Heavy Tails of Discounted Aggregate Claims in the Continuous-time Renewal Model
|
|
- Alyson Merry Peters
- 5 years ago
- Views:
Transcription
1 Heavy Tails of Discouned Aggregae Claims in he Coninuous-ime Renewal Model Qihe Tang Deparmen of Saisics and Acuarial Science The Universiy of Iowa 24 Schae er Hall, Iowa Ciy, IA 52242, USA February 8, 27 Absrac We sudy he ail behavior of discouned aggregae claims in a coninuous-ime renewal model. For he case of Pareo-ype claims, we esablish a ail asympoic formula, which holds uniformly in ime. Keywords: Asympoics, exended regular variaion, renewal process, uniformiy. Inroducion and he Main Resul Consider a coninuous-ime renewal model, in which claim sizes X k, k = ; 2; : : :, consiue a sequence of independen, idenically disribued (i.i.d.), and nonnegaive random variables wih common disribuion F, while heir arrival imes k, k = ; 2; : : :, consiue a renewal couning process N = # fk = ; 2; : : : : k g ; : (.) We assume ha fx k ; k = ; 2; : : :g and fn ; g are muually independen. To avoid rivialiy, we menion ha X and are no degenerae a. We allow o possibly have a posiive probabiliy a no for pracical usefulness bu for heoreical compleeness. Suppose ha here is a consan ineres force >. Tha is o say, afer ime one dollar becomes e dollars. Then, he aggregae claims form ino a sochasic process of he form A () = X k e ( k) ( k ); ; where for an even E he symbol E denoes is indicaor funcion. Since A () almos surely as, we insead sudy he ail behavior of he discouned process D () = X k e k ( k ); : (.2)
2 We shall derive for he ail probabiliy of D (),, an asympoic formula, which holds uniformly for all for which he renewal funcion = EN = Pr ( k ) is posiive. For his purpose, de ne = f : > g. Wih = inff : > g = inff : Pr ( ) > g, i is clear ha [; ] if Pr ( = = ) > (.3) (; ] if Pr ( = ) =. We shall assume ha he disribuion F on [; ) is exended-regularly-varying ailed, hence heavy ailed. Tha is o say, F (x) = F (x) > holds for all x and here are some consans and, < <, such ha v lim inf x F (vx) F (x) lim sup x F (vx) F (x) v for all v : (.4) We use F 2 ERV( ; ) o signify he regulariy propery in (.4). The class ERV is he union of all classes ERV( ; ) over he range < <. This class has been used o he sudy of precise large deviaions by many people since he work of Klüppelberg and Mikosch (997). I is well known ha ERV is a subclass of he class S of subexponenial disribuions; see Theorem of Goldie (978). The subexponenialiy of a disribuion F is characerized by he relaions F (x) > for all x and F F (x) lim x F (x) Clearly, he class ERV covers he famous class R of disribuions wih regularly-varying ails characerized by he relaions F (x) > for all x and = 2: F (vx) lim x F (x) = v for some > and all v : (.5) I is usually easier o handle disribuions from he class R because of he well-developed Karamaa heory. Alhough he class ERV is marginally larger han he class R, we expec ha asympoic resuls for he ERV case provide more insigh o he sudy in he subexponenial case. For more deails of heavy-ailed disribuions, he reader is referred o Bingham e al. (987) and Embrechs e al. (997). Hereafer, all limi relaionships are for x unless saed oherwise. For wo posiive funcions a() and b(), we wrie a(x) b(x) if lim a(x)=b(x) =. Furhermore, for wo posiive bivariae funcions a(; ) and b(; ), we say ha he asympoic relaion a(x; ) b(x; ) holds uniformly over all in a nonempy se if lim sup a(x; ) x b(x; ) = : 2 2
