RISK-NEUTRAL VALUATION UNDER FUNDING COSTS AND COLLATERALIZATION
|
|
- Melina Thomas
- 5 years ago
- Views:
Transcription
1 RISK-NEUTRAL VALUATION UNDER FUNDING COSTS AND COLLATERALIZATION Damiano Brigo Dep. of Mahemaics Imperial College London Andrea Pallavicini Dep. of Mahemaics Imperial College London Crisin Buescu Dep. of Mahemaics King s College London Marco Francischello Dep. of Mahemaics Imperial College London Marek Rukowski School of Mahemaics and Saisics Universiy of Sydney June 25, 216 Absrac The expeced cash flows approach leading o an exended version of he classical risk-neural valuaion formula was proposed in he recen works by Pallavicini e al. [6] and Brigo e al. [3] who sudied he problem of valuaion of conracs under differenial funding coss and collaeralizaion. The crucial difference beween he approach developed in [3, 6] and he presen noe is ha he pricing formula via condiional expecaion of discouned adjused cash flows is posulaed in [3, 6] and i is jusified using financial argumens, whereas in his noe he formula is esablished in more general se-up and i is shown o be a consequence of he sandard replicaion-based argumens. Keywords: risk-neural valuaion, hedging, funding coss, collaeral, margin agreemen Mahemaics Subjecs Classificaion 21: 91G4, 6J28 1
2 2 Brigo Buescu Francischello Pallavicini Rukowski 1 Simple Trading Model We firs provide an informal explanaion how he cash flows adjusmens are moivaed in [3, 6]. To show how he adjused cash flows originae, we assume ha he hedger buys a call opion on an equiy asse S T wih srike K. In oher words, he eners ino a conrac A, C where A = 1 {=T } S T K +. We assume ha collaeral is re-hypohecaed and, as in [6], we denoe he cash in he collaeral accoun a ime by C. When C > hen he cash collaeral is received as a guaranee and remuneraed by he hedger a he rae c b and when C is negaive, hen he cash collaeral is posed by he hedger and remuneraed by he counerpary a he rae c l. We analyze he hedger s operaions wih he reasury, he repo marke and he counerpary, in order o fund his rade. The following seps in each small inerval [, + d] are presened from he poin of view of he hedger buying he opion. Le us firs describe he hedger s iniial rades a ime : 1. The hedger wishes o buy a call opion wih mauriy T whose curren price is V <. The price is negaive for he hedger since he is he buyer and hus has o pay he price a ime. He borrows V cash from he reasury and buys he call opion. 2. To hedge he call opion he bough, he akes he synheic shor posiion ξ < in he sock in he repo marke. Specifically, he borrows H = ξ S from he reasury and lends cash a repo marke and ges ξ > shares of he sock as collaeral. 3. He immediaely sells he sock jus obained from he repo o he marke, geing back H in cash and gives H back o he reasury. Hence his ousanding deb a ime o he reasury is V. 4. He poss when C = C < or receives when C = C + > he cash collaeral, which can be rehypohecaed in oher ransacions. We now proceed o he descripion of he hedger s erminal rades a ime + d: 1. To close he repo, he hedger needs o buy and deliver ξ sock. For his rade, he needs ξ S +d cash and hus he borrows ha amoun of cash from he reasury. He buys ξ sock and gives he sock back o close he repo. Hence he ges back he cash H deposied a ime wih ineres h H d, ha is, 1 + h dh cash. Thus he ne value of hese rades is 1 + h dh + ξ S +d = 1 + h dξ S + ξ S +d = ξ ds h ξ S d He sells he call opion in he marke and hus ges V +d cash recall ha V +d < for he buyer of he opion. He pays back his ousanding deb V o he reasury plus ineres f V d using he cash V +d jus obained, he ne effec being V +d + V 1 + f d = dv + f V d He eiher pays he ineres c b C + d or receives he ineres c l C d on he collaeral amoun C and pays he ineres f C d o he reasury or receives he ineres f C + d from he reasury. Hence he ne cash flow a ime + d due o he margin accoun equals dĉ := c l f C d c b f C + d The oal value of cash flows from equaions equals ξ ds h S d dv + f V d + dĉ = 1.4 where he equaliy is a consequence of he self-financing propery, which underpins he hedger s rades. Equaliies 1.3 and 1.4 will be formally esablished in Lemma 2.1.
