GiaiTichNgauNhien

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Hanoi Center for Financial and Industrial MathematicsTrung Tm Ton Ti Chnh v Cng Nghip H Ni

NHP MN

TON TI CHNH

QUYN 1

GS. c Thi

GS. Nguyn Tin Dng

H Ni Toulouse, 2011


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2

Bn tho ny: Ngy 19 thng 1 nm 2011

cHanoi Center for Financial and Industrial Mathematics


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Chng 1

Gii tch ngu nhin

Theo ngn ng ton hc, s bin ng theo thi gian ca gi c (nh gi vng, gi

du ha, gi c phiu ca cng ty Intel, v.v.), cng nh ca cc s liu khc (v d nh

mc tng trng kinh t, t l tht nghip, v.v.) c gi l cc qu trnh ngu nhin

(random process), bi v ni chung khng ai c th bit trc c mt cch chnh xc gi

tr ca chng trong tng lai s ra sao. nghin cu cc qu trnh ngu nhin ny, chng

ta s cn dng n mt b phn ca ton hc gi l gii tch ngu nhin (stochastic

culculus). Gii tch ngu nhin tc l gii tch ton hc (cc php tnh gii hn, vi tchphn, v.v.) p dn vo cc qu trnh ngu nhin, da trn c s ca l thuyt xc sut

thng k.

Trong chng ny chng ta s tm hiu s lc mt s kin thc quan trng nht v

gii tch ngu nhin, cn thit cho ton ti chnh. Bn c mun nghin cu su thm

v gii tch ngu nhin c th tm c cc sch chuyn kho, v d nh quyn sch ca

Karatzas v Shreve [12] hoc quyn sch ca tc gi Nguyn Duy Tin [17].

1.1 Mt s m hnh bin ng gi chng khon

phn ny, chng ta s coi gi S ca mt c phiu (hay ni mt cch tng qut

hn, ca mt chng khon c gi dng) nh l mt qu trnh ngu nhin nhn gi tr

trong tp hp cc s thc dng, v chng ta s xt mt s m hnh h ng lc ngu

nhin mt chiu n gin m t chuyn ng ca S theo thi gian. Ch rng, do ch

c 1 chiu, nn cc m hnh ny tng i th: s tng tc gia cc thnh phn ca th

trng khng a c vo m hnh, v m hnh ch da trn cc phng trnh bc 1,thay v phng trnh bc 2 nh trong vt l. Tuy l cc m hnh tng i th, nhng

3


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4 CHNG 1. GII TCH NGU NHIN

chng vn rt quan trng trong vic phn tch s bin ng gi ca cc c phiu.

Trc ht, chng ta s nh ngha mt cch hnh thc ton hc th no l mt qutrnh ngu nhin.

1.1.1 Qu trnh ngu nhin

Cc qu trnh bin i theo thi gian, v d nh gi c phiu, lng nc ma trong

thng, s ngi mc bnh cm, v.v., m ta khng th d on c trc mt cch chnh

xc, th c gi l cc qu trnh ngu nhin. m t mt qu trnh ngu nhin

theo ngn ng ton hc, ta cn cc yu t sau:- Thi gian. Theo qui c, c mt mc thi gian ban u, l 0. Thi gian t c th

l bin i lin tc, t R+, hoc ri rc, tc l ta ch xt mt dy cc mc thi im0 = t0 < t1 < t2 < . . . no . Trong trng hp ri rc, cho n gin, ta s gi s

thm l cc bc thi gian l bng nhau, t l ti ti1 = l mt hng s khng phthuc vo i. Nhiu khi, ta s dng dy s nguyn khng m 0, 1, 2, . . . k hiu cc mc

thi gian, thay v dng cc thi im t0, t1, t2, . . .

- Khng gian xc sut. Vi mi mc thi gian t, c mt khng gian t tt c cc tnh

hung c th xy ra t thi im ban u cho n thi im t. Khng gian ny l khnggian xc sut, vi mt o xc sut Pt i km (tc l xc sut ca cc tnh hung c

th xy ra cho n thi im t). Nu s v t l hai mc thi im no vi s t, th tac mt php chiu t nhin

t,s : (t, Pt) (s, Ps) (1.1)Khi t l mt tnh hung c th xy ra cho n thi im t, th s,tt l tnh hung

nhng ch tnh n thi im s, b qua nhng g xy ra sau thi im s. Cc php chiu

s,t tha mn cc tnh cht t nhin sau:a) Ton nh (surjective), tc l mi tnh hung c th xy ra cho n thi im s th phi

c th tip din tr thnh tnh hung c th xy ra cho n thi im t.

b) t,t l nh x ng nht trn t.

c) Bc cu: r,s s,t = r,t vi mi r s t.d) Bo ton xc sut, c ngha l l nu A (s, Ps) l tp o c (tc l tn ti xcsut Ps(A)), th nh ngc ca n trong (t, Pt) c cng xc sut vi n:

Pt(1s,t (A)) = Ps(A). (1.2)

Mt dy cc khng gian xc sut (t, Pt) vi cc php chiu s,t tha mn cc tnh


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1.1. MT S M HNH BIN NG GI CHNG KHON 5

cht pha trn s c gi l mt h lc cc khng gian xc sut (filtered family of

probability spaces).

Cc khng gian xc sut (t, Pt) c th c gp chung li thnh mt khng gian xc

sut (, P) tt c cc tnh hung c th xy ra (cho mi thi gian): mi phn t ng vi mt h cc phn t t t thch hp vi nhau, c ngha l s,tt = s vi mis < t. Ta c th vit:

(, P) = limt

(t, Pt), (1.3)

vi cc php chiu t nhin

t : (, P)

(t, Pt), (1.4)

cng tha mn cc tnh cht ton nh, bc cu, v bo ton xc sut nh pha trn.

Nhc li rng (xem Chng 1 ca [5]), i km vi mi mt khng gian xc sut l mt

sigma-i s cc tp con o c ca n, tc l cc tp con m nh ngha c xc sut

ca n. Sigma-i s cc tp o c trn (, P) l

F=t

Ft (1.5)

trong

Ft l nh ngc ca sigma-i s trn (t, Pt) qua php chiu t: mt phn t

ca Ft l mt tp con ca c dng 1t (At) trong At t sao cho tn ti Pt(At),v khi ta c

P(1t (At)) = Pt(At), (1.6)

c ngha l cc nh x t bo ton xc sut.

T cc tnh cht trn ca h lc (t, Pt), d thy rng Fs Ft vi mi s t. H Ftcc sigma-i s con ca F vi tnh cht ny v tnh cht F = t Ft c gi l mtlc (filtration) ca F. B ba (, Ft, P), trong (, P) l mt khng gian xc sut v

(Ft) l mt lc ca sigma-i s ca P, c gi l mt khng gian xc sut c lc(filtered probability space).- Bin ngu nhin thay i theo thi gian. Nu ta c mt qu trnh lc cc khng gian

xc sut (t, Pt), v vi mi mc thi gian t ta c mt bin ngu nhin St thc vi khng

gian xc sut tng ng l (t, Pt), c ngha l mt hm o c

St : t R, (1.7)

(xem Chng 2 ca [5] v cc khi nim c bn v bin ngu nhin), th ta ni rng ta c

mt qu trnh ngu nhin (stochastic process) S trn m hnh xc sut (t, Pt). HmSt : St : t R chnh l hm gi tr ca qu trnh ngu nhin S ti thi im t.


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Ta c th coi St nh l bin ngu nhin trn qua cc php chiu t :

St t : R. (1.8) cho tin, ta cng s k hiu St t l St, khi n l hm s trn t v o c theosigma-i s Ft. T , ta c nh ngha sau v qu trnh ngu nhin, l nh ngha cdng trong cc ti liu ton:

nh ngha 1.1. Gi s ta c mt khng gian xc sut c lc (, Ft, P), v mt h cchm s St : R, sao cho St l o c theo sigma-i s Ft vi mt t (trong tp ccmc thi gian ca lc). Khi h St c gi l mt qu trnh ngu nhin vi m

hnh xc sut (, Ft, P) v tng thch (compatible) vi lc Ft.

Trong nh ngha 1.1, cc khng gian (t, Pt) b b qua. Nhng cho tin, trong

quyn sch ny, khi xt cc qu trnh ngu nhin, ta s lun coi l khng gian xc xut

lc (, Ft, P) c sinh bi mt h lc cc khng gian xc sut (t, Pt), v mi qu trnhngu nhin S u c nh ngha qua mt h cc bin ngu nhin St : (t, Pt) R.Cc qu trnh ngu nhin nh vy tt nhin u l cc qu trnh ngu nhin tng thch

vi lc Ft.

Khi ta gi s rng tnh hung xy ra, th qu trnh ngu nhin S tr thnh mthm s xc nh theo bin thi gian: t St(). Hm s S(t) := St() ny c gi lmt ng i (sample path) ca S, ng vi tnh hung .

Nu s < t, v ta bit l tnh hung s xy ra cho n thi im s, th ta bit gi

tr S(s) = Ss(s) ca qu trnh ngu nhin ti thi im s (v cc thi im trc ),

nhng cha thng tin bit gi tr ca S ti thi im t. Ni cc khc, nu t > s

th St cng l bin ngu nhin ti thi im s, tuy bit tnh hung no xy ra cho n

thi im s. Nhng khi bit s, th khng gian xc sut ca St khng cn l khng

gian (t, pt), m l khng gian xc sut c iu kin

(t|s := {t t | s,t(t) = s}, Pt|s) (1.9)

vi xc sut c iu kin Pt|s. Trong trng hp m Ps(s) > 0 th xc sut c iu kinPt|s c th c nh ngha theo cng thc thng thng:

Pt|s(A) = Pt(A|s) =Pt(A)

Ps(s)(1.10)

vi mi A o c trong t|s. Trong trng hp m Ps(s) > 0 th nh ngha xc sutc iu kin phc tp hn, phi thng qua cc gii hn; chng ta s coi rng cc xc sut


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c iu kin ny tn ti v tha mn cc tnh cht thng dng (xem [5] v xc sut c

iu kin cho bin ngu nhin).

Hon ton tng t nh trn, ta c th nh ngha qu trnh ngu nhin vi gi tr l

vector, hoc tng qut hn, qu trnh ngu nhin trn mt a tp hay mt khng gian

metric no .

1.1.2 M hnh mt bc thi gian

Trong m hnh mt bc thi gian, ta ch quan tm n gi c phiu ST ti mt thi

im T trong tng lai, v ta mun d on ST. V ST c tnh ngu nhin, nn vic don ST khng c ngha l d on 1 con s duy nht cho ST, m l d on theo

ngha xc sut: Ci m chng ta c th lm l, da trn cc thng tin c c, xy

dng mt khng gian xc sut (T, PT) cc tnh hung c th xy ra n thi im T, v

biu din ST nh l mt bin ngu nhin, vi m hnh khng gian xc sut l (T, PT):

ST : (T, PT) R+ (1.11)

Nhc li rng (xem Chng 2 ca [5]), mi bin ngu nhin Y : (, P)

R trn mt

m hnh khng gian xc sut (, P) cho mt phn b xc sut PY trn R theo cngthc push-forward:

PY(A) = P(Y1(A)) (1.12)

cho mi on thng A R, v ta c th nh ngha cc i lng c trng ca Y, vd nh cc moment bc k:

Mk(Y) =

YkdP =

R

ykdPY(y). (1.13)

Tuy rng Y l ngu nhin, nhng mt khi ta bit phn b xc sut ca n, th cci lng c trng ca n l khng ngu nhin, c xc nh, v cho ta cc thng tin

v Y.