3 Asympoic formulae ha hold wih such a uniformiy feaure are usually of higher heoreical and pracical ineress. Recall (.2) and (.3). Our main resul is given below: Theorem.. Consider he renewal model inroduced above. If F 2 ERV, hen he relaion holds uniformly for all 2. 2 Some Remarks Pr (D () > x) F xe s d s (.6) Remark 2.. When =, he sum D () reduces o D () = X k e k : (2.) For F 2 ERV( ; ), from inequaliy (3.) below wih x xed, we see ha F (y) = O(y ) for all, < <. Hence, EX < for all, < <. Using his fac we may furher verify ha E (D ()) ^ <. This means ha D () converges almos surely as. Likewise, using he fac EX < and he elemenary renewal heorem, i is easy o verify ha R F e s d s <, irrespecive of wheher or no has a nie mean. Therefore, boh sides of (.6) are well de ned. Remark 2.2. Suppose ha premiums are colleced coninuously a a consan rae c >. Then, he surplus process is S () = xe + c e ( s) ds A () ; ; where x denoes he iniial surplus. De ne he probabiliy of ruin by ime as he probabiliy ha he surplus process ever becomes negaive by ime. Denoe his probabiliy by (x; ). The limi (x; ) = lim (x; ) represens he probabiliy of ulimae ruin. Alhough he pracical relevance of ruin probabiliies is quesionable, hey do provide a good risk measure for insurance business. Clearly, for all 2, Hence, (x; ) = Pr inf S (v) < <v = Pr sup <v D (v) Z v c e s ds > x : (x; ) Pr (D () > x) and (x; ) Pr (D () > x + c=) : Noe ha, by (.4), F (x + c=)e s F xe s holds uniformly for all s 2 [; ). Applying Theorem., we immediaely obain ha he relaion (x; ) 3 F xe s d s (2.2)
4 holds uniformly for all 2. Klüppelberg and Sadmüller (998) rs obained a resul similar o (2.2) wih = for he special case ha fn ; g is a homogeneous Poisson process and F belongs o he class R. For his special case, Tang (25) obained he uniformiy of (2.2). Recenly, Chen and Ng (27) exended he asympoic relaion (2.2) wih = o he case of negaively dependen claims. Remark 2.3. We saed Theorem. in erms of he renewal model, where he innovaions X k, k = ; 2; : : :, denoe claim sizes and hence are nonnegaive. However, in mos siuaions considered in pracice, hese innovaions appearing in (.2) could be real valued. For his more general case, we may insead sudy he ail behavior of he running maximum process We show ha he asympoic formula fd () := sup D (s); : s Pr fd () > x F xe s d s (2.3) holds uniformly for all 2 as long as he righ ail of F is sill exended regularly varying as described in (.4). Acually, for his case, he proof given in Secion 4 unil (4.3) is valid for boh D () and D + () := P X+ k e k ( k ),. Since for all, D () f D () D + (); we see ha (2.3) holds uniformly for all 2 \ [; T ] for an arbirarily xed number T 2. The remaining proof of he uniformiy on of (2.3) can be given by simply copying he par afer (4.3) of he proof of Theorem. wih all D replaced by D f. Remark 2.4. Relaion (.6) unforunaely involves he renewal funcion,, so do relaions (2.2) and (2.3). If he i.i.d. iner-arrival imes have a common nie mean E = =, hen as by he elemenary renewal heorem. This emps us o consider o replace s in (.6) by s. However, his is no feasible in general. Acually, under he condiion F 2 ERV, he di erenial d s in he inegral is on an equal fooing. Thus, we can no ignore an inegral par in he righ neighborhood of. If fn ; g is a homogeneous Poisson process wih inensiy >, hen s = s for all s >. Oher cases where he explici form of he renewal funcion s is available can be found in he lieraure. For example, le have a phase-ype disribuion (of which he Erlang disribuion is a special case) wih densiy given by g(s) = e Ts ; s ; where is a row vecor, T is a marix, and = T wih = (; : : : ; ). The vecor T and he marix T should be chosen such ha (; ) is he iniial disribuion and 4