3 Risk-neural valuaion under funding coss and collaeralizaion 3 Assume ha Q h is a probabiliy measure such ha E Q hds h S d F = or, more formally, ha he process S := B h 1 S is a local maringale under Q h where db h = h B h d wih B h = 1. Then we obain an explici represenaion under Q h for he price V for < T V = B f E Q h B f T 1 S T K + Bu f 1 dĉu F where db f = f B f d wih B f = 1. I is naural o refer o 1.5 as he exended risk-neural valuaion formula wih funding coss. To derive anoher version of he risk-neural valuaion formula, we now assume ha here exiss a risk-neural measure Q r associaed wih a locally risk-free bank accoun numeraire wih he risk-free ineres rae r, so ha E Q rds r S d F =. This is a formal posulae, which is saisfied in a ypical model, and i does no mean ha funding from he risk-free bank accoun is available o he hedger or ha he model a hand is arbirage-free in any sense. Then, using 1.4, we obain dv r V d = f r V d + ξ ds r S d h r H d + dĉ. Hence he price V admis also he following implici represenaion under Q r called he risk-naural valuaion formula wih adjused cash flows V =B r E Q r B r T 1 S T K + B r u 1 dĉu + + B r E Q r Bu r 1 h u r u H u du F where B r = r B r d wih B r = 1 and Ĉ equals see 1.3 Ĉ = c l u f u C u du Bu r 1 r u f u V u du F c b u f u C + u du. 1.7 The formal derivaion of 1.5 and 1.6 is given in Proposiion 2.1. I is clear ha 1.5 and 1.6 are equivalen and hey reduce o he classic risk-neural valuaion formula when f = h = r. Noe ha 1.6 coincides wih he resul given earlier in [6]. I will be more formally derived in Corollary 2.1 in a more general framework. Observe ha he financial inerpreaion of he process r was employed in he derivaion of 1.6 given in [6], bu in fac i is no relevan a all for he validiy of Exended Risk-Neural Valuaion Formulae We fix a finie rading horizon dae T > for our model of he financial marke. Le Ω, F, F, P be a filered probabiliy space saisfying he usual condiions of righ-coninuiy and compleeness, where he filraion F = F [,T ] models he flow of informaion available o all raders. For convenience, we assume ha he iniial σ-field F is rivial. Moreover, all processes inroduced in wha follows are implicily assumed o be F-adaped and, as usual, any semimaringale is assumed o be càdlàg. Also, we will assume ha any process Y saisfies Y := Y Y =. Le us inroduce he noaion for he prices of all raded asses in our model. Traded risky asses. We denoe by S 1,..., S d he collecion of he prices of a family of d risky asses, which do no pay dividends. We assume ha he processes S 1, S 2,..., S d are coninuous semimaringalea.
4 4 Brigo Buescu Francischello Pallavicini Rukowski Treasury raes. The lending respecively, borrowing cash accoun B l respecively, B b is used for unsecured lending respecively, borrowing of cash from he reasury. When he borrowing and lending cash raes are equal, we denoe he single cash accoun by B f. We assume ha db l = fb l l d, db b = f b B b d and db f = f B f d. Repo marke. We denoe by B i,l respecively B i,b he lending respecively borrowing repo accoun associaed wih he ih risky asse. In he special case when B i,l = B i,b, we will use he noaion B i. We assume ha db i,l = h i,l B i,l d, db i,b = h i,b B i,b d and db i = h i B i d. 2.1 Valuaion in a Linear Model wih Funding Coss We will now examine a special case of he linear model wih funding coss inroduced in [1]. We assume ha we have he cash funding accoun B f and d risky asses raded in he repo marke wih he asse prices S i and he corresponding as funding accoun B i for i = 1, 2,..., d. Recall from [1] ha he rading consrain ψb i i + ξs i i = means ha he posiions in sock are funded using exclusively he accoun B i wih he repo rae h i. The value process of a rading sraegy ϕ hus equals ϕ := ψ f B f + ψb i i + ξs i i = F + where F sands for he funding par and H = d Hi represens he hedging par of he value process ϕ, specifically, F := ψ f B f + H i ψb i, i H i := ξs i i = ψb i. i Under he posulae ha ψ i B i + ξ i S i =, we also have ϕ = ψ f B f and hus ϕ coincides wih he cash funding process F f := ψ f B f. Assuming ha A represens he cash flows sream also known as he dividends sream of a given conrac, he self-financing condiion inclusive of he sream of cash flows A reads ϕ = ψ f db f + = ψ f db f ψ i db i + ξ i ds i + da ξs i B i i 1 db i + where he second equaliy is he consequence of he rading consrain. ξ i ds i + da Le us now consider he case of a collaeralized conrac A, C. The margin accoun C is assumed o be any adaped process of finie variaion such ha C T =. We do no consider C o be a par of he hedger s rading sraegy, bu raher as a par of cash flows of a conrac. This is moivaed by wo reasons. Firs, he margin accoun C will always be presen no maer wheher a given conrac is hedged or no. Second, he process C is assumed here o be exogenously given, so i does no depend on he hedger s rading sraegy. Therefore, he value process of a rading sraegy ϕ is sill defined as he sum F + H, raher han F + H + C. A he same ime, we assume ha he cash collaeral C is rehypohecaed, ha is, i is used for he hedger s rading purposes and hus i is implici in F and H hrough he self-financing condiion
