrown jBW. nGBk, ,jILB&d,qXMR|w, n,Cn&yB\, B&djJ".

16. !\A: U::‘Z,X 10 lj8’ j, :=, 2?,U:: ‘Z,XA, 100 [j8’ 'j, :=, jh\A, j!\A, 5.'- xH\j0. ;_!\An^4?Fj$. tKR!\AHL0\, bHiF,jnJ,lT\:3zT5.j-j?G0,. -jL!\AL% , 8:‘tR&zFlXj8’, bH:n,lT\Y'zj-j?G0, ;L:Ya xRFljzjA, 3-j0\. R T j0\, J0\7, T = n J X1, · · · , Xn 3. 3 Yn = Xn∧T , k, jw\A T 0:=. bHYn − Yn−1 = Xn∧T −X(n−1)∧T

= 1T≥n · (Xn −Xn−1)

Um zqh 48J0\ ,

1T≥n = 1− 1T≤n−1J X1, · · · , Xn−1 x. JB , Yn∧T jB.

17. *t:=: <RlnB7U,UAI;L p *,2C p =O 1/2. 5+, t:=, ,` K [, tA,88, tz,k` ,, -`;, bHtl88\, l7j3A K [, JL,UT% !, 0\Aj!\A, >a-4., 'jjn"!A,UJ[I;L p !p", mJ[s`To\A, -jLq (q-LV). ÆUB \ G. ^4K-G, ;LB j*J,l (LV h\A) j\, t:=P*jYV\A. ^4Kt:=&tJ[r)U.G (1) Yq2; (2) Yq\A. tXMp2jLV*`LKV, bH%% !'j-bn (WÆ&TZr). 3LBJlL<gYq, Xn'YI Las Vegas |wKq2Cjkq1000 .

18. t:=S: R Xn jC*"!#*,t:=^4Kj,U!#4:K0\, 3Yn − Yn−1 = K2n−1(Xn −Xn−1);

T = infn ≥ 1 : Xn −Xn−1 = 1.bHw T 0!. Yn∧T jjKRt:=, 'jB, P(T < ∞) = 1 YT = K, 3ÆE-# O!\A T-J". JO0\!& , XMK-UiO0\!4.

Um zqh 49

19. s`Ee\A: ;nCKEG, L8lW,U=j X0, 3;xGEWk,UT= X0 QE&. R X0, X1, · · · , Xn, · · · j"34.!., Xn j n U8T=. 3T = infn ≥ 1 : Xn > X0,m3 T > n =OX0 ≥ maxX1, · · · , Xn X0 j n+1 T=<Q=, JH , JU8T=<QI;j,%, ;_t X0 4.j&, bH

P(T > n) =1

n + 1.J_;T E[T ] =

∑n P(T > n) = ∞.

20. ÆE-# : qjBnn0\K-,LÆE-# , n$>tW)\1b(wu. ^4KBn,U0\KLILÆE-# j1C!&ÆfW), &Wn0\KLÆE-# Æ&Fw.

21. f): 9KR0T&Cn!K Xn s`q\A T ,W: Yn = X2n − n, n ≥ 0, 'jB, 3W Yn n0\ T KILÆE-# .

22. :znW)BU': j,UCn#`Hh*W), ;2)U:3CJ`,[zA*, ;2Vi, a [ b [7:*, A-?W (jjY>T5>2), *w8zn+, W;znI;j M? F Xn &d;n n l

Um zqh 50*Xk, bH X0 = a, F T &d8zn\A,

T = infn ≥ 1 : Xn ∈ 0, a+ b,XM& P(XT = 0). m3, ;L XT L+ 0 a + b ) j, fE[XT ] = 0 · P(XT = 0) + (a+ b)P(XT = a+ b),;_

P(XT = 0) = 1− E[XT ]

a+ b,fXMÆ E[XT ] j,. tÆE-# n T KJ", bH

E[XT ] = E[X0] = a, %;TP(XT = 0) =

b

a+ b.XM3WÆE-# n T KJ".

23. f): L,U^IL m+1UIL, 3 0, 1, · · · , m-`!~. ;a 0 ~LT&, ,UC, oC(Rq8x=MIL,35C, 5q, ,w8$LIL"8T, Wa i ~ILTI;j M? 1 ≤ i ≤ m.

24. &qÆEM (,O): &WKRÆE-# n T KJ", &W\&qUÆE3M? ;LJ Doob ,

Xn∧T jB, 3LE[Xn∧T ] = E[X0]9 n J". 2?limn→∞

Xn∧T = XT ,

Um zqh 51f#&)3M, bHE[X0] = lim

nE[Xn∧T ] = E[lim

nXn∧T ] = E[XT ].Xs4.3 (LVq') t!. Xn q'O X bnC C az9 n L |Xn| ≤ C, bH

E[limnXn] = lim

nE[Xn]. FU, KRMj3,, ;L Xn∧T jLV.

25. f): RnKR:znW), ;A2AI;-3, VL pq = 1− p, mFB,'1;znI;.

26. f): R Xn jCn!K, Hn ∈ σ(X1, · · · , Xn) bnU n ω YiC C az |Hn(ω)| ≤ C, mR T j0\ E[T ] <∞,

Yn j H iO X !#4, W: Yn n T K'LÆE-# :

E[YT ] = E[Y0]. _)jOt:=U!4,U,OW. (d: ?Fq'.

27. f): FB,''0T&1H (p 6= 1/2) !K Sns`d k \A Tk 4.3*s`)0\A T 4..

28. ^zUB : B,'^z,'TFjg. Rk > 0, z ∈ (0, 1), ;L

E[zξn ] = pz + qz−1, E[z−ξn ] = pz−1 + qz3 (pz + qz−1)−nzSn

U (pz−1 + qz)−nz−Sn

jB. Jg:, t p > q \, Sn (O +∞, 3 P(Tk < ∞) = 1; #t q > p\, Sn (O −∞. onLÆE-# E[(pz + qz−1)−Tk∧nzSTk∧n

]= 1.

Um zqh 52t n → ∞ \, D Tk < ∞, STk∧n &qj STk, (Æ&qj −∞.2?= ,~*LV3jM&qÆE. &

pz+ qz−1 ≥ 1 STk∧n kLV, qj STk∧n jKLV#-kLV, ;_XMKR+B, -, ?G+2,UB(pz−1 + qz)−nz−Sn

,\& pz−1 + qz > 1 j3,, #t p > q \, pz−1 + qz n

(0, 1) KgO 1, JÆE-# E[(pz−1 + qz)−Tk∧nz−STk∧n

]= 1m4 n (OY%z

1 = z−kE[(pz−1 + qz)−Tk ;Tk <∞

]= z−kE

[(pz−1 + qz)−Tk

],+ 0 < y < 1, U,L pz−1 + qz = y−1 z

E[yTk]=

(1−

√1− 4pqy2

2qy

)k

. p, q Sjzw T−k ^z. L, T ^z, 4!K8,/, w 1 −1, \ T j4'stO0T& T−1 T1,

E[zT ] = pzE[zT−1 ] + qzE[zT1 ] = 1−√

1− 4pqz2.

W? ^#x%=. !4_nY>?FsljnÆ.=,R,zXNkCn%vÆ.\, =A% , 62, Æ., o, PY>IpS, (L=B;3*B&dQyIp~wY>57, &&tj Doob B , 2uzf, V?G KU,B57. nQYY>ÆXt(g9j^fV3\k, qEa-jA,3-&3LXMKjY>, jtB4.iF^

• % • Æ.• 5>2:=• =B;• P• A=

53

Us Xt ebu 54

1. ,`: Ito3LY, qjaILTt?FOY>1W. <o>Gg#j 1900n's'Y9 H. Poincare *eV Bachelier *e>U, >UaUNvT, Brown j,0I;O, on2C:Lj A. Einstein [b'Y>Uo, 8 n, /j Bachelier >Unt\uE, ;Y8U#/F, ,w 60 n38!|w, 3on:LY>fO 1973n=M F.Black, M.Scholes, R.Merton,g.Z?, \, Ito!4_>1bAK. _l_, JOÆ&QTZr, O&\A!4_Y>j-4U, 3 1979 n Cox, Ross, Rubinstein %>wG\AB, azY8Wk, # XMU!4_LY%(8.

2. T% : % >Uj arbitrage, jIL9nq8. 'jjY>nBsH#, ,-:=. T, bn,U:=Hn, az H iO X !#4 Yn F9 Yn − Y0 ≥ 0 (&dIL9n) P(Yn − Y0 > 0) > 0 (&dLq8). % j9,UBAnB-j?.

3. UI;: O,U!oy, XMCC=<!&I;;. CC4, ^L,UI;n. UI;OjjU\, j3,U8Cj\#2,U8C-\,%. H#; g#, ,# kj-ty~, qnY>1W11CL57. n-3I;;, =I;;L&yx7. RL%A Ω, P jK,UI;;, S ΩgG&Vj g. t Q j Ω K2?,UI;, 9gG A, P(A) > 0 t \t Q(A) > 0, bH P U Q j

Us Xt ebu 55=.

4. ! U*: U=I;;-j,G?4g#, I;;j!gGj , *'r, =I;j, *'r, q-H#! , 'jj-!gG#J2 , '-2 gG#J!. !#* ξ I;+ 1,−1, qn=;k, +j 1,−1, -uI;H#,, #J, p, q. -!gG#J2, jp, q j, ;Lt p = 0, bHjO 1, on1#z-j, H#,! ; -2gG#J!, j p+ q = 1, ;Lt p+ q < 1, bH3U-j+w#zj+w, 'H#,! .

5. =B;: tL,U!#*. Xn, n\I; P kj-jB, qjnUU P =I; Q kjUB, bH XnL=B;. R Yn j"34.!., ÆEgO 0, bHXn = Y1 + · · · + Yn jkB. LIL=B;? 3|T, tYn +1C\ (\% bn), -jL=B;, qjtP(Yn < 0) > 0 \, ,L=B;. (m3t Xn j1C"34.!.K#\LÆT>.) WHW)h? t Yn |J*zA, bH Yn 1Cj_8X-Lzj , #t P(Yn < 0) > 0 \, -lÆE g, bnzj .'jjtzAj bn\, =B;bn, ;r-bn. |ÆCn, ^4KjY>=, U. q, ,>gIbU 1980 nj3<AI.Xs5.1 (, ) % !-U=B;^b.