Trong cc i lng c trng, c hai i lng quan trng nht, l k vng v phng

sai. K vng E(Y) ca Y l:

E(Y) =

R

ydPY(y), (1.14)

v phng sai 2(Y) ca Y l:

2(Y) =

R

(y E(Y))2dPY(y) = E(Y2) (E(Y))2. (1.15)


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Cn bc hai ca phng sai, (Y), c gi l lch chun ca Y. Khi m phng

sai cng nh, th tc l cc gi tr ca Y cng gn gi tr k vng ca n, c ngha l

ngu nhin (bt xc nh) ca Y cng nh. Bi vy phng sai (hay lch chun) chnh

l mt thc o ngu nhin, bt xc nh.

Trong trng hp m bin ngu nhin l gi c phiu ST, i lng

=E(ST) S0

S0, (1.16)

trong S0 l gi c phiu ti thi im 0, chnh l mc li nhun k vng ca c

phiu S cho khong thi gian t 0 n T, cn

=(ST)

S0, (1.17)

l mt i lng o bt xc nh ca gi c phiu, theo m hnh d on.

Ta c th coi S nh l mt qu trnh ngu nhin vi ch c 2 mc thi gian 0 v T, v

khng gian xc sut chnh l (T, PT). H ng lc ngu nhin m t chuyn ng ca S

sau 1 bc thi gian, t 0 n T, c th c vit di dng phng trnh sai phn:

S = S+ SE, (1.18)

trong :

S = S0 l gi c phiu ti thi im 0,

S = ST S0 l thay i gi c phiu t thi im 0 n thi im T,

l mc li nhun k vng, cn c gi l h s drift ( chuyn dch) ca mhnh,

l h s o bt xc nh ca gi ST, hay cn gi l h s volatility ( dgiao ng) ca m hnh,

E = (STE(ST))/S0 l phn ngu nhin chun ha ca m hnh: k vngca E bng 0 v lch chun ca E bng 1.

V d 1.1. Gi s mt cng ty cng ngh sinh hc nh, ang tp trung nghin cu mt

loi thuc chng ung th, c gi c phiu ngy hm nay l 10$. Sau gi ng ca th

trng ngy hm nay, cng ty s cng b kt qu nghin cu loi thuc chng ung th. Gi s ta bit rng s c mt trong hai tnh hung xy ra:


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a) Tnh hung thuc c tc dng, vi xc sut xy ra l 60%, v nu xy ra th gi ca

phiu ngy hm sau s tng ln thnh 16$.

b) Tnh hung thuc khng c tc dng, vi xc sut xy ra l 40%, v nu xy ra th

gi ca phiu ngy hm sau s gim cn 5$.

K vng gi c phiu ca ngy hm sau ca cng ty bng 60% 16 + 40% 5 = 11.6 la, phng sai bng 60% (16 11.6)2 + 40% (5 11.6)2 = 29.04, v lch chunbng

29.04 5.4. Ta c m hnh chuyn ng gi c phiu 1 bc

S = S+ SE, (1.19)

vi cc tham s sau: S0 = 10, = (E(S1) S0)/S0 = 0.16, = (S1)/S0 = 0.54, v E lmt bin ngu nhin ch nhn hai gi tr, v c k vng bng 0 v lch chun bng 1.

V d 1.2. C phiu Coca-Cola (m chng khon: KO) t gi 80$ vo u nm 1998.

Vi cc cng thc c lng gi tr thc ca c phiu da trn li nhun v tng trng,

vo thii im u nm 1998, c th c lng l gi tr ca KO vo thi im u nm

2003 s khng qu 50$/c phiu. (Xem V d ??). Tm coi n l 50$. V yu t con cng

ca th trng s mt dn i theo thi gian khi m cng ty Coca-Cola khng cn pht

trin nhanh c na nn ta gi thit l gi c phiu s i v gi tr thc sau 5 nm,

trong giai on 1998-2003. Khi , vo u nm 1998, m hnh d on gi KO cho thi

im u nm 2003 ca ta s l:

KO2003 = 50 + .KO1998.E, (1.20)

trong KO1998 = 80, E l mt bin ngu nhin no chun ha (k vng bng 0,

phng sai bng 1), l mt s no cn c lng. Theo m hnh ny th mc li

nhun k vng cho 5 nm s bng (50 80)/80 38%, tc l k vng l gi c phiu s

gim gn 40% sau 5 nm. Ta s tm thi b qua vic chn E v y. (C th tm coil E c phn b normal chun tc N(0, 1) da trn nh l gii hn trung tm trong xc

sut, v c lng da trn giao ng lch s (historical volatility) ca KO). Thc

t xy ra l KO2003 = 40, kh gn vi d bo ca m hnh.

1.1.3 M hnh vi thi gian ri rc

Tng t nh l trong m hnh vi mt bc thi gian, trong cc m hnh vi thi

gian ri rc (h ng lc vi thi gian ri rc) cho gi c phiu, vi gi s l gi c phiulun lun dng, ta c th vit chuyn ng ca qu trnh ngu nhin S theo phng


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1.1.4 M hnh cy nh thc

Ging nh trc, ta k hiu bc thi gian l , v gi thi im n l thi im thn (thi im th 0 l thi im 0, tc l thi im ban u). Gi ca c phiu ti thi

im th n c k hiu l S(n) hay Sn. Ta coi Sn (n Z+) l mt qu trnh ngu nhinvi thi gian ri rc, v ta s vit phng trnh m t chuyn ng ca n.

Ta s gi s l bc thi gian nh n mc, t thi im th n 1 n thi imth n gi c phiu ch kp thay i 1 ln, ph thuc vo 1 tin xy ra trong khong thi

gian . Tin y s ch l tt (k hiu l g) hoc xu (k hiu l b), v gi c phiu s

thay i, ph thuc vo tin tt hay tin xu, theo cng thc sau:

Sn =

Sn1(1 + un) nu tin ttSn1(1 + dn) nu tin xu . (1.22)

Cc i lng un v dn khng nht thit phi c nh, m c th ph thuc vo tnh

hung xy ra cho n thi im th n 1 (v c bit vo thi im th n 1 khitnh hung c bit). Ni cch khc, chng l cc hm s trn khng gian xc sut

(n1, Pn1) cc tnh hung c th xy ra cho n thi im th n

1 :

un, dn : n1 R. (1.23)

Chng ta s gi s rng S0 > 0, v cc i lng un v dn tha mn cc bt ng thc

un > dn > 1, (1.24)

c ngha l gi c phiu lun lun dng, v gi c phiu khi tin tt th cao hn gi c

phiu khi tin xu.

Khng gian xc sut (n, Pn) cc tnh hung c th xy ra n thi im th n l

mt tp hu hn gm c 2n phn t: mi phn t c th c k hiu bi 1 dy n ch

ci n = (a1, . . . , an), trong mi ch ci nhn mt trong hai gi tr g (tin tt) hoc b

(tin xu), v ch ci th i trong dy ng vi tin xy ra t thi im th i 1 n thiim th i. Ta c th vit:

n = {g, b}n. (1.25)

M hnh c gi l cy nh thc v mi tnh hung n1 n thi im n 1 cr lm hai nhnh, thnh 2 tnh hung n thi im n, k hiu l (n1, g) v (n1, b).

Ti thi im 0 ban u th cy ch c 1 nhnh, n thi im th 1 th thnh 2 nhnh,n thi im th 2 th thnh 4 nhnh, v.v.


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Hnh 1.1: Cy nh thc

Xc sut r nhnh t tnh hung n1 thnh tnh hung (n1, g) n c khiu l pn(n1), v n c th c tnh theo cng thc xc sut c iu kin:

pn(n1) = P((n1, g)|n1) = Pn(n1, g)Pn1(n1)

. (1.26)

Ngc li, khi ta bit cc xc sut r nhnh pn, th ta cng c th tm li c phn b

xc sut trn n theo cng thc sau: vi mi n = (a1, . . . ,n )

n, ta c

P(n) =ni=1

P(i)

P(i1), (1.27)

trong i = (a1, . . . , ai),P(i)P(i1)

= pn(i1) nu ai = g vP(i)P(i1)

= 1 pn(i1) nuai = b. Chng ta s gi s cc phn b xc sut y l khng suy bin, c ngha l cc

xc sut r nhnh tha mn bt ng thc 0 < pn < 1.

Cc i lng un, dn v pn c th c coi nh l cc qu trnh ngu nhin, v n

khng nhng ph thuc vo n, m cn c th ph thuc vo tnh hung xy ra. Cc qutrnh ngu nhin ny c gi l d on c, v t thi im th n 1 bit ccc gi tr ca un, dn v pn.

V d 1.3. Mt m hnh nh thc hai bc, tc l vi n 2:S0 = 100 (gi thi im 0 l 100)

S1(g) = 125 (gi thi im 1 l 125 nu tin tt)

S1(b) = 105 (gi thi im 1 l 105 nu tin xu)

p1 = 0.5 (xc sut tin u tin l tt bng 0.5)

S2(g, g) = 150 (gi thi im 2 l 150 nu tin u tt tin sau cng tt)S2(g, b) = 115 (gi thi im 2 l 115 nu tin u tt tin sau xu)


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p2(g) = 0.4 (nu tin u tt, th xc sut tin th hai cng tt l 0.4)

S2(b, g) = 130 (gi thi im 2 l 130 nu tin u xu tin sau tt)

S2(b, b) = 90 (gi thi im 2 l 115 nu tin u xu tin sau cng xu)

p2(b) = 0.7 (nu tin u xu, th xc sut tin th hai tt l 0.7)

Theo m hnh ny, ti thi im 0, S2 l mt bin ngu nhin nhn 4 gi tr 150, 115,

130 v 90, vi cc xc sut tng ng l: 0.5 0.4 = 0.2, 0.5 (1 0.4 ) = 0.3, v(1 0.5) 0.7 = 0.35, (1 0.5) (1 0.7) = 0.15. Ti thi im 1, th S2 vn l binngu nhin, nhng n ch cn nhn 2 gi tr, v ph thuc vo tnh hung xy ra cho n

thi im 1. V d, nu tin u tin l tt, th khi S2 l bin ngu nhin vi hai gi

tr 150 v 115, vi cc xc sut tng ng l 0.4 v 1 0.4 = 0.6.Mt trng hp c bit ca cy nh thc hay c dng n l khi un = u, dn = d

v pn = p l nhng hng s, khng ph thuc vo n cng nh l vo cc tnh hung xy

ra. Ta s gi m hnh cy nh thc m trong un, dn, pn l cc hng s l m hnh cy

nh thc bt bin (invariant binary tree model), phn bit vi m hnh cy nh thc

tng qut. M hnh cy nh thc bt bin tt nhin l tnh ton d hn so vi m hnh

nh thc tng qut v c t tham s hn, v bi vy hay c dng, nhng b li n khng

c chnh xc bng m hnh tng qut.