5 Risk-neural valuaion under funding coss and collaeralizaion 5 saed in Definiion 2.1. As in [1], he process ϕ := F + H is he value process of he hedger s rading sraegy, whereas he process V ϕ := ϕ C = F +H C represens he hedger s wealh under rehypohecaion alhough he symbols F and H were no used in [1]. Due o he erminal condiion C T =, we always have ha V T ϕ = T ϕ, bu V ϕ ϕ for < T, in general. In he case of a collaeralized conrac wih he margin process C and remuneraion raes c l and c b for he margin accoun, o compue he price and hedge for a collaeralized conrac, i suffices o replace he cash flow sream A by he process A C given by he following expression A C := A + C + = A + C + Cu Bu c,l 1 dbu c,l c l uc u du C u + Bu c,b 1 dbu c,b c b uc + u du. 2.1 Hence he conrac A, C can be formally idenified wih he cash flows sream A C Dynamics of he Value Process of a Trading Sraegy We will now examine he dynamics of he value process of a self-financing rading sraegy inclusive of he cash flows sream A C and he concep of replicaion of he conrac A C. The following definiion summarizes hese noions. Definiion 2.1 Assume ha a conrac has cash flows given by a process A C of finie variaion and a replicaing sraegy ϕ for he cash flow sream A C exiss, meaning ha here exiss a rading sraegy ϕ such ha V T ϕ = T ϕ = where V ϕ = ϕ C and he process ϕ saisfies he self-financing condiion ϕ = ψ f db f ξs i B i i 1 db i + ξ i ds i + da C 2.2 Then he ex-dividend price equals π A C = V ϕ = ϕ C for all [, T ]. Since db i = h i B i d, he self-financing condiion can be represened as follows ϕ = f ϕ d + ξds i i h i S i d + da C. 2.3 We are in a posiion o formally esablish equaions 1.3 and 1.4, which were informally posulaed in Secion 1 Lemma 2.1 Assume ha a single sock S is raded and he conrac A, C has null cash flows A before ime T. Then he self-financing condiion reduces o he following equaliy for he hedger s wealh process V ϕ dv ϕ = f V ϕ d + ξ ds h S d + dĉ 2.4 where Ĉ is given by 1.3. Proof. Equaliy 2.4 is an immediae consequence of equaions 2.1 and 2.3 and he relaionship V ϕ = ϕ C. Remark 2.1 We have implicily assumed ha he iniial endowmen of he hedger equals zero. This assumpion can be made here wihou loss of generaliy, since we deal wih a linear model, bu i is no longer rue when dealing wih a non-linear se-up examined in Secion 2.2. Neverheless, for simpliciy of presenaion, we will sill assume in Secion 2.2 ha he iniial endowmen of he hedger equals zero.
6 6 Brigo Buescu Francischello Pallavicini Rukowski Linear BSDE Recall ha replicaion of a conrac wih cash flows A C by a self-financing rading sraegy ϕ means ha V T ϕ = T ϕ =. Moreover, we assume ha ha he hedger s iniial endowmen is null. Using 2.3, we hus obain he following linear BSDE for he porfolio s value and hedging sraegy dy = f Y Zh i i S i d + Z i ds i + da C 2.5 wih he erminal condiion Y T =. Under mild echnical assumpions, he unique soluion o his BSDE exiss and i is given by an explici formula of course, provided ha he dynamics of S i are known. We hus henceforh assume ha he processes V ϕ and ϕ are well defined and we will examine heir properies Absrac Risk-Neural Valuaion Formulae Le us define noe ha, by convenion, we se B γi = 1 B γi := exp γu i du where γ i is an arbirary adaped and inegrable process. Le γ = γ 1, γ 2,..., γ d and le Q γ be a probabiliy measure such ha he process S i = B γi 1 S i is a Q γ local maringale. This is equivalen o he propery ha he process S i γ i us i u du is a Q γ local maringale for i = 1, 2,..., d, meaning ha Q γ is an equivalen local maringale measure ELMM for he asse price S i discouned wih he process B γi. Le ϕ be a self-financing rading sraegy, in he sense of Definiion 2.1, and le η be an arbirary adaped and inegrable process. We define B η := exp η u du. Lemma 2.2 Le η be an arbirary F-adaped process and le he process V η ϕ be given by V η ϕ := ϕ+b η α u F f u B η u 1 du+ B η β i uh i ub η u 1 du B η,] B η u 1 da C u. 2.6 If α = η f, β i = h i γ i, 2.7 hen he process V η := B η 1 V η is a local maringale under Q γ Proof. Equaion 2.6 implies ha dv η ϕ = ϕ + α F f d + β i H i d + V η η d da C.