6. CnY>nB: ,Ubk ;zL r nB,Un\

Us Xt ebu 56 n =TL Sn enB. XM<Re=Tj"34.!#*K#, +Sn = S0ξ1ξ2 · · · ξn, (5.1) ξn "34., S0 je&=T. t ξn +)U

d, u, d < u I;4'L q, p, p+ q = 1, t ξn + u &den\ n |, ;r&d. bHXMj,U%v\. <RXM3a<Yqbk, Oe'35DE. ;L Sn !, f#J6e3bng*6!L. 6!U!oyB.

7. Y% : slXMW, <RnBY% , rLd < 1 + r < u.g^K, t 1 + r ≥ u, 'jj ;uQ, bH8M3lET

(D) 1 ee, z2Y S0, bA<, nk,\+TbDe, L (1 + r)S0 − S1, ;L 1 + r ≥ u > d, 3_!#*j1C, # 3I;gO 0, ;_nBbn% . 3%XM3Wt ;u~ 1 + r ≤ d, bH8M3k-De#zY9nq8, fnB'L% !.

8. Æ.: Æ.j!V0, ;Lj-EOe%0*o=. r*, Æ.j,6d, 3v DAÆ.L!, WRd8nUJ\A3J=T`De. . RJ\Aj m,J=TL K, tw\e=TQOJ=T, bHd8jL , L (Sm −K), (Æ, Y 9, 0. , 3Æ.=j!, OVm := (Sm −K)+. (5.2)

Us Xt ebu 57K DAÆ.L-3, Rd8nJ\AÆ9\A3-J=T K DAe, 3=LVm := min

1≤i≤m(Si −K)+. (5.3),, ,U n\wÆ!V0jj,UiO S1, · · · , Sn ;!#* Vn.

9. fZV: Æ.j, . , 53-jQ3, ?GE M53jL57W), Ijzv Æ.j!_, OM, <53jIL99nzvÆ.3z,6jr3. %:=j;bn? Æ.<Sj?L, Y>\;T, &&_jL~nFe\9n. 8|e, ;_g*Tve3mn~=\DA# , qL,-eB,K|i9n, b5D,6||Æ.d, %3n|8 # k-OYg. qj&wW, Y 6!FOY:6!#J0g,9n. -nY>H!3jjJÆ.Y>\=.

10. o: ;LL ;L r nBbn, n \,[stOn + 1 \ 1 + r [, 3Inon|Æ&o, +n + 1 \ K [n n \j K(1 + r)−1 [. oY1j,Ud, -jHg#, tÆ&, XM3<R r = 0,-R$>pJ9-3. 2?, ng # k,

m = 1 '9aWW),.

11. =: Æ.dj,U!, B!A/Æ. 3sltw=,'jg:, +=T?Gj!

Us Xt ebu 58ÆEo(1 + r)−mE[Vm]. (5.4)nIL'# k, jK,q, =tCn, qjkxqg9n, Y'V, _l9n1%* (`D*) t:#DM, Æ.j!_Sj-=. 2?U=UI; p Li, n^4# k, UI;f021. bHkRXNÆ.=tyj1C+.

12. tCnS: Æ.=tj2uÆ.d 32ue62z.

13. sH: <XLSb2Y x0, Xj-j2u62e-wÆ.= Vm, -.a m = 1 b. XD x ee, , xS0 [, 3Xk x0 − xS0 bA<, %jj,U62:=, x0, x jm7. %w\ 1, Xe= xS1 #bA<= (1 + r)(x0 − xS0), , X62=L

xS1 + (1 + r)(x0 − xS0).X_j-Æ.=W`D, 3XÆE-lnH# k, Lφ(S1) = xS1 + (1 + r)(x0 − xS0), (5.5) φ(S1) jjÆ.=, nv ||#, φ(S1) = (S1 −K)+.

14. U,L: KR,L (5.5) j-jjJ"h? ,j-j, ;LXM&'jnLj# kJ". qjnXMU%v\&'<Rkj. S1 = S0ξ1 j+)U S0d S0u,;_Lφ(S0d) = xS0d+ (1 + r)(x0 − xS0),

Us Xt ebu 59

φ(S0u) = xS0u+ (1 + r)(x0 − xS0).UT)UQ *x =

φ(S0u)− φ(S0d)

(u− d)S0

,

x0 = (1 + r)−1 (u− (1 + r))φ(S0d) + ((1 + r)− d)φ(S0u)

u− d. (5.6)'jj, &,bL x0 [, jJj32u3K62zUÆ.d3%=, 3UÆ.dM?GEH . %-lnB#, jJ-L9Y. 3 x0 ?GjÆ.=T, (5.6) jjÆ.=\ .

15. AKTl: qjXM?G|w, g#_AK\;j%v\<R, 1C&y.

16. !4:<Reon=Tj 50[, kU\3 3/4I;| 20%,

1/4 I; 20%. UgD,6k,\wÆJ=TL52 [v ||Æ., <R ;3=-1, bHÆ.g:=j

(60− 52)× 3/4 + 0× 1/4 = 6[, #KR=\ , Æ.=Tj(60− 52) · 20%

40%= 4[, KR\ RXMH>P9n, l-<W 16 [, :KEÆ.z 4 [D 2/5 ee01, wk,U\,tew 40 [, bHÆ.-o, XMEez 16 [W<; te|w 60 [, bEez 24 [, W<

16 [, mWÆ.ML8 8 [z.

Us Xt ebu 60

17. Æ.PL?F: XWKRU!4Ul,kÆ.PL?F.8LZB[, Le|, tW_g-,, tDejD 1000 e, teK|\, |T, -,B[, wzM,N-tW, j,k, DÆ., 3D 12500 6Æ., te|,, J6Æ.- 8 [, ^ZB[, 'jj-ZB[. jjÆ.O857, qj'F,, tYe,, bHÆ.JL,2, ZB[Yq; #DlDje, lL B[n.3Æ.jPL, 30g,nq8'0g,9n, 7Æ there is

no free lunch.

18. k=: 3jk=Æ.=\ (5.6), XM&oU\ U p ILig, j,U , ;L p j,U3875, f2, j&19n. 2?\ L)UO 1 u− (1 + r)

u− d,1 + r − d

u− d,2L p′. )ULT?F, XM3!7I; P az

P(ξn = u) = p′, P(ξn = d) = 1− p′,bH P j P ,U=I;;azE(ξn) = d

u− (1 + r)

u− d+ u

1 + r − d

u− d= 1 + r,voe=T (1 + r)−nSn n P kJL,UB. 3

P j P n 57k,U=B;, # XM'&|w, P jaz (1 + r)−nSn JLBK,=;. jjY>=>% .Xs5.2 (% ) Æ.LY9n=t \t=B;K,bn.

Us Xt ebu 61

19. m|=\ : F=;, =\ (5.6) 3JLx0 = (1 + r)−1[q′φ(S0d) + p′φ(S0u)] = (1 + r)−1E[V1], V1 = φ(S1), jÆ.dq8nI;kÆEo,Ug:=\ (5.4) s3, jI;-3. Y,,3\A m LJ\A!V0 Vm n 0 \v=?Gj

x0 = (1 + r)−mE[Vm]. (5.7)kRXMp1W.

20. 62: 62x:5jjxn\ k ` MDe. <RnL Xk 4D, `D (E) Hk 6e, XkbA (k)<. bHnk,4D7LXk+1 = HkSk+1 + (1 + r)(Xk −HkSk), (5.8),U62. Hk ?1,U4D. Xk. IK Cz(1 + r)−k−1Xk+1 − (1 + r)−kXk

= Hk

((1 + r)−k−1Sk+1 − (1 + r)−kSk

).XM x (1 + r)−kXk U (1 + r)−kSk 4'j4De=To., j Hk iO!#4. ;LiOI; P jB Hk iO S1, · · · , Sk ;, 3Xp Doob B (1 + r)−kXk iOI; P 'jB, +o4Dn;k'j,UB.

21. 5>2: LU!&Ip"XN,k, 5>25>- j5W, n62uLV,,b2Y?-m:A+T2

Us Xt ebu 62Y. |,L (5.8), 3 Bk := Xk − HkSk, j62eP"0A<, nk,\, Bk+1 = (1 + r)Bk, ;_Xk+1 −Xk = Hk(Sk+1 − Sk) + (Bk+1 −Bk), (5.9)'jj4Dt:5et< , IL_,jj5>2.

22. =: tXMLSb2Y x0, t2u62 Hk : 0 ≤ k < m nJ\A m dwUÆ.d Vm ,%4D, + Xm = Vm, bHBÆE-# zx0 = (1 + r)−mE[Xm] = (1 + r)−mE[Vm].%W,XM=\ .

23. A=: 3jj|T, KR\ jL,Ulx.G, 'jjbn Hk : 0 ≤ k < m az Xm = Vm. L,Cn, 4XMn r = 0<RkUlUW). t Hkbn, bHXMjL,U^bn62,'nenK-w Vm, 3%='HLA=, ;LÆ.-_%32ue62^o. bH Hk bn5R1Hh? jRB&dW). Snn; P kjB, B52L Fn. ;L Vm jZl xÆ.n m \=, 3Vk = E(Vm|Fk),bH (Vk : 1 ≤ k ≤ m) jiOB5B, Vk jjÆ.n k \Y9n=T, J Vm K,2. tiOB5B3B&d, bHbn Hk azs?!#4 Xk = Vk L 1 ≤ k ≤ m J", W)jUx,.

Us Xt ebu 63

24. r*8?: j<R r = 0, ;rl4o Sn t> Sn Zj3,. a m \b,

Vm − Vm−1 = Hm−1(Sm − Sm−1), Vm j S1, · · · , Sm z, Hm−1 Sm−1 j S1, · · · , Sm−1 z. <RXMn m − 1 \, \ S1, · · · , Sm−1 j1 , XM&U Vm−1 Hm−1. ;L Sm sO Sm−1 L) j Sm−1u Sm−1d, 3XMzw)U,LVm(S1, · · · , Sm−1, uSm−1)− Vm−1 = Hm−1Sm−1(u− 1);

Vm(S1, · · · , Sm−1, dSm−1)− Vm−1 = Hm−1Sm−1(d− 1).)U,L)UQ *, UzHm−1 =

Vm|u − Vm|dSm−1(u− d)

;

Vm−1 =(u− 1)Vm|d + (1− d)Vm|u

u− d= E[Vm|S1, · · · , Sm−1].

25. =U62:=: ,, 0 < k ≤ m LHk−1 =

Vk|u − Vk|dSk−1(u− d)

;

Vk−1 =(u− 1)Vk|d + (1− d)Vk|u

u− d= E[Vm|S1, · · · , Sk−1]. Vk|u = Vk|ξk=u, Vk|d = Vk|ξk=d.