Hnh 1.2: Cy nh thc bt bin 3 bc

Bi tp 1.1. Vit li phng trnh chuyn ng (1.22) ca m hnh cy nh thc di

dng chunSn Sn1

Sn1= n + nEn: tnh n, n v tm phn b xc sut ca En t

cc bin un, dn v pn.

Bi tp 1.2. i) Chng minh rng, trong m hnh cy nh thc bt bin, vo thi im 0

ban u, Sn l mt bin ngu nhin nhn n + 1 gi tr (1 + u)i(1 + d)niS0, i = 0, 1, . . . , n

(thay v c th nhn n 2n gi tr nh trong m hnh cy nh thc tng qut), v migi tr (1 + u)i(1 + d)niS0 c xc sut l Cinpi(1 p)ni, trong Cin = n!/(i!(n i)!) l


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nh thc Newton. (Phn b xc sut vi cc xc sut nh vy c gi l phn b nh

thc, xem [5]).

ii) Tnh k vng E(Sn) ca gi c phiu sau n bc trong m hnh cy nh thc bt bin.

1.1.5 M hnh vi thi gian lin tc

Trong ton hc, cc m hnh vi thi gian lin tc c th c xy dng nh l gii

hn ca cc m hnh vi thi gian ri rc, khi m bc thi gian tin ti 0. i vi cc

qu trnh ngu nhin m t gi c phiu cng vy: mt qu trnh vi thi gian lin tc

c th nhn c bng cch ly gii hn mt qu trnh vi thi gian ri rc, khi m

bc thi gian tin ti 0. Phng trnh m t chuyn ng ca mt qu trnh ngu nhin

trong trng hp thi gian lin tc s l phng trnh vi phn ngu nhin (stochastic

differential equation). M hnh vi thi gian lin tc n gin nht, 1 chiu, m t s

thay i ca gi c phiu, c dng sau:dStSt

= (t, St)dt + (t, St)dBt, (1.28)

hay cn vit l

dSt = (t, St)Stdt + (t, St)StdBt, (1.29)

trong (t, St) l h s drift, (t, St) l h s volatility, l cc hm s vi hai bin s

thc, c cho bi m hnh, (mi khi bit gi tr ca t v St th cng bit gi tr ca

v ), Bt l mt qu trnh ngu nhin gi l chuyn ng Brown (Brownian motion).

Chuyn ng Brown Bt ny c sinh ra bng cch ly gii hn phn ngu nhin trong

m hnh ri rc, khi khi bc thi gian tin ti 0.

Phng trnh trn hiu ngha nh sau:

Khi m thi gian t dch chuyn i mt i lng t = t

t > 0 rt nh, th gi

c phiu cng dch chuyn i mt i lng St = St St, bng tng ca hai phn,mt phn l d on c ti thi im t, v mt phn l khng d on c. Phn

d on c xp x bng (t, St).St.t, cn phn khng d on c xp x bng

(t, St).St.(Bt Bt).Chng ta s ngha chnh xc chuyn ng Brown, v nghin cu phng trnh vi phn

ngu nhin , trong cc phn pha sau ca chng ny. Cc tnh cht quan trng nht ca

mt chuyn ng Brown Bt l:

Bc chuyn ng Bt Bt t thi im t n thi im t > t l c lp theo nghaxc sut vi mi iu xy ra cho n thi im t.


15/44


Phn b xc sut ca bin ngu nhin Bt Bt (ti thi im t) l phn b normalN(0, t

t) vi k vng bng 0 v phng sai bng t

t.

Cc tnh cht trn ca thnh phn chuyn ng Brown Bt trong phng trnh vi phn

ngu nhin (1.1.5) c th c gii thch mt cch trc gic nh sau:

Bc chuyn ng Bt Bt t thi gian t n thi gian t > t khng ph thuc vobt c iu g xy ra cho n thi im t, bi v nhng ci g m ph thuc vo

nhng chuyn xy ra ti t v cc thi im trc th coi l d on c v

c th chuyn sang phn d on c trong m hnh. Phn hon ton khng d

on c ca m hnh l phn khng h ph thuc vo nhng iu xy ra trc

.

Phn khng d on c c th coi l c k vng bng 0, bi v bn thn k vngl i lng d on c, nu khc 0 c th chuyn sang phn d on c ca

m hnh.

Phn khng d on c trong khong thi gian t = t t t t n t c thchia thnh tng ca N phn khng d on c cho cc khong thi gian c

di t/N t t + (i 1)t/N n t + it/N. Nh vy n l tng ca N bin ngunhin, m ta c th coi l c lp (do tnh hon ton khng d on c va nu

trn) v c phn b xc sut tng t nhau. Theo nh l gii hn trung tm (xem

Chng 4 ca [5]) th mt tng nh vy, khi N tin ti v cng, phi tin ti mt

bin ngu nhin c phn b xc sut l phn b normal. Chnh bi vy m phn

khng d on c Bt Bt trong m hnh c phn b normal.

xc nh mt phn b xc sut normal, ta ch cn bit k vng v phng saica n. y ta bit k vng bng 0. Cc bin ngu nhin c lp c phng

sai ca tng bng tng ca cc phng sai. V tnh cht cng tnh ca phng sai

theo thi gian (khi chia mt bc chuyn ng ngu nhin Bt Bt thnh nhiu

bc nh theo thi gian), nn phng sai ca Bt Bt c t bng ng t

t,sau khi ta chun ha n bng cch a h s volatility vo m hnh.


16/44


1.2 Chuyn ng Brown

Chuyn ng Brown (Brownian motion) l mt lp cc qu trnh ngu nhin mang

tn nh thc vt hc Robert Brown (17731858)(1), ngi quan st chuyn ng ca

cc ht bi (phn hoa) trong nc thy chng i hng lin tc (mi khi va p phi cc

phn t khc th li i hng). N cn c gi l qu trnh Wiener, theo tn nh ton

hc Robert Wiener (18941964), ngi c nhiu cng trnh nghin cu v cc qu trnh

ngu nhin v nhiu(2). T nm 1900, ng Louis Bachelier t c s cho ton ti chnh

hin i bng vic dng chuyn ng Brown m hnh ha cc qu trnh bin ng gi

chng khon trong lun n tin s ca mnh, tuy rng lun n ca ng ta thi khngc my ai quan tm, v phi n na sau th k 20 ngi ta mi thc s quan tm n

n. Chuyn ng Brown l mt trong nhng lp qu trnh ngu nhin quan trng nht,

v phn ln cc chuyn ng ngu nhin c tnh lin tc trong thc t c th c m

hnh ha da trn chuyn ng Brown v cc php bin i gii tch. y, chng ta s

nh ngha v mt ton hc th no l mt chuyn ng Brown, v nghin cu mt s

tnh cht quan trng nht ca n.

1.2.1 nh ngha chuyn ng Brown

C th nh ngha chuyn ng Brown trn cc khng gian nhiu chiu. Tuy nhin,

trong khun kh quyn sch ny, chng ta s ch nh ngha chuyn ng Brown 1 chiu,

trn tp hp cc s thc R.

nh ngha 1.2. Mt qu trnh ngu nhin Bt vi thi gian lin tc (tp cc mc thi

gian lR+) nhn gi tr thc, tng thch vi mt m hnh xc sut (, Ft, P), c gil mt qu trnh Wiener hay chuyn ng Brown chun tc 1 chiu, nu n tha

mn cc iu kin sau:

i) Xut pht im l 0: B0 = 0.

ii) Bt l mt qu trnh lin tc. C ngha l, vi hu ht mi , hm s B(t) :=Bt(), tc l qu o ca Bt trong tnh hung , l hm lin tc theo bin thi gian t.

iii) Vi mi 0 s < t, bin ngu nhin Bt Bs (gi l bc i, hay gia s, ca qutrnh ngu nhin t s n t) khng ph thuc vo tnh hung xy ra cho ti thi im

s, hay ni cch khc, n c lp vi sigma-i s Fs. C ngha l, vi mi A Fs v(1)Xem: http://en.wikipedia.org/wiki/Robert_Brown_(botanist)(2)Xem: http://en.wikipedia.org/wiki/Norbert_Wiener . Cc nhiu m c m hnh bi chuyn ng

Brown gi l nhiu trng (white noise).


17/44

1.2. CHUYN NG BROWN 17

[a, b] R ta c

P(A (Bt Bs [a, b])) = P(A).P(Bt Bs [a, b]). (1.30)

iv) Vi t > s 0 bt k, phn b xc sut ca Bt Bs (bc i trong khong thi giant s n t ca chuyn ng) l phn b normal N(0, t s) vi k vng bng 0 v phngsai bng t s.

Khi ni mt qu trnh no l chuyn ng Brown m khng ni c th thm, chng

ta s lun hiu l chuyn ng Brown chun tc 1 chiu.

Trong nh ngha trn, iu kin i) l chun ha, gi im xut pht ca chuynng l 0. iu kin ii) l tnh cht lin tc ca chuyn ng Brown. ngha ca iu

kin iii) cng kh hin nhin: nhng bc chuyn ng trong tng lai khng h ph

thuc vo nhng g xy ra trong qu kh. iu kin iv) xut pht t tng sau:

bc chuyn ng theo thi gian t s n t, vi di thi gian bng ts, c th chia nhthnh tng ca N bc chuyn ng c lp, mi bc c gii thi gian l (t s)/N(vi mi s t nhin N). Khi N tin n v cng, th theo nh l gii hn trung tm

(xem Chng 4 ca [5]), tng ca N bin ngu nhin c lp cng phn b xc sut s

c phn b xc sut tin n mt phn b normal (sau khi chun ha). Bi vy, mtcch trc gic, phn b xc sut ca Bt Bs phi l phn b normal. Vic t phn bnormal y bng N(0, t s) cng l chun ha.

T nh ngha trn, suy ra ngay c rng, nu Bt l mt chuyn ng Brown v

0 < t1 < . . . < tn th b n bin ngu nhin (Bt1, . . . , Btn) c phn b xc sut chung l

phn b normal n chiu (xem Chng 3 ca [5] v phn b normal nhiu chiu).

Bi tp 1.3. Chng minh rng, nu B(t) l mt chuyn ng Brown, th cc qu trnh

B(t), B(t + t0)

B(t0) (trong t0 > 0 l mt hng s), v aB(t/a2) (trong a

= 0

l mt hng s) cng l cc chuyn ng Brown.

Bi tp 1.4. t Zt = a + t + Bt trong a,, l cc hng s. Tm phn b xc sut

ca cc gia s Zt Zs, v chng minh rng qu trnh Zt cng l qu trnh lin tc v cgia s c lp, tc l n tha mn cc tnh cht iii) v iv) trong nh ngha chuyn ng

Brown. (Qu trnh Zt ny c th c gi l mt chuyn ng Brown khng chun

tc, vi xut pht im l a, h s volatility l v h s drift l ).

Bi tp 1.5. Gi s Bt l mt chuyn ng Brown, v a > 0 l mt hng s. Xy dng

mt qu trnh ngu nhin Wt sau, gi l gng phn (reflection) ca Bt theo a:- Nu Bs() < a vi mi s < t, th Ws() = Bt(). (Tc l khi cha i ln chm vo n


18/44


a, th qu trnh Wt trng vi Bt).