5 is he inensiy marix of a coninuous-ime Markov jump process wih nie sae space in which one sae is absorbing and he ohers are ransien. In his case, he derivaive of he renewal funcion s, called he renewal densiy, is given by d s ds = e(t+)s : As anoher example, le have a uniform disribuion on (; a). Then, he renewal densiy is given by d s ds = X a es=a e k (k s=a)k : k k: ks=a These formulae are copied from pages 88 and 48 of Asmussen (23). Remark 2.5. We now propose a resul in a general siuaion, in which he renewal funcion s can indeed be simpli ed o s. Consider he discouned process (.2). For any x, denoe by T x = inf f : D () > xg he rs ime when D () up-crosses he level x, where inf? = by convenion. The following is a corollary of Theorem., giving explici approximaions for he ail probabiliy of T x. Corollary 2.. In addiion o he assumpions of Theorem., we assume ha he i.i.d. iner-arrival imes have a non-laice disribuion and a nie mean E = =. Then, Pr ( < T x < ) lim lim sup x R s ) ds = lim lim inf x Pr ( < T x < ) R s ) ds = : (2.4) If F 2 R as de ned in (.5) wih some >, hen relaion (2.4) can be srenghened o lim lim Pr ( < T x < ) x e F (x) = : (2.5) The proof of Corollary 2. is lef o Secion 4. From he proof one sees ha he same resul holds for he case discussed in Remark Lemmas Lemma 3.. Le F 2 ERV( ; ). Then for any and, < <, < <, here are posiive consans c i and d i, i = ; 2, such ha he inequaliy F (y) F (x) c (y=x) (3.) 5
6 holds whenever y x d, and ha he inequaliy F (y) F (x) c 2 (y=x) (3.2) holds whenever y x d 2. Proof. This lemma is a consequence of Proposiion 2.2. of Bingham e al. (987). Acually, wih f = =F we see ha (3.) and (3.2) above are, respecively, (2.2. ) and (2.2.) of Bingham e al. (987). Lemma 3.2. Consider he renewal process fn ; g de ned in (.). I holds for all T 2 and all v > ha lim sup EN v (N>x) = : x 2\[;T ] Proof. Follow he proof of Lemma 5.3 of Tang (24) wih sligh modi caions. Lemma 3.3. Le fx ; X 2 ; : : : ; X n g be n i.i.d. random variables wih common disribuion F 2 S. Then for arbirarily xed numbers a and b, < a b <, he relaion Pr c k X k > x F (x=c k ) holds uniformly for all (c ; : : : ; c n ) 2 [a; b] [a; b]. Proof. See Proposiion 5. of Tang and Tsisiashvili (23). 4 Proofs 4. Proof of Theorem. To be more precise, we assume F 2 ERV( ; ). In he rs half of his subsecion, we prove ha relaion (.6) holds uniformly for all 2 \ [; T ] for an arbirarily xed number T 2. We spli he probabiliy Pr (D () > x) ino wo pars as mx Pr (D () > x) = + Pr X k e k > x; N = n := I + I 2 ; n=m+ n= where m is a emporarily xed ineger. Firs we deal wih I. Recall ha F 2 ERV( ; ). As menioned in he proof of Lemma 4.5 of Tang (25), using a resul of Nagaev (979) we may prove ha for an arbirarily xed number v >, here is some c v > such ha for all n = ; 2; : : : and all x, Pr X k > x c v n v F (x) : 6
