THE TERM STRUCTURE OF INTEREST RATES IN A MARKOV SETTING
|
|
- Emory Jones
- 5 years ago
- Views:
Transcription
1 THE TERM STRUCTURE OF INTEREST RATES IN A MARKOV SETTING Rober J. Ellio Haskayne School of Business Universiy of Calgary Calgary, Albera, Canada rellio@ucalgary.ca Craig A. Wilson College of Commerce Universiy of Saskachewan Saskaoon, Saskachewan, Canada cwilson@commerce.usask.ca May 8, 2003 Absrac An ineres rae model is described in which randomness in he shor-erm ineres rae is due enirely o a Markov chain. We model randomness hrough he mean-revering level raher han hrough he ineres rae direcly. The shor-erm ineres rae is modeled in a risk-neural seing as a coninuous process in coninuous ime. This allows he valuaion of ineres rae derivaives using he maringale approach. In paricular, a closed-form soluion is found for he value of a zero-coupon bond. We also show how o incorporae he iniial erm srucure o calibrae he model for arbirage-free bond prices. The model is shown o encapsulae several empirically observed feaures associaed wih ineres rae ime series. Key Words: Ineres Rae Modeling, Term Srucure, Derivaives, Markov Chain 1
2 1 Inroducion Curren models of he shor-erm ineres rae ofen involve reaing he shor rae as a diffusion or jump diffusion process in which he drif erm involves exponenial decay oward some value. The basic models of his ype are Vasiček [1] and Cox, Ingersoll and Ross [2], where he disincion beween hese wo ineres rae models ress wih he diffusion erm. The drif erm, (of boh models), ends o cause he shor rae process o decay exponenially owards a consan level. This feaure is responsible for he mean-revering propery exhibied by hese processes. An exension o hese models has come in he form of allowing he drif o incorporae exponenial decay oward a manifold, raher han a consan. This is known as he Hull and Whie [3] model, and i allows he shor rae process he endency o follow he iniial erm srucure of ineres raes. This is an imporan exension, because wih a judicious choice of he manifold, he iniial erm srucure prediced by he model can exacly mach he exising erm srucure. In general, his canno be done wih a consan mean-revering level. Alhough here are many oher exensions o he basic models incorporaing sochasic volailiy, non-linear drif (so decay is no longer exponenial), and jumps, for example he Hull-Whie exension is he mos applicable o our sudy. The Hull-Whie model is described under he risk-neural probabiliy by he sochasic differenial equaion dr = (){ r() r } d + σ()r ρ dw, (1) where r represens he shor-erm, coninuously compounded ineres rae, and {W } is a Brownian moion under he risk-neural probabiliy. The parameer ρ akes one of he wo values 0 or 1/2, depending on wheher i exends he Vasiček or Cox-Ingersoll-Ross model. The parameer funcions (), r(), and σ() exend he basic models, in which hese parameers are jus consans. The randomness in his model comes from he Brownian moion, and for he exended Vasiček model when ρ = 0, i can be inerpreed as adding whie noise o he shor rae. For he exended Cox-Ingersoll-Ross model he noise is muliplicaive, bu i is sill applied direcly o he shor rae process. 2
3 The main problem wih his model is in he way i handles he cyclical naure of ineres raes. A ime series of ineres raes ends o appear cyclical because he supply and demand for money is closely relaed o income growh, which flucuaes wih he business cycle. This has implicaions for real (adjused for inflaion) ineres raes. For example, a a business cycle peak shor-erm raes should be rising and a a rough raes should be falling. This also has implicaions for he slope of he erm srucure i should be seeper a a peak and flaer a a rough. Roma and Torous [4] find ha his propery of real ineres raes canno be explained by a simple addiive noise ype model, such as Vasiček. The Hull-Whie exension can provide a correcion for his problem o a degree, bu since he parameer funcions are deerminisic, i implies ha he business cycle effecs are known wih cerainy, which does no allow for he possible variaion in lengh and inensiy. In addiion, when he cenral bank arges a consan rae of inflaion, his flucuaion is ransferred o nominal ineres raes, so