26. f): R ξ j1C!#*, W: P(ξ > 0) > 0 t \t Eξ > 0.

27. f): RL,eon\ 0 =Tj 50 [, nJU\A33/4 I;K| 20%, 3 1/4 I;k 20%, <R ;3=,

Us Xt ebu 64UgYn0j\`Dn\ 2 wÆJ=TL 52 [v |Æ., 'jjLn\ 2 3 52 [=TEWj\,ee. , W6Æ.?GE M? j\?GsH62P9n, t\ 1 ek, 6Æ. M?j\N?GsH8??

28. f): <RK)bk ;j r = 5%, s?W)?GsH>?

29. f): t (5.1) ξn + 0 < u1 < u2 < u3 I;4'Lp1, p2, p3, p1, p2, p3 j L 1, WbnI;;azo=TuL (1 + r)−nSn JLB.GjH? K,C?

Wu? Mr<Z=. aG\A!uLw&\A!uLntU,',RY:Qy, q&\A!uLn\k\*j\m3Y53. Brown j?L\kJV\9 R.

Brownn 1927nk=w(S5G#n+*&R!jj<!&!uL, J A. Einstein n 1905nn$>7$oy\WTS(J N. Wiener n1918 n<WTbn W3, ,j! f. zXN Brown j7* , 2u,b!4XNt,'.iF^

• Brown j• "Vt*• LqM4.• sd\4.• 5sÆ • Donsker -#\

65

Ua Q`|V 66

1. !uL: aG\A!.w&\A!uL, kK>- , q,'-3, ;L\*j,j&, 3&\A!uL2CjY\kf!oy.

2. &\A!uL: nU\A)AK&28,U!p"TtHL&\A!uL. r*, %A Ω K!#*:X = (Xt : t ∈ [0, T ]),HL!uL, T 3j +∞. *5 Xt = X(t, ω), t ≤ T, ω ∈

Ω j)U#*z, t j\A#*, ω j%#*, h t j!#*, h ω j [0, T ] Kz, HL% ω rx, %rx.%rxU^y, 28,Ue8ij,.%rx, 285F*j,.%rx, 28wd,U<8j%rx, . !#*?j,U, #!uL?j,Uz, 3!uL'3UL!z.

3. =~: !uL%0'j X = (Xt : t ∈ [0, T ]), X jU!uLY6, Xt j!uLn\A t +, L\XMQ!uL X (Xt) Cn Xt , -,,U.

4. LqM4.: 9+LqU\A 0 ≤ t1 < t2 < · · · < tn, !x*(Xt1 , Xt2 , · · · , Xtn)4.HL!uLn\A (t1, · · · , tn) KLqM4., LLqM4./*HLLqM4.:. t)U!uLnLs3\AKLqM4.s3, bHXMM=.

5. <CE<!&)U!uLj Poisson (+) uLU Brown (.) j, S\A [0, t] wd>8, 'HLx

Ua Q`|V 67uL, S5n+*&RJO44.=jr$, Js`S oyV\9 BrownY6ZY, q^4K\H"&&q[O\9 A. Einstein 9 N. Wiener.

Poisson uL%rxj/f, Brown j%rxj&. UNgY6%g1, t# , )U!uLj H!&.

6. 5: \A t Æ σ(Xs : 0 ≤ s ≤ t) 2L Ft, 3xYq U!#*, qjk57UG\AÆ. (Ft) HL 5. *5 Ft x-Y% U!#*, Y:QyYfU, jOU!4_ji!&. 4XMa-U-3N>,UUl, dEja(U. sl, XM3W,U!#* ξ iO;t \t3a+T,U!.az Xsn az ξ 3JU.z. 2?!#* ξ UFt "j ξ U9LqU!#*;J!x*".

7. iO 5.GÆE: sHU!#* ξ iO.GÆEE(ξ|Ft)? kK, RFKiO!#*.GÆE,%, j1 .Gk<8Z;. r*1\, j,UiOFt ;!#* η, F99LqU!#* Xs1, · · · , Xsnz g, L

E[ξg(Xs1, · · · , Xsn)] = E[ηg(Xs1, · · · , Xsn)].2CXM-Æ&-1!#*iO 5.GÆE, Y j?F . FKRiO.GÆEb n 5# kjJ", ?GU(2), RbCFw.

Ua Q`|V 68

(a) E(1|Ft) = 1;

(b) E(a1Y1 + a2Y2|Ft) = a1E(Y1|Ft) + a2E(Y2|Ft);

(c) t Y1 ≤ Y2 bH E(Y1|Ft) ≤ E(Y2|Ft);

(d) |E(Y |Ft)| ≤ E(|Y ||Ft);

(e) E[E(Y |Ft)] = E[Y ];

(f) t s ≤ t, bH E[E(Y |Ft)|Fs] = E(Y |Fs);

(g) t Y j Ft ;, bH E(Y |Ft) = Y ;

(h) t H j Ft ;, bHE(H · Y |Ft) = H · E(Y |Ft).

8. &\AB: ÆOG\A, H!uL X = (Xt) jB, t95 t > s ≥ 0 LE(Xt|Fs) = Xs. (6.1)=Ot*.GÆEO0

E(Xt −Xs|Fs) = 0.3%'LKBkBIp.

9. "t*: R X = (Xt) j!uL, Xt 3+nQMA, Brown jS5n+*&Rj, ;_?Gj%M, qjL,Cn, XM2C<RMj,M. O t > s ≥ 0, Xt −XsHLt*. ,U!uLHLj"t*jt*jY26, t* Xt −Xs U\A s Fs ", 'jj, 95+ s \A 0 < s1 < · · · < sn ≤ s, !#* Xt −Xs U!x* (Xs1 , · · · , Xsn) ". Poisson uLU Brown jj"t*.

Ua Q`|V 69

10. Bm : Bm jWonS, IUu-j". Bm j!uL<!& Æ,, "t*!uL,jLBm . Bm j3!s9 Markov Y6ZY, n 150ns`QyTonTBm . LBm !uLHLBmuL. "t*kxBm , ;L Xt − Xs U s Æ", ;_n x Xs Xt U s ,j", FV&d: 9t > s > 0 U x ∈ R, L

P(Xt ≤ x|Fs) = P(Xt ≤ x|Xs). (6.2)

11. Vt*: tt* Xt−Xs U Xt−s−X0 34., bHXM!uLjVt*. Vt*jt*4.-O\A> t−s,U\A s, t Yi. Poisson uLt*4.j7L λ(t− s) Poisson 4., # Brown jt*4.j 4. N(0, t−s), jVt*uL.

12. ;_ Poisson uLU Brown jjV "t*uL. L,,", XM<RMa 0 T&, + X0 = 0. F9"t* UVt* !uL2HL Levy uL, Levy ,,g!uL, 3F\k !oy. nY>nBK, 8M1b-F9OF Brown j\ke=T, ;L2u,p4_,,:Le8ixL/oy, e8i4.Æ-?Gj!P, ) oy-= Brown j, 3dEj3=)oy<Cn!uL\ke8i, jjLevy uLVuL.

13. .j: F9kR.G!uL X = (Xt) HL\T&Brown j,

Ua Q`|V 70

(a) X0 = 0;

(b) "t*(c) Vt* Xt −Xs ∼ N(0, t− s);

(d) %rx&.tXM)U<\, )U<(R-3, qjM4.,%; 3%, tXMk=)U5j\, Mr$-3,qjMj Brown j, ;M4.s3. 2? X = (Xt)ja x T& Brown j, t (Xt − x) ja0T& Brownj. ,, n M Brown jj n U" Brown j?L4*;J!x*uL. 2?, Brown j (eXt) HLj- Brown j.

14. bn : 2C!uLL,Ubn W), O Brown jI_, bn W)ubÆ59Z n, <j N. Wiener n1923 nWRW, 3 Brown j'HL Wiener uL. ', Brown jbn -3L3, :L Brown j?LÆ5n+*&Rjr$53bn1, q^4KXM Brown jjV\ Brown j,U\, jF9W.G,UIp, bn zw?',#-jj3V\k=. ,UIpHL Brown jj;L:L\k,V\T Brown j.iO Brown jbn W, 7 [4].

15. .jLqM4.: R X = (Xt) j0T& Brown j,J"t* ;T9 0 < t1 < t2 < · · · < tn, t*!x*(Xt1 , Xt2 −Xt1 , · · · , Xtn −Xtn−1

)

Ua Q`|V 71" j 4., 3%Ozj n M 4.pt1(x1)pt2−t1(x2) · · · ptn−tn−1

(xn), pt(x) &dÆE 0 ,> t ,M 4.Oz. JO!x* (Xt1 , · · · , Xtn) jK!x*,Ur #, 3'j 4., Ozjpt1(x1)pt2−t1(x2 − x1) · · · ptn−tn−1

(xn − xn−1)

=1√

(2π)nt1(t2 − t1) · · · (tn − tn−1)

· exp(−1

2

(x21t1

+(x2 − x1)

2

t2 − t1+ · · · (xn − xn−1)

2

tn − tn−1

)).;_ Brown jLqM4.jÆEL0 4., j GaussuL (LLqM4.j 4.!uLHLj,U GaussuL). OÆE0 4., x,>mj9a,, i j .[L

E[XtiXtj ] = ti ∧ tj .

16. ,>: sH,> E[XtXs]? F"t* , t t > s, bHE[XtXs] = E[(Xt −Xs)Xs] + E[X2

s ] = s.3H, &IOF"t* , Brown W) K3Ux.

17. f): R X = (Xt) j\T& Brown j, (1) 9t1, t2, t3, t4 ' E[Xt1Xt2Xt3 ] U

E[Xt1Xt2Xt3Xt4 ];

(2) R s < t, ' E[Xs|Xt]; (3) R s < t < u, ' E[Xt|Xs, Xu].

Ua Q`|V 72

18. f): Einstein &o Brown j Xt Oz u(t, x) = pt(x) ?GF97[v,L∂u

∂t=

1

2

∂2u

∂x2,WÆ.

19. : ;_&",U!uLj Brown j, ,U'j"j Gauss uL, ÆEL0 ,>L E(XtXs) = t∧ s j3,, 3kR-U ?4".

(a) H : (−Xt) 'j Brown j;

(b) sÆ : 9^ c 6= 0, Yt = c−1Xc2t 'j Brown j;

(c) m : \AN,u, Yt = tX1/t 'j Brown j.

(d) A : t T j0\, bH Yt := Xt+T − XT ?L!uL'j0T& Brown j.