7 Therefore, I Pr n=m+ X k > x Pr (N = n) c v F (x) EN v (N>m): By inequaliy (3.2), for some > and all x d 2, Hence by Lemma 3.2, for all x d 2, F (x) c 2 e T F xe T : lim inf m sup 2\[;T ] I R s ) d s c v T e c 2 lim inf sup F m 2\[;T ] xe T EN v (N>m) T ) = : (4.) We urn o I 2. Under he condiion N = n, all k appearing in I 2 are no larger han T. Using Lemma 3.3, i holds uniformly for all 2 \ [; T ] ha mx I 2 = Pr X k e k > x N = n Pr (N = n) = Clearly, n= mx n= Pr X k e k > x N = n Pr (N = n) Pr X k e k > x; N = n := I 2 I 22 : n= n=m+ I 2 = = Pr X k e k > x; N = n n=k Pr X k e k > x; k = F xe s d s : Noe ha I 22 F (x) n=m+ Hence, similar o he proof of (4.), for all x d 2, lim sup m 2\[;T ] n Pr (N = n) : I 22 R s ) d s = : We conclude ha he asympoic relaion (.6) holds uniformly for all 2 \ [; T ]. 7
8 In he second half of his subsecion, we exend he uniformiy of (.6) o he whole inerval. For arbirarily xed numbers and, < <, < <, again by inequaliies (3.) and (3.2), i holds for all x maxfd ; d 2 g and all 2 [; ) ha R F xe s d s R = s ) d s R R F(xe s ) d F (x) s F(xe s ) F (x) c R e s d s R : (4.2) d c 2 e s s d s The righ-hand side of he above ends o as. Therefore, for any " >, here is some T 2 such ha he inequaliy Z F xe s Z T d s " F xe s d s (4.3) T holds for all x maxfd ; d 2 g. Recall (2.). Using Theorem 3. of Tang and Tsisiashvili (24), we obain ha Pr (D () > x) Pr X k e k > x = Z F xe s d s : (4.4) Hence, relaion (.6) holds for =. We are ready o exend he uniformiy of (.6) o he whole inerval. On he one hand, i holds uniformly for all 2 (T ; ] ha Z T Pr (D () > x) Pr (D (T ) > x) F xe s d s Z F xe s d s ( ") T F xe s d s ; where in he second sep we used relaion (.6) wih replaced by T, while in he las sep we used (4.3). On he oher hand, likewise, i holds uniformly for all 2 (T ; ] ha Z Pr (D () > x) Pr (D () > x) F xe s d s Z + F xe s d s ( + ") T F xe s d s ; where in he second sep we used relaion (4.4), while in he las sep we used (4.3). Hence, i holds for all 2 (T ; ] and all large x, say x > x >, ha ( 2") F xe s d s Pr (D () > x) ( + 2") F xe s d s : (4.5) From he rs half of his proof we see ha (4.5) sill holds for all 2 \ [; T ] and all large x, say x > x 2 >. Therefore, (4.5) holds for all 2 and all x > maxfx ; x 2 g. Since " > is arbirary, we have obained he uniformiy of relaion (.6) over all 2. 8
9 4.2 Proof of Corollary 2. Since every rajecory of D () is piecewise consan wih only upward jumps, we have Pr (T x ) = Pr (D () > x) for all 2 \ [; ) and Pr (T x < ) = Pr (D () > x). Hence by Theorem., for all 2 \ [; ), Pr ( < T x < ) = Pr (D () > x) Pr (D () > x) Z F xe s d s F xe s d s = Z F xe s d s ; (4.6) where in he second sep we used he asympoic relaion (.6). There is no problem wih his sep because, wih arbirarily xed, similar o (4.2), lim inf x R F xe s d s R + c 2 s ) d s R e s d s > : e s d s c R For any ` >, by he well-known Blackwell renewal heorem, lim ( s+` s ) = `: s I follows ha, for any " > and all large s, say s > s = s ("; `) >, Therefore, for all x 2 [; ) and > s, Z F xe s d s = ( ")` s+` s ( + ")`: +k` ( + ") F xe s d s +(k )` +(k )` +(k 2)` F F xe [+(k )`] ( + ")` xe s ds = ( + ") Z F xe (s `) ds: Using he de niion in (.4), i holds for all x 2 [; ) and all large s, say s > s 2 >, ha F xe (s `) ( + ")e `F xe s : I follows ha, for all x 2 [; ) and > maxfs ; s 2 g, Z F xe s d s ( + ") 2 e ` Z F xe s ds: A similar lower bound for he inegral R F xe s d s can also be esablished. Hence by he arbirariness of he consans ` and ", he relaion Z F xe s d s Z F xe s ds; ; (4.7) 9