he same characerisics could apply o hem. We approach his problem by modeling he mean-revering level direcly as a random process, and have he shor rae chase he mean-revering level in a linear drif ype model. The mean-revering level is assumed o follow a finie sae, coninuous ime Markov chain. The swiching of he Markov chain o differen levels produces a cyclical paern in he shor rae ha is consisen wih he above effec, and he randomness inheren in he Markov chain prevens he business cycle lenghs and inensiies from being compleely predicable. The remainder of his aricle is planned ou as follows. Secion 2 discusses our model for he shor rae. We look a deails of he Markov chain and how i relaes o he shor rae. In Secion 3 we use our shor rae model o value a zero-coupon bond. A closed-form soluion is provided. This also provides a closed-form soluion for longer-erm ineres raes by invering he bond price. We also show ha he model can be calibraed using he iniial erm srucure of zero-coupon bond prices. Finally Secion 4 concludes, and some mahemaical deails are included in he appendix. 3
4 2 The Model We model uncerainy in he economy wih a probabiliy space denoed (Ω, F, P ), where Ω is a se of possible saes of he economy ha is assumed o be large enough o suppor he Markov chain defined below, F is a sigma field over subses of Ω ha is assumed o be large enough o suppor he filraion generaed by he Markov chain, and P is a risk-neural probabiliy measure. Several properies of he Markov chain are described in he following subsecion, and afer ha we describe how he shor-erm ineres rae is o be modeled. 2.1 The Markov Chain We denoe by {X } a homogeneous, righ coninuous Markov chain aking values in he sae space of uni vecors S = {e 1,..., e N }, where e i is a vecor in R N wih 1 in he i h coordinae and 0 elsewhere. We denoe he righ coninuous compleion of he filraion generaed by he Markov chain as {F X }. The filraion models he informaion abou he realised sae of he economy, ω Ω, ha is revealed by observing he pah of he Markov chain. The Markov chain, X, is assumed o change saes according o a ransiion inensiy marix A, in which he columns sum o zero and he offdiagonal enries are non-negaive. Therefore he risk-neural probabiliy of going from sae e i o e j beween ime and + s is he enry from he j h row and he i h column of he marix exponenial e As. We also assume ha he ransiion inensiy marix, A, is symmeric. This is a resricive assumpion because i implies ha he risk-neural probabiliy of swiching from one sae o anoher is equal o he probabiliy of swiching back over any equivalen ime inerval. We mainain his assumpion because i allows for convenien closed-form soluions of he zero-coupon bond price and oher quaniies of ineres. Our specificaion of he sae space allows he sochasic process X o have he following semi-maringale represenaion X = 0 AX s ds + M, (2) 4
5 where he vecor process {M } is a righ coninuous, square inegrable, zeromean maringale wih respec o he filraion {F X } and he measure P, (see Ellio [5]). From Equaion 2 we can wrie he dynamics of X as dx = AX d + dm, (3) and by applying Fubini s heorem he condiional expecaion of X +s given is F X E[X +s F X ] = X + s AE[X u X ] du = e As X. (4) This condiional expecaion is he condiional probabiliy disribuion vecor ha he Markov chain will be in each sae. 2.2 The Shor-Term Ineres Rae The shor-erm ineres rae is modeled by he sochasic process denoed by {r }, which is assumed o have dynamics given by he following ordinary differenial equaion dr = (){ r(, X ) r } d. (5) The quaniy r(, X ) is he level ha he process ends oward, and (), which is assumed o be a posiive valued funcion, is he rae a which he mean-revering level is approached. In general, hese dynamics do no describe a saionary process, because he parameers may vary wih ime, (his is also rue of he Hull-Whie model); however, he process can be made saionary by requiring ha he parameers depend on ime only hrough he Markov chain. Jus as in he Hull-Whie model, he ime dependence is allowed so ha he iniial erm srucure can be mached exacly. In paricular we can choose how r(, e i ) follows ime in each sae in order o mach he iniial erm srucure of ineres raes and () so ha he erm srucure of volailiies is mached. The main difference beween he sochasic process described in Equaion 5 and he Hull-Whie dynamics of Equaion 1 is ha insead of incorporaing noise ino he shor rae using a diffusion erm, randomness eners he shor rae hrough he mean-revering level. This is a subsanial change because i means ha ineres rae sample pah is differeniable. We use a 5
