Một số quá trình ngẫu nhiên có bước nhảy trong tài chính

I HC QUC GIA H NI TRNG I HC KHOA HC T NHIấN _ Hong Th Phng Tho MT S QU TRèNH NGU NHIấN Cể BC NHY D THO LUN N TIN S TON HC H Ni 2015 I HC QUC GIA H NI TRNG I HC KHOA HC T NHIấN _ Hong Th Phng Tho MT S QU TRèNH NGU NHIấN Cể BC NHY Chuyờn ngnh: Lý thuyt xỏc sut v thng kờ toỏn hc Mó s: 62460106 LUN N TIN S TON HC NGI HNG DN KHOA HC: PGS TS TRN HNG THAO H Ni - 2015 Li cam oan Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca riờng tụi Cỏc s liu, kt qu nờu lun ỏn l trung thc v cha tng c cụng b bt k cụng trỡnh no khỏc Nghiờn cu sinh Hong Th Phng Tho Li cm n Trong quỏ trỡnh hc nghiờn cu hon thnh c lun ỏn Tin s ny tụi ó nhn c rt nhiu s giỳp t cỏc thy cụ giỏo, bn bố ng nghip v gia ỡnh tụi Ngi u tiờn tụi mun gi li cm n chõn thnh nht l PGS TS Trn Hựng Thao, ngi Thy ó v ang hng dn, o to tụi nghiờn cu khoa hc rt nhit tỡnh Thy khụng ch giỳp tụi ngy cng cú thờm nim say mờ nghiờn cu khoa hc, thy cũn cho tụi rt nhiu li khuyờn cuc sng Tip theo tụi mun by t nhng li cm n ti cỏc thnh viờn B mụn Xỏc sut Thng kờ , Khoa Toỏn C Tin hc ó thng xuyờn giỳp tụi, cho tụi nhng li khuyờn chõn thnh quỏ trỡnh lm bn lun ỏn ny c bit tụi ó c tham gia xờ mi na ca B mụn Xỏc sut Thng kờ, qua xờ mi na tụi ó trau di, m rng thờm kin thc v cỏc thy b mụn ó luụn cho tụi nhng li nhn xột quý bỏu quỏ trỡnh hc v nghiờn cu ca mỡnh ng thi, tụi xin gi li cm n sõu sc n Ban giỏm c i hc Quc gia H Ni, Ban giỏm hiu Trng i hc Khoa hc t nhiờn, Ban ch nhim Khoa Toỏn-C-Tin hc, Phũng sau i hc ó to nhng iu kin thun li tụi nghiờn cu tt hn v giỳp tụi hon thnh th tc bo v lun ỏn Cui cựng, tụi xin gi li cỏm n n nhng ngi thõn gia ỡnh, h hng, bn bố thõn thit, nhng ngi ó luụn bờn cnh ng viờn giỳp tụi, tụi hon thnh lun ỏn ny H ni, 01/2015 NCS: Hong Th Phng Tho Mc lc Li cam oan Li cm n Bng ký hiu M u Cỏc kin thc chun b 1.1 12 Quỏ trỡnh im 12 1.1.1 Quỏ trỡnh im mt bin 13 1.1.2 Quỏ trỡnh im nhiu bin 13 1.1.3 Quỏ trỡnh Poisson ngu nhiờn kộp hay quỏ trỡnh Poisson cú iu kin 14 c trng Wantanabe 15 1.2 Quỏ trỡnh Poisson 16 1.3 Quỏ trỡnh Poisson phc hp 18 1.4 Tớch phõn ngu nhiờn i vi quỏ trỡnh cú bc nhy 21 1.5 Cụng thc Itụ i vi quỏ trỡnh cú bc nhy 22 1.1.4 1.6 1.5.1 Cụng thc Itụ i vi quỏ trỡnh Poisson tiờu chun 23 1.5.2 Cụng thc Itụ i vi quỏ trỡnh Poisson phc hp 23 1.5.3 Trong trng hp tng quỏt 24 Quỏ trỡnh ngu nhiờn phõn th 26 1.6.1 26 Chuyn ng Brown