- Nu tn ti s < t sao cho Bs() = a, th W

t() = 2a

Bt(). (K t khi bt u chm

vo a, th Wt l gng phn ca Bt qua a).

Chng minh rng qu trnh Wt xy dng nh trn cng l mt chuyn ng Brown.

Bi tp 1.6. Chng minh cng thc xc sut vt ro sau y ca chuyn ng Brown:

P{ max0tT

Bt a} = 2a

ex2/2T

2T

dx. (1.31)

Hng dn: vit s kin max0tT Bt a di dng hp khng gian nhau ca hai s kinBT

a v WT > a, trong Wt l gng phn ca Bt qua a nh trong bi tp trc.

1.2.2 Phn b xc sut ca chuyn ng Brown

Gi s St l mt qu trnh ngu nhin ty , tng thch vi mt m hnh xc sut

(, Ft, P), v gi T l tp cc mc thi gian ca qu trnh ny. (Trng hp T = R+ ltrng hp thi gian l lin tc, cn trng hp T = {0 = t0 < t1 < t2 < . . .} l trnghp vi thi gian ri rc; mt qu trnh ngu nhin St vi thi gian ri rc cng c th

c coi l qu trnh vi thi gian lin tc bng cch t St = Stn nu t kp gia hai mc

thi gian ri rc tn v tn+1 : tn t < tn+1). Ta c th coi qu trnh ngu nhin St nh lmt nh x

S : RT (1.32)t khng gian cc tnh hung vo khng gian RT cc hm s thc trn T : nh camt tnh hung theo nh x ny chnh l qu o S(.) = S.() ca S trong tnh hung

. Qu trnh ngu nhin St cng s c k hiu l S hay S(t), nu k hiu nh th tin

hn.

Nu X l mt bin ngu nhin (hay mt vector ngu nhin n chiu), th c mt phnb xc sut PX tng ng trn R (hay trn Rn), c nh ngha bng push-forward:

PX(A) = P(X A) (1.33)

trong A l on thng bt k trn R (hay mt hnh hp bt k trong Rn). Khi lm

cc php tnh vi cc bin ngu nhin hay cc vector ngu nhin ra cc con s c

ngha, th m hnh khng gian xc sut ban u ni chung khng quan trng, m ci

quan trng chnh l phn b xc sut ca n trn R hay Rn. Tng t nh vy, khi tnh

ton vi mt qu trnh ngu nhin St, th m hnh phn b xc sut ban u (, Ft, P)khng quan trng trng bng phn b xc sut PS trn RT, nhn c t phn b xc


19/44


sut trn qua push-forward ca nh x S, v gi l phn b xc sut ca qu trnh

ngu nhin St

trn RT: Sigma-i s trn RT l sigma-i s BorelB

, sinh bi cc

tp con c dng

CAt1,...,tn = {f : T R; (f(t1), . . . , f (tn)) A}, (1.34)

trong n N, ti l cc phn t ca T v A Rn l mt tp Borel. Cc tp c dngnh vy c gi l cc tp hnh tr (cylinder). Xc sut theo PS ca tp hnh tr l:

PS(CAt1,...,tm

) = P{(St1, . . . , S tn) A}, (1.35)

Sigma-i s Borel B trn RT

c mt lc t nhin cc sigma-i s con B, gi l lcBorel: vi mi t T, Bt c sinh bi cc tp hnh tr CAt1,...,tn tha mn iu kin ti tvi mi i = 1, . . . , n .

S tng thch ca mt qu trnh St vi m hnh xc sut(, Ft, P) tng ng viiu kin sau: SBt Ft, trong Bt l lc Borel trn RT v SBt l nh ngc ca ntrn theo S. Ta c th ly lun SBt lm lc cho m hnh khng gian xc sut ca Strn , nu lc trn cha c nh. Lc SBt c tnh cht ti u sau: mi lc khc trn sao cho S l tng thch phi cha lc ny. Ta s gi SBt l lc sinh bi S trn khnggian xc sut .

Ch rng, mi phn b xc sut trn RT, vi sigma i s l sigma-i s Borel sinh

bi cc tp hnh tr, u l phn b xc sut ca mt qu trnh ngu nhin tng thch

St no . Tht vy, ta c th xy dng v d nh sau: t khng gian cc tnh hung

bng chnh RT vi phn b xc sut ny, t lc Ft cc sigma-i s con bng chnh lcBorel Bt, v t St() = (t), tc l khi tnh hung xy ra th qu o ca qu trnhngu nhin St chnh l hm s . Khng nh ny c gi l nh l Kolmogorov v

s tn ti ca cc qu trnh ngu nhin vi phn b xc sut cho trc.

Trong trng hp m St = Bt l mt chuyn ng Brown, th theo nh ngha, vi

mi 0 t0 < t1 . . . < tn t v cc on thng Di R, ta c:

P(Bti Bti1 Di i = 1, . . . n) =ni=1

P(Bti Bti1 Di) =ni=1

Di

12

ex2/2dx,

(1.36)

v do xc sut ca cc tp hnh tr trn RR+ theo phn b xc sut ca chuyn ng

Brown l:

PS(CAt1,...,tm

) =1

(

2)n

A

ex21/2e(x2x1)

2/2 . . . e(xnxn1)2/2dx1 . . . d xn (1.37)


20/44


chng t s tn ti v mt ton hc ca chuyn ng Brown, ta c th xy dng

v d hon ton tng t nh trong trng hp tng qut. Ch c iu khc l, l thay

v t = RR+ l khng gian tt c cc hm s thc trn na ng thng R+, ta t

= { C0(R+,R) | (0) = 0} l khng gian cc hm s lin tc trn R+ v c gi trbng 0 ti 0, m bo mi qu o u l lin tc. Cc tp hnh tr, v cc sigma-i

s, nh ngha ht nh c, ch thm iu kin l cc phn t u l cc hm lin tc. V

d m hnh chuyn ng Brown ny cho thy, qu o ca mt chuyn ng Brown c

th l mt hm s lin tc bt k. Tuy nhin, nh chng ta s thy, hu ht cc qu o

ca mt chuyn ng Brown tha mn mt s tnh cht c trng nh: khng kh vi ti

bt c im no, v c bin phn v hn.Ghi ch 1.1. iu kin iii) trong nh ngha ca chuyn ng Brown hay c thay bng

iu kin sau:

iii) Vi mi 0 t0 < t1 < .. . < tn, b n bin ngu nhin (Bt1Bt0, Bt2Bt1, . . . , BtnBtn1) l mt b bin ngu nhin c lp. Ni cch khc, cc bc i ca chuyn ng

Brown l c lp vi nhau.

Mt qu trnh ngu nhin tha mn iu kin iii) pha trn th c gi l mt qu

trnh c gia s c lp (independent increments).

D thy rng iu kin iii) l h qu ca iu kin iii). Trong trng hp m lc Fttrong m hnh xc sut chnh l lc sinh bi qu trnh ngu nhin, th iu kin iii) tng

ng vi iu kin iii).

1.2.3 i do ngu nhin

Chuyn ng Brown c dng nhiu trong thc t, chnh l v n l gii hn (lin

tc ha) ca cc qu trnh ngu nhin vi thi gian ri rc c dng gi l i do ngunhin, khi ta cho di thi gian ca mi bc i tin ti 0. Cc quan st ca Robert

Brown dn n chuyn ng mang tn ng cng chnh l quan st s i do ngu nhin

ca cc ht bi trong nc (tc l thc ra trong cc khong thi gian rt nh, gia 2 ln

va p vo cc phn t khc, th chuyn ng ca mt ht bi l c hng nht nh,

ch khng hon ton v hng (k vng ca gia s bng 0) nh trong nh ngha ca qu

trnh Wiener).

Ni mt cch c th hn, xt mt qu trnh ngu nhin X vi thi gian ri rc, c

bc thi gian bng > 0. Gi s l X0 = 0, cc gi s Xn X(n1) l c lp vi nhauv c cng phn b xc sut, l phn b Bernoulli sau: xc sut Xn X(n1) = a l


21/44


50% v xc sut Xn X(n1) = a cng l 50%, trong a l mt hng s dngc ph thuc vo tham s m chng ta s xc nh sau. Mt qu trnh ngu nhin nh

vy c gi l mt qu trnh i do ngu nhin (random walk) mt chiu, vi di

ca bc thi gian bng v di ca bc i do bng a (tc l c sau mi khong

thi gian th li dch chuyn ton ton ngu nhin, hoc l sang tri hoc l sang phi,

mt on c di a). Qu trnh i do ngu nhin ny l mt trng hp c bit ca

m hnh cy nh thc.

Hnh 1.3: Mt s v d th ng i do ngu nhin ( = a = 1)

Xt qung ng i c Xt Xs ca mt qu trnh i do ngu nhin t mt thiim s = M n mt thi im t = (M + N). Ta c th vit

Xt Xs =Ni=1

YM+i (1.38)

trong YM+i = X(M+i) X(M+i1) l tng bc i mt. Theo gi thit, cc bc

i YN+i l c lp vi nhau v c cng phn b xc sut l phn b Bernoulli, tc l

Xt Xs l tng ca N bin ngu nhin cng phn b xc sut. Khi N ln (tc l nh,v ta gi s s v t c nh, v t s = N ), ta c th s dng nh l gii hn trung tm nghin cu phn b xc sut ca Xt Xs . Trc ht, nhc li nh l gii hn trung


22/44


tm (xem Chng 4 ca [5]; trng hp c bit ca n, cho cc phn b Bernoulli, c

chng minh bi de Moivre v Laplace t th k 18):

nh l 1.1 (nh l gii hn trung tm). Gi s Y1, Y2, . . . , Y n, . . . l mt dy cc

bin ngu nhin c lp c cng phn b xc sut vi k vng bng 0 v lch chun

bng hu hn. t ZN =Ni=1

Yi

N. Khi vi mi on thng [a, b] R, ta c:

limN

P(a ZN b) =ba

12

ex2/2dx. (1.39)

Ni cch khc, phn b xc sut ca ZN tin ti phn b normal chun tc N(0, 1) khiN tin ti v cng.

Ch rng, trong nh l gii hn trung tm, c i lng

N xut hin. s dng

nh l gii hn trung tm cho Xt Xs, ta s t YM+n = Yn/

N/(t s) = Yn, hayYn = Y

M+n/

t. Ta s coi l cc bin YM+n/

t c cng phn b xc sut v phn b ny

khng ph thuc vo . Nhc li rng, phn b xc sut ca YM+n l phn b Bernoulli,

nhn hai gi tr a, vi xc sut 50% cho mi gi tr. Ni rng phn b xc sut caYM+n/

t khng ph thuc vo c ngha l a/

t l hng s khng ph thuc vo .

cho tin, ta s ta =

t. (1.40)

Khi lch chun ca YM+n/

t ng bng 1, v ta c Xt Xs =N

i=1 YM+i =

t sZN. Nh vy, ta c h qu sau ca nh l gii hn trung tm:

nh l 1.2. Gi Xt l qu trnh i do ngu nhin 1 chiu vi bc thi gian bng

v bc dch chuyn bng

. Gi st > s > 0 l hai mc thi gian bt k. Khi phn

b xc sut ca Xt

Xs tin ti phn b normal N(0,

t

s) khi m tin ti 0.