6 Markov chain o generae randomness raher han a Brownian moion because i seems inuiively beer suied o deal wih he business cycle effec. The soluion o Equaion 5 saring a ime s is obained by variaion of consans, ( ){ ( u ) } r = exp (u) du r s + exp (v) dv (u) r(u, X u ) du. (6) s s s We wrie he filraion generaed by his sochasic process as {F r }, and by noicing ha r only depends on X u for u, we can see ha {r } is adaped o {F X }. Noice ha because he mean-revering rae, (), does no depend on he Markov chain, he exponenial erm in Equaion 6 is a deerminisic quaniy. We will make use of his observaion for valuing zero-coupon bonds under he model. The expeced fuure shor rae can be easily deermined from Equaions 4 and 6. Applying Tonelli s heorem o ake he expecaion operaor hrough he inegral sign and wriing r(u, X u ) = r(u) T X u, where r(u) = ( r(u, e 1 ),..., r(u, e n )) T, gives ( E[r Fs r ] = exp s { r s + ) (u) du (7) ( u ) } exp (v) dv (u) r(u) T e A(u s) E[X s Fs r ] du. s s We analyse his expression in he conex of a long-erm mean-revering propery in Subsecion 3.2 in he case of consan parameer funcions. 3 Zero-Coupon Bonds This secion provides he deails of deermining he value of a defaul free zero-coupon bond ha pays $1 wih cerainy a a fixed mauriy ime, T, wih no oher cash flows. We denoe he value a ime of a zero-coupon bond mauring a ime T by B(, T ), and under he risk-neural measure, is value is deermined by solving [ ( T B(, T ) = E exp r u du ) F r ]. (8) 6
7 We firs proceed o solve for his quaniy in he general case, and hen we specialise he general resuls o he paricular case when he parameer funcions do no vary direcly wih ime. This special case will be imporan when we calibrae he model o he iniial erm srucure of zero-coupon bond prices. 3.1 The General Case To solve for he quaniy in Equaion 8, we mus firs deermine he inegral of he shor rae process. This can be done by inegraing Equaion 6; however i will also be convenien o have he erm r(, X ) in he ouer inegral he reason for his is made clear in he appendix where Resul 1 is proven. We can accomplish his by inerchanging he order of inegraion according o Tonelli s heorem. This gives T r u du = r T where { T β(, u, T, X u ) = u ( u ) T exp (v) dv du + β(, u, T, X u ) du, (9) ( v ) } ( u ) exp (w) dw dv exp (w) dw (u) r(u, X u ). (10) The firs erm in Equaion 9 is F r -measurable and i can be deermined a leas by numerical inegraion i can be deermined analyically for some special cases, in paricular when () is consan. This means our ask in solving Equaion 8 is reduced o finding E[exp( T β(, u, T, X u) du) F r ]. We find his value in hree seps. Firs we consruc a vecor sochasic process by pos muliplying he exponenial erm by he Markov chain, X, hen we find he expeced value of ha vecor process by making use of he semi-maringale represenaion in Equaions 2 and 3, and from ha we can deermine he expeced value of he exponenial erm as desired. The soluion we find is in erms of a linear sysem of ordinary differenial equaions, bu he approach is somewha echnical, so we summarise his in he following resul and leave he deails o he appendix. Resul 1 If P is a risk-neural probabiliy and he risk free shor-erm ineres rae is characerised by Equaion 5, hen he value a ime of a 7