7 Risk-neural valuaion under funding coss and collaeralizaion 7 Since = ψ f B f = F f, we obain dv η η V η d = ψ f db f = α + f η F f d + ξs i B i i 1 db i + β i h i + γh i i d + ξ i ds i + α F f d + ξ i ds i γs i i d. I is now clear ha if α and β saisfy 2.7, hen V η is a local maringale under Q γ. βh i i d η F f d The following resul is a simple consequence of Lemma 2.2. We sress ha he financial inerpreaion of processes η and γ i for i = 1, 2,..., d, if any, is no relevan in he derivaion of an absrac risk-neural valuaion formulae Proposiion 2.1 Assume ha a conrac A, C can be replicaed by a rading sraegy ϕ and he associaed process V η ϕ is a rue maringale under Q γ. Then he ex-dividend price of A, C saisfies π A, C = V ϕ = ϕ C where ϕ = B η E Q γ Bu η 1 da C u + η u f u Fu f Bu η 1 du F B η E Q γ,t ] h i u γuh i ub i u η 1 du F or, equivalenly, V ϕ = B η E Q γ Bu η 1 da u + c u f u C u Bu η 1 du F,T ] + B η E Q γ η u f u V u ϕbu η 1 du + h i u γ u HuB i u η 1 du F where we denoe 2.9 c := c l 1 {C<} + c b 1 {C }. 2.1 Proof. Equaliy 2.8 is an immediae consequence of he maringale propery of V η ϕ and he equaliies T ϕ = and ϕ = F f. To derive 2.9 from 2.8, we noe ha F C = c u C u du and we apply he inegraion by pars formula and he equaliy C T =, o ge Bu η 1 dc u = C T B η T 1 C B η 1 C u dbu η 1,T ] This complees he derivaion of 2.9. = C B η 1 + η u Bu η 1 C u du. As special cases, we obain equaions 1.5 and 1.6. To ge 1.5, i suffices o se η = f and γ i = h i for i = 1, 2,..., d so ha ϕ = B f E Q h Bu f 1 da C u F.,T ]
8 8 Brigo Buescu Francischello Pallavicini Rukowski Similarly, upon seing η = γ i = r, we obain he following formula ϕ = B r E Q r Bu η 1 da C u + r u f u Fu f Bu r 1 du F,T ] + B r E Q r h i u r u HuB i u r 1 du F, which can be referred o as he risk-neural valuaion wih adjused cash flows. 2.2 Valuaion in a Non-Linear Model wih Funding Coss A non-linear exension of he framework inroduced in he preceding secion is obained when a single cash rae f is replaced by he lending and borrowing raes, denoed as f l and f b and, similarly, by inroducing differen repo raes for he long and shor posiions in sock, denoed as h i,l and h i,b, respecively. Then we have he following generic decomposiion of he value process ϕ of a rading sraegy ϕ where in urn ϕ = ψ l B l + ψ b B i,b + F := ψ l B l + ψ b B b + ψ i,l B i,l ψ i,l + ψ i,b B i,b + ξs i i = F + B i,l H i = F + H + ψ i,b B i,b, H i := ξs i. i Moreover, we posulae ha ψ l, ψ b and ψψ l b = for all. Finally, we assume ha ψ i,l, ψ i,b, ψ i,l ψ i,b = and he following repo funding condiion holds ψ i,l B i,l + ψ i,b B i,b + ξs i i =. Consequenly, ϕ = F f := ψ l B l + ψ b B b. As before, we se V ϕ = ϕ C where C is he margin accoun Dynamics of he Value Process of a Trading Sraegy We are now in a posiion o derive he non-linear dynamics of he value process and hus also obain a generic non-linear pricing BSDE for he conrac A, C. Lemma 2.3 We have ψ l = B l 1 ϕ +, ψ b = B b 1 ϕ and ψ i,l Proof. Noe ha = B i,l 1 ξs i i = B i,l 1 H i, ψ i,b ψ l B l + ψ b B b = ϕ, ψ i,l B i,l = B i,b 1 ξs i i + = B i,b 1 H i +. + ψ i,b B i,b = ξs i. i I hus suffices o use he posulaed naural condiions ψ l, ψ b, ψψ l b = and ψ i,l, ψ i,b, ψ i,l ψ i,b =.
9 Risk-neural valuaion under funding coss and collaeralizaion 9 Using Lemma 2.3, we obain he unique dynamics for he value process of a self-financing rading sraegy ϕ inclusive of cash flows sream A C. We posulae ha and hus ϕ = ψ l db l + ψ b db b + ψ i,l db i,l ϕ = B l 1 ϕ + db l B b 1 ϕ db b + B i,b 1 ξ i S i + db i,b + + ψ i,b db i,b + ξ i ds i + da C 2.11 ξ i ds i + da C. B i,l 1 ξ i S i db i,l When all accoun processes are absoluely coninuous, we obain he following lemma. Lemma 2.4 The value process of a self-financing rading sraegy ϕ saisfies ϕ = f l ϕ + d f b ϕ d+ where H i = ξ i S i. ξ i ds i +h i,l H i d h i,b H i + d +da C To formally linearize he problem, we will now inroduce he effecive raes f and h i, which depend on he hedger s rading sraegy. Lemma 2.5 The value process ϕ saisfies ϕ = f F f d + where he effecive raes f and h i are given by ξ i ds i h i S i d + da C 2.13 f := f l 1 {F f } + f b 1 {F f ϕ<} = f l 1 {Vϕ+C } + f b 1 {Vϕ+C <} 2.14 and h i := h i,l 1 {H i } + h i,b 1 {H i ϕ>} Proof. From 2.12, we obain ϕ = f l ϕ + d f b ϕ d + = f F f ϕ d + ξ i ds i + h i,l H i d h i,b H i + d + da C ξ i ds i h i S i d + da C 2.16 where f and h i are given by 2.14 and 2.15, respecively.