Trong nh l trn, t v s khng nht thit phi chia ht cho (bng nhn vi

mt s nguyn), v mi qu trnh ngu nhin Xt vi thi gian ri rc u c th c

coi l qu trnh ngu nhin vi thi gian lin tc, qua mt cng thc ni suy. V d, nu

tn < t < tn+1, trong tn v tn+1 l hai mc thi gian lin tip ca qu trnh ri rc,

th ta c th t Xt = Xtn (coi n l bt bin trn tng khc thi gian), hoc l t

Xt =tn+1ttn+1tn

Xtn +ttn

tn+1tnXtn+1 ( bin n thnh qu trnh ngu nhin lin tc tuyn

tnh tng khc). nh l trn ng vi c hai cch t .

nh l trn cho thy cc qu trnh i do ngu nhin Xt (vi bc thi gian bng v bc dch chuyn bng

) tin ti (theo ngha phn b xc sut) chuyn ng Brown


23/44


chun tc khi tin ti 0, v n cng gii thch v sao ta yu cu rng gia s Bt Bs cachuyn ng Brown phi c phn b xc sut l phn b N(0,

t

s). V l qu trnh

i do ngu nhin Xt c bc dch chuyn

tin ti 0 khi tin ti 0, nn, mt cch

trc gic, gii hn ca Xt khi tin ti 0 phi l qu trnh lin tc (tc l hu ht mi

ng i u l lin tc). Do vy m ta c iu kin lin tc trong nh ngha ca chuyn

ng Brown.

1.2.4 Mt s tnh cht ca chuyn ng Brown

Mt trong nhng tnh cht quan trng nht ca chuyn ng Brown l tnh cht

martingale. pht biu tnh cht ny bng cng thc ton hc, ta cn khi nim k

vng theo sigma-i s con: Nu X l mt bin ngu nhin trn khng gian xc sut

(, F, P), v G l mt sigma-i s con ca F, th k vng ca X theo G (nu tn ti),theo nh ngha, l mt bin ngu nhin Y trn khng gian xc sut (, G, P), (tc lhm Y : R l hm o c khng nhng theo Fm cn o c theo G) sao chovi mi tp con A o c theo G (tc l A G) ta c

A

XdP = A

Y dP. (1.41)

hiu ngha ca ng thc (1.41), hnh dung l P(A) > 0, nhng A nh sao cho

Y l hng s (hoc gn nh l hng s) trn A. Chia c hai v ca ng thc cho P(A),

ta c

E(X|A) = Y(A), (1.42)trong v tri l k vng c iu kin ca X trong iu kin A, v v phi l gi tr

(trung bnh) ca Y trong (tp) tnh hung A. V A G l ty , nn ta ni rng k vng

ca X theo G bng Y, v ta vitE(X|G) = Y. (1.43)

nh l 1.3. (Tnh cht martingale). Vi mi 0 s t ta c

E(Bt|Fs) = Bs, (1.44)

hay cn c th vit l:

E(Bt Bs|Fs) = 0, (1.45)

tc l k vng (c iu kin) ca mi bc i Bt Bs (d tnh hung xy ra cho n thiim s) u bng 0.


24/44


nh l trn chng qua l h qu trc tip ca cc iu kin ii) v iii) trong nh ngha

chuyn ng Brown.

nh l 1.4. Gi s Bt l mt chuyn ng Brown. Khi tn ti mt tp con trong khng gian cc tnh hung, c xc sut bng 1, sao cho vi mi tnh hung th qu o B(t) := Bt() tha mn cc tnh cht sau:

i) Vi mi a R, c v hn cc thi im t R+ sao cho B(t) = a.ii) Vi mi > 0, tp hp cc khng im ca B (tc l cc im t sao cho B(t) = 0)

trn on thng [0, ] l mt tp v hn.

iii) B khng n iu theo t trn bt c on thng no.

iv) B khng kh vi ti bt c im no theo t.Ni cch khc, hu ht cc qu o ca mt chuyn ng Brown tha mn cc tnh cht

trn.

Chng minh. i) Ta s gi s a > 0 (cc trng hp khc chng minh tng t). D thy

rng, nu Bt l chuyn ng Brown v > 0 l hng s, th 1B2t cng l chuyn ng

Brown. (Xem bi tp 1.3). Tnh cht ny gi l tnh cht scaling (phng to thu nh)

ca chuyn ng Brown. Do tnh cht scaling nn ta c

P(Bt < x|Bs < y) = P(B2t < x|B2s < y) (1.46)

vi mi hng s x,y,,s v t tha mn > 0, 0 < s < t. Trng hp ring ca ng thc

trn l

P(B4n < 2na|B4n1 < 2n1a) = P(B4n+1 < 2n+1a|B4n < 2na) = c, (1.47)

vi c l mt s nh hn 1, ln hn 0, v khng ph thuc vo n Z. Bi vy

P(B4n < 2n

0

n

N) = P(B

1< a)

N

n=1

P(B4n < 2na

|B4n1 < 2n1a) = cNP(B

1< a)

(1.48)

tin ti 0 khi N tin ti v cng, v do vy P(B4n < 2na n Z+) = 0. C nghal, vi hu ht mi tnh hung , tn ti t nht mt s nguyn khng m n sao cho

B(4n) 2na a. V (vi hu ht mi ) B l hm lin tc theo t, c gi tr bng 0

ti 0 v gi tr ln hn hoc bng a ti mt thi im no , nn n cng phi nhn a

lm gi tr ti thi im no (theo nh l gi tr trung gian cho hm lin tc). Nh

vy ta chng minh c rng, trong hu ht mi tnh hung , th tn ti t nht 1

thi im sao cho gi tr ca B ti thi im bng a. Tng t nh vy, ta c thchng minh rng, vi mi M R+, hu nh chc chn rng qu o ca chuyn ng


25/44


Brown c nhn gi tr bng a ti mt thi im ln hn M. V M l ty , nn t suy

ra hu nh chc chn rng qu o ca chuyn ng Brown c nhn gi tr bng a ti v

hn cc thi im. Tht vy, gi Ak,M l tp cc tnh hung tha mn iu kin: trn

on thng [0, M[ c t nht k thi im m gi tr ca B bng a. V hu nh chc chn

rng trn na ng thng [M, [ c t nht 1 thi im m gi tr ca B bng a, nnP(Ak+1,) P(Ak,M) vi mi M. Do P(Ak+1,) limM P(Ak,M) = P(Ak,), vtheo qui np, ta c P(Ak,) P(A0,) = 1, tc l P(Ak,) = 1. V tp cc tnh hung tha mn iu kin c v s thi im m gi tr ca qu o l a l tp

k=1 Ak,,

nn tp ny cng c xc sut bng 1.

ii) Tng t nh pha trn, do tnh cht scaling nn P(B4n < 2

n

n Z+) = 0,do hu nh chc chn rng tn ti n Z+ sao cho B4n 2n > 0. Tng t nhvy, hu nh chc chn rng tn ti m Z+ sao cho B4m 2m < 0. V (vi hu htmi ) qu o B l lin tc v nhn c gi tr m ln gi tr dng trn on thng

na m ]0, ], nn n phi c t nht mt khng im trn on thng ]0, ] ny. V l

ty , nn t suy ra l hu ht mi quB o phi c v hn khng im trong on

[0, ], cng bng l lun tng t nh l trong chng minh tnh cht i).

iii) Chng minh ht nh l chng minh tnh cht ii). (Ch cn chng minh cho cc

on thng c iu u v im cui l s hu t l , v tp hp cc on thng nhvy l tp m c).

iv) Tnh khng kh vi ti bt c im no cng c chng minh bng tnh cht

scaling ca chuyn ng Brown v nhng l lun tng t nh trong chng minh cc tnh

cht i) v ii). Chng ta s b qua chng minh ca tnh cht iii) y. (N l mt tnh

cht th v, nn c a vo tham kho, ch chng ta cng s khng cn dng n

n trong quyn sch ny).

1.2.5 Bin phn v bin phn bnh phng

Theo nh ngha, bin phn ca mt hm s f trn mt on thng [a, b] l i lng

Vba (f) = sup{ni=1

|f(xi) f(xi1)| ; n N, x0 = a < x1 < .. . < xn = b}. (1.49)

Nu i lng bng + th ta ni rng f c bin phn v hn trn on [a, b], cnnu n nh hn + th ta ni rng f c bin phn hu hn trn on [a, b]. V d,


26/44


nu f l hm kh vi lin tc, th n c bin phn hu hn, bng

Vba (f) =ba

|f(t)|dt. (1.50)

Tng qut hn, mi hm s f tha mn iu kin Lipschitz (tc l tn ti mt hng s K

sao cho |f(x)f(y)| K|xy| vi mi x v y s c bin phn hu hn: Vba (f) K(ba).Nh ta thy trong mc trc, chuyn ng Brown l khng kh vi. Hn na, n

cn c bin phn v hn:

nh l 1.5. Gi sBt l mt chuyn ng Brown. Khi hu ht mi qu o B ca

Bt u c bin phn v hn trn mi on thng [a, b] (0

a < b).

Chng minh. N l h qu trc tip ca nh l 1.6 di y, bi v nu mt hm lin

tc c bin phn hu hn trn mt on thng no , th bin phn bnh phng ca

n trn on bng 0, trong khi i vi chuyn ng Brown, bin phn bnh phng l

khc 0 trn mi on thng.

Theo nh ngha, bin phn bnh phng (quadratic variation) ca mt hm s f

trn mt on thng [a, b] l gii hn:

QVba (f) = lim0

ni=1

|f(xi) f(xi1)|2, (1.51)

trong a = x0 < x1 < . . . < xn = b l mt phn hoch ca on [a, b], v =

maxi(xi xi1) l mn (mesh) ca phn hoch, nu nh gii hn tn ti. (Nu giihn khng tn ti, th ta c th thay lim bng lim sup, nhng i vi chuyn ng Brown,

vn ny khng t ra, v nh ta s thy, gii hn ny tn ti cho hu khp mi qu

o ca chuyn ng Brown).

D thy rng, nu hm s tha mn iu kin Lipschitz, th n c bin phn vnhphng bng 0. Tng qut hn, nu mt hm s l lin tc v c bin phn hu hn, th

bin phn bnh phng ca n bng 0. (Khng nh ny dnh cho bn c nh l mt

bi tp).

i vi chuyn ng Brown, th bin phn bnh phng l khc 0 nhng hu hn trn

cc on thng thi gian. Hn na, n bng ng di ca on thng:

nh l 1.6. Gi sBt l mt chuyn ng Brown. Khi hu ht mi qu o B ca

n c bin phn bnh phng trn on thng [a, b] bt k bng ng b

a :

QVba (B) = b a. (1.52)


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Chng minh. Chng ta s chng minh khng nh sau y, hi yu hn nh l 1.6 mt

cht, nhng cho thy bn cht vn : vi hu ht mi tnh hung ta c

limN

Ni=1

(B(i/N) B((i 1)/N))2 = 1. (1.53)

t YN,i =1

N(B(i/N) B((i 1)/N))2 . Khi cc bin ngu nhin YN,i u c phn

b xc sut bng phn b xc sut ca (B1)2, tc l phn b ki bnh phng 21 (vi

1 bc t do xem chng 4 ca [5] v phn b ki bnh phng). Nhc li rng, phn b

21 c k vng bng 1. Ta cn chng minh rng

limN

1N

Ni=1

YN,i = 1

hu khp mi ni. Th nhng y chnh l lut s ln (dng mnh) p dng vo phn

b 21, v cc bin ngu nhin YN,1, . . . , Y N,N c lp vi nhau v c cng phn b xc

sut 21. (Xem Chng 3 ca [5] v dng mnh ca lut s ln cch chng minh ca n

tnh hung y hi khc nhng chng minh vn th). Bi vy ta c iu phi chng

minh.

