About the Pricing Equation in Finance

Transcription

1 Abou he Pricing Equaion in Finance Séphane Crépey Déparemen de Mahémaique Univerié d Évry Val d Eonne Évry Cedex, France Thi verion: July 10, Inroducion In [16], we conruced a raher generic Markovian model Jump Diffuion Seing wih Regime and, uing abrac reul of [18], we howed ha relaed Markovian refleced and doubly refleced BSDE are well-poed in he ene ha hey have unique oluion, which depend coninuouly on heir inpu daa. In hi paper we derive he relaed variaional inequaliy approach, working in a uiable pace of vicoiy oluion o he aociaed yem of parial inegro-differenial obacle problem. An alernaive weak Sobolev oluion approach will be deal wih in [17]. 1.1 Ouline of he Paper In Secion 2 we fix he general e-up, noaion and definiion of he refleced BSDE problem o be conidered in hi paper. In Secion 3 we preen he Jump Diffuion Seing wih Regime model X conruced in [16], and we define a relaed Markovian decoupled Forward Backward SDE FBSDE, for hor. Thi FBSDE i moivaed by numerou applicaion in financial modeling whence he ile of hi paper, ee e.g. [13, 11, 16, 18, 17, 7]. The conrucion of a model wih he required properie and he reoluion of he relaed BSDE are reaed in deail by a uiable change of meaure approach in [16]. In hi paper, we work under he anding Aumpion 3.5 ha he FBSDE of inere ha a oluion wih uiable coninuiy and Markov properie a i i for inance he cae under he aumpion of [16], and we derive he relaed variaional inequaliy approach. In Secion 4 we inroduce he yem of parial inegro-differenial variaional inequaliie formally aociaed o our refleced BSDE problem, and we ae uiable definiion of diconinuou vicoiy emioluion and oluion for hee problem. In Secion 5 we how ha he ae-procee fir componen Y of he oluion o our refleced BSDE can be characerized in erm of he value funcion o relaed opimal opping or game problem, given a vicoiy oluion wih polynomial growh o he relaed obacle problem. We eablih in Secion 6 a diconinuou vicoiy emioluion comparion principle or maximum principle for our problem, which implie in paricular uniquene of vicoiy oluion for hee problem. Thi maximum principle i alo ued in Secion 7 for proving he convergence of able, monoone and conien approximaion cheme cf. Barle and Souganidi or [6] Briani, La Chioma and Naalini [14] o our value funcion. The relaed reul hu exend o model wih regime whence yem of PIDE he reul of [6, 14], among oher. To he be of our knowledge, he laer reul are he fir convergence reul in he lieraure regarding approximaion cheme of vicoiy oluion o

2 2 Abou he Pricing Equaion in Finance yem of PIDE. In Secion 8 we provide an exenion of our reul needed for applicaion in finance ee [11] in he cae where he running co funcion g in he problem i no coninuou on he whole pace, bu only on pecific ime-rip pariioning he whole pace. 2 Abrac Se-Up We fir recall he general e-up of [16]. Le u hu be given an iniial ime convenionally aken in hi Secion a 0, a finie ime horizon T > 0, a probabiliy pace Ω, F, P and a filraion F = F [0,T ] wih F T = F, aifying he uual condiion of righ-coninuiy and compleene. By defaul we declare ha a random variable i F-meaurable, and ha a proce i defined on he ime inerval [0, T ] and F-adaped. All emimaringale are aumed o be càdlàg, wihou rericion. Le B = B [0,T ] be a d-dimenional Brownian moion. Given an auxiliary meaured pace E, B E, ρ, where ρ i a non-negaive σ-finie meaure on E, B E, le µ = µde, d [0,T ],e E be an ineger valued random meaure on [0, T ] E, B[0, T ] B E ee Jacod Shiryaev [23, Definiion II.1.13 p. 68]. Denoing by P he predicable igma field on Ω [0, T ] and eing P = P B E, we aume ha he compenaor of µ i defined by ζ ω, eρded, for a P-meaurable non-negaive bounded random ineniy funcion ζ. We refer he reader o he lieraure [23, 8] regarding he definiion of he inegral proce of P-meaurable inegrand wih repec o random meaure uch a µde, d or i compenarix compenaed meaure µde, d = µde, d ζ ω, eρed. By defaul in he equel, all inequaliie beween random quaniie are o be underood dp almo urely, dp d almo everywhere or dp ρde d almo everywhere, a uiable in he iuaion a hand. For impliciy we omi all dependence in ω of any proce or random funcion in he noaion. We denoe by: X, he d-dimenional Euclidean norm of a vecor or row vecor X in or R 1 d ; M = up MX over he uni ball of, for M in d ; M ρ = ME, B E, ρ; R, he e of meaurable funcion from E, B E, ρ o R endowed wih he opology of convergence in meaure, and for v M ρ and [0, T ] : v = [ ve 2 ζ eρde ] 1 2 R + {+ } ; 1 E BO, he Borel igma field on O, for any opological pace O. Le u now inroduce ome Banach or Hilber, in cae of H 2 d or H2 µ pace of procee or random funcion, for a real p aumed o lie in [2, in all he paper: L p, he pace of real valued F T -meaurable random variable ξ uch ha ξ L p := [ E ξ p] 1 p < + ; S p d, for any real p 2 or Sp, in cae d = 1, he pace of -valued càdlàg procee X uch ha [ X S p := E d up [0,T ] X p] 1 p < + ; H p d or Hp, in cae d = 1, he pace of R 1 d -valued predicable procee Z uch ha [ T ] p 1 Z H p := E Z d 2 2 p d 0 < + ;