10 holds uniformly for all x 2 [; ). Clearly, he uniformiy of relaion (4.7) indicaes ha R F xe s d s lim lim sup x R From (4.6) and (4.8) we have Pr ( < T x < ) lim lim sup x R s ) ds = lim s ) ds = lim lim x lim inf x R F xe s d s R s ) ds Pr ( < T x < ) R lim sup s ) d s x = : (4.8) R F xe s d s R s ) ds The derivaion above wih lim sup replaced by lim inf is sill valid. This proves (2.4). Likewise, when F 2 R wih some >, R lim lim Pr ( < T x < ) Pr ( < T x < ) F xe s d s x e = lim lim R lim F (x) x s ) d s x e F (x) Z F xe s = lim e lim d s x F (x) Z = lim e s d s e = lim e Z e s ds where in he second sep we used (4.6), in he hird sep we used he dominaed convergence heorem jusi ed by (3.) and (.5), and in he fourh sep we used he argumen of he Blackwell renewal heorem as we did in proving (4.7). This proves (2.5). Acknowledgmens. The auhor wishes o hank he referee for his/her very helpful commens. The suppor of he Old Gold Summer Fellowship from he Universiy of Iowa is acknowledged. References [] Asmussen, S. Applied probabiliy and queues. Second ediion. Springer-Verlag, New York, 23. [2] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variaion. Cambridge Universiy Press, Cambridge, 987. [3] Chen, Y.; Ng, K. W. The ruin probabiliy of he renewal model wih consan ineres force and negaively dependen heavy-ailed claims. Insurance Mah. Econom. (27), o appear. [4] Embrechs, P.; Klüppelberg, C.; Mikosch, T. Modelling exremal evens for insurance and nance. Springer-Verlag, Berlin, 997. = ; = :
11 [5] Goldie, C. M. Subexponenial disribuions and dominaed-variaion ails. J. Appl. Probabiliy 5 (978), no. 2, [6] Klüppelberg, C.; Mikosch, T. Large deviaions of heavy-ailed random sums wih applicaions in insurance and nance. J. Appl. Probab. 34 (997), no. 2, [7] Klüppelberg, C.; Sadmüller, U. Ruin probabiliies in he presence of heavy-ails and ineres raes. Scand. Acuar. J. 998, no., [8] Nagaev, S. V. Large deviaions of sums of independen random variables. Ann. Probab. 7 (979), no. 5, [9] Tang, Q. Uniform esimaes for he ail probabiliy of maxima over nie horizons wih subexponenial ails. Probab. Engrg. Inform. Sci. 8 (24), no., [] Tang, Q. The nie-ime ruin probabiliy of he compound Poisson model wih consan ineres force. J. Appl. Probab. 42 (25), no. 3, [] Tang, Q.; Tsisiashvili, G. Randomly weighed sums of subexponenial random variables wih applicaion o ruin heory. Exremes 6 (23), no. 3, [2] Tang, Q.; Tsisiashvili, G. Finie- and in nie-ime ruin probabiliies in he presence of sochasic reurns on invesmens. Adv. in Appl. Probab. 36 (24), no. 4,
A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails
A Uniform Asympoic Esimae for Discouned Aggregae Claims wih Subeponenial Tails Xuemiao Hao and Qihe Tang Deparmen of Saisics and Acuarial Science The Universiy of Iowa 241 Schae er Hall, Iowa Ciy, IA 52242,
More informationA Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails
A Uniform Asympoic Esimae for Discouned Aggregae Claims wih Subeponenial Tails Xuemiao Hao and Qihe Tang Deparmen of Saisics and Acuarial Science The Universiy of Iowa 241 Schaeffer Hall, Iowa Ciy, IA
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
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 informationMODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE
Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS
More informationInternational Journal of Scientific & Engineering Research, Volume 4, Issue 10, October ISSN
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 900 FUZZY MEAN RESIDUAL LIFE ORDERING OF FUZZY RANDOM VARIABLES J. EARNEST LAZARUS PIRIYAKUMAR 1, A. YAMUNA 2 1.