20. -#\: R ξn "34.!#*, Eξn = 0, Var(ξn) = 1. 3Sn = ξ1 + · · ·+ ξn, Fr (n, Sn) -`s&, +

St := (t− n)(Sn+1 − Sn) + Sn, n ≤ t < n + 1, n ≥ 0.bH St j!uL. 3\AA>UNK#, 9 n, 3X

(n)t =

Snt√n, (6.3)t neeg\, uL (X

(n)t )|K-ew,U Brownj. Oh t, J&qj3|T X

(n)t -4.q'O 4. N(0, t), #YgTtk.Xs6.1 (Donsker -#\) t n (OY%\, X

(n)t q'O,U

Brown j.

Ua Q`|V 73

21. f): Brown jk, qjngKY'Q^o, XM3F Donsker -#\\k Brown j, jgÆZ. ,UL\k Brown j.

22. s\: R Ta ja 0 T& Brown j X sd a ∈ R \A,a2Ta(ω) := inft > 0 : Xt(ω) = a,3 Brownja 0T&, qj7, &W T0 QO 0. nCn!KBk, XMT,4.( W, P(Ta <

∞) = 1, bHU,';JO Brown jh? Brown 'LH , + (−Xt) j (Xt) 4.s3 Brown j, 3'3?F(O\. R a > 0, n Ta(ω) < t \, %rx Xs(ω) ?L s zn t Æw a, bHnw a ÆrxDK8Dk8jj, ;T Xt > a+ (b− a) Xt < a− (b− a) I;s3.Xs6.2 ((O\) R a > 0, b ∈ R, t > 0, bHP(Ta < t,Xt > b) = P(Ta < t,Xt < 2a− b).U(O\3F Ta 4.,, *5 T−a U Ta 34.. + b = a, z

P(Ta < t,Xt > a) = P(Ta < t,Xt < a),bH Ta 4.zLP(Ta < t) = 2P(Ta < t,Xt > a)

= 2P(Xt > a)

= 2

∫ ∞

a

1√2πt

e−|x|2

2t dx

Ua Q`|V 74

= 2

∫ ∞

a/√t

1√2π

e−|x|2

2 dx

= 2P(X1 > a/√t),t t→ ∞ \,

P(Ta <∞) = 2P(X1 > 0) = 1. t 'v, z Ta OzLφa(t) =

a√2πt3

exp

(−a

2

2t

).

23. q\A: KR,' a = 0Y. 53tw+&q, t a ↓ 0\, j;L Ta ↓ T0? t, bH9 t > 0,

P(T0 < t) = 2P(X1 > 0) = 1.;_ T0 = 0. qKR&q-,, sHh? + t > 0, F T t &d t \As`dw 0 \A, bH limt→0 Tt = T0. J Brown j0-# , t Xt = x \, Æsd 0 stOa 0 T&sd −x, l.GÆE

P(T t < u|Xt = x) = P(T−x < u− t)

= 2P(X1 > |x|/√u− t),m+ÆEz

P(T t < u) =

∫ ∞

−∞

1

2πte

−|x|2

2t 2P(X1 >|x|√u− t

)dx

=

∫ ∞

−∞

1

2πe

−|x|2

2 2P(X1 >|x|√t(u− t)

)dx.3 t ↓ 0, ;T P(T0 < u) = 1 9 u > 0 J", ;_ T0 = 0. W X rxT&\Y% `U x- $sM. ^4K X rxU x-$sM)IL ", qj7DL0.

Ua Q`|V 75

24. f): R X = (Xt) j\T& Brown j, 7Yt(ω) :=

∫ t

0

Xs(ω)ds, t > 0, ω ∈ Ω.U#4j2C Riemann #4, W: Y = (Yt) j Gauss uL,',>z cov(Ys, Yt).

W? Mr<ZU-IdS=. Brown j%rxj&z, ;L&! , rx#z&-n , zI=A$>zn *: ,`#>U%`#>. W,%rxILLq,`#>, qjL%`#>, MWBrown jrx-j72C#4, n 19 f6\, VL Weierstrass _p,U&YZvz#`, , -J.rxj%.iF^

• B , B• %rx• ,`#>• %`#>• Riemann-Stieltjes #4

76

Ud Q`|VTwO[S 77

1. Wiener _p: Brown j_p,'L , Wiener -\slW, Wiener ; ('jj Brown j) bn , F" >a%0 4.!. gn : n ≥ 0 32?, ,'_p, Brown j. F Fourier ,, 7Xt =

t√πg0 +

∑

n≥1

2n−1∑

k=2n−1

√2

π

sin kt

kgk,M ^4Kn [0, π]Kj3I; 1,q', ;_j Brown j.

2. ! W): RXMu!#* , |I;ILig, qjI;" 57-g, 3XMI;[:-$>!#*! W), ^4Kj4.! W), jYL57. 4.x! j-3, C! ILC4! . ,D!#* Y j X z, XM X Y , ' X 4.zY!. tM3s&d, bH!#* ,%, M4.z! ,%. qj-3 !#*3Ls3! , ;L! =,I;0 . Brown jxL M! ? | , ;LxbH !#*, JU t > 0, Xt j , q Wiener _pRXM^4KH"n U"%0 4.!#*K, # U%0 4.!#*j32u [0, 1] Kzi4.!#*_pT, 3 Brown jx! [0, 1] Kzi4.!#*! j,%.

3. U Brown jsiB: R X = (Xt) j Brown j.

(a) (Xt) jB;

(b) !uL (X2t − t) jB;

Ud Q`|VTwO[S 78

(c) !uL (exp(aXt − a2t/2)) jB, W- Brown jexp(Xt) ,-jB, #Dk0 Brown j Xt − t/2 3jB.&W X2

t − t jB, + t > s > 0,

E[(Xt −Xs)2|Fs] = E[X2

t − 2XtXs +X2s |Fs]

= E[X2t |Fs] +X2

s − 2XsE[Xt|Fs]

= E[X2t −X2

s |Fs],+E[X2

t −X2s |Fs] = t− s,|T X2

t − t jB. iOB, ;L Xt − Xs U Fs " >aN(0, t− s), ;_

E(ea(Xt−Xs)|Fs

)= E

(ea(Xt−Xs)

)= ea

2(t−s)/2,;T (3) J".

4. !: n&\A#, Doob !0;3J". UG\A#Æ, !\A T : Ω → [0,∞] j0\j9 t ≥ 0 LT ≤ t ∈ Ft.Xs7.1 t M = (Mt) jB T j,U0\, bH (Mt∧T ) 'jB. LÆE-#

E[M0] = E[Mt∧T ]9 t > 0 J". &jM&qÆE, B3Ux W),R T = Ta (a > 0) 4., XMFB,'>. FaUB

Ud Q`|VTwO[S 79>? )UILF, FB. U7,

ebXt− 1

2b2tjB, bHJÆE-#

E

(ebXt∧T− 1

2b2(t∧T )

)= 1.t t → ∞ \, Xt∧T → XT = a, W)j3Ux,. q&qUÆE3MC? Jq', &ÆER*,UU t U

ω YiVj3Fq',. ;L t ∧ T ≤ T , 3Xt∧T ≤ a, KLV, f#& b > 0,

ebXt∧T ≤ eba,2,04, e−b2(t∧T )/2 ≤ 1, ;_n b > 0 \3M, zE

(e−

1

2b2T)= e−ab,># θ = b2/2, ;T T Laplace #L

E(e−θTa

)= e−a

√2θ.aK|, ,U1C!#* Laplace #K,2,4..

5. %rx: k=,U5#4jr$jj,.%rx, %AjLj%rx/*, I;RXMa%4rx&Vj g. a41N|, ,.%rxj-W MW), k= %3j|w%rx4.# . e8ij,.%rx, T+91:L5.3a,.r

Ud Q`|VTwO[S 80x|T4.# I8ia>K| ML ". -uXMUK7jn<R,8i4.# k?F!4_,O<Rj;-j!4_jUxW).

6. n U-n : 4XMlaK Brown j%rx 4_. ab, XMQYz|K-2Cjn , ;L2CSzjY%`G, XMn z_|e,3O&_p,UZZ-vz(-?4, qj Brown jwo2,U&, L,|"WUW), XMslÆ&t,tsH*zn L? n (7^?Gj-n ? a Donsker-#\|, Brown j3&, qj&-n ,,-n LazXMY'FST, ;L8r-j53wb L. bXMsHS &-n h?

7. -ZZ: nI;>, g9&F-ZZtW).ndI;, I;0gGjj-j, qjn,I;A, I;0j--j, n [0, 1] K+U, jLI;-j-j; n nMA" 4.+ nUx*, bHMr siI;j--j. qjjTtn%Abn1. O Brown j X = (Xt : t ∈ [0, 1]),h% ω, bH Xt(ω) iO t j&z, HL%rx, XM&W-L%rxj-n .

8. z9wD: 9 [0, 1] K&z: x = x(t). (^XM399)AKz, [0, 1] jL,,".) Oz9w, ,U53IpjD, ,U&z9w?L&*r?Gj3;*D, qjsH;*h? t3jFY%,',

Ud Q`|VTwO[S 819954 ∆:

0 = t0 < t1 < · · · < tn = 1.r<\r, z9wD L(x) -OkrD, +L(x) ≥

∑

i

√(ti − ti−1)2 + (x(ti)− x(ti−1))2.m FFN- , M ,j14#j#t, 3XMDj7L

L(x) = sup∆

∑

i

√(ti − ti−1)2 + (x(ti)− x(ti−1))2. (7.1)t9wDYq, 3tyzS,-n , ;LC8gI-jÆT,.DYq*r, 3XMjaDLqjYq$>zn L. kRTtjG#4,U'Y\ .4s7.1 tz x = x(t) n [0, 1] K&v, bH L(x) < ∞

L(x) =

∫ 1

0

√1 + x′(t)dt.

9. sH+4: ;LP*#4, XMRbCL+4Æ&. a>KK#4bn ?GU4Yi, 3*RP*4'j_. qL,?4UQ<, XM3pnR4#j(O0, j)A [0, T ] +,.&'4: %44, n ≥ 1, 7

∆n := 0, T2n,2T

2n, ·, (2

n − 1)T

2n, T,

Ud Q`|VTwO[S 82'jjDL T2−n 4, %+4Zj,`4xn,`4, 4eej, t n (OY%\, 4D(O0. qjL,,", XM7jCn “4 ∆ = ti#j(O0”.