8 zero-coupon bond paying $1 a ime T is T ( u ) } B(, T ) = exp { r exp (v) dv du 1 T Φ(, T )E[X F r ], (11) where 1 is an N-dimensional column vecor wih 1 in each enry, and Φ(, v) = diag [ e v β(,u,t,e 1)du,..., e v β(,u,t,e N )du ] Qdiag[e λ 1(v ),..., e λ N (v ) ]Q 1 (12) is he fundamenal marix saring a ime for he N-dimensional linear sysem of ordinary differenial equaions, y (u) = {A diag[β(, u, T, e 1 ),..., β(, u, T, e N )]}y(u) (13) evaluaed a ime u = T, and β(, u, T, e i ) is given in Equaion 10. Here Q is an N N marix whose columns are eigenvecors of he ransiion inensiy marix A, and he λ i s are he corresponding eigenvalues. From Resul 1 we can see ha our model of he shor-rae leads o an affine erm srucure. To deermine longer-erm ineres raes we jus solve for he yield of he zero-coupon bonds, R(, T ) = ln{b(, T )}/(T ). 3.2 Consan Parameers An imporan special case occurs when he parameer funcions do no vary direcly wih ime. As menioned earlier his resuls in a saionary shor rae process, which is an imporan feaure for analysing ineres rae ime series, paricularly for real ineres raes ha have been adjused for inflaion. This secion replaces he main resuls from he general case wih inegraion performed when possible. We rewrie he shor rae dynamics as we rewrie he soluion o Equaion 14 as dr = { r(x ) r } d, (14) r = e {r ( s) s + e (u s) r(x } u ) du, (15) s 8
9 and he funcion β defined in Equaion 10 akes he form β(, u, T, X u ) = β(t u, X u ) = {1 e (T u) } r(x u ). (16) Taking he condiional expecaion of Equaion 15 gives { } E[r Fs r ] = e ( s) r s + r T e (u s) e A(u s) due[x s Fs r ]. (17) Noe ha he above inegral exiss in any case; however if he marix A+I is inverible, (which i will be for all choices of alpha excep N paricular values ha depend on he ransiion rae marix A), we can wrie he soluion as follows E[r F r s ] = e ( s) {r s + r T (A + I) 1 {e ( s) e A( s) I}E[X s F r s ]}. (18) Furhermore, if we assume ha he Markov chain is recurren and ergodic, hen aking he limi gives s lim E[r F r s ] = r T (A + I) 1 π, (19) where π is he limiing probabiliy disribuion vecor for he Markov chain. Therefore in he consan parameer case, he expeced ineres rae converges o a consan ha is independen of boh he iniial ineres rae and he iniial sae of he Markov chain. where Applying Resul 1 gives a bond price of 1 e B(, T ) = exp { r (T ) } 1 T Φ(, T )E[X F r ], (20) [ Φ(, v) = diag exp {( v + e (T v) (T ) }] e ) r i Qdiag[e λ1(v ),..., e λ N (v ) ]Q 1. (21) Here again Q is an N N marix whose columns are eigenvecors of A, he λ i s are he corresponding eigenvalues and r i = r(e i ). 9
10 3.3 Maching he Iniial Term Srucure Since he erm srucure of zero-coupon bond prices can be exacly mached by a one dimensional sysem, such as he Hull-Whie model, i seems clear ha he general form of our model will possess oo much freedom o have he funcions r(, e i ) be uniquely deermined by he iniial erm srucure. In fac, whenever he Markov chain and he associaed funcion r(, X ) have a non-rivial sae space, (so he Markov chain has more han one sae and r(, e i ) r(, e j ) for some saes e i and e j and for all in some subse of ime wih posiive Lebesgue measure), here is very lile srucure imposed on he mean-revering manifold. Inuiively, his is because whaever one r(, e i ) funcion is, we can choose anoher o cancel i ou on average when calculaing zero-coupon bond prices. Because of his freedom, in order o make use of he iniial erm srucure as a useful inpu o he model, we have o specify some relaionship beween mean-revering funcions associaed wih differen saes. One simple possibiliy for accomplishing his is o have he various mean-revering funcions shifed from each oher by consan amouns. This is he approach we presen here. Suppose ha he mean-revering rae () is a consan as in he previous subsecion, and ha he mean-revering level is he sum of wo quaniies, r(, X ) = r() + r T X. (22) Here he r wihou an argumen is a column vecor of consans, and he r() is a real-valued funcion. We assume ha he consans and r i are known, bu he funcion r() is no. The goal is o find r() as a funcion of he oher parameers and he curren erm srucure. We can wrie he funcion β from Equaion 10 as β(, u, T, X u ) = {1 e (T u) } r(u) + {1 e (T u) } r T X u. (23) This means ha he erms from he fundamenal marix Φ(, v) in Equaion 12, e v β(,u,t,e i) = e v {1 e (T u) } r(u) du e v {1 e (T u) } r i du, (24) ake he form of he produc of wo exponenial funcions, he firs of which is independen of e i, and he second involves only he consan parameers. 10