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
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 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 informationConvergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
More informationA proof of Ito's formula using a di Title formula. Author(s) Fujita, Takahiko; Kawanishi, Yasuhi. Studia scientiarum mathematicarum H Citation
A proof of Io's formula using a di Tile formula Auhor(s) Fujia, Takahiko; Kawanishi, Yasuhi Sudia scieniarum mahemaicarum H Ciaion 15-134 Issue 8-3 Dae Type Journal Aricle Tex Version auhor URL hp://hdl.handle.ne/186/15878
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
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 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 informationHomogenization of random Hamilton Jacobi Bellman Equations
Probabiliy, Geomery and Inegrable Sysems MSRI Publicaions Volume 55, 28 Homogenizaion of random Hamilon Jacobi Bellman Equaions S. R. SRINIVASA VARADHAN ABSTRACT. We consider nonlinear parabolic equaions
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 informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
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 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 information4 Sequences of measurable functions
4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences
More informationarxiv: v1 [math.pr] 19 Feb 2011
A NOTE ON FELLER SEMIGROUPS AND RESOLVENTS VADIM KOSTRYKIN, JÜRGEN POTTHOFF, AND ROBERT SCHRADER ABSTRACT. Various equivalen condiions for a semigroup or a resolven generaed by a Markov process o be of
More informationThe consumption-based determinants of the term structure of discount rates: Corrigendum. Christian Gollier 1 Toulouse School of Economics March 2012
The consumpion-based deerminans of he erm srucure of discoun raes: Corrigendum Chrisian Gollier Toulouse School of Economics March 0 In Gollier (007), I examine he effec of serially correlaed growh raes
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationStochastic models and their distributions
Sochasic models and heir disribuions Couning cusomers Suppose ha n cusomers arrive a a grocery a imes, say T 1,, T n, each of which akes any real number in he inerval (, ) equally likely The values T 1,,
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 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 informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationarxiv: v1 [math.fa] 9 Dec 2018
AN INVERSE FUNCTION THEOREM CONVERSE arxiv:1812.03561v1 [mah.fa] 9 Dec 2018 JIMMIE LAWSON Absrac. We esablish he following converse of he well-known inverse funcion heorem. Le g : U V and f : V U be inverse
More informationExpert Advice for Amateurs
Exper Advice for Amaeurs Ernes K. Lai Online Appendix - Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
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 informationHomework 4 (Stats 620, Winter 2017) Due Tuesday Feb 14, in class Questions are derived from problems in Stochastic Processes by S. Ross.
Homework 4 (Sas 62, Winer 217) Due Tuesday Feb 14, in class Quesions are derived from problems in Sochasic Processes by S. Ross. 1. Le A() and Y () denoe respecively he age and excess a. Find: (a) P{Y
More informationINDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE
INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE JAMES ALEXANDER, JONATHAN CUTLER, AND TIM MINK Absrac The enumeraion of independen ses in graphs wih various resricions has been a opic of much ineres
More informationLecture 2 April 04, 2018
Sas 300C: Theory of Saisics Spring 208 Lecure 2 April 04, 208 Prof. Emmanuel Candes Scribe: Paulo Orensein; edied by Sephen Baes, XY Han Ouline Agenda: Global esing. Needle in a Haysack Problem 2. Threshold
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 informationInequality measures for intersecting Lorenz curves: an alternative weak ordering
h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 Absrac Inequaliy measures for
More informationOptimality Conditions for Unconstrained Problems
62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x
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 informationTransform Techniques. Moment Generating Function
Transform Techniques A convenien way of finding he momens of a random variable is he momen generaing funcion (MGF). Oher ransform echniques are characerisic funcion, z-ransform, and Laplace ransform. Momen
More informationarxiv:math/ v1 [math.nt] 3 Nov 2005
arxiv:mah/0511092v1 [mah.nt] 3 Nov 2005 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON AND S. M. GONEK Absrac. Le πs denoe he argumen of he Riemann zea-funcion a he poin 1 + i. Assuming
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
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 information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
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 informationEndpoint Strichartz estimates
Endpoin Sricharz esimaes Markus Keel and Terence Tao (Amer. J. Mah. 10 (1998) 955 980) Presener : Nobu Kishimoo (Kyoo Universiy) 013 Paricipaing School in Analysis of PDE 013/8/6 30, Jeju 1 Absrac of he
More informationRENEWAL PROCESSES. Chapter Introduction
Chaper 5 RENEWAL PROCESSES 5.1 Inroducion Recall ha a renewal process is an arrival process in which he inerarrival inervals are posiive, 1 independen and idenically disribued (IID) random variables (rv
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 informationSample Autocorrelations for Financial Time Series Models. Richard A. Davis Colorado State University Thomas Mikosch University of Copenhagen
Sample Auocorrelaions for Financial Time Series Models Richard A. Davis Colorado Sae Universiy Thomas Mikosch Universiy of Copenhagen Ouline Characerisics of some financial ime series IBM reurns NZ-USA
More informationPOSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION
Novi Sad J. Mah. Vol. 32, No. 2, 2002, 95-108 95 POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial
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 informationAn random variable is a quantity that assumes different values with certain probabilities.