10. z#>: JO- ∑

i

|x(ti)− x(ti−1)| ≤∑

i

√(ti − ti−1)2 + (x(ti)− x(ti−1))2

≤ 1 +∑

i

|x(ti)− x(ti−1)|,3 L(x) LqYq ∑i |x(ti)−x(ti−1)| Li, HLjz x n4 ∆ K#>, 7

V (x) = sup∆

∑

i

|x(ti)− x(ti−1)|, (7.2)HLj x = x(t) ,`#>. m3 L(x) < ∞ t \t V (x) <

∞. 3W, ,Uz,`#>Lqt \t3JL)Utz>.

11. rx#4: t y = y(t) j [0, 1] K&z, bHXM?4∆ KiOz x = x(t) Riemann

∑

i

y(si)(x(ti)− x(ti−1)), (7.3) si ∈ [ti, ti−1]. <Rz x rLLq,`#>, bHt4#j(O0\, Riemann &qbn# U si +'Yi, 2CXM+= ti−1, 2U&qL∫ t

0

y(t)dx(t),

Ud Q`|VTwO[S 83HLj Riemann-Stieltjes#4. KR#4bn 1C-O xLq,`#> , ;r#4j-jbn. t x &v\, KR#4#J2C Riemann #4∫ 1

0

y(t)dx(t) =

∫ 1

0

y(t)x′(t)dt.kRWW)n, qW&Fw+z4_^X.Xs7.2 t4 ∆ #j(O0\, Riemann (7.3) 95&zLLq&q, bH x LLq,`#>.

12. G#4 : R f j R K&vz, x LLq,`#>, bHG#4 J"f(x(t))− f(x(0)) =

∫ t

0

f ′(x(s))dx(s), t ∈ [0, 1]. (7.4)

13. %`#>: t ∑i(x(ti)− x(ti−1))

2 HLj x n4 ∆ K%`#>, t4#j(O0\,

lim∆→0

∑

i

(x(ti)− x(ti−1))2bn, bHXM x %`#>bn, 2L 〈x〉, tÆ&, XMnk%K)A3djU)AK%`#>.4s7.2 t x & LLq,`#>, bH 〈x〉 = 0.W: 3

x(∆) := max|x(ti)− x(ti−1)| : 1 ≤ i ≤ n,bHt4#j(O0\, x(∆) (O0, ;_〈x〉 ≤ x(∆)V (x) −→ 0.

Ud Q`|VTwO[S 84

14. z,q': !. ξn q'O ξ L -357, <!&L-ZZq'Uz,q', JUh% ω, ξn(ω) j., tq'O ξ(ω), bH ξn n% ω Kq'O ξ. -ZZq'q'>- , j-q'b%/*I;O0. tE[ξn − ξ]2 −→ 0,bHXM ξn z,q'O ξ. z,q'-ZZq'ILkxig, Mj-357q'. qjz,q'3;T-I;q', # 3;TL,U4.j-ZZq'.

15. Brown j%`#>: L,,", _XM&'F B = (Bt)&d Brown j, ∆ j)A [a, b] 4,Xs7.3 %`#> B(∆) z,q'O b− a.W: 6iO 4.m1\ E[(Bt − Bs)

2n] = (2n− 1)!!(t− s)n.m Ft*" E[(B(∆)− (b− a))2]

= E[(∑

i

(Bti − Bti−1)2 − (b− a))2]

= E[∑

i

(Bti − Bti−1)2]2

− 2(b− a)E

[∑

i

(Bti −Bti−1)2

]+ (b− a)2

= E

[∑

i

(Bti − Bti−1)4

]

Ud Q`|VTwO[S 85

+ E

[∑

i 6=j

(Bti − Bti−1)2(Btj −Btj−1

)2

]− (b− a)2

= 3∑

i

(ti − ti−1)2 +

∑

i 6=j

(ti − ti−1)(tj − tj−1)− (b− a)2

= 2∑

i

(ti − ti−1)2,;_L

lim∆→0

E(B(∆)− (b− a))2 = lim∆→0

2∑

i

(ti − ti−1)2 = 0.U"aRXM, Brown jn9)AK%`#>j, ;_,`#>jYq, 'jjrx--n .

16. %`#>uL: 7 〈X〉t j!uL X n)A [0, t] K%`#>, ?!uL 〈X〉 = (〈X〉t, t ≥ 0),HL X %`#>uL,;_KR Brown j%`#>uLj t, + 〈X〉t = t,j1C+,UTt.

17. -ZZ57: KR%`#>jz,q'57k, qj-fWn+&y4Æk, '3W-ZZq'. +)A [0, 1] %4 4∆n = k · 2−n : 0 ≤ k ≤ 2n.bHE[B(∆n)− 1)2] = 2 · 2−n,J Chebyshev -

P(|B(∆n)− 1| > 1/n) ≤ 2n2

2n,

Ud Q`|VTwO[S 86;T ∑n P(|B(∆n) − 1| > 1/n) < ∞, <J Borel-Cantelli =z%`#> B(∆n) -ZZq'O 1.

18. !: R f j R K&vz, B = (Bt) j Brown j, '!uL Xt = f(Bt) %`#>uL. + [0, t] K4 ∆, J∑

i

(f(Bti)− f(Bti−1))2

=∑

i

[(f ′(Bsi))2(Bti − Bti−1

)2 − (f ′(Bsi))2(ti − ti−1)]

+∑

i

(f ′(Bsi))2(ti − ti−1),t ∆ #j(O 0 \, ;L f ′ LV, 3,UIz,(O 0%Uj Riemann , (O2C#4, ;_

〈f(B)〉t =∫ t

0

[f ′(Bs)]2ds.

19. f): R a < 0 < b, Ta, Tb 4'j\T& Brown js`w a, b \A. FB,'' P(Ta < Tb), E[T ] 3* T Laplace#, T = Ta ∧ Tb.

20. f): R|w Brown j,w9,.r. bHw,.r x = at + b C? m3t a, b :~\,3~\C? ÆQ. + a, b > 0, bH x = at+ b j,.n\ÆK, ;r, 3 T j Brown j X s`w.r\A, +T = inft > 0 : Xt = at+ b.' P(T < ∞). X|un,e8i4_,', L8FrsM,'xDAET, ÆjFwÆTt. XM

Ud Q`|VTwO[S 87nXN,U/, ,jS4W. ;L exp(cXt−c2t/2)jB, ;_J Doob B , 9 t > 0 LE(exp(cXt∧T − c2(t ∧ T )/2)

)= 1,;L Xt∧T ≤ a(t∧ T ) + b, f& ac− c2/2 ≤ 0 c > 0 \, t 3(OY%, \

E

[exp

((ac− c2

2)T

)]= e−bc,,L ac− c2/2 = −λ t λ > 0 \LX c = a +

√a2 + 2λ, bH

E(e−λT

)= exp(−b(a +

√a2 + 2λ)).

21. f): R Xt = tBt, ' X %`#> 〈X〉. W: Xt := tBt −∫ t

0Bsds jB.

WE? fh`=. !#4j Ito =A, 3'HL Ito #4, j!4_<!&Ip, |y Riemann-Stieltjes#4, qm3-j2C57k, bn j F,#BAA W, t3U Brown j%`#>uLOsi.iF^

• ,#B• %`#>uL• %`#>• += Riemann • !#4• !#4Æ

88

UM n\Y 89

1. nz, R B = (Bt) j Brown j, VJ 52L (Ft),

Ft |ÆUI;Yi, q^4K-*oI;5jIL57, 3XM&L P- I;O0)*4):AwJU Ft -. bu%Z5 (Ft) HL Brown 5.

2. t: Q1XMt7,U!uL X = (Xt) iO Brown jRiemann-Stieltjes #4, qj Brown jj-n , t%rx7#4

∫ t

0

Xs(ω)dBs(ω)j:. qjXM3nz,q'57k (nYE-I;q'57k) 7 Riemann q', j Ito !#4t.

3. ,#B: -.9Lq)A [0, T ], ,U&k?!uLM = (Mt) HL,#B, t9 t ∈ [0, T ], EM2

t < ∞,

Mt ∈ Ft, 9 T ≥ t > s ≥ 0 LE[Mt|Fs] =Ms.5j Brown 5, t3iO Brown 5B'j2CB. 2,UBjJ MT x. ,#B/*jr A,RB3,Ux*, L53DIp,

||M || := E[M2T ]1/2,XM3WLA , 'jj Cauchy .L&q. O7!#4i!&.

UM n\Y 90Xs8.1 D M (n) j,#B.azt n,m (OY%\,∣∣∣∣M (n) −M (m)

∣∣∣∣ (O0, bHbn,U,#B M = (Mt) az∣∣∣∣M (n) −M∣∣∣∣ (O0.

4. &WA , iFjW&qB M = (Mt) j&uL, \Doob - |B,.

E[maxM2s : 0 < s < t] ≤ 1

4E[M2

t ]. (8.1)WU- jj&Fw<S Doob B , quLLAl, M,7 [4].

5. ,`#>: U Brown j,%, V,OC (iO t), ,#B,`#>7jYq. W,`#>Yqj F E[M2

T −M20 ] = E

[∑

i

(M2ti−M2

ti−1)

]

= E

[∑

i

(Mti −Mti−1)2

].M -Eu

E[maxi

|Mti −Mti−1| · V (M)], V (M)(ω) j Mt(ω), t ∈ [0, T ] ,`#>. t,`#>Lq, t4#j(O0\, _*J& (O0, ;_

E[(MT −M0)2] = E[M2

T −M20 ] = 0,;T MT =M0, #9 t L Mt =M0.

6. %`#>: ,#B%`#>uLbnK,, j!4_<!&. K, -fW, qbn -j!4_<f

UM n\Y 91,U, jY, Doob-Meyer &y# , Doob-MeyerjG\A Doob 4U&\A.Xs8.2 t [0, T ] 4#j(O0\, %`#>∑

i

(Mti∧t −Mti−1∧t)2q'O 〈M〉t, jK,,Uaz 〈M〉0 = 0 az M2

t − 〈M〉t jB&tuL.Wt: [0, T ] K4 ∆ 7X∆

t :=M2t −

∑

i

(Mti∧t −Mti−1∧t)2, t ∈ [0, T ],W X∆ j,#B t ∆ #j(O0\j,#BA

Cauchy ., bH&qjj M2 − 〈M〉. 7 [4].