3 S. Crépey 3 H p µ, he pace of P-meaurable funcion V : Ω [0, T ] E R uch ha V H p µ := [ T ] 1 E V e p p ζ eρded 0 E < +, o in paricular cf. 1 V H 2 µ = [ T ] 1 E V 2 2 d ; 0 A 2, he pace of finie variaion coninuou procee K wih coninuou and non decreaing Jordan componen K ± S 2 null a ime 0; A 2 i, he pace of non-decreaing procee in A2. So in paricular: K = K + K and K ± define muually ingular meaure on R +, for any K A 2 ; Z db and V e µde, d are rue maringale, for any Z Hd 2 and V H2 µ. 0 0 E Noe ha we hall alo abuively ue he noaion X H p for [ T E 0 ] p 1 X 2 2 p d merely progreively meaurable no necearily predicable real-valued procee X. in he cae of 2.1 Abrac BSDE Le u now be given a erminal condiion ξ, and a driver coefficien g : Ω [0, T ] R R 1 d M ρ R, uch ha: H.0 ξ L 2 ; H.1.i g y, z, v i a progreively meaurable proce, and g y, z, v H 2 <, for any y R, z R 1 d, v M ρ ; H.1.ii g i uniformly Λ Lipchiz coninuou wih repec o y, z, v, in he ene ha Λ i a conan uch ha for every [0, T ], y, y R, z, z R 1 d, v, v M ρ : idenically. g y, z, v g y, z, v Λ y y + z z + v v, Remark 2.1 Given he Lipchiz coninuiy propery of g in H.1.ii, he requiremen ha g y, z, v H 2 < for any y R, z R 1 d, v M ρ in H.1.i reduce of coure o g 0, 0, 0 H 2 <. We alo inroduce he barrier or obacle L and U uch ha: H.2.i L and U are càdlàg quai-lef coninuou procee in S 2 ; H.2.ii L U, [0, T and L T ξ U T, P-a.. Regarding H.2.i, we recall ha for a càdlàg proce X, quai-lef coninuiy i equivalen o he exience of equence of oally inacceible opping ime which exhau he jump of X, whence p X = X Jacod Shiryaev [23, Propoiion I.2.26 p. 22 and I.2.35 p. 25]. Definiion 2.2 a An Ω, F, P, B, µ-oluion Y o he doubly refleced backward ochaic differenial equaion R2BSDE for hor wih daa g, ξ, L, U i a quadruple Y = Y, Z, V, K, uch