phõn th 1.6.2 Xp x L2 -semimartingale 27 1.6.3 Tớch phõn ngu nhiờn phõn th v phng trỡnh vi phõn ngu nhiờn phõn th 28 Quỏ trỡnh cú bc nhy v bi toỏn ri ro tớn dng 2.1 Mụ hỡnh cú bc nhy iu khin bi mt martingale Poisson 32 Phỏ sn ti thi im t cụng ty cú mt khon n L 33 Phỏ sn cú n khon n L1 , L2 , , Ln 34 Mụ hỡnh cú bc nhy iu khin bi mt chuyn ng Brown v mt quỏ trỡnh Poisson 36 2.2.1 Xỏc sut phỏ sn cụng ty cú mt khon n 38 2.2.2 Phỏ sn cụng ty cú nhiu khon n 39 Mụ hỡnh cú bc nhy iu khin bi mt chuyn ng Brown v mt quỏ trỡnh Poisson phc hp 42 2.3.1 Cụng ty cú mt khon n 44 2.3.2 Trng hp cụng ty cú nhiu khon n 47 2.1.1 2.1.2 2.2 2.3 Quỏ trỡnh cú bc nhy v quỏ trỡnh phõn th 3.1 Cỏc quỏ trỡnh phõn th cú bc nhy 3.1.1 55 55 Chuyn ng Brown phõn th hỡnh hc cú bc nhy 56 Quỏ trỡnh Ornstein-Uhlenbeck phõn th cú bc nhy 59 Phng trỡnh vi phõn ngu nhiờn phõn th cú bc nhy 61 c lng bin ng ngu nhiờn phõn th vi quan sỏt l quỏ trỡnh cú bc nhy 66 3.2.1 67 3.1.2 3.1.3 3.2 30 3.2.2 Xp x ngu nhiờn phõn th c lng Vt ,1 70 3.2.3 c lng Vt ,2 v Vt 73 3.2.4 3.2.5 S hi t ca Vt ti nghim Vt c lng bin ng Vt 74 75 Danh mc cỏc cụng trỡnh khoa hc ca tỏc gi liờn quan n lun ỏn 78 Ti liu tham kho 79 Bng ký hiu P- h.c.c S hi hu chc chn L2 (, F, P ) Tp hp cỏc lp tng ng cỏc hm bỡnh phng kh tớch () N (0, 1) L2 lim Chun khụng gian L2 (, F, P ) Hm Gamma Bin ngu nhiờn cú phõn phi chun tc S hi t L2 C(S) Khụng gian cỏc hm ngu nhiờn liờn tc trờn khụng gian S Khụng gian cỏc hm ngu nhiờn b chn trờn S Phn nguyờn ca x C b (S) [x] M u Mt quỏ trỡnh cú bc nhy l mt quỏ trỡnh ngu nhiờn m cỏc qu o ca nú b giỏn on bi cỏc bc nhy V mt lch s thỡ u tiờn, ngi ta nghiờn cu cỏc h ng lc ngu nhiờn iu khin bi chuyn ng Brown m li gii l cỏc quỏ trỡnh cú qu o liờn tc Tuy nhiờn cỏc ng dng thc t thỡ nhiu cỏc h ng lc y khụng phn ỏnh ỳng s thc nhng s kin quan sỏt c Thay vo ú ngi ta nhn thy cỏc quỏ trỡnh cú bc nhy ỏp ng c tt hn s mụ t cỏc hin tng ú Chng hn, cỏc quỏ trỡnh cú bc nhy úng vai trũ ht sc quan trng tt c cỏc lnh vc ti chớnh úng gúp cho s phỏt trin ca cỏc mụ hỡnh ngu nhiờn cú bc nhy phi k n nhng thnh tu ca lý thuyt Semimartingale v c nng lc tớnh toỏn hin i ca cụng ngh thụng tin Quỏ trỡnh cú bc nhy n gin nht l quỏ trỡnh cú mt bc nhy Gi T l mt thi im ngu nhiờn, thụng thng ú l mt thi im dng ng vi mt b lc (Ft , t 0) no ú Xt = 1{T t} , (1) quỏ trỡnh ny cú giỏ tr bng trc mt s kin no ú xy ti thi im T v bng sau ú Nú cng