Probabiliy The probabiliy PrA) of an even A is a number in [, ] ha represens how likely A is o occur. The larger he value of PrA), he more likely he even is o occur. PrA) means he even mus occur. PrA)
More informationA FAMILY OF MARTINGALES GENERATED BY A PROCESS WITH INDEPENDENT INCREMENTS
Theory of Sochasic Processes Vol. 14 3), no. 2, 28, pp. 139 144 UDC 519.21 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET A FAMILY OF MARTINGALES GENERATED BY A PROCESS WITH INDEPENDENT INCREMENTS An explici procedure
More informationdi Bernardo, M. (1995). A purely adaptive controller to synchronize and control chaotic systems.
di ernardo, M. (995). A purely adapive conroller o synchronize and conrol chaoic sysems. hps://doi.org/.6/375-96(96)8-x Early version, also known as pre-prin Link o published version (if available):.6/375-96(96)8-x
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
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 informationOn Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature
On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check
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 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 informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationOn R d -valued peacocks
On R d -valued peacocks Francis HIRSCH 1), Bernard ROYNETTE 2) July 26, 211 1) Laboraoire d Analyse e Probabiliés, Universié d Évry - Val d Essonne, Boulevard F. Mierrand, F-9125 Évry Cedex e-mail: francis.hirsch@univ-evry.fr
More informationA note to the convergence rates in precise asymptotics
He Journal of Inequaliies and Alicaions 203, 203:378 h://www.journalofinequaliiesandalicaions.com/conen/203//378 R E S E A R C H Oen Access A noe o he convergence raes in recise asymoics Jianjun 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 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 informationSolutions Problem Set 3 Macro II (14.452)
Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.
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 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 informationExample on p. 157
Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =
More informationWe just finished the Erdős-Stone Theorem, and ex(n, F ) (1 1/(χ(F ) 1)) ( n
Lecure 3 - Kövari-Sós-Turán Theorem Jacques Versraëe jacques@ucsd.edu We jus finished he Erdős-Sone Theorem, and ex(n, F ) ( /(χ(f ) )) ( n 2). So we have asympoics when χ(f ) 3 bu no when χ(f ) = 2 i.e.
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
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 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 informationRUIN PROBABILITIES FOR RISK PROCESSES WITH NON-STATIONARY ARRIVALS AND SUBEXPONENTIAL CLAIMS. 1. Introduction
RUIN PROBABILITIES FOR RISK PROCESSES WITH NON-STATIONARY ARRIVALS AND SUBEXPONENTIAL CLAIMS LINGJIONG ZHU Absrac. In his paper, we obain he finie-horizon and infinie-horizon ruin probabiliy asympoics
More informationLecture 6: Wiener Process
Lecure 6: Wiener Process Eric Vanden-Eijnden Chapers 6, 7 and 8 offer a (very) brief inroducion o sochasic analysis. These lecures are based in par on a book projec wih Weinan E. A sandard reference for
More informationBoundedness and Exponential Asymptotic Stability in Dynamical Systems with Applications to Nonlinear Differential Equations with Unbounded Terms
Advances in Dynamical Sysems and Applicaions. ISSN 0973-531 Volume Number 1 007, pp. 107 11 Research India Publicaions hp://www.ripublicaion.com/adsa.hm Boundedness and Exponenial Asympoic Sabiliy in Dynamical
More informationSTRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN
Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN 001-004 OBARA, Takashi * Absrac The
More informationOn Gronwall s Type Integral Inequalities with Singular Kernels
Filoma 31:4 (217), 141 149 DOI 1.2298/FIL17441A Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma On Gronwall s Type Inegral Inequaliies
More informationSemi-Competing Risks on A Trivariate Weibull Survival Model
Semi-Compeing Risks on A Trivariae Weibull Survival Model Jenq-Daw Lee Graduae Insiue of Poliical Economy Naional Cheng Kung Universiy Tainan Taiwan 70101 ROC Cheng K. Lee Loss Forecasing Home Loans &
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 informationAsymptotic Analysis of Multivariate Tail Conditional Expectations