10 1 Brigo Buescu Francischello Pallavicini Rukowski Nonlinear BSDE Under he sanding assumpion ha he iniial endowmen of he hedger is null, equaion 2.16 leads o he following nonlinear BSDE for he porfolio s value and he hedging sraegy dy = f Y h i ZS i i d + wih he erminal condiion Y T = where from Z i ds i + da C 2.17 f := f l 1 {Y } + f b 1 {Y<} and h i := h i,l 1 {Z i S i } + h i,b 1 {Z i S i>}. We henceforh assume ha BSDE 2.17 has a unique soluion in a suiable space of sochasic processes Absrac Risk-Neural Valuaion Formulae Recall ha and B η := exp η u du B γi := exp γu i du where η and γ i for i = 1, 2,..., d are arbirary adaped and inegrable processes. Le Q γ be a probabiliy measure such ha he processes B γi 1 S i, i = 1, 2,..., d are Q γ -local maringales. Then Lemmas 2.2 and 2.5 yield he following resul, which exends Proposiion 2.1 and agrees wih he pricing formula posulaed in [3]. For he validiy of his resul, one needs o impose some mild inegrabiliy assumpions. Proposiion 2.2 Assume ha a collaeralized conrac A, C can be replicaed by a rading sraegy ϕ. Then is ex-dividend price a ime equals V ϕ = ϕ C where ϕ = B η E Q γ Bu η 1 da C u + η u f u Fu f Bu η 1 du F + B η E Q γ,t ] h i u γuh i ub i u η 1 du F or, equivalenly, V ϕ = B η E Q γ Bu η 1 da u + c u f u C u Bu η 1 du F 2.18,T ] + B η E Q γ η u f u V u ϕbu η 1 du + h i u γuh i ub i u η 1 du F where c := c l 1 {C<} + c b 1 {C }. Proof. The firs assered formula follows from The second is easy consequences of he firs one.
11 Risk-neural valuaion under funding coss and collaeralizaion 11 References [1] Bielecki, T. R. and Rukowski, M.: Valuaion and hedging of conracs wih funding coss and collaeralizaion. SIAM Journal on Financial Mahemaics 6 215, [2] Brigo, D., Liu, Q., Pallavicini, A., and Sloh, D.: Nonlineariy valuaion adjusmen. In: Grbac, Z., Glau, K, Scherer, M., and Zags, R. Eds, Innovaion in Derivaives Markes Fixed Income Modeling, Valuaion Adjusmens, Risk Managemen, and Regulaion. Springer, 216. [3] Brigo, D., Francischello, M., and Pallavicini, A.: Analysis of nonlinear valuaion equaions under credi and funding effecs. In: Grbac, Z., Glau, K, Scherer, M., and Zags, R. Eds, Innovaions in Derivaives Markes Fixed Income Modeling, Valuaion Adjusmens, Risk Managemen, and Regulaion. Springer, Heidelberg, 216. [4] Brigo, D. and Pallavicini A.: Nonlinear consisen valuaion of CCP cleared or CSA bilaeral rades wih iniial margins under credi, funding and wrong-way risks. Journal of Financial Engineering , 1 6. [5] Burgard, C. and Kjaer, M.: Parial differenial equaion represenaions of derivaives wih bilaeral counerpary risk and funding coss. The Journal of Credi Risk , [6] Pallavicini, A., Perini, D. and Brigo, D.: Funding, collaeral and hedging: uncovering he mechanics and he subleies of funding valuaion adjusmens. Working paper, December 13, 212.