mụ t thi im phỏ sn ca mt cụng ty vic mụ hỡnh húa ri ro tớn dng Tip theo l cỏc quỏ trỡnh cú giỏ tr nguyờn v cú c bc nhy ch bng 1, gi l quỏ trỡnh m (Xt , t 0) ú l quỏ trỡnh mụ t s cỏc bin c xy khong thi gian t n t Quỏ trỡnh m in hỡnh l quỏ trỡnh Poisson (Nt , t 0), ú Nt cú phõn phi Poisson vi tham s t Ngi ta cng cú th mụ t quỏ trỡnh ú bng cỏch cho khong thi gian gia hai bc nhy l bin ngu nhiờn c lp cựng phõn b m vi tham s S m rng tip theo l cỏc quỏ trỡnh Poisson phc hp (Xt , t 0), tc l cỏc quỏ trỡnh vi gia s c lp, dng v cú c bc nhy khụng phi l na m l cỏc bin ngu nhiờn cú phõn b xỏc sut no ú Nt Yk , Xt = (2) k=1 ú (Y1 , Y2 , ) l dóy cỏc bin ngu nhiờn c lp cựng phõn phi Mt ng dng in hỡnh ca quỏ trỡnh Poisson phc hp l mụ t tng s tin m cụng ty bo him phi tr cho khỏch hng ti thi im t, ti thi im y s khỏch hng ũi tr bo him l bin ngu nhiờn cú phõn b Poisson Bờn cnh ú ngi ta cng chỳ ý n quỏ trỡnh i trng ca Xt , tc l quỏ trỡnh Xt E[Xt ] Nu phõn phi cú k vng hu hn thỡ vỡ Xt cú gia s c lp, dng nờn ta cú E[Xt ] = tE[X1 ] v ú ta cú biu din Xt = (Xt E[Xt ]) + tE[X1 ] (3) Quỏ trỡnh i trng (Xt E[Xt ]) l mt martingale nờn tng ca (3) l tng ca mt martingale v mt dch chuyn tuyn tớnh tE[X1 ] Biu din (3) trờn gi ý n mt nh ngha tng quỏt v quỏ trỡnh semimartingale Xt = X0 + Vt + Mt , (4) ú V = (Vt , t 0) l mt quỏ trỡnh thớch nghi, cdlg v cú bin phõn hu hn, cũn M = (Mt , t 0) l mt martingale a phng Cng cú nhng quỏ trỡnh khụng phi l semimartingale, mt vớ d quan trng ú l quỏ trỡnh chuyn ng Brown phõn th H thc (4) núi chung khụng phi l nht, nú s l nht vi Ti liu tham kho [1] Alũs E., Mazet O., and Nualart D (2000), "Stochastic calculus with respect to fractional Brownian motion with Hurst paramenter less than 21 , Stochastic Processes and Their Applications 86(1), pp 121139 [2] Berg T (2010), "From actual to risk-neutral default probabilities: Merton and Beyond", T he Journal of Credit Risk 6(1), pp 55-86 [3] Biagini F., Hu Y., ỉksendal B., Sulem A (2002), "A stochastic maximum principle for processes driven by a fractional Brownian motion", Stoch Proc Appl 100, pp 233-254 [4] Bielecki T., Jeanblan M and Rutkowski M (2009), Credit Risk Modeling, Center for Study of Insurance and Finance, Osaka University [5] Bystrom H (2007), "Merton for Dummies: A Flexible Way of Modelling Default Risk", Research Paper Series, 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