Asympoic Analysis of Mulivariae Tail Condiional Expecaions Li Zhu Haijun Li Ocober 0 Revision: May 0 lzhu@mah.wsu.edu, Deparmen of Mahemaics, Washingon Sae Universiy, Pullman, WA 9964, U.S.A. lih@mah.wsu.edu,
More informationRichard A. Davis Colorado State University Bojan Basrak Eurandom Thomas Mikosch University of Groningen
Mulivariae Regular Variaion wih Applicaion o Financial Time Series Models Richard A. Davis Colorado Sae Universiy Bojan Basrak Eurandom Thomas Mikosch Universiy of Groningen Ouline + Characerisics of some
More information11!Hí MATHEMATICS : ERDŐS AND ULAM PROC. N. A. S. of decomposiion, properly speaking) conradics he possibiliy of defining a counably addiive real-valu
ON EQUATIONS WITH SETS AS UNKNOWNS BY PAUL ERDŐS AND S. ULAM DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, BOULDER Communicaed May 27, 1968 We shall presen here a number of resuls in se heory concerning
More informationOptima and Equilibria for Traffic Flow on a Network
Opima and Equilibria for Traffic Flow on a Nework Albero Bressan Deparmen of Mahemaics, Penn Sae Universiy bressan@mah.psu.edu Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 1 / 1 A Traffic
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 informationIntroduction to Probability and Statistics Slides 4 Chapter 4
Inroducion o Probabiliy and Saisics Slides 4 Chaper 4 Ammar M. Sarhan, asarhan@mahsa.dal.ca Deparmen of Mahemaics and Saisics, Dalhousie Universiy Fall Semeser 8 Dr. Ammar Sarhan Chaper 4 Coninuous Random
More informationLecture 4 Notes (Little s Theorem)
Lecure 4 Noes (Lile s Theorem) This lecure concerns one of he mos imporan (and simples) heorems in Queuing Theory, Lile s Theorem. More informaion can be found in he course book, Bersekas & Gallagher,
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 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 information10. 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 informationAverage Number of Lattice Points in a Disk
Average Number of Laice Poins in a Disk Sujay Jayakar Rober S. Sricharz Absrac The difference beween he number of laice poins in a disk of radius /π and he area of he disk /4π is equal o he error in he
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 informationA remark on the H -calculus
A remark on he H -calculus Nigel J. Kalon Absrac If A, B are secorial operaors on a Hilber space wih he same domain range, if Ax Bx A 1 x B 1 x, hen i is a resul of Auscher, McInosh Nahmod ha if A has
More informationLecture 4: Processes with independent increments
Lecure 4: Processes wih independen incremens 1. A Wienner process 1.1 Definiion of a Wienner process 1.2 Reflecion principle 1.3 Exponenial Brownian moion 1.4 Exchange of measure (Girsanov heorem) 1.5
More informationA Note on Goldbach Partitions of Large Even Integers
arxiv:47.4688v3 [mah.nt] Jan 25 A Noe on Goldbach Pariions of Large Even Inegers Ljuben Muafchiev American Universiy in Bulgaria, 27 Blagoevgrad, Bulgaria and Insiue of Mahemaics and Informaics of he Bulgarian
More informationTHE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX
J Korean Mah Soc 45 008, No, pp 479 49 THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX Gwang-yeon Lee and Seong-Hoon Cho Reprined from he Journal of he
More informationExistence of positive solution for a third-order three-point BVP with sign-changing Green s function
Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp://www.mah.u-szeged.hu/ejqde/ Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion
More informationBOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS
BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS XUAN THINH DUONG, JI LI, AND ADAM SIKORA Absrac Le M be a manifold wih ends consruced in [2] and be he Laplace-Belrami operaor on M
More informationCERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS
SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22 (2014, 67 76 DOI: 10.5644/SJM.10.1.09 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ Absrac. This paper presens sufficien
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 informationAsymptotic Analysis of Tail Conditional Expectations
Asympoic Analysis of Tail Condiional Expecaions Li Zhu Haijun Li June Absrac Tail condiional expecaions refer o he expeced values of random variables condiioning on some ail evens and are closely relaed
More information