11 This means ha we can rea he firs erm as a scalar muliple and pull i ou of he diagonal marix so ha he fundamenal marix akes he form Φ(, v) = e v {1 e (T u) } r(u) du Φ cons (, v), (25) where Φ cons (, v) is he fundamenal marix for he consan case described in Equaion 21. Therefore he bond price in his siuaion can be wrien as where B(, T ) = A(, T )B cons (, T ), (26) { T } A(, T ) = exp (1 e (T u) ) r(u) du, (27) and B cons (, T ) is he bond price calculaed from he consan parameer case given in Equaion 20. Since he consan parameers are presumed o be known, he value B cons (, T ) is known, so o find he bond price, we only have o deermine A(, T ). We do his by examining wo quaniies associaed wih he iniial erm srucure he raio of wo zero-coupon bonds and he slope of he erm srucure. We leave he derivaion o he appendix, and jus quoe he resul. Resul 2 If P is a risk-neural probabiliy and he risk free shor-erm ineres rae is characerised by Equaion 5, where () = is a consan and r(, X ) is as given in Equaion 22, hen he value a ime of a zero-coupon bond paying $1 a ime T is given by Equaion 26, where ln{a(, T )} = ln { B(0, T ) B(0, ) } { B cons (0, T )} ln B cons (0, ) ) 1 e (T { ln{b(0, )} B cons (, T ) is given in Equaion 20, and ln{b cons (0, )} ln{b cons(0, )} }, { = r 0 e 1 + B cons (0, ) exp 1 e } r 0 1 T {A diag[(1 e ) r i ]}Φ cons (0, )E[X 0 F r 0 ], where Φ cons (0, ) is given in Equaion 21 wih T replaced by. 11 (28) (29)
12 From Equaion 27 he funcion r() can be deermined quie easily by differeniaion r(t ) = 1 A(, T ) A(, T ) T 1 T { 1 A(, T ) A(, T )}, (30) T for any [0, T ], and where A(, T ) is obained from Resul 2. As a cavea o his procedure, we should menion ha he well-known problem of inerpolaing erm srucure observaions, (see Jordan [6] for a brief overview of he problem), can cause considerable error in esimaing r() because of he insabiliy of he differeniaion operaion. However, a discussion of his issue is bes lef o an empirical es of he model. 4 Conclusion We modeled he shor-erm ineres rae using a mean-revering equaion wih a random mean-revering level. A Markov chain is used o model he randomness in he mean-revering level. We argue ha his model can explain a business cycle effec in he shor rae ha encompasses boh regulariy in he business cycle and randomness in boh he duraion and severiy of business cycles. From his shor-erm ineres rae model we deermine he value of zero-coupon bonds and we show how o calibrae he model using he iniial erm srucure o obain arbirage-free bond prices. A Appendix A.1 Derivaion of he Zero-Coupon Bond Price According o he discussion following Equaions 9 and 10 we need only deermine E[Z,T F r ], where ( T ) Z,T = exp β(, u, T, X u ) du, (31) and he funcion β is defined in Equaion 10. Now for T fixed, we define a new sochasic process ( v ) Z,T (v) = exp β(, u, T, X u ) du (32) 12
13 for v T. Thus Z,T = Z,T (T ), and since T eners he above expression in a deerminisic way, Z,T (v) is F X v -measurable. The dynamics of Z,T (v) are given by dz,t (v) = β(, v, T, X v )Z,T (v) dv. (33) Applying Iô s inegraion by pars for general semi-maringales and subsiuing he dynamics from Equaions 3 and 33 gives Z,T (v)x v Z,T ()X = = v v X u dz,t (u) + β(, u, T, X u )Z,T (u)x u du + + [Z,T ( ), X] v, v v Z,T (u ) dx u (34) Z,T (u ){AX u du + dm u } where f(u ) is he lef limi of f as u, and he square bracke erm is he general quadraic covariaion. Equaion 34 can be simplified in a number of ways. Firs, because he Markov chain akes values as uni vecors, he funcion β can be wrien in he form β(, u, T ) T X u, where β(, u, T ) = (β(, u, T, e 1 ),..., β(, u, T, e N )) T. Also, Z,T (v) is differeniable in v, so i is of finie variaion. This means ha he square bracke process jus adds up he producs of he jumps of Z and X. Bu Z,T (v) is coninuous, so i has no jumps. Thus he square bracke erm is idenically zero and he lef limi of Z akes is value, Z,T (u ) = Z,T (u) almos surely. Since X is righ coninuous wih lef limis exising, i has a mos a counable number of disconinuiies, all of he jump ype, so X u = X u for Lebesgue-almos every u [, T ], and he lef limi can be replaced by is righ limi inside he inegral wihou affecing he value. Finally, we can wrie ( β(, u, T ) T X u )X u = diag[ β(, u, T )]X u, again because of our choice of he Markov chain s sae space. (This is why i was imporan for us o apply Tonelli s heorem in Equaion 9 o ge β as a funcion of X u raher han a funcion of he inegral of X u.) This means ha Equaion 34 akes he form Z,T (v)x v Z,T ()X = v (A diag[ β(, u, T )])Z,T (u)x u du+ v Z,T (u) dm u. (35) Because Z,T (v) is lef coninuous and adaped, hence a predicable process, and also bounded on [, T ], and M is a square inegrable maringale, he 13