7. %`#>: R M = (Mt) N = (Nt) j,#B, %`#>〈M,N〉t :=

1

2(〈M +N〉t − 〈M〉t − 〈N〉t)j ∑

i

(Mti∧t −Mti−1∧t)(Nti∧t −Nti−1∧t)t4(O0\&q, j)UtuL> azuL MtNt −〈M,N〉t jB, 'jF9)U.GK,uL. 〈M,N〉 LÆi# :

(a) H : 〈M,N〉 = 〈N,M〉;

(b) r : 〈cM + c′M ′, N〉 = c〈M,N〉+ c′〈M ′, N〉;

(c) t M,N ", bH 〈M,N〉 = 0.

UM n\Y 92

8. = Riemann : lCn<R F = (Ft) j&LV iOBrown 5k?!uL. ∆ j)A [0, T ] 4, 7+= Riemann !uL

(F∆.B)t :=

n∑

i=1

Fti−1(Bti∧t − Bti−1∧t), t ≤ T, F∆ j47!uL,

F∆t := Fti−1

, t ∈ (ti, ti−1].%ZjXM3|JL n U!uL.4s8.1 uL (F∆.B) j,#B L Ito sE(F∆.B)2T = E

(∫ T

0

(F∆s )2ds

). (8.2)

9. W: (1) ,#B: XM&W9 i,

M(i)t : = Fti∧t(Bti∧t − Bti−1∧t) (8.3)

=

0, t ≤ ti−1;

Fti−1(Bt −Bti−1

), t ∈ (ti−1, ti);

Fti−1(Bti − Bti−1

), t ≥ tij,#Bj3,. & jm3, ,#jILW),;L F jLV. (2) Ito s: (F∆.B)2T 4J,vM<v, O,v, F"t* ;T Fti−1U Bti −Bti−1

j",fLE

(∑

i

F 2ti−1

(Bti − Bti−1)2

)

UM n\Y 93

=∑

i

E

[F 2ti−1

]E[(Bti − Bti−1

)2]

=∑

i

E[Fti−1

](ti − ti−1)

= E

(∫ T

0

(F∆t )2dt

).OM<v, ' F.GÆE 3WMO 0. 4?f).

10. q': ;Llim∆→0

∫ T

0

(F∆s − Fs)

2ds = 0,3t ∆ #j(O0\ (F∆.B)t z,q'OU,#!#*, 2L (F.B)t ∫ t

0

FsdBs,HLj F iO B n)A [0, t] K!#4. J,#BAA , ?L,U!uLj,#B L Ito sE

(∫ t

0

FsdBs

)2

= E

(∫ t

0

F 2s ds

). (8.4)t-LV , R F j&k?!uLF9# .G

E

(∫ T

0

F 2t dt

)<∞, (8.5)\9 n, F ∧ n iO B !#41b7,,

E

(∫ T

0

(Ft − Ft ∧ n)2dt)

−→ 0,;_J Ito s (F ∧ n).B L&q, jj F iO B !#4F.B. 3!#4!uL3mg, qXM-tH

UM n\Y 94>,. 2?KR#4.Gj!#4j,#B.G.

11. !#4: XMKRL:, ^&U7, jÆ& xG#4 , _. B7L)U&: I;;5. ,#Bj,ULi#r A, F H 2 &d,JI;;3*5x, U2Cv ALÆ,, x*LD, x*ÆALN, 3LIp, 2?jA,3j Hilbert A, R&,#B/*j,U4A, F H 2c . 5jj H 2

c .+&qnR.F9KR# .Gk?uL F = (Ft) /00n,, 2LL 2, 'j,U Hilbert A, 'jJI;;5x. on!#4jjA L 2 uL F #J H 2

c uL F.B,

F 7→ F.B,-.HL Ito DO, j,Ur DO#. r*sH>wh?l L 2 CnuL/* L0 7 Ito DO, 3H" Ito sE[(F.B)2T ] = E

(∫ T

0

F 2t dt

),<F&q,'#7W>w L0 L 2 KjjT!#4,. 3KUs#jnO, qQjFO, n<,KXMW, t5j Brown 5, bHjjFO,jTB&d.

12. jpr: 3K#4bn 3*%`#>bn W-j?4Ug#, I X-, -uM'-n_u g?, Æ&| Mj'rj3,, 3O2C

UM n\Y 95#4,%, 1+)V'j xsH#4, aI-tuLH. O9F9# E

(∫ T

0

F 2t dt

)<∞&k?!uL F = (Ft), iO Brown j!#4j,UiO\A t !uL

∫ t

0

FsdBs,j+= Riemann ∑

i

Fti−1(Bti∧t −Bti−1∧t)t4#j(O0\z,q' (-I;q') &q, U!#4uL'Cn2L F.B = (F.B)t : t ∈ [0, T ]. 2?%`#>uLj

〈F.B〉t =∫ t

0

F 2s ds. (8.6)

13. (8.6) kx1 Ito s\ (8.4), M1C!&, &WU, ÆWkR), 4?f).

(a) (8.3) 7 M (i) L〈M (i)〉t = F 2

ti−1(t ∧ ti − t ∧ ti−1),M jj2C#4

∫ t

0

F 2ti−1

1s∈(ti−1,ti]ds.

UM n\Y 96

(b) -3 i, j L〈M (i),M (j)〉 = 0.

14. iO,#B!#4: Æ37 F iO,#BM !#4 ∫ t

0

FsdMs F.M , ;3j,U,#B L〈F.M〉t =

∫ t

0

F 2s d〈M〉s,M j,U2C#4, ;L 〈M〉 jt&z. \#uL

F = (Ft) ?G&k? F9E

(∫ T

0

F 2t d〈M〉t

)<∞,# RXM7j_<R. kR!#4Æ1Cg1CLF.Xs8.3 !#4 F.M jF9kR.GK,,#B: 9,#B N L

〈F.M,N〉t =∫ t

0

Fsd〈M,N〉s.*5M j2C#4.

15. : !#4U2C#4-,%, ^4K-jziOz#4, J2C#4%4jv^U, qÆ3HÆ3HL#4, j;L w2C#4, jr ∫ t

0

(c1F(1)s + c2F

(2)s )dBs = c1

∫ t

0

F (1)s dBs + c2

∫ t

0

F (2)s dBs, (8.7)

UM n\Y 97 c1, c2 jC. 2?, tL)U&#4, bH∫ t

0

Gsd

∫ s

0

FudBu =

∫ t

0

GsFsdBs. (8.8))U 3CnF!#4ÆW. W (8.8),9+,#B N ,

〈G.(F.B), N〉t =∫ t

0

Gsd〈F.B,N〉s

=

∫ t

0

Gsd

∫ s

0

Fud〈B,N〉u

=

∫ t

0

GsFsd〈B,N〉s

= 〈GF.B,N〉t,JK, , G.(F.B) = GF.B, GF = (GtFt : t ∈ [0, T ]) &d)U!uLK#. kR!4W2C#4-,%,R.

16. *l: ^RiO!#4> W)&Fl0BIp, ;r T>jY';o, l0BjÆ&F0\7, iO T>'jF0\UZrW, 1bET,K7*J.

17. !: '!#4 ∫ t

0

BsdBs.-.R ∆ j [0, t] 4, bHn∑

i=1

Bti(Bti − Bti−1)

UM n\Y 98

=

n∑

i=1

B2ti−

n∑

i=1

BtiBti−1

=n∑

i=1

[B2ti−1

− BtiBti−1] +B2

t − B20

=

n∑

i=1

Bti−1(Bti−1

− Bti) +B2t − B2

0 ,;_= Riemann 2

n∑

i=1

Bti−1(Bti − Bti−1

)

= −n∑

i=1

(Bti − Bti−1)2 +B2

t − B20 ,+&q, = &qj!#4, M &qj%`#>, z

2

∫ t

0

BsdBs = B2t − B2

0 − t.J_|T!#4U%`#>Osi, 2?|TG#4 nY, ;LG#4 e?Gj2

∫ t

0

BsdBs = B2t − B2

0 .

18. f): R 0 ≤ t1 < t2 ≤ t3 < t4, !#* ξ, η 4'iO Ft1 U Ft3;, W:

E[ξ(Bt2 − Bt1)η(Bt4 − Bt3)] = 0.

19. f): R F,G j)U&k?uL, W〈F.B,G.B〉t =

∫ t

0

FsGsds.

UM n\Y 99

20. f): R B(1) U B(2) j)U" Brown j, W: MK#jB, a#W 〈B(1), B(2)〉 = 0.

21. f): R M j,#B, t M2 = (M2t ) 'jB, W: 9

t, Mt =M0.

22. f): R M j,#B, t9,#B N L 〈M,N〉 =0, W: 9 t, Mt =M0.

Wn? 2b=. L,!#4, jLsH1!#4W), 2uÆ, !#4 ÆO2C#4, qj40#4\ U#*+\ -3O2C#4, HL.'\ , KOG#4Ng Newton-Leibniz\ , zXN!#4 3*40#4\ U.'\ .iF^

• &ÆB• ÆB4U• 40#4\ • .'\ • B

100

U_ yoZj 101

1. R B = (Bt) j Brown j, (Ft) j Brown 5. 6,Uk?!uL X j9 t, Xt iO Ft ;, 'jj Xt J Brownj t \A 2. *5XMkR(w!uLj&.

2. Lq,`#>uL: j,Uk?!uL V = (Vt), -L%rxj& LLq,`#>. %O95&k?!uL F = (Ft) XM372C#4∫ t

0

FsdVsj-%rx+= Riemann &q7, zw;3j,ULq,`#>uL, '2L F.V .

3. &ÆB: &ÆB=AjL,,". t,U!uLX 3J,U,#B M ,ULLq,`#>&k?!uL V ,

Xt =Mt + Vt,bHXM X j&ÆB. XM7j3&' V0 = 0 '7j_&', ;L%3KR%4UjK,, ;rjs>,UC. 4U X =M + V HLÆB4U.

4. !#4: R F j&k?uL, 537 F iOÆB X !#4(F.X)t =

∫ t

0

FsdXs =

∫ t

0

FsdMs +

∫ t

0

FsdVs, (9.1)'jjF.X = F.M + F.V.

U_ yoZj 102j F iO X += Riemann (Mj-I;q') &q. ;L F.M jB, F.V LLq,`#>, 3 F.X ;3jÆB.m|%`#>.4s9.1 R K = (Kt) j&k?uL, bHLq,`#> V U%`#> 〈K, V 〉 = 0.W: ;Ln4 ∆ K#>∑

i

|(Kti −Kti−1)(Vti − Vti−1

)|

≤ maxi

|Kti −Kti−1| · V (∆),# V (∆) LV, K &, 3t4#j(O0\, M (O0. =W V U9&k?uL%`#>O0, 3LkR .

(a) 〈X〉 = 〈M〉.