C nh l ngc li sau y ca Paul Lvy(3)

, cho thy v tr quan trng ca chuynng Brown trong l thuyt cc qu trnh ngu nhin:

nh l 1.7 (Lvy). Mi qu trnh martingale(4) vi thi gian lin tc, tha mn tnh

cht lin tc (tc l hu ht mi qu o u lin tc), v c bin phn bnh phng hu

hn, u l chuyn ng Brown sau mt php bin i thi gian.

C th xem chng minh ca nh l Lvy ny trong Chng 2 ca quyn sch [ 12]

ca Karatzas v Shreve.

Bi tp 1.7. Th chng minh trc tip nh l 1.5 m khng cn dng nh l 1.6

1.2.6 Chuyn ng Brown hnh hc

Gi ca c phiu khng th m (thm ch ta s coi n l lun dng, tuy trong thc t

n c th v 0 khi cng ty ph sn), nn n khng th l chuyn ng Brown, bi v cc(3)Paul Lvy (18861971) l nh ton hc Php ni ting v cc cng trnh v xc sut, xem:

http://en.wikipedia.org/wiki/Paul_Pierre_Lvy(4)Mt qu trnh St c gi l martingale nu E(

|St

|) 0), th qu trnh ngu nhinexp(a+bt+Bt) c gi l mtchuyn ng Brown

hnh hc (geometric Brownian motion). Ni cch khc, mt qu trnh ngu nhin Gt

c gi l mt chuyn ng Brown hnh hc khi v ch khi ln Gt l mt chuyn ng

Brown (khng nht thit chun tc).

Mt bin ngu nhin X ch nhn gi tr dng c gi l c phn b xc sut log-

normal nu nh ln X c phn b xc sut normal. T nh ngha trn, ta c ngay h

qu sau: nu Gt l mt chuyn ng Brown hnh hc, v t > s, th Gt/Gs c phn b xc

sut log-normal.

Ch rng, tuy chuyn ng Brown Bt l martingale, nhng exp(Bt) khng phi l

martingale. Tht vy, ta c

E(exp(Bt)) =12t

exex2/2tdx = 1

2t

et/2e(xt)2/2tdx = et/2 (1.54)

tng theo thi gian, ch khng bt bin, v do n khng th l martingale. Tuy nhin,

ng thc trn cng cho thy E(exp(Bt t/2)) = 1 l bt bin theo thi gian. Hn na,ta c:

nh l 1.8. (nh l v nh ngha). Nu Bt l chuyn ng Brown th qu trnh ngu

nhin exp(Bt t/2) l martingale. Qu trnh ngu nhin exp(Bt t/2) ny c gi l

chuyn ng Brown hnh hc chun tc.

Chng minh. Pha trn ta kim tra rng E(Gt|F0) = G0 = 1, trong Gt = exp(Bt t/2) l chuyn ng Brown hnh hc chun tc. Vic kim tra ng thc E(Gt+s|Fs) = Gsvi mi s 0, t > 0 hon ton tng t, da trn tnh cht ca chuyn ng Brown Bt.Tht vy, khi s c nh, th Gt+s/Gs = exp(Bt+s Bs t/2) cng l mt chuyn ngBrown hnh hc chun tc, v Bt+s Bs cng l mt chuyn ng Brown hnh hc, do ta c ng thc trn.

Bi tp 1.8. Gi s l mt hng s ty . Chng minh rng qu trnh exp(Bt 2t2 ),

trong Bt l chuyn ng Brown chun tc, l mt qu trnh martingale.


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1.3. VI PHN CA CC QU TRNH NGU NHIN 29

1.3 Vi phn ca cc qu trnh ngu nhin

Tng t nh i vi cc hm s, chng ta cng mun lm cc php vi tch phn i

vi cc qu trnh ngu nhin, c th tm li c cc qu trnh ngu nhin t vi phn

ca n, qua vic tnh tch phn, gii cc phng trnh vi phn ngu nhin (theo bin thi

gian). Tuy nhin, do cc qu trnh ngu nhin ni chung khng kh vi theo bin thi gian

(chuyn ng Brown l mt v d tiu biu), nn cc php tnh vi tch phn ca chng

phc tp hn v mt k thut so vi gii tch thng thng.

1.3.1 Vi phn ca chuyn ng Brown hnh hc hiu vi phn ngu nhin, trc ht chng ta xt mt v d c th: vi phn ca

exp(Bt), trong Bt l mt chuyn ng Brown chun tc.

Nhc li rng, nu f(t) l mt hm kh vi theo t, th ta c cng thc sau:

d exp(f) = exp(f)df = exp(f)fdt, (1.55)

trong f = df /dt l o hm ca f theo t. Th nhng, Bt khng kh vi theo t, v khi

thay f(t) bng Bt th cng thc trn khng ng na ! hiu ti sao, chng ta quaytr li nh ngha th no l vi phn.

Vi phn df ca mt hm s (hay nh x) f l mt k hiu ton hc, ch thay

i ca f khi cc bin ca n thay i ( y ch c 1 bin, l bin thi gian t): khi t thay

i mt i lng t, th f thay i mt i lng l f = f(t + t) f(t), v t lthay i gia f v t bng

f

t=

f(t + t) f(t)t

. (1.56)

Trong trng hp m t l trn tin ti m s no khi t tin ti 0, th s c

gi l o hm ca f theo t, thng k hiu l f(t), v ta vit

df

dt= f(t) = lim

t0

f(t + t) f(t)t

, (1.57)

hay

df = fdt. (1.58)

ng thc cui cng c ngha l khia m t thay i th f cng thay i, vi tc c thay

i bng f ln tc thay i ca t.

Vi phn dBt cng l khi nim o thay i ca Bt khi m t thay i. Th nhng,v Bt khng kh vi ti bt c im no, nn khng th vit dBt di dng Htdt (trong


30/44


Ht l mt qu trnh ngu nhin). Bi vy, cch n gin l ta s nguyn dBt, v hiu

n nh l mt k hiu ton hc ch mt bin ng v cng nh dng chuyn ng

Brown. V chuyn ng Brown l qu trnh ngu nhin n gin nht, quan trng nht,

c nghin cu k nht, trong cc loi qu trnh ngu nhin lin tc khng kh vi, nn

ta s s dng dBt nh mt vi phn c s tng t nh dt, v tm cch biu din vi phn

ca cc qu trnh ngu nhin khc di dng t hp tuyn tnh Htdt + FtdBt ca dt v

dBt nu c th (trong Ht v Ft l hai qu trnh ngu nhin). Khi vit

dGt = Htdt + FtdBt (1.59)

th c ngha l

G(t + t) G(t) = Ht.t + Ft.(B(t + t) B(t)) + , (1.60)

trong l i lng rt nh so vi t, c th b qua: /t tin ti 0 khi t tin ti 0.

Phn FtdBt l phn khng kh vi (theo ngha thng thng) v c k vng bng 0 (v k

vng ca (B(t + t) B(t)) bng 0) ca dGt, v phn Htdt l phn kh vi. Tt nhin,nu c th vit dGt = Htdt + FtdBt, th Ht v Ft c xc nh duy nht theo Gt, v tng

ca mt phn kh vi v mt phn khng kh vi khng th bng 0 tr khi c hai phn

bng 0.

p dng tng trn vo trng hp Gt = exp(Bt). Ta c:

exp(Bt)

exp(Bt)=

exp(B(t + t)) exp(B(t))exp(B(t))

= exp(B(t + t) B(t)) 1

= exp(Bt) 1 = Bt + (Bt)2

2!+

(Bt)3

3!+ . . .

Nhc li rng, bin phn bnh phng ca Bt trn mt on thi gian t bng chnh

t. Do , (Bt)2 xp x bng t khi m t nh (tri ngc vi trng hp kh vi: nu

f kh vi th (f)2

rt nh so vi t, c th b qua). Cc i lng (Bt)n

vi n t 3tr ln rt nh so vi t, c th b qua. Bi vy ta c

exp(Bt)

exp(Bt)= Bt +

t

2+ , (1.61)

trong /t tin ti 0 khi t tin ti 0 (trong hu khp mi tnh hung), t ta suy

ra cng thc vi phn sau:d exp(Bt)

exp(Bt)= dBt +

dt

2, (1.62)

hay cn c th vit l:

d exp(Bt) =1

2exp(Bt)dt + exp(Bt)dBt. (1.63)


31/44

1.3. VI PHN CA CC QU TRNH NGU NHIN 31

Mt cch hon ton tng t, d dng kim tra rng

d exp(Bt t2 ) = exp(Bt t2)dBt, (1.64)c ngha l vi phn ca chuyn ng Brown hnh hc chun tc bng chnh n nhn vi

vi phn ca chuyn ng Brown chun tc. Ch rng, trong v bn phi ca ng thc

trn khng c s tham gia ca dt, ch c s tham gia ca dBt l phn c k vng bng 0,

v iu ny cng gii thch ti sao exp(Bt t2) li l martingale (vi phn ca n lun ck vng bng 0, nn cc bc chuyn ng cng u c k vng bng 0).

Bi tp 1.9. Tnh vi phn ca mt chuyn ng Brown hnh hc ty exp(a + bt + Bt),

trong a,b, l cc hng s, t suy ra rng cc nghim ca phng trnh vi phnngu nhin

dStSt

= dt + dBt,

trong v l cc hng s, chnh l cc chuyn ng Brown hnh hc.

1.3.2 B It

Mt trong nhng cng thc quan trng m chng ta dng trong phn tch pha trn

v vi phn ca chuyn ng Brown hnh hc l:

(dBt)2 = dt, (1.65)

Cng thc ny c th hiu l

limt0+

(Bt)2 t

t= 0, (1.66)

v n ng vi cng thc bin phn bnh phng ca chuyn ng Brown. Ngoi cng thc

(dBt)2

= dt, cn c vit l dBt.dBt = dt, chng ta cn c cc cng thc sau, tng ihin nhin, cho vic tnh vi phn cc qu trnh ngu nhin:

dt.dBt = dt.dt = 0. (1.67)

Cc cng thc ny c hiu tng t nh trn, tc l cc i lng (t).(Bt) v (t)2

l rt nh so vi t (thng ca chng chia cho t tin ti 0 khi t tin ti 0), c th

b qua trong cc php tnh ton vi phn.