14 sochasic inegral of Z wih respec o dm is a zero-mean maringale wih respec o P and he filraion {F X }. Thus he condiional expecaion of he hird erm of Equaion 35 given he informaion F X is zero. Taking condiional expecaion and applying Fubini s heorem o ake he expecaion hrough he inegral sign gives E[Z,T (v)x v F X ] Z,T ()X = v (A diag[ β(, u, T )])E[Z,T (u)x u F X ] du. (36) This is he inegral version of he linear sysem of ordinary differenial equaions described in Equaion 13. Since he coefficien marix, A diag[ β(, u, T )], saisfies he Lappo- Danilevskiĭ condiion of being muliplicaively commuaive wih is own inegral, (a consequence of he ransiion inensiy marix being consan and symmeric), he fundamenal marix of his sysem is jus he marix exponenial of he inegral of he coefficien marix. (For more deails on his resul see Adrianova [7] 4.2 for example.) Therefore he fundamenal marix, call i Φ(, v), akes he form described in Equaion 12, which is clearly deerminisic. The soluion o Equaion 36 is E[Z,T (v)x v F X ] = Φ(, v)z,t ()X. (37) Now because he Markov chain sae space consiss of uni vecors, Z,T (v) = 1 T Z,T (v)x v ; also F r F X and Z,T () = 1, so E[Z,T (T ) F r ] = 1 T Φ(, T )E[X F r ], (38) and he zero-coupon bond value is as given in Equaion 11. This concludes he proof of Resul 1. A.2 Maching he Iniial Term Srucure In his subsecion we prove ha A(, T ) has he form given in Equaion 28. Firs, from Equaions 26 and 27 ln { B(0, T ) B(0, ) } T = (1 e (T u) ) r(u) du (1 e ( u) ) r(u) du + ln { B cons (0, T ) B cons (0, ) 14 }
15 and Thus ln{b(0, )} T = (1 e (T u) ) r(u) du (39) {1 e (T ) } ln{a(, T )} = ln { B(0, T ) B(0, ) 0 e ( u) r(u) du + ln { B cons (0, T ) B cons (0, ) = 0 (1 e ( u) ) r(u) du + ln{b cons(0, )} = e ( u) r(u) du + ln{b cons(0, )}. (40) 0 } { B cons (0, T )} ln B cons (0, ) ) 1 e (T { ln{b(0, )} ln{b cons(0, )} }. }, (41) Using Equaion 20 we can see ha he derivaive of he consan parameer bond price wih respec o he mauriy is B cons (0, ) From Equaion 21 we find ha Φ cons (0, ) = r 0 e 1 e exp { r 0 1 e + exp { r } 0 [ = diag exp [ +diag exp {( + 1 e {( + 1 e } 1 T Φ cons (0, )E[X 0 F r 0 ] 1 T Φ cons(0, ) E[X 0 F r 0 ]. (42) ) r i }{ 1 + e } r i ] Qdiag[e λ i ]Q 1 ) r i }] Qdiag[e λ i λ i ]Q 1. (43) Since a diagonal marix wih produc enries is equal o he produc of he diagonal marices, he marix Qdiag[λ i ] = AQ, and A is symmeric, he above expression akes he following form Φ cons (0, ) = {A diag[(1 e ) r i ]}Φ cons (0, ). (44) 15
16 Puing his quaniy ino he consan parameer bond price derivaive gives B cons (0, ) = r 0 e 1 e B cons (0, ) + exp { r } 0 (45) 1 T {A diag[(1 e ) r i ]}Φ cons (0, )E[X 0 F0 r ] so ln{b cons (0, )} { = r 0 e 1 + B cons (0, ) exp 1 e } r 0 1 T {A diag[(1 e ) r i ]}Φ cons (0, )E[X 0 F r 0 ] (46) and rearranging hese equaions shows ha A(, T ) is given as in Equaion 28. This ends he proof of Resul 2. References [1] O. Vasiček, An equilibrium characerizaion of he erm srucure, Journal of Financial Economics, 5, 1977, [2] J. C. Cox, J. E. Ingersoll and S. A. Ross, A heory of he erm srucure of ineres raes, Economerica, 53(2), 1985, [3] J. Hull and A. Whie, Pricing ineres-rae derivaive securiies, Review of Financial Sudies, 3(4), 1990, [4] A. Roma and W. Torous, The cyclical behavior of ineres raes, Journal of Finance, 52(4), 1997, [5] R. J. Ellio, New finie dimensional filers and smoohers for noisily observed Markov chains, IEEE Transacions on Informaion Theory, 39(01), 1993, [6] J. V. Jordan, Term srucure modeling using exponenial splines: Discussion, Journal of Finance, 37(2), 1982, [7] L. Ya. Adrianova, Inroducion o Linear Sysems of Differenial Equaions (Rhode Island, American Mahemaical Sociey, 1995). 16
10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationCash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationThe expectation value of the field operator.
