DISCRETE TIME MODELS OF FORWARD CONTRACTS INSURANCE

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1 G Tstsshvl DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE (Vol) 008 September DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE GSh Tstsshvl e-ml: Vldvosto Rdo str 7 sttute for Appled Mthemtcs Fr Ester Brch of RAS troducto ths pper fcl mgemet model of forwrd cotrcts surce suggested [] s cosdered by mes of rs theory d hevy tled techque Ths model s bsed o compesto prcple t ttrcted lrge terest d clled ctve dscusso mog ecoomsts So ts mthemtcl lyss s tted s ecoomsts so mthemtcs Suppose tht there re two surce compes surg both prtcpts of some forwrd cotrct d worg dscrete tme wth the et pyouts ξ + η ξ η durg oe step Here ξ wth Eξ = < 0 s commo rdom summd for et pyouts of these two compes The η wth Eη = 0 s dvdul rdom clm of the frst prtcpt of the cotrct d η s dvdul clm of the secod prtcpt The clm η (the clm η ) my be cosdered s premum for η < 0 (for η > 0 ) Suppose tht dstrbuto fuctos (df`s) P( ξ ) = H( ) P( η ) = S( ) P( η ) = L( ) d H( ) = o( S( ) ) H( ) = o( L( ) ) () F = F Assume tht ξ η re depedet rdom vrbles (rv`s) d ξ wth η η re subepoetl rv`s Deote the oe-step ru probbltes of the compes wth the tl cptl surg the frst d the secod prtcpts of the cotrct by ( ) = P( ξ + η > ) ( ) = P( ξ η > ) d the oe-step ru probbltes of both compes d oe of them by ( ) = P ( ξ + η > ) ( ξ η > ) Here ( ) ( ) chrcterze dvdul rss of the surce compes d chrcterzes ther group rs troduce c( ) = P( ξ + η+ ξ η > + ) = H( ) oe step ru probblty of these two compes ggregto Here c( ) chrcterzes s dvdul so group rss The ggregto of these two compes llows to decrese dvdul rss ( ) ( ) to smll H( ) d to coserve the group rs ( ) t smll level ( ) S( ) ( ) L( ) H : c () M purpose of ths pper s to obt symptotcl comprsos logous to () for dvdul d group rss seprte d ggregted surce models We spe bout fte horzo dscrete tme rs models wthout terest force wth costt terest force d wth stochstc terest force

2 G Tstsshvl DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE (Vol) 008 September Prelmres Clsses of dstrbutos Throughout for gve rv X cocetrted o ( ) wth df F the ts rght tl = P( X > ) For two df's F d F cocetrted o ( ) we * wrte by F F ( ) F d wrte by F = F F the covoluto of F d the covoluto of F wth tself All lmtg reltoshps uless otherwse stted re for Let ( ) 0 d b( ) > 0 be two ftesmls stsfyg ( ) ( ) + l lm f lm sup l b( ) b( ) We wrte ( ) = O b( ) f l + < ( ) = o b( ) f l + = 0 d ( ) b( ) > b( ) f l = d ( ) b( ) f both troduce the followg clsses of df`s cocetrted o [ 0 ) : * F ( ) S = :lm = t F( ) < f l + = F t L = : lm = F( θ ) : θ 0 lm θ R - = > = 0 < < R= U R - 0< < F( θ ) R - = : θ > lm = 0 S = : F( y) F( y) dy m+ m+ = d 0 0 S s clled the clss of subepoetl df`s L s clled the clss of log tled df`s R (or R - ) s clled the clss of regulr vryg df`s (wth de ) R - s clled the clss of rpdly vryg tled df`s Proposto The clsses R S L stsfy the formul [3] R S L Proposto f F S the [4] F F ( ) S wth m + F = F y dy More geerlly df F cocetrted o ( ) s lso sd to belog to these clsses f ts rght-hd dstrbuto F ( ) = ( > 0 ) does Proposto 3 Let F d F = O F the [5Lemm 3] F F F be two df`s cocetrted o ( ) S d F F ( ) F ( ) F ( ) f F S F L d + Proposto 4 Suppose tht X Y re depedet rdom vrbles wth the df`s F F cocetrted o( ) d F L F( = ) the [6] P( X Y > t) F ( t) Dscrete tme rs model uder stochstc terest force Cosder rs model wth dscrete tme = d deote X the surer`s et loss - the totl clm mout mus the totl comg premum wth perod d Y the dscout fctor from tme to tme Here X