4 4 Abou he Pricing Equaion in Finance ha: i Y S 2, Z H 2 d, V H2 µ, K A 2, ii Y = ξ + T T iii L Y U and T 0 g Y, Z, V d + K T K T Z db V e µde, d for any [0, T ], P-a.. Y L dk + = E for any [0, T ], P-a.., T 0 U Y dk = 0, P-a.. b An Ω, F, P, B, µ-oluion Y o he refleced BSDE RBSDE, for hor wih daa g, ξ, L i a quadruple Y = Y, Z, V, K uch ha: i Y S 2, Z H 2 d, V H2 µ, K A 2 i ii Y = ξ + iii L Y and T 0 T T g Y, Z, V d + K T K T Z db V e µde, d for any [0, T ], P-a.. E for any [0, T ], P-a.., Y L dk = 0, P-a.. c Auming ha ξ i F τ -meaurable for ome [0, T ]-valued opping ime τ, a oluion o he opped RBSDE wih daa g, ξ, L, τ i a quadruple Y, Z, V, K which olve he RBSDE wih daa 1 τ g, ξ, L τ, and uch ha Y i conan on [τ, T ] whence K conan and Z, V = 0 on [τ, T ], ee [18]. d When here i no barrier, we define likewie oluion o he BSDE wih daa g, ξ, or, in cae when ξ i F τ -meaurable for ome [0, T ]-valued opping ime τ, oluion o he opped BSDE wih daa g, ξ, τ. Remark 2.3 i All hee definiion admi obviou exenion o problem in which he driving erm conain a furher finie variaion proce A no necearily aboluely coninuou. ii Acually he problem we are mo inereed in from he poin of view of applicaion coni of a more general τ-r2bsde ee [16, 11, 9]. However i come ou from he reul of [16] ha he oluion of uch a τ-r2bsde i eenially given a he oluion of a relaed opped RBSDE before τ, appropriaely paed a τ wih he oluion of a relaed andard R2BSDE afer τ. So we effecively reduced in [16] he udy of τ-r2bsde o hoe of refleced RBSDE and R2BSDE. Thi i he reaon why in hi paper we do no need o formally inroduce τ-r2bsde even if our reul are applicable o τ-r2bsde, giving in paricular way o compue heir oluion in wo piece, before and afer τ, by he reul of [16]. 3 Markovian decoupled Forward Backward SDE Given ineger d and k, we define he following linear operaor G acing on regular funcion u = u i, x for, x, i E = [0, T ] I wih I = {1,, k}, and where u rep. Hu denoe he

5 S. Crépey 5 row-gradien rep. Heian of u, x, i = u i, x wih repec o x : Gu i, x = u i, x Tr[ai, xhu i, x] + u i, xβ i, x 2 + u i, x + δ i, x, y u i, x f i, x, ymdy + λ i,j, x u j, x u i, x j I wih β i, x = b i, x δ i, x, yf i, x, ymdy. 3 Aumpion 3.1 In 2 3, mdy i a finie jump meaure no charging {0}, and all he coefficien are Borel meaurable funcion uch ha: he a i, x are d-dimenional covariance marice, wih a i, x = σ i, xσ i, x T, for ome d- dimenional diperion marice σ i, x; he b i, x are d-dimenional drif vecor coefficien; he ineniy funcion f i, x, y are bounded, and he jump ize funcion δ i, x, y are bounded w.r. y a fixed, x, locally uniformly in, x in he ene ha he bound w.r. y may be choen uniformly a, x varie in a compac e; he [λ i,j, x] i,j I are ineniy marice uch ha he funcion λ i,j, x are non-negaive and bounded for i j, and λ i,i, x = j I\{i} λ i,j, x. We hall ofen find convenien o denoe v, x, i, raher han v i, x, for a funcion v of, x, i,, and λ, x, i, j, for λ i,j, x. For inance, he noaion f, X, N, y or even f, X, y, wih X = X, N below will ypically be ued raher han f N, X, y. Alo noe ha a funcion u on [0, T ] I i equivalenly referred o in he equel a a yem u = u i i I of funcion u i = u i, x on [0, T ]. Definiion 3.2 A model wih generaor G and iniial condiion, x, i i a riple Ω, F, P, B, χ, ν, X = X, N, where he upercrip and in reference o he iniial condiion, x, i E, uch ha Ω, F, P i a ochaic bai on [, T ], relaive o which he following procee and random meaure are defined: i A d-dimenional andard Brownian moion B aring a, and ineger-valued random meaure χ on [, T ] and ν on [, T ] I, uch ha χ and ν canno jump ogeher a opping ime; ii An I-valued proce X = X, N on [, T ] wih iniial condiion x, i a and uch ha for [, T ] : { dn = j I j N dνj dx = b, Xd + σ, X db + δ, X R,, y χ 4 dy, d. d hu in paricular νj coun he number of raniion of N o ae j beween ime and, and he P -compenarice of ν and χ are uch ha { d ν j = dνj 1 {j N } λ, X, j d χ dy, d = χ dy, d f, X, 5 ymdyd wih λ, X, j = λ N,j, X, f, X, y = f N, X, y.