An 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 informationArbitrage-Free Pricing Of Derivatives In Nonlinear Market Models
Arbirage-Free Pricing Of Derivaives In Nonlinear Marke Models Tomasz R. Bielecki a, Igor Cialenco a, and Marek Rukowski b Firs Circulaed: January 28, 217 This Version: April 4, 218 Forhcoming in Probabiliy,
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 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 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 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 informationFINM 6900 Finance Theory
FINM 6900 Finance Theory Universiy of Queensland Lecure Noe 4 The Lucas Model 1. Inroducion In his lecure we consider a simple endowmen economy in which an unspecified number of raional invesors rade asses
More informationOn the Timing Option in a Futures Contract
On he Timing Opion in a Fuures Conrac Francesca Biagini, Mahemaics Insiue Universiy of Munich Theresiensr. 39 D-80333 Munich, Germany phone: +39-051-2094459 Francesca.Biagini@mahemaik.uni-muenchen.de Tomas
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 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 informationMean-Variance Hedging for General Claims
Projekbereich B Discussion Paper No. B 167 Mean-Variance Hedging for General Claims by Marin Schweizer ) Ocober 199 ) Financial suppor by Deusche Forschungsgemeinschaf, Sonderforschungsbereich 33 a he
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationDEFAULTABLE GAME OPTIONS IN A HAZARD PROCESS MODEL
DEFAULTABLE GAME OPTIONS IN A HAZARD PROCESS MODEL Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Séphane Crépey Déparemen de Mahémaiques Universié
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationarxiv: v2 [math.pr] 8 Dec 2014
arxiv:141.449v2 [mah.pr] 8 Dec 214 BSDEs DRIVEN BY A MULTI-DIMENSIONAL MARTINGALE AND THEIR APPLICATIONS TO MARKET MODELS WITH FUNDING COSTS Tianyang Nie and Marek Rukowski School of Mahemaics and Saisics
More informationGeneralized Snell envelope and BSDE With Two general Reflecting Barriers
1/22 Generalized Snell envelope and BSDE Wih Two general Reflecing Barriers EL HASSAN ESSAKY Cadi ayyad Universiy Poly-disciplinary Faculy Safi Work in progress wih : M. Hassani and Y. Ouknine Iasi, July
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 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 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 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 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 informationModelling of the Defaultable Term Structure: Conditionally Markov Approach
Modelling of he Defaulable Term Srucure: Condiionally Markov Approach Tomasz R. Bielecki Marek Rukowski Absrac The paper provides a deailed echnical descripion of he Bielecki and Rukowski 2a,b approach
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 informationVALUATION AND HEDGING OF DEFAULTABLE GAME OPTIONS IN A HAZARD PROCESS MODEL
VALUATION AND HEDGING OF DEFAULTABLE GAME OPTIONS IN A HAZARD PROCESS MODEL Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Séphane Crépey Déparemen
More informationA general continuous auction system in presence of insiders
A general coninuous aucion sysem in presence of insiders José M. Corcuera (based on join work wih G. DiNunno, G. Farkas and B. Oksendal) Faculy of Mahemaics Universiy of Barcelona BCAM, Basque Cener for
More informationBackward stochastic dynamics on a filtered probability space
Backward sochasic dynamics on a filered probabiliy space Gechun Liang Oxford-Man Insiue, Universiy of Oxford based on join work wih Terry Lyons and Zhongmin Qian Page 1 of 15 gliang@oxford-man.ox.ac.uk
More informationLecture Notes 2. The Hilbert Space Approach to Time Series
Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship
More informationMartingales Stopping Time Processes
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765. Volume 11, Issue 1 Ver. II (Jan - Feb. 2015), PP 59-64 www.iosrjournals.org Maringales Sopping Time Processes I. Fulaan Deparmen
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 informationProblem 1 / 25 Problem 2 / 20 Problem 3 / 10 Problem 4 / 15 Problem 5 / 30 TOTAL / 100
eparmen of Applied Economics Johns Hopkins Universiy Economics 602 Macroeconomic Theory and Policy Miderm Exam Suggesed Soluions Professor Sanjay hugh Fall 2008 NAME: The Exam has a oal of five (5) problems
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 informationAffine credit risk models under incomplete information
Affine credi risk models under incomplee informaion Rüdiger Frey 1, Cecilia Prosdocimi 2, Wolfgang J. Runggaldier 2 1 Deparmen of Mahemaics, Universiy of Leipzig 04081 Leipzig, Germany (ruediger.frey@mah.uni-leipzig.de).
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 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 informationArbitrage-Free XVA. August 9, 2016
Arbirage-Free XVA Maxim Bichuch Agosino Capponi Sephan Surm Augus 9, 216 Absrac We develop a ramework or compuing he oal valuaion adjusmen XVA o a European claim accouning or unding coss, counerpary credi
More informationDEFAULTABLE OPTIONS IN A MARKOVIAN INTENSITY MODEL OF CREDIT RISK
DEFAULTABLE OPTIONS IN A MARKOVIAN INTENSITY MODEL OF CREDIT RISK Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Séphane Crépey Déparemen de Mahémaiques
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 informationINSIDER INFORMATION, ARBITRAGE AND OPTIMAL PORTFOLIO AND CONSUMPTION POLICIES
INSIDER INFORMATION, ARBITRAGE AND OPTIMAL PORTFOLIO AND CONSUMPTION POLICIES Marcel Rindisbacher Boson Universiy School of Managemen January 214 Absrac This aricle exends he sandard coninuous ime financial
More informationDynamic Conic Finance via Backward Stochastic Difference Equations
Dynamic Conic Finance via Backward Sochasic Difference Equaions Tomasz R. Bielecki bielecki@ii.edu Igor Cialenco igor@mah.ii.edu Deparmen of Applied Mahemaics, Illinois Insiue of Technology, Chicago, 60616
More informationOptimal Investment under Dynamic Risk Constraints and Partial Information
Opimal Invesmen under Dynamic Risk Consrains and Parial Informaion Wolfgang Puschögl Johann Radon Insiue for Compuaional and Applied Mahemaics (RICAM) Ausrian Academy of Sciences www.ricam.oeaw.ac.a 2
More informationOnline Appendix to Solution Methods for Models with Rare Disasters