3 G Tstsshvl DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE (Vol) 008 September s clled surce rs d Y s clled fcl rs These rdom vrbles re depedet wth F t G t reltvely df`s () () Let { X = } be sequece of depedet d detclly dstrbuted (d) rv`s wth geerc rdom vrble X let { Y = } be other sequece of d postve rv`s wth geerc rdom vrble Y d let the two sequeces be mutully depedet Deote Ψ ( ) = P sup X < = > j = The Ψ ( ) s fte-tme ru probblty of the rs model uder stochstc terest force wth tl cptl Proposto 5 Suppose tht F [8] Proposto 6 f F R 0 Proposto 7 Suppose tht 0 < < the [9] S d EX m 0 = < d Ψ ( ) F ( ) m δ + δ < < d E { Y Y } PY= = the [7] m < for some 0 < δ < the Ψ ( ) PY= + r = for some 0 < r < f F R for some Ψ ( ) f F S R the Ψ ( ) F ( + r) ( r) + Asymptotc comprso of dvdul d group rss for forwrd cotrcts surce Suppose tht t the step the et pyouts of both prtcpts of the step forwrd cotrct re ξ + η ξ η Here rdom sequeces { ξ = ξ0 ξ } d { η = η0 η } re depedet Ech of these two rdom sequeces cossts of drv`s wth ther ow commo P t H t P t S t Lt S wth df`s ( ξ ) = ( η ) = correspodgly d H( t ) S( t ) () Lt () = P( η ) Defe dvdul rss of the compes wth tl cptls surg et pyouts ξ η ξ η ξ η ξ η seprtely by { + + } { } A ( ) = P sup ( ξ + η) > < = A ( ) = P sup ( ξ η) > j = < = j = Alogously defe group rs of seprtely worg surce compes by A( ) = P sup ( ξ + η) > sup ( ξ η) > < = j= < = j=

4 G Tstsshvl DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE (Vol) 008 September d commo dvdul d group rs of ggregted compy by C( ) = P sup ( ξ + η + ξ η) > = P sup ξ > < = j= < = j= Lemm Suppose tht the codto () s true the P( ξ + η > ) S( ) P( ξ η > ) L( ) Proof Ths sttemet rses from the proposto 3 Lemm The formul P( ξ η > ) H( ) s true Proof Ths sttemet rses from the propostos 4 Lemm 3 The followg equlty tes plce A( ) R( ) R ( ) = P sup ( ξ η ) < = > j = Proof Ths sttemet rses from the equltes whch re true for ll rel Theorem Suppose tht the codto () s true d H( t ) S( t ) Lt () S the ( ) A ( ) PY= = S L H A ( ) H C( ) R( ) Eη Proof Ths sttemet rses from the propostos 5 d from the lemms Theorem Suppose tht the codto () s true d for some 3 0 < < 3 δ( ) + δ( ) S L H R Em Y Y < for df`s R R 3 some < δ ( ) < 3 the 0 { } 3 A ( ) S( ) A ( ) L( ) C ( ) R( ) H( ) 3 Proof Ths sttemet rses from the propostos 6 d from the lemms Theorem 3 Suppose tht the codto () s true d PY ( = + r) = for some 0 < r < f for some S L H R the < < df`s R R 3 S( ) ( ) ( + r) L( ) A ( + r) H( ) C R ( r) 3 A + f df`s S( ) L( ) H( ) S R the A ( ) S ( + r) A ( ) L ( + r) ( ) C R H + r Proof Ths sttemet rses from the propostos 7 d from the lemms Theorem 4 f the codtos of the theorem (the theorem or the theorem 3) re true the for some 0 < < A( ) > C( ) C( ) = o ( A ) C( ) = o( A ( ) ) Proof Ths sttemet rses from the lemm 3 d from the theorems

5 G Tstsshvl DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE (Vol) 008 September Refereces [] Afsev AA Commercl Bs t Mret of Dervtve Fcl strumets Fr Ester Stte Acdemy of Ecoomcs d Cotrol Vldvosto 00 ( Russ) [] Rudo-Selvov VV Fedoseev DA Mret of dervtve fcl strumets: loo to future Moey d Credt 006 Vol ( Russ) [3] Embrechts P Kluppelberg C Mosch T Modellg Etreml Evets Fce d surce 997 Sprger [4] Kluppelberg C O subepoetl dstrbutos d tegrted tls Appl Probb 988 Vol [5] Tg Q; Tstsshvl G Precse estmtes for the ru probblty fte horzo dscrete-tme model wth hevy-tled surce d fcl rss Stochst Process Appl 003 Vol [6] Tstsshvl GSh Mrov NVAsymptotc chrcterstcs of put flows queueg etwors Fr Ester Mthemtcl Jourl 003 Vol ( Russ) [7] Embrechts P Ververbee N Estmtes for the probblty of ru wth specl emphss o the possblty of lrge clms surce Mth Ecoom 98 Vol 55-7 [8] Tg Q Tstsshvl G Fte d fte Tme Ru Probbltes the Presece of Stochstc Returs o vestmets Adv ApplProbb 004 Vol [9] Tg Q The ru probblty of dscrete tme rs model uder costt terest rte wth hevy tls Scd Actur J 004 Vol