6 6 Abou he Pricing Equaion in Finance The conrucion of a model wih uch muual dependence beween N and X i acually a non-rivial iue. I i reaed in deail in Crépey [16], reoring o a uiable Markovian change of probabiliy approach. Remark 3.3 i If Ω, F, P, B, χ, ν, X = X, N i a model wih generaor G, we have he following varian of he Iô formula ee, e.g., Jacod [22, Theorem 3.89 p. 109]: du, X = Gu, Xd + uσ, XdB + u, X + δ, X, y, N u, X χ dy, d + j I u, X, j u, X d ν j, [, T ] 6 for any yem u = u i i I of funcion u i = u i, x of cla C 1,2 on [0, T ]. In paricular Ω, F, P, X i a oluion o he ime-dependen local maringale problem wih generaor G and iniial condiion, x, i ee Ehier Kurz [19, ecion 7.A and 7.B]. ii The pure jump proce N define he o-called regime of he coefficien b, σ, δ and f, whence he name of Jump Diffuion Seing wih Regime for hi model. We refer he reader o [16] for all commen on hi model. In paricular, hi model admi veraile applicaion in financial modeling ee Crépey e al. [16, 18], Bielecki e al. [11, 13]. More pecific ub-cae or relaed model are numerou in he lieraure. So: Barle e al. [4] and Harraj e al. [21] conider jump in X wihou regime N, bu for a general Lévy jump meaure mdy; Pardoux e al. [26] conider a diffuion model wih regime, which correpond o he pecial cae of our model in which f i equal o 0, and he regime are driven by a andard Poion proce wih conan ineniy; Becherer and Schweizer [7] conider a diffuion model wih regime which correpond o he pecial cae of our model in which f i equal o 0. For impliciy we do no conider he infinie aciviy cae, ha i, he cae when he Lévy jump meaure f m i infinie. Noe however ha our approach could be exended o general Lévy jump meaure wihou major change if wihed ye a he co of a ignificanly heavier formalim in he cae of unbounded meaure, ee he eminal paper by Barle e al. [4], recenly complemened by Barle and Imber [5]. We pecify henceforh he noaion of Secion 2 o he preen model wih uing he noaion inroduced in [18, 17], o which we refer he reader for he deailed erm-by-erm correpondence beween he model X and he abrac e-up of Secion 2: E, B E, ρ = I, B B I, mdy 1, where B I i he igma field of all par on I, and µ = χ ν on [0, T ] E, B[0, T ] B E, viewed a a random meaure relaive o he ochaic bai Ω, F, P. Noe ha H 2 µ can be idenified wih he produc pace H 2 χ H 2 ν, and ha M ρ = ME, B E, ρ; R can be idenified wih he produc pace M, B, mdy; R R k. Thee idenificaion will be ued freely in he equel. Le ṽ denoe a generic pair v, w M ρ M, B, mdy; R R k. We denoe accordingly, for cf. 1: ṽ 2 = vy 2 f, X, ymdy + j I\{N } wj2 λ, X, jd. 7 We denoe by P q he cla of funcion u on E uch ha u i i Borel-meaurable wih polynomial growh of exponen q 0 in x, for any i I. Here by polynomial growh of exponen q in x we

7 S. Crépey 7 mean he exience of a conan C which may depend on u uch ha for any, x, i E : u i, x C1 + x q. Le alo P = P q denoe he cla of funcion u on E uch ha u i i Borel-meaurable wih polynomial growh in x for any i I. Le u furher be given a yem C of real-valued coninuou co funcion, namely a running co funcion g i, x, u, z, r where u, z, r R k R 1 d R, a erminal co funcion Ψ i x, and lower and upper co funcion l i, x and h i, x, uch ha: M.0 Ψ lie in P q ; M.1.i, x, i g i, x, u, z, r lie in P q, for any u, z, r R k R 1 d R; M.1.ii g i uniformly Λ Lipchiz coninuou wih repec o u, z, r, in he ene ha Λ i a conan uch ha for every for any, x, i E and u, z, r, u, z, r R k R 1 d R : g i, x, u, z, r g i, x, u, z, r Λ u u + z z + r r ; M.1.iii g i non-decreaing wih repec o r ; M.2.i l and h lie in P q ; M.2.ii l h, lt, Ψ ht,. Fixing an iniial condiion, x, i E for X = X, N, we define for any, y, z, ṽ [, T ] R R 1 d M ρ, wih ṽ = v, w M ρ M, B, mdy; R R k : g, X, y, z, ṽ = g, X, ũ, z, r where ũ = ũ y, w and r = r v are defined a { ũ j = y, j = N y + w j, j N j I\{N } w j λ, X, j, 8, r = vyf, X, ymdy. 9 Le finally τ denoe an F -opping ime parameerized by he iniial condiion, x, i of X. Definiion 3.4 a A oluion o he Markovian decoupled Forward Backward SDE wih daa G, C and τ i a parameerized family of riple Z = Ω, F, P, B, χ, ν, X, Y, Ȳ, where he upercrip and in reference o he iniial condiion, x, i E, uch ha: i Ω, F, P, B, χ, ν, X = X, N i a model wih generaor G and iniial condiion, x, i; ii Y = Y, Z, V, K, wih V = V, W H 2 µ = H 2 χ H 2 ν, i an Ω, F, P, B, µ oluion o he R2BSDE on [, T ] wih daa g, X, y, z, ṽ, ΨX T, l, X, h, X 10 where ṽ denoe a generic pair v, w M ρ ; iii Ȳ, Z, V, K, wih V = V, W Hµ 2 o he opped RBSDE on [, T ] wih daa = H 2 χ H 2 ν, i an Ω, F, P, B, µ oluion where Y i he ae-proce of Y in ii. g, X, y, z, ṽ, Y τ, l, X, τ 11 b The oluion i aid o be conien, if: i Y =: u i, x define a, x, i varie in E a coninuou value funcion of cla P on E, and we