The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationMath Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.
Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need
More information13.3 Term structure models
13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationUnit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationDiebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationModeling Economic Time Series with Stochastic Linear Difference Equations
A. Thiemer, SLDG.mcd, 6..6 FH-Kiel Universiy of Applied Sciences Prof. Dr. Andreas Thiemer e-mail: andreas.hiemer@fh-kiel.de Modeling Economic Time Series wih Sochasic Linear Difference Equaions Summary:
More informationAn Introduction to Backward Stochastic Differential Equations (BSDEs) PIMS Summer School 2016 in Mathematical Finance.
1 An Inroducion o Backward Sochasic Differenial Equaions (BSDEs) PIMS Summer School 2016 in Mahemaical Finance June 25, 2016 Chrisoph Frei cfrei@ualbera.ca This inroducion is based on Touzi [14], Bouchard
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationWhat Ties Return Volatilities to Price Valuations and Fundamentals? On-Line Appendix
Wha Ties Reurn Volailiies o Price Valuaions and Fundamenals? On-Line Appendix Alexander David Haskayne School of Business, Universiy of Calgary Piero Veronesi Universiy of Chicago Booh School of Business,
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationE β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.
Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationIntroduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.
Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationLecture 10: The Poincaré Inequality in Euclidean space
Deparmens of Mahemaics Monana Sae Universiy Fall 215 Prof. Kevin Wildrick n inroducion o non-smooh analysis and geomery Lecure 1: The Poincaré Inequaliy in Euclidean space 1. Wha is he Poincaré inequaliy?
More informationA Dynamic Model of Economic Fluctuations
CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationTesting for a Single Factor Model in the Multivariate State Space Framework
esing for a Single Facor Model in he Mulivariae Sae Space Framework Chen C.-Y. M. Chiba and M. Kobayashi Inernaional Graduae School of Social Sciences Yokohama Naional Universiy Japan Faculy of Economics
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationUtility maximization in incomplete markets
Uiliy maximizaion in incomplee markes Marcel Ladkau 27.1.29 Conens 1 Inroducion and general seings 2 1.1 Marke model....................................... 2 1.2 Trading sraegy.....................................
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More informationf(s)dw Solution 1. Approximate f by piece-wise constant left-continuous non-random functions f n such that (f(s) f n (s)) 2 ds 0.
Advanced Financial Models Example shee 3 - Michaelmas 217 Michael Tehranchi Problem 1. Le f : [, R be a coninuous (non-random funcion and W a Brownian moion, and le σ 2 = f(s 2 ds and assume σ 2
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationProblem Set on Differential Equations
Problem Se on Differenial Equaions 1. Solve he following differenial equaions (a) x () = e x (), x () = 3/ 4. (b) x () = e x (), x (1) =. (c) xe () = + (1 x ()) e, x () =.. (An asse marke model). Le p()
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More information4.1 - Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
More informationLECTURE 1: GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS
LECTURE : GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS We will work wih a coninuous ime reversible Markov chain X on a finie conneced sae space, wih generaor Lf(x = y q x,yf(y. (Recall ha q
More informationStochastic Modelling in Finance - Solutions to sheet 8
Sochasic Modelling in Finance - Soluions o shee 8 8.1 The price of a defaulable asse can be modeled as ds S = µ d + σ dw dn where µ, σ are consans, (W ) is a sandard Brownian moion and (N ) is a one jump
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationT L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More informationStability and Bifurcation in a Neural Network Model with Two Delays
Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy
More informationThe Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite
American Journal of Operaions Research, 08, 8, 8-9 hp://wwwscirporg/journal/ajor ISSN Online: 60-8849 ISSN Prin: 60-8830 The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationStochastic Model for Cancer Cell Growth through Single Forward Mutation
Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle 31 5-1-2017 Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationThe Strong Law of Large Numbers
Lecure 9 The Srong Law of Large Numbers Reading: Grimme-Sirzaker 7.2; David Williams Probabiliy wih Maringales 7.2 Furher reading: Grimme-Sirzaker 7.1, 7.3-7.5 Wih he Convergence Theorem (Theorem 54) and
More informationt 2 B F x,t n dsdt t u x,t dxdt
Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion.