(b) R2,UÆB Y = N + U , bH 〈X, Y 〉 = 〈M,N〉.Lq,`#>04n>#>\3=.

5. %XM3iOÆB!#4>w,2C#4, tB04O0\, iOÆB#4jj2C#4, LtB04To\, 3j!#4.

6. !#440#4\ : t U , V jLq,`#>uL, bH40#4\ 3JUtVt − U0V0 =

∫ t

0

UsdVs +

∫ t

0

VsdUs. (9.2)

U_ yoZj 103?F 'rUtVt =

1

2[(Ut + Vt)

2 − V 2t − U2

t ],KR\ 3J U = V \ &dU2t − U2

0 = 2

∫ t

0

UsdUs.O Brown j,

B2t − B2

0 = 2

∫ t

0

BsdBs + t,R t j B %`#>. A/Æ3W, t M = (Mt)j,#B\, bHM2

t −M20 = 2

∫ t

0

MsdMs + 〈M〉t. (9.3)j/UW, XMzwkRiOÆB40#4\ .Xs9.1 R X j&ÆB, bHX2

t −X20 = 2

∫ t

0

XsdXs + 〈X〉t. (9.4) F 'r3;T, 9&ÆB X, Y

XtYt −X0Y0 =

∫ t

0

XsdYs +

∫ t

0

YsdXs + 〈X, Y 〉t. (9.5)t X , Y Æ,ILB04, bH 〈X, Y 〉 = 0, K jj2C40#4\ .

7. /2C#4, XM'3FG[', \ (9.5) JdXtYt = XtdYt + YtdXt + d〈X, Y 〉t,

U_ yoZj 104%FL\,". L\kR'Y,"XY −X0Y0 = X.Y + Y.X + 〈X, Y 〉, X0Y0 |JO!#* X0Y0 !uL.

8. !: !uL Xt = tBt, j)UÆBK#, F40#4\ , zwÆB4UtBt =

∫ t

0

sdBs +

∫ t

0

Bsds,bH%`#>j〈X〉 =

∫ t

0

s2ds =1

3t3.Æ, !uL Xt = tB2

t ÆB4UtB2

t =

∫ t

0

sdB2s +

∫ t

0

B2sds

= 2

∫ t

0

sBsdBs +

∫ t

0

(s+B2s )ds,%`#>

〈X〉t =∫ t

0

s2B2sds.'ÆE

E〈X〉t =∫ t

0

s2E[B2s ]ds =

1

4t4.

9. !: B2 jÆB, 4Uj B2 = 2B.B + t, t &d t \O t !uL. Xp (9.5)

B3 = B2.B +B.B2 + 〈B2, B〉

U_ yoZj 105

= B2.B + 2B2.B +B.t + 〈2B.B + t, B〉

= 3B2.B + 3B.t'jj B3 'jÆB, 3B2.B jB04, 3B.t jLq,`#>04. Fq '3*40#4\ 3WBn

t −Bn0 = n

∫ t

0

Bn−1s dBs +

n(n− 1)

2

∫ t

0

Bn−2s ds.\, 2uk=( Xp!#4r 3TW, t f j,U v , bH

f(Bt)− f(B0) =

∫ t

0

f ′(Bs)dBs +1

2

∫ t

0

f ′′(Bs)ds.jj'Y.'\ ÆU.

10. .' (Ito) \ :Xs9.2 R f j%`&v, bHf(Bt)− f(B0) =

∫ t

0

f ′(Bs)dBs +1

2

∫ t

0

f ′′(Bs)ds. (9.6)ÆG[ df(Bt) = f ′(Bt)dBt +

1

2f ′′(Bt)dt. (9.7)Æ, ,#B.'\ j

df(Mt) = f ′(Mt)dMt +1

2f ′′(Mt)〈M〉t, (9.8)UÆB.'\

df(Xt) = f ′(Xt)dXt +1

2f ′′(Xt)〈X〉t. (9.9)M A/,, tÆB X -xB04\, 〈X〉 = 0, jj2C Newton-Leibniz \ .

U_ yoZj 106

11. .'\ W: Ito \ W,'2Cj) , , j2C v ^, -bHT, bn,U v . fn, &3,Rv%Rv4',q'O f *,RvU%Rv. %j;T Ito \ . qj v q'jnLq)AK, 3&TAJKWÆ&Zr. % ,'j F%R, n a, b )AKbn ξ azf(b)− f(a) = f ′(a)(b− a) +

1

2f ′′(ξ)(b− a)2,bH

f(Bt)− f(B0) =n∑

i=1

(f(Bti)− f(Bti−1))

=

n∑

i=1

[f ′(Bti−1

)(Bti − Bti−1) +

1

2f ′′(Bsi)(Bti −Bti−1

)

], si ∈ [ti−1, ti], jJX2. t n (OY%\, ,vz,(O!#4, ,vz,(O%`#>#4. &W,v&q, &

E

(∑

i

f ′′(Bsi)(Bti − Bti−1)2 −

∑

i

f ′′(Bsi)(ti − ti−1)

)2q'O0ja,, #K -Eum2 · E

(∑

i

(Bti − Bti−1)2 − t)

)2 mj%Rv f ′′ n)A [0, t]KyKV, 3 F Brownj%`#>z,q'j3,.

12. ÆB8 : ÆBGjL8 , slÆBjr A, 3LqUÆBK#jÆB, J Ito \ |T, ÆBA,Un zjÆB.

U_ yoZj 107

13. B\ : Rn Brown jb,K1bu Brown jB , +eaBt−at/2jB, J Ito \ ?4W, Xt = aBt − a2t/2 t>ÆBF

Ito \ eaBt−a2t/2 − 1 =

∫ t

0

eaBs−a2s/2d(aBs − a2s/2)

+1

2

∫ t

0

eaBs−a2s/2d〈aBs − a2s/2〉

= a

∫ t

0

eaBs−a2s/2dBs,UBHLj Brown jB. ,LkR, WÆ.Xs9.3 t M j,#B, bHε(M)t := exp

(Mt −

〈M〉2

)(9.10)jB, HL M B, # jj5SiO M !#4

ε(M)t = eM0 +

∫ t

0

ε(M)sdMs.k, BjW Brown jk!G4 dXt = aXtdBt (9.11)U. Æ%!G4 HL!G4,L. !G4,LÆOCG4,L, L,%iOUEUbnK, >, qjj%m UT,Lj&M, # fLHg4,'.

U_ yoZj 108

14. !: 'F9S X0 = x (Langevin) !G4,LdXt = −αXtdt+ dBt,U X . 3 Yt := eαtXt, bH

dYt = eαt(αXtdt+ dXt) = eαtdBt,;_Yt = Y0 +

∫ t

0

eαudBu,J Xt zXt = e−αt

(x+

∫ t

0

eαudBu

).j'Y Orstein-Uhlenbeck uL.

15. f): Xt := B3t − 3tBt, W: (Xt) jB.

16. f): (Ito ,UTt) W:

n!

∫

0≤u1≤···≤un≤t

dBu1· · ·dBun

= tn2 hn

(Bt√t

), hn jT Hermite v

hn(x) = (−1)nex2

2

dn

dxn

(e−

x2

2

).

17. f): 1 Orstein-Uhlenbeck uL,>E[(Xt − EXt)(Xs − EXs)].

W? CK6CQ\=. n<,K, XMIXN!4_)U<!&, B&dU;#. B&dj,UiO Brown 5,#B,3JiOBrownj!#4, ,Æ.=Y9n62:=bn , ;#<oHL Cameron-Martin, jYqM4_,U!&, ;ow,I;AK, 'HL Girsanov , UÆ.=, si. <XMF)U;vT Black-

Scholes Æ.=\ .iF^• B&d• ;#• Levy Brown jÆ• Girsanov \ • Black-Scholes \• =\

109

Ug PkzRW 110

1. n,KXMI1!$>B;bnW)3*B&dW), 3G\A$>3|T, )Y>Æ.=LO%g.t3, nG\A>, )UTt-jm3, W1CCn. lI|w, )UTtn&\Ak-jbH?4W, LW04W=,. Fw&&ZrjjK,KH"!#4 Ito \ 40#4\ .

2. 6B&d: R%vÆ.^4K1b$>uB&dW), ^4K, t Xn : 0 ≤ n ≤ N jCn!K, bHjB, RYn jiOB5 Fn B, 'jj Yn iO X1, · · · , Xn ;. B&dW)jWj;bn Hn−1 ∈ Fn−1 az

Yn − Yn−1 = Hn−1(Xn −Xn−1)?2 ξn = Xn −Xn−1, + 1,−1 I;V 1/2. <R Yn 1b x,, bHXML,LYn|ξn=1 − Yn−1 = Hn−1

Yn|ξn=−1 − Yn−1 = −Hn−1,;_Hn−1 = (Yn|ξn=1 − Yn|ξn=−1)/2,*-O X1, · · · , Xn−1, ;L ξn W,. B&dJ"&&\;j ξn +)U. t9B, bH;L

E[2ξn ] = 5/4,3 Zn = (4/5)n2Xn j Xn ,UB, bHt33B&dkZn − Zn−1 = 3/5 · Zn−1(Xn −Xn−1),

Ug PkzRW 111;_iOB5B Yn 3FB&dYn − Yn−1 =

3Hn−1

5Zn−1

(Xn −Xn−1).4s10.1 BUBj3s&d.

3. B&d: iO Brown 5B3F Brown j!#4#4&d, sljJ Ito T. RW Brown j B = (Bt) U:A0I;)3*4) Brown 5 (Ft), M = (Mt) j,#B,-.R M0 = 0, ;r9,#B Mt −M0 j3,. tXMjWbnk?uL F = (Ft), azMT =

∫ T

0

FsdBs,bH;L) jB, JK, zB&dMt =

∫ t

0

FsdBs.# F = (Ft) j M K,2. XM x,#B MT ;JAjUiO FT ;,#!#*;JA L2(FT ) s3_. ;_^4KXM&WkR.Xs10.1 (Ito) R!#* ξ ,#, iO FT ;, bHbnK,k?!uL F = (Fs) azξ − E[ξ] =

∫ T

0

FsdBs.