8 8 Abou he Pricing Equaion in Finance have for every [0, T ], P -a..: wih in 14: Y = u, X, [, T ] 12 For any j I : Wj = u j, X u, X, [, T ] 13 [ gζ, Xζ, Yζ, Zζ, Vζdζ = gζ, Xζ, uζ, Xζ, Zζ, r ζ 14 j I λζ, X ζ, j u j ζ, X ζ uζ, X ζ ] dζ, [, T ] uζ, Xζ := u j ζ, Xζ j I, r ζ = V ζ yfζ, Xζ, ymdy cf. 9; ii Ȳ =: v i, x define a, x, i varie in E a coninuou value funcion of cla P on E, and we have for every [0, T ], P -a..: Ȳ = v, X, [, τ ] 15 For any j I : W j = v j, X v, X, [, τ ] [ 16 gζ, Xζ, Ȳ ζ, Z ζ, V ζdζ = gζ, Xζ, vζ, Xζ, Z ζ, r ζ 17 j I λζ, X ζ, j v j ζ, X ζ vζ, X ζ ] dζ, [, τ ] wih in 17: vζ, X ζ := v j ζ, X ζ j I, r ζ := r ζ V ζ = V ζ yfζ, X ζ, ymdy 18 cf. 9. In he equel we work under he anding Aumpion 3.5 The Markovian SDE wih generaor G, co funcion C and parameerized opping ime τ ha a conien oluion Z = Ω, F, P, B, χ, ν, X, Y, Ȳ. Remark 3.6 Needle o ay, hi i a raher heroic Aumpion, covering variou iue a Lipchiz coninuiy properie of he coefficien b, σ, δ w.r.. x, maringale repreenaion, ome kind of coniency beween he driver B, χ, ν a varie, and almo ure coninuiy of he random funcion τ of, x, i on E. We refer he reader o [16] for a complee e of echnical aumpion including in paricular he baic aumpion explicily made above in hi paper enforcing Aumpion 3.5. Noe ha RBSDE and R2BSDE have originally been developed in connecion wih opimal opping problem and opimal opping game problem or Dynkin game, repecively. In paricular, Aumpion 3.5 ha he following immediae conequence ee Crépey [16]. Propoiion 3.1 A addle-poin τ, θ of he Dynkin game relaed o Y ee Crépey [16] i given by: τ = inf{ [, T ] ; X E } T, θ = inf{ [, T ] ; X E + } T,

9 S. Crépey 9 wih E = {, x, i [0, T ] I ; u i, x = h i, x} E + = {, x, i [0, T ] I ; u i, x = l i, x}. An opimal opping ime θ of he opimal opping problem relaed o Ȳ ee Crépey [16] i given by: wih θ = inf{ [, τ ] ; X E + } T, 19 E + = {, x, i [0, T ] I ; v i, x = l i, x}. 4 Vicoiy Soluion of Obacle Problem Our nex goal i hen o eablih he connecion beween Z and relaed yem of obacle problem aociaed o he daa G, C, τ problem denoed by V1 and V2 below. In hi paper we hall conider hi iue from he poin of view of vicoiy oluion o he relaed yem of