Online Appendix o Soluion Mehods for Models wih Rare Disasers Jesús Fernández-Villaverde and Oren Levinal In his Online Appendix, we presen he Euler condiions of he model, we develop he pricing Calvo block,
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 informationSatisfying Convex Risk Limits by Trading
Saisfying Convex Risk Limis by rading Kasper Larsen Deparmen of Accouning and Finance Deparmen of Mahemaics and Compuer Science Universiy of Souhern Denmark DK-53 Odense M kla@sam.sdu.dk raian Pirvu Deparmen
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 informationRANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY
ECO 504 Spring 2006 Chris Sims RANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY 1. INTRODUCTION Lagrange muliplier mehods are sandard fare in elemenary calculus courses, and hey play a cenral role in economic
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 informationResearch Article Defaultable Game Options in a Hazard Process Model
Hindawi Publishing Corporaion Journal of Applied Mahemaics and Sochasic Analysis Volume 29, Aricle ID 695798, 33 pages doi:1.1155/29/695798 Research Aricle Defaulable Game Opions in a Hazard Process Model
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 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 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 informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationOptimal Consumption and Investment Portfolio in Jump markets. Optimal Consumption and Portfolio of Investment in a Financial Market with Jumps
Opimal Consumpion and Invesmen Porfolio in Jump markes Opimal Consumpion and Porfolio of Invesmen in a Financial Marke wih Jumps Gan Jin Lingnan (Universiy) College, China Insiue of Economic ransformaion
More informationarxiv: v1 [math.ca] 15 Nov 2016
arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy
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 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 informationRobust XVA. August 14, 2018
Robus XVA Maxim Bichuch Agosino Capponi Sephan Surm Augus 14, 2018 Absrac We inroduce an arbirage-free framework for robus valuaion adjusmens. An invesor rades a credi defaul swap porfolio wih a defaulable
More informationGeneralized Chebyshev polynomials
Generalized Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, 39 86 Roma, Ialy email: c.cesarano@unineunouniversiy.ne ABSTRACT
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 informationLocal Strict Comparison Theorem and Converse Comparison Theorems for Reflected Backward Stochastic Differential Equations
arxiv:mah/07002v [mah.pr] 3 Dec 2006 Local Sric Comparison Theorem and Converse Comparison Theorems for Refleced Backward Sochasic Differenial Equaions Juan Li and Shanjian Tang Absrac A local sric comparison
More informationRisk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles
Finance and Sochasics manuscrip No. (will be insered by he edior) Risk assessmen for uncerain cash flows: Model ambiguiy, discouning ambiguiy, and he role of bubbles Bearice Acciaio Hans Föllmer Irina
More information1. Consider a pure-exchange economy with stochastic endowments. The state of the economy
Answer 4 of he following 5 quesions. 1. Consider a pure-exchange economy wih sochasic endowmens. The sae of he economy in period, 0,1,..., is he hisory of evens s ( s0, s1,..., s ). The iniial sae is given.
More informationNumerical Approximation of Partial Differential Equations Arising in Financial Option Pricing
Numerical Approximaion of Parial Differenial Equaions Arising in Financial Opion Pricing Fernando Gonçalves Docor of Philosophy Universiy of Edinburgh 27 (revised version) To Sílvia, my wife. Declaraion
More informationAcademic Editor: Mogens Steffensen Received: 24 September 2017; Accepted: 16 October 2017; Published: 23 October 2017
risks Aricle Opional Defaulable Markes Mohamed N. Abdelghani 1, *, and Alexander V. Melnikov 2, 1 Machine Learning, Morgan Sanley, New York Ciy, NY 119, USA 2 Mahemaical and Saisical Sciences, Universiy
More informationand Applications Alexander Steinicke University of Graz Vienna Seminar in Mathematical Finance and Probability,
Backward Sochasic Differenial Equaions and Applicaions Alexander Seinicke Universiy of Graz Vienna Seminar in Mahemaical Finance and Probabiliy, 6-20-2017 1 / 31 1 Wha is a BSDE? SDEs - he differenial
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 informationCounterparty Risk and Funding: Immersion and Beyond
Noname manuscrip No. will be insered by he edior Counerpary Risk and Funding: Immersion and Beyond Séphane Crépey Shiqi Song he dae of receip and accepance should be insered laer Absrac In Crépey 215,
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 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 informationOn Option Pricing by Quantum Mechanics Approach
On Opion Pricing by Quanum Mechanics Approach Hiroshi Inoue School of Managemen okyo Universiy of Science Kuki, Saiama 46-851 Japan e-mail:inoue@mskukiusacp Absrac We discuss he pah inegral mehodology
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 information2 Models of Forward LIBOR and Swap Raes 2 Preliminaries Le T > be a horizon dae for our model of economy. In his preliminary secion, we focus on LIBOR
Models of Forward LIBOR and Swap Raes MAREK RUTKOWSKI Insyu Maemayki, Poliechnika Warszawska, -66 Warszawa, Poland Absrac The backward inducion approach is sysemaically used o produce various models of
More informationAsymptotic instability of nonlinear differential equations
Elecronic Journal of Differenial Equaions, Vol. 1997(1997), No. 16, pp. 1 7. ISSN: 172-6691. URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp (login: fp) 147.26.13.11 or 129.12.3.113 Asympoic insabiliy
More informationA New Approach for Solving Boundary Value Problem in Partial Differential Equation Arising in Financial Market
Applied Mahemaics, 6, 7, 84-85 Published Online May 6 in cires. hp://.scirp.org/journal/am hp://dx.doi.org/.436/am.6.7975 A Ne Approach for olving Boundary Value Problem in Parial Differenial Equaion Arising
More informationNotes for Lecture 17-18
U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up
More informationCooperative Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS. August 8, :45 a.m. to 1:00 p.m.