Cc cng thc trn l c s ca cng thc sau, cho php chng ta tnh vi phn ca

cc qu trnh ngu nhin c dng hm s f(t, Bt) theo hai bin t v Bt trong Bt lchuyn ng Brown:


32/44


nh l 1.9. Nu Ft = f(t, Bt) trong f(t, x) l mt hm kh vi lin tc theo bin t

v kh vi lin tc 2 ln theo bin x, th

dFt =f(t, Bt)

tdt +

f(t, Bt)

xdBt +

1

2

2f(t, Bt)

x2dt. (1.68)

Chng minh ca cng thc trn hon ton tng t nhng chng minh ca cng thc

vi phn ca exp(Bt). So vi trng hp kh vi, th thnh phn cn thm vo trong cng

thc l1

2

2f(t, Bt)

x2dt, v l do chnh l bi v (dBt)2 = dt. Tht vy, theo khi trin

Taylor-Lagrange, ta c

f(t, Bt) = f(t + t, B(t + t)) f(t, Bt) == [f(t + t, B(t + t)) f(t, B(t + t))] + [f(t, B(t + t)) f(t, Bt)] =

=f(t, Bt + t)

tt + 1 +

f(t, Bt)

xBt +

1

2

2f(t, Bt)

x2(Bt)

2 + 2,

trong 1 v 2 u rt nh (c th b qua) so vi t, t suy ra cng thc trn.

V d 1.4. Ta c d(tB3t ) = (B3t + 3tBt)dt + 3tB

2t dBt, bi v (tx

3)/t = x3, (tx3)/x =

3x2, v 2(tx3)/x2 = 6tx.

Mt cch tng qut hn, ta c cng thc sau, gi l cng thc It, hay b It:

nh l 1.10 (B It). Gi s f(t, x) l mt hm s kh vi lin tc theo bin t v

kh vi lin tc 2 ln theo binx, vXt l mt qu trnh It tha mndXt = tdt + tdBt

trong Bt l chuyn ng Brown. Khi ta c:

df(t, Xt) =

f

t+ t

f

x+

2t2

2f

x2

dt + t

f

xdBt (1.69)

Trong nh l trn c khi nim qu trnh It. Mt qu trnh ngu nhin Xt c gil mt qu trnh It nu n l qu trnh ngu nhin vi thi gian lin tc, v vi phn

ca n c th vit c di dng (hay ni cch khc, n tha mn phng trnh vi phn

dng):

dXt = tdt + tdBt, (1.70)

trong t v t l cc cc qu trnh ngu nhin tng thch (hoc l cc hm s theo t)

tha mn iu kin

E(

t0

|s|2ds) < +, E(t0

|s|2ds) < + (1.71)


33/44

1.4. TCH PHN IT 33

vi mi t 0. Cc qu trnh ngu nhin tng thch tha mn iu kin ny c gi lcc qu trnh ngu nhin c bnh phng kh tch (tc l c tch phn bnh phng

hu hn), hay cn gi l thuc lp H2.

Chng minh ca nh l 1.10 hon ton tng t nh chng minh cng thc (1.68).

iu kin bnh phng kh tch c dng kim sot cc i lng nh khi tnh gii

hn tm vi phn.

Bi tp 1.10. Chng minh cng thc (1.69) trong trng hp t v t l cc hm s lin

tc (khng ngu nhin) theo t.

1.4 Tch phn It

Nu ta bit o hm f(t) ca mt hm s kh vi f(t), th ni chung ta c th tm li

c f t f bng cch ly tch phn:

f(t) = f(0) +

t0

f(s)ds. (1.72)

Tng t nh vy, nu ta bit vi phn ca mt qu trnh It

dXt = tdt + tdBt, (1.73)

th v nguyn tc, ta cng phi tm li c qu trnh ngu nhin Xt t cc qu trnh

ngu nhin t v t bng cch ly tch phn:

Xt = X0 +

t0

sds +

t0

sdBs. (1.74)

Vn l nh ngha cc tch phn trn sao cho thch hp. Tch phnt0

sds c th

c nh ngha theo cch c in (theo nh ngha tch phn Riemann). Tuy nhin, nh

chng ta s thy, vic nh ngha tch phnt0 sdBs phc tp hn, do Bt c bin phn

v hn. Trong phn ny, chng ta s bn cch nh ngha tch phnt0

tdBt theo Kioshi

It (19152008), mt trong nhng cha ca gii tch ngu nhin, v tch phn ny c

gi l tch phn It.

1.4.1 Tch phn RiemannStieltjes

Nhc li rng, nu f v g l hai hm s trn mt on thng [0, t] R, th tch phnt0

fdg, (1.75)


34/44


c nh ngha nh sau, v gi l tch phn RiemannStieltjes: t

U(f, g) =ni=1

supxi[ai1,ai]

f(xi)(g(ai) g(ai1)) (1.76)

v

L(f, g) =ni=1

infxi[ai1,ai]

f(xi)(g(ai) g(ai1)), (1.77)

trong l mt phn hoch ca on thng [0, 1] cho bi mt dy cc s 0 = a0 a1 . . . an = t, ri t

t0

f dg = limmesh()0

L(f, g) = lim

mesh()0U(f, g). (1.78)

nu nh cc gii hn tn ti v bng nhau. ( y mesh() = supi(ai ai1) l khiu nh ca phn hoch ). Nu tch phn RiemannStieltjes tn ti, th mi tng

c dngn

i=1 f(xi)(g(ai) g(ai1)), trong xi [ai1, ai] c th chn ty , u tintit0

f dg khi nh mesh() ca phn hoch tin ti 0, bi v tng b kp gia

U(f, g) v L(f, g). Trng hp c bit, khi m g(t) = t, th nh ngha trn trng vi

nh ngha tch phn Riemann

t

0f(s)ds.

Nu chng hn g l mt hm kh vi lin tc, v f l hm b chn v lin tc tngkhc, th tch phn Riemann-Stieltjest0

f dg tn ti, v ta c cng thc chuyn i sau:t0

f dg =

t0

f gds, (1.79)

trong g l o hm ca g.

Nu chng hn f v g khng phi l hm s, m l cc qu trnh ngu nhin, sao

cho f lin tc v g c bin phn phu hn, th ta vn c th nh ngha c tch phn

RiemannStieltjes t0 f dg nh trn. (Bn thn tch phn cng s l mt qu trnh ngunhin, v ta nh ngha n cho tng tnh hung v tng mc thi gian). Th nhng khim g = Bt l mt chuyn ng Brown, th nh ngha tch phn RiemannStieltjes ni

chung khng cn p dng c na, nh v d n gin sau y cho thy.

V d 1.5. Gi s ta mun nh ngha tch phnt0

BsdBs, trong Bt l mt chuyn

ng Brown. Lm theo phng php RiemannStieltjes, ta vit

U(B, B) =ni=1

supti[ai1,ai]

B(ti)(B(ai) B(ai1)), (1.80)

L(B, B) =ni=1

infti[ai1,ai]

B(ti)(B(ai) B(ai1)), (1.81)


35/44

1.4. TCH PHN IT 35

trong 0 = a0 a1 . . . an1 an = t l mt phn hoch ca on thng [0, t],ri xt hiu U

(B, B)

L(B, B). D thy rng, vi mi i ta c

supti[ai1,ai]

B(ti)(B(ai) B(ai1)) infti[ai1,ai]

B(ti)(B(ai) B(ai1)) (B(ai) B(ai1))2,(1.82)

bi vy

U(B, B) L(B, B) ni=1

(B(ai) B(ai1))2. (1.83)

V phi ca bt ng thc trn, khi mesh() tin ti 0, chnh l bin phn bnh phng

ca chuyn ng Brown trn on thng thi gian [0, t]. Th nhng, ta bit rng, bin

phn bnh phng ca chuyn ng Brown trn on thng [0, t] bng t, ln hn 0. Do

, U(B, B) L(B, B) khng tin ti 0 khi mesh() tin ti 0, v bi vy khng thnh ngha c tch phn

t0

BsdBs theo kiu RiemannStieltjes, v U(B, B) v L(B, B)

khng th c cng gii hn khi mesh() tin ti 0.

V d n gin trn cho thy, v bin phn bnh phng ca Bt khc 0, nn chng ta

khng th nh ngha c tch phnt0

BsdBs theo kiu RiemannStieltjes. Do chng

ta cn mt nh ngha tch phn khc, gi l tch phn It.

1.4.2 nh ngha tch phn It

Chng ta s nh ngha cc tch phn It c dngt0

sdBs, (1.84)

trong B l mt chuyn ng Brown chun tc, v l mt qu trnh ngu nhin c

bnh phng kh tch, tc lEt

0 |(s)|2

ds

< vi mi t R+.

Trc ht, ta s nh ngha tch phn It cho cc qu trnh s cp. Mt qu trnh ngu

nhin t c gi l s cp (elementary), hay cn gi l n gin (simple), trn on

thng thi gian [0, t] nu nh tn ti mt phn hoch c nh 0 = t0 < t1 < .. . < tn = t

ca on thng [0, t], sao cho trn mi on thng na m [ai1, ai[ th (s, ) l hng s

theo thi gian (trong hu ht mi tnh hung ), tc l (s, ) = (ai1, ) vi miai1 s < ai. Vic nh ngha tch phn It cho cc qu trnh s cp c suy ra trctip t tnh cht cng tnh ca tch phn v ng thc hin nhin cn phi ng:b

a

dBs = Bb Ba. (1.85)


36/44


Nu l mt qu trnh s cp theo phn hoch 0 = t0 < t1 < . . . < tn = t ca on

thng [0, t], th ta c:t0

(s)dB(s) =ni=1

titi1

(s)dB(s) =ni=1

titi1

(ti1)dB(s) =ni=1

(ti1)(B(ti)B(ti1)),

hay vit gn li l t0

(s)dB(s) =ni=1

(ti1)(B(ti) B(ti1)). (1.86)

ng thc trn chnh l nh ngha tch phn It trong trng hp m s l mt qu

trnh s cp. D thy rng, nh ngha ny khng ph thuc vo phn hoch on thng[0, t] so cho qu trnh ngu nhin l s cp theo phn hoch . Hn na, c th thy

rng, khi m s l qu trnh s cp, th tch phn RiemannStieltjes cng c ngha, v

trng vi tch phn It.

Khi m l mt qu trnh ngu nhin tng qut tha mn cc iu kin nu

trn, tch phn It c nh ngha bng cch ly gii hn, thng qua mt dy cc qu

trnh s cp hi t n . S tn ti ca dy ny c cho bi nh l sau:

nh l 1.11. C nh mt sT > 0 ty . Gi s l mt qu trnh ngu nhin tngthch c bnh phng kh tch. Khi tn ti n cc qu trnh ngu nhin (n N) tngthch v c bnh phng kh tch, v s cp trn on thng [0, T], sao cho

limn

E

T0

|n(t) (t)|2dt

= 0. (1.87)

Dy qu trnh ngu nhin n tha mn iu kin ca nh l trn c gi l hi t

n theo chun L2[0, T]. Theo nh ngha, chun L2[0, T] ca l:

L2[0,T] :=E

T

0

|(t)|2dt

. (1.88)

Khng gian cc qu trnh ngu nhin tng thch c chun L2[0, T] hu hn l mt

khng gian Hilbert (sau khi ta coi rng hai qu trnh ngu nhin l bng nhau, nu chun

L2[0, T] ca hiu ca chng bng 0). nh l trn tng t nh nh l mi hm s c

bnh phng kh tch u c th c xp x bng cc hm s dng bc thang mt cch

chnh xc ty theo chun L2. Ta s b qua chng minh ca n y.