4. WI: -.R E[ξ] = 0. WtO,UL X8-fU. XM3%!#*HL3&d. sl&

Ug PkzRW 112"3&d!#*/*j,Ur 4A. 9B, ;LeaBt− a2t

2 − 1 = a

∫ t

0

eaBs− a2s2 dBs,3= %!#*j3&d. m9Y,B,+ f j [0, T ] K4&z, B

(if.B)t :=

∫ t

0

if(s)dBsBjεt(f) := exp

(i

∫ t

0

f(s)dBs +1

2

∫ t

0

f 2(s)ds

),F9kR,L

εt(f)− 1 =

∫ t

0

εs(f)d(if.B)s = i

∫ t

0

εs(f)fdBs.'jj, = %!#*;3j3&d. <Æ&W%!#*/* εT (f) − 1 : f n L2(FT ) ROj9a,. 'jjtLUÆEL0,#!#* η UU)M, +9% f LE[η · εT (f)] = E[η · (εT (f)− 1)] = 0, (10.1)bH η = 0. #<,/nKj<f, Æ&aF Fourier #K, t.

5. WjS: (10.1) =OE

[η exp

(i

∫ t

0

f(s)dBs

)]= 0.

Ug PkzRW 1139+ 0 < t1 < · · · < tn = T , x1, · · · , xn ∈ R, jAf(t) =

∑

i

xi1t∈(ti−1,ti]zE[η exp

(ix1Bt1 + ix2(Bt2 −Bt1) + · · ·+ ixn(Btn − Btn−1

))]

= 0,JO x1, · · · , xn j95, ;T9 x1, · · · , xn ∈ R LE [η exp (ix1Bt1 + ix2Bt2 + · · ·+ ixnBtn)] = 0,Jn%KUliO&zK, (7 17 () z, Rz3F9LVz+j;T

E [η · g(Bt1, · · · , Btn)] = 0,;L g(Bt1, · · · , Btn) %!#*/*n L2(FT ) RO (sWt'-j?4g#), f η = 0 a.s.

6. *l: G\AW-3, bXMj3&d Hn an = N − 1 bu1,/,/T, 'j^S&wux!G4,L<St. qKR Ito &dWj,Ubn W, -j_p W, ;LXM^4K<(IL F _pT. n^4?F, &'jY>nB?F, jY!&j F 1T. UWiOB5,#B,j&.

7. LH3&d: nG\A#, XMB3&dj;L\j%v\, &d,LU. &\A#Ul&Alz , aW|, iF,j Brown j%`#>uLj2

Ug PkzRW 114IL! qjj/~ .G, Y,jj;LBrown j4WiO Brown 55J- -M. 2? Brownjj,#B %`#>uLj2 uL t, U Æ, Brown j, jjkR Levy Æ.Xs10.2 (Levy) t,U!uL X j,#B# %`#> 〈X〉t = t, bH X ,j Brown j.W: XpB\ , !uL

eizXt+1

2z2tjB, ;_9 t > s U^ z, L

E

(eizXt+

1

2z2t|Fs

)= eizXs+

1

2z2s,%=O

E(eiz(Xt−Xs)|Fs

)= e−

1

2z2(t−s).#j Brown j& , W, Xt−Xs "Ou- Fs >a,>L (t− s) 4., '+ X = (Xt) j Brown j. W'Fw Fourier # .

8. B;: B;W)jj$>,U-jB!uLj-j32u#,U=;, #JB. XM9<Cn, # , R B = (Bt) jnI;A (Ω,F ,P) k Brown j,

Xt = Bt + bt, b j,U10^. bH X = (Xt) -jB, onXMj-j3~w,UU P =I; P az X n;kjB? tjB, bH;L 〈X〉t = t, 3Xp Levy Æ, X 3jj Brown j.

Ug PkzRW 115

9. ;#: nZKXM1b;#W)uA:I;;j!gGj , *'r, =I;j, *'r, q-H#! , 'jj-!gG#J2 , '-2 gG#J!.

10. KsH>: nG# k, ;H#j?4, n&# k, XM2C2uOz, H#;, +,UT F ;!#* ζ , 3P(A) := E(ζ1A) = E(ζ ;A).t E[ζ ] = 1 \, P j,U=O P I;;, =nKj P(A) = 0 t \t P(A) = 0, !j!, 2j2. F E &dI; P ÆE, bH9!#*

X , LE[X ] = E[ζX ].

11. onw Brown jbUI;A, F jj FT , ζ 3?LO.Gj: T, iO FT ; E[ζ2] <∞, E[ζ ] = 1. 3Zt := E[ζ |Ft],bH Z = (Zt) jT,#B. R XT j FT ;!#*, bH Xt := E[XT |Ft] jiOI; P B. 9 Ft ;!#* A, J.GÆE z

E[XtZtA] = E[XtA]

= E[E[XT |Ft]A]

= E[XT1A] = E[XTZTA]

Ug PkzRW 116

= E[E[XTZT |Ft]A],;TXtZt = E[XTZT |Ft],'jjK#uL XZ jiO P B.4s10.2 X = (Xt) iOI; P jBt \t XZ = (XtZt) iOI; P jB.

12. B?LO: Yr*, Rζ = exp

(aBT − 1

2a2T

),bH ζ =?LO.G, Z = (Zt) jjB, F9

dZt = aZtdBt,J ζ zwI;; P. on X = (Xt) j P Bt \t XZ j PB, qjJ40#4\ XtZt −X0Z0 =

∫ t

0

ZsdXs +

∫ t

0

XsdZs + 〈X,Z〉t.&', t X j0 Brown j Xt = Bt + bt, bH〈X,Z〉t =

∫ t

0

aZsd〈B〉s =∫ t

0

aZsds,;_XtZt −X0Z0 =

(∫ t

0

ZsdBs +

∫ t

0

XsdZs

)+

∫ t

0

(a+ b)Zsds.;L B Z jiOI; P ,#B, fR04j P B,Oj XZ j P Bt \t a + b = 0. 'jj, XMW,kR

Ug PkzRW 117, HL Cameron-Martin Girsanov . k,

Brown jjH, tL,0, bH3H#I;;a!JL Brown j.Xs10.3 Xt = Bt − at nI;; P kjB, OjnI;;kj Brown j.

13. Æ.=>I: <XMWT!4_<+?F: Black-

Scholes \ . !#4k57n3O4wo. -l&\AjG\A, Æ.= >jÆ, -3j\=\ . R S = (St) jen t \=T, r > 0 j ;,&1 . ,U T \wÆ J=Tj K v ||Æ.n T\=m3jVT = (ST −K)+ = max(0, ST −K).,!V0n T \=j,U FT ;1C!#* VT .

14. 5>2: ,U62:=L H = (Ht) 628n\ t 4DLXt = HtSt + bt, HtSt j62neK, bt j0n<, < ;L

r, &A dbt = rbtdt. JG\ (5.9) |T, 5>25j4Dt:\5et< dXt = HtdSt + dbt, (10.2)3GB,%, 9o, J40#4\ ;T

d(e−rtXt) = e−rt(−rXtdt+ dXt)

Ug PkzRW 118

= e−rt[−r(HtSt + bt)dt +HtdSt + dbt]

= Htd(e−rtSt),+o4Dt:5Oeot, ;_Sb62L

X0 = x0 5>2\p4DoLe−rtXt = X0 +

∫ t

0

Hud(e−ruSu),o4Dj H iOoe=T!#4. !#4k zw,/_*o.

15. =t: tXMja~w,UI;; P, azoe=Tn_I;kjB, bH!V0n 0 \=T?GjV0 = e−rT E[VT ].tB3&d, bHJ VT 2B

e−rtXt := e−rT E[VT |Ft]3J H iOB (e−rtSt) &d, H jj3`P;LET,6Æ.i9n62:=, \KR=THLA=T. -A=T^4KjIL57.

16. Black-Scholes \: Black U Scholes pjjn<Re=TF9kR!G4 .GkWTÆ.=Tmd&d ,

dSt

St= σdBt + µdt, (10.3) σ > 0 HL);, µ ∈ R HLq8;. J40#4\

d(e−rtSt) = e−rtdSt − re−rtStdt

Ug PkzRW 119;_oe=TuL (e−rtSt) F9!G4 d(e−rtSt)

e−rtSt=

dSt

St− rdt

= σdBt + (µ− r)dt

= σ

(dBt +

µ− r

σdt

).3

Bt = Bt − (r − µ)σ−1t.tXMja~,UI; P az B = (Bt) n_I;kjB, bHoe=TnI;kjB. J;#, &7I;;P(A) := E

[exp(aBT − 1

2a2T );A

],j3>w,, a = (r − µ)σ−1. +nI; P k, B j Brownj.

17. Æ.=\ : !G4 (10.4) j3U, J Ito \ ,

d log St =dSt

St

− 1

2S2t

d〈S〉t

= σdBt + µdt− 1

2S2t

σ2S2t dt

= σdBt + (µ− σ2/2)dt,;_LSt = S0 exp

(σBt + (µ− σ2/2)t

).on

V0 = e−rT E[VT ],

Ug PkzRW 120t9v || VT = (ST −K)+, bHV0 = e−rT E

((ST −K)+

)

= e−rT E

(S0e

σBT+(µ−σ2/2)T −K)+

= e−rT E

(S0e

σBT+(r−σ2/2)T −K)+

,\L!#* BT nI; P k>a 4. N(0, T ), ;_Æ.=Tm &d LV0 = e−rT

∫

R

(S0e

σx√T+(r−σ2/2)T −K

)+φ(x)dx, φ(x) j%0 4.Oz. zYW2, F%0 4.z Φ(x) &d, jj Black-Scholes \

V0 = S0Φ(d + σ√T )−Ke−rTΦ(d), (10.4)

d =log(S0/K) + (r − σ2/2)

σ√T

.

18. f): W Xt = |Bt| jÆB.

l 8Black-Scholes \ *^ 30 n&wY>=>j!4_<+?F, ÆOB, !#4, Ito \ , B&d, =B;rVOKUf6KÆ*&QyIpe3nU1W#z1Ck, 3Qz, Riemann G4-w Einstein o7s>, #OQyj&fz. j;LUZ?, .'".'\ Y#Y>V, ';LUZ?, Merton Scholes (J\ Black1b-f) z, Nobelb0J. q-A/jg, Y>VjUfVr, b8 !YJ0g, `Y>H!&VUÆ.Y>PL=-,Kg, T\Y>Y>VY(a>MFH2U", MT47jL5Y5U"4Dw5.l,3X- x!4_n%1WL_+?Fjj-, #9|w\4srV,%.

121

Op![1] A. Etheridge, A Course in Financial Calculus, Cambridge Uni-

versity Press, 2002 (LU)

[2] . M. Ross, Introduction to probability models, Academic Press,

1997, (LU)

[3] ??N, , FJK, AogTQ, Kw, 2005

[4] ??N, YNN, LIGDHE, AogTQ, Kw, 2005

122

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