Cooperaive Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS Augus 8, 213 8:45 a.m. o 1: p.m. THERE ARE FIVE QUESTIONS ANSWER ANY FOUR OUT OF FIVE PROBLEMS.
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 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 information1 Consumption and Risky Assets
Soluions o Problem Se 8 Econ 0A - nd Half - Fall 011 Prof David Romer, GSI: Vicoria Vanasco 1 Consumpion and Risky Asses Consumer's lifeime uiliy: U = u(c 1 )+E[u(c )] Income: Y 1 = Ȳ cerain and Y F (
More informationHazard rate for credit risk and hedging defaultable contingent claims
Finance Sochas. 8, 45 59 24 DOI:.7/s78-3-8- c Springer-Verlag 24 Hazard rae for credi risk and hedging defaulable coningen claims Chrisophee Blanche-Scallie, Monique Jeanblanc 2 C.B. Laboraoire J.A.Dieudonne,
More informationSeminar 4: Hotelling 2
Seminar 4: Hoelling 2 November 3, 211 1 Exercise Par 1 Iso-elasic demand A non renewable resource of a known sock S can be exraced a zero cos. Demand for he resource is of he form: D(p ) = p ε ε > A a
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 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 informationNonlinear expectations and nonlinear pricing
Nonlinear expecaions and nonlinear pricing Zengjing Chen Kun He Deparmen of Mahemaics Shandong Universiy, Jinan 250100 China Reg Kulperger Deparmen of saisical and acuarial Science The Universiy of Wesern
More informationANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c
John Riley December 00 S O EVEN NUMBERED EXERCISES IN CHAPER 6 SECION 6: LIFE CYCLE CONSUMPION AND WEALH Eercise 6-: Opimal saving wih more han one commodiy A consumer has a period uiliy funcion δ u (
More informationMacroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3
Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has
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 informationLocal risk minimizing strategy in a market driven by time-changed Lévy noises. Lotti Meijer Master s Thesis, Autumn 2016
Local risk minimizing sraegy in a marke driven by ime-changed Lévy noises Loi Meijer Maser s Thesis, Auumn 216 Cover design by Marin Helsø The fron page depics a secion of he roo sysem of he excepional
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 informationarxiv: v1 [math.pr] 21 May 2010
ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES, AND PASSAGE TIME PROBLEMS arxiv:15.498v1 [mah.pr 21 May 21 GERARDO HERNÁNDEZ-DEL-VALLE Absrac. In his work we relae he densiy of he firs-passage
More informationarxiv: v1 [math.pr] 6 Oct 2008
MEASURIN THE NON-STOPPIN TIMENESS OF ENDS OF PREVISIBLE SETS arxiv:8.59v [mah.pr] 6 Oc 8 JU-YI YEN ),) AND MARC YOR 3),4) Absrac. In his paper, we propose several measuremens of he nonsopping imeness of
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 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 informationIMPLICIT AND INVERSE FUNCTION THEOREMS PAUL SCHRIMPF 1 OCTOBER 25, 2013
IMPLICI AND INVERSE FUNCION HEOREMS PAUL SCHRIMPF 1 OCOBER 25, 213 UNIVERSIY OF BRIISH COLUMBIA ECONOMICS 526 We have exensively sudied how o solve sysems of linear equaions. We know how o check wheher
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 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 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 informationSupplementary Material
Dynamic Global Games of Regime Change: Learning, Mulipliciy and iming of Aacks Supplemenary Maerial George-Marios Angeleos MI and NBER Chrisian Hellwig UCLA Alessandro Pavan Norhwesern Universiy Ocober
More informationStochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management
Sochasic conrol under progressive enlargemen of filraions and applicaions o muliple defauls risk managemen Huyên PHAM Laboraoire de Probabiliés e Modèles Aléaoires CNRS, UMR 7599 Universié Paris 7 e-mail:
More informationOption pricing for a partially observed pure jump price process
Opion pricing for a parially observed pure jump price process Paola Tardelli Deparmen of Elecrical and Informaion Engineering Universiy of L Aquila Via Giovanni Gronchi n. 18, 674 Poggio di Roio (AQ),
More informationKEY. Math 334 Midterm III Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Fall 28 secions and 3 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More information