Khi ta tm c mt dy qu trnh ngu nhin n s cp trn on [0, T] v hit n theo chun L2[0, T], th vi mi t T ta c th nh ngha tch phn It bng


37/44

1.4. TCH PHN IT 37

gii hn sau:

t0 (s)dB(s) := limn

t0 n(s)dB(s). (1.89)

Gii hn y cng c hiu theo chun L2[0, T] : nu t Fn,t =t0

n(s)dB(s) v Ft =t0

(s)dB(s), th ni rng Fn,t tin ti Ft c ngha l limn ET

0|Fn(t) F(t)|2dt

= 0.

chng t rng nh ngha trn c ngha, ta phi chng minh l gii hn v phi ca

cng thc trn tn ti (tc l dy cc tch phn It Fn,t =t0

n(s)dB(s) l mt dy

Cauchy theo chun L2[0, T]), v khng ph thuc vo s la chn dy n. Cc khng nh

ny c suy ra t mt ng thc gi l ng c It (xem nh l 1.12). Chng ta s

b qua chng minh y.

Trong trng hp m qu trnh l lin tc bn tri v b chn a phng (locally

bounded tc l tp hp cc tnh hung sao cho qu o chy ra v cng trong khong

thi gian hu hn c xc sut bng 0 cc qu trnh ngu nhin m chng ta quan tm

n trong sch ny u b chn a phng), ta c th xy dng dy qu trnh s cp ntrn [0, T] hi t n theo chun L2[0, T] nh sau: t

n(t) = ([nt/T]T /n) (1.90)

vi mi t

T, trong [nt/T] l k hiu phn nguyn ca s nt/T, c ngha l nu

kT/n t < (k + 1)T /n th n(t) = (kT/n), hay ni cch khc, gi tr ca n trn onthng [kT/n, (k + 1)T /n[ l bt bin theo t v bng gi tr ca ti im u ca on

thng , tc l im kT/n. (Ch l ta khng th ly gi tr ca ti im cui hay

cc im khc ca on thng [kT/n, (k + 1)T /n[ lm gi tr ca n trn on thng

, v nu ly nh vy th qu trnh n ni chung s khng phi l mt qu trnh tng

thch). Theo cch chn n ny, ta c cng thc

T

0

tdBt = limn

n1

k=0

(kT

n)(B(

(k + 1)T

n) B(kT

n)). (1.91)

V d 1.6. Ta s chng minh rngt0

BsdBs =B2t2

t2

, (1.92)

vi mi t, trong Bt l chuyn ng Brown. Tht vy, theo cng thc (1.91) ta ct0

BsdBs = limn

n1k=0

B(kT

n)(B(

(k + 1)T

n) B(kT

n))

= limn

12

(B(t))2

n1k=0

(B( (k + 1)Tn

) B(kTn

))2

.


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Hnh 1.4: Mt qu o ca chuyn ng Brown Bt v tch phn Itt0

BsdBs

Theo tnh cht bin phn bnh phng ca chuyn ng Brown, ta c

limn

n1k=0

(B((k + 1)T

n) B( kT

n))2 = t,

t ta c cng thc tch phn cn chng minh. Ch rng, nu Ft c bin phn bnh

phng bng 0 v F0 = 0, th ta ct0

FsdFs = F2t (nh ngha tch phn mt cch tng

t, hoc theo nh ngha RiemannStieltjes), trong khi trong cng thc (1.92) c thm

thnh phn t/2 v phi. V d n gin ny cho thy nh hng ca bin phn bnh

phng vo gi tr ca tch phn It.

1.4.3 Mt s tnh cht c bn ca tch phn It

Tng thch: Tch phn It l mt qu trnh tng thch vi lc sigma-i s ban u.

Tuyn tnh: Tng t nh tch phn thng thng, tch phn It c tnh cht tuyn

tnh: nu a v b l hai hng s tht0

(as + bs)dBs = a

t0

sdBs + b

t0

sdBs. (1.93)

Lin tc: Nu mt qu trnh ngu nhin Ft vit c di dng tch phn It Ft =t0

sdBs, th hu ht mi qu o ca Ft l lin tc.


39/44

1.4. TCH PHN IT 39

Tch phn It l php tnh ngc ca vi phn: Tng t nh tch phn thng thng,

tch phn It tha mn tnh cht c bn sau, lin h gia php tnh vi phn v php tnh

tch phn: nu Ft =t0

sdBs, trong s l mt qu trnh ngu nhin c tch phn bnh

phng hu hn, th dFt = tdBt, v ngc li, nu dFt = tdBt, th Ft = F0 +t0

sdBs.

Tng qut hn, nghim ca phng trnh vi phn ngu nhin

dXt = tdt + tdBt (1.94)

l

Xt = X0 + t

0

sds + t

0

sdBs. (1.95)

V d, nghim ca phng trnh dXt = BtdBt l Xt = X0+t0

BsdBs = X0+B2t /2t/2.

Tnh ngc li, theo b It, d dng thy rng d(B2t /2 t/2) = BtdBt.ng c It: ng c It l mt cng thc gii tch cho php nh gi chun ca cc

qu trnh ngu nhin (c dng chng hn trong vic chng minh s hi t ca mt dy

qu trnh ngu nhin nh ngha theo tch phn It).

nh l 1.12 (ng c It). Nu l mt qu trnh ngu nhin c bnh phng hu hn

th ta cE

t0

sdBs

2= E

t0

|s|2ds

(1.96)

vi mi t 0.

Vic chng minh ng c trn l mt bi tp dnh cho bn c trong trng hp m

l mt qu trnh c s. Trng hp tng qut suy ra t trng hp cc qu trnh c s

bng php ly gii hn.

Mt h qu trc tip ca ng c It l: tch phn It Ft =t0 sdBs cng l mt qutrnh c bnh phng kh tch (nu gi s l s c bnh phng kh tch).

Bin phn bnh phng: Ta c ng thc sau cho bin phn bnh phng ca tch

phn It:

QVt0 (F) =

t0

|s|2ds (1.97)

Tng t nh ng c It, chng minh ca cng thc trn trong trng hp m t l

mt qu trnh c s tng i hin nhin v suy ra trc tip t bin phn bnh phng

ca chuyn ng Brown.Tnh cht martingale:


40/44


nh l 1.13. Tch phn It Ft = t

0

sdBs l mt qu trnh martingale (a phng),

c ngha lE (Ft|Ft) = Fs (1.98)

vi mi T > t.

Chng minh. Trong trng hp m l mt qu trnh s cp, th nh l trn kh hin

nhin, v E(Bt Bt Ft) = Bt Bt = 0 vi mi t, t > t. Trng hp tng qut suy ra

t trng hp ring ny bng cch ly gii hn trong nh ngha tch phn It.

Khng nh ngc li cng ng, v n c bit di tn gi nh l biu dinmartingale: Nu Mt l mt qu trnh martingale, lin tc, tng thch vi lc sigma-i

s sinh bi chuyn ng Brown Bt, v c bnh phng kh tch, th n vit c di

dng tch phn It.

Cng thc tch phn tng phn. Trong tch phn

tdBt, thnh phn Bt c gi l

integrator (bi ly tch phn). nh ngha tch phn It p dng ng khng nhng

ch cho integrator l chuyn ng Brown, m cn cho cc integrator tng qut hn, c

dng semi-martingale. Theo nh ngha, mt qu trnh ngu nhin Xt c gi l semi-

martingale nu n vit c di dng tng ca hai thnh phn, Xt = At + Mt, trong At c bin phn hu hn, cn Mt l martingale a phng (tc l tha mn tnh cht

martingale (1.98)). Tng t nh trng hp tch phn RiemannStieltjes c in, ta c

cng thc tch phn tng phn sau: nu Xt v Yt l hai semimartingale th

XtYt = X0Y0 +

t0

Xs dYs +

t0

Ys dXs + [X, Y]t, (1.99)

trong Xs l k hiu gii hn bn tri, tc l Xs = limrs Xr, v [X, Y]t l k hiu

qu trnh hip bin phn bnh phng ca X v Y, nh ngha nh sau

[X, Y]t = limmesh()0

ni=1

(Xai Xai1)(Yai Yai1), (1.100)

(vi gi s l gii hn tn ti), trong = {0 = a0 a1 . . . an = t} l k hiumt phn hoch ca on thng [0, t]. S khc nhau gia cng thc tch phn tng phn

cho tch phn Riemann-Stieltjes v cng thc tch phn tng phn cho tch phn It nm

chnh thnh phn hip bin phn bnh phng ny.

V d 1.7. Trong trng hp c bit, khi Yt = Xt l cng mt qu trnh ngu nhin, th[X, Y]t = [X, X]t chnh l qu trnh bin phn bnh phng ca Xt, v ta c cng thc


41/44

1.4. TCH PHN IT 41

sau:

t0

XsdX

s=

1

2(X2

t X2

0 [X, X]

t). (1.101)

Bi tp 1.11. Tnh cc tch phn It sau:

a)t0

B2sdBs

b)t0

BsdB2s

c)t0

exp(Bs)d(Bs + s)


42/44



43/44

Ti liu tham kho

[1] Robert Buchanan, An undergraduate introduction to financial mathematics, World

Scientific, 2006

[2] M. Capinski and T. Zastawniak, Mathematics for finance - An introduction to finan-

cial engineering, Springer, 2003.

[3] Kiryakos Chourdakis, Financial engineering - a brief introduction using the Matlab

system, 2008.

[4] , Davies, Glyn. A History of money from ancient times to the present day, 3rd. ed.

Cardiff: University of Wales Press, 2002. 720p.[5] Nguyn Tin Dng v c Thi, Nhp mn hin i Xc sut Thng k, 2010.

[6] R. Elliott and K. Kopp, Mathematics of financial markets, 2nd edition, Springer,

2005.

[7] S. Focardi and F. Fabozzi, The mathematics of financial modeling a nd investment

management, Wiley, 2004.

[8] Joel Greenblatt, You Can Be a Stock Market Genius: Uncover the Secret HidingPlaces of Stock Market Profits, 1997.

[9] Joel Greenblatt, The little book that beats the market, John Wiley & Sons, 2006.

[10] M. Harrison and P. Waldron, Mathematical economics and finance, 1998.

[11] John Hull, Options, futures, and other derivatives, 5th edition.

[12] I. Karatzas and S. Shreve, Brown motion and stochastic calculus, 2nd ed., 1991.

[13] Dalih Neftci, Principples of financial engineering, 2nd edition, Academic Press, 2008.

43


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44 TI LIU THAM KHO

[14] Stanley Pliska, Introduction to mathematical finance - discrete time models, 2001.

[15] Martin Pring, Technical analysis explained, 4th ed., 2002.

[16] Sheldon Ross, An introduction to mathematical finance, Cambridge University Press,

1999.

[17] Nguyn Duy Tin, Cc m hnh xc sut v ng dung, Phn III: Gii tch ngu

nhin, NXB i hc Quc gia H ni, 2005.

[18] P. Wilmott et al., The mathematics of financial derivatives - A student introduction,

1996.

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