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationVectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1
Vecorauoregressive Model and Coinegraion Analysis Par V Time Series Analysis Dr. Sevap Kesel 1 Vecorauoregression Vecor auoregression (VAR) is an economeric model used o capure he evoluion and he inerdependencies
More informationRecursive Least-Squares Fixed-Interval Smoother Using Covariance Information based on Innovation Approach in Linear Continuous Stochastic Systems
8 Froniers in Signal Processing, Vol. 1, No. 1, July 217 hps://dx.doi.org/1.2266/fsp.217.112 Recursive Leas-Squares Fixed-Inerval Smooher Using Covariance Informaion based on Innovaion Approach in Linear
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationChapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws
Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationChapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
More informationTHE MYSTERY OF STOCHASTIC MECHANICS. Edward Nelson Department of Mathematics Princeton University
THE MYSTERY OF STOCHASTIC MECHANICS Edward Nelson Deparmen of Mahemaics Princeon Universiy 1 Classical Hamilon-Jacobi heory N paricles of various masses on a Euclidean space. Incorporae he masses in he
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationChapter 14 Wiener Processes and Itô s Lemma. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull
Chaper 14 Wiener Processes and Iô s Lemma Copyrigh John C. Hull 014 1 Sochasic Processes! Describes he way in which a variable such as a sock price, exchange rae or ineres rae changes hrough ime! Incorporaes
More informationEssential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems
Essenial Microeconomics -- 6.5: OPIMAL CONROL Consider he following class of opimizaion problems Max{ U( k, x) + U+ ( k+ ) k+ k F( k, x)}. { x, k+ } = In he language of conrol heory, he vecor k is he vecor
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationThe Brock-Mirman Stochastic Growth Model
c December 3, 208, Chrisopher D. Carroll BrockMirman The Brock-Mirman Sochasic Growh Model Brock and Mirman (972) provided he firs opimizing growh model wih unpredicable (sochasic) shocks. The social planner
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More information12: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME. Σ j =
1: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME Moving Averages Recall ha a whie noise process is a series { } = having variance σ. The whie noise process has specral densiy f (λ) = of
More informationThis document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC
his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,
More information( ) = b n ( t) n " (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2.
Andrei Tokmakoff, MIT Deparmen of Chemisry, 3/14/007-6.4 PERTURBATION THEORY Given a Hamilonian H = H 0 + V where we know he eigenkes for H 0 : H 0 n = E n n, we can calculae he evoluion of he wavefuncion
More informationOrdinary Differential Equations
Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationBias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé
Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070
More informationEmpirical Process Theory
Empirical Process heory 4.384 ime Series Analysis, Fall 27 Reciaion by Paul Schrimpf Supplemenary o lecures given by Anna Mikusheva Ocober 7, 28 Reciaion 7 Empirical Process heory Le x be a real-valued
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XI Control of Stochastic Systems - P.R. Kumar
CONROL OF SOCHASIC SYSEMS P.R. Kumar Deparmen of Elecrical and Compuer Engineering, and Coordinaed Science Laboraory, Universiy of Illinois, Urbana-Champaign, USA. Keywords: Markov chains, ransiion probabiliies,
More informationACE 562 Fall Lecture 4: Simple Linear Regression Model: Specification and Estimation. by Professor Scott H. Irwin
ACE 56 Fall 005 Lecure 4: Simple Linear Regression Model: Specificaion and Esimaion by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Simple Regression: Economic and Saisical Model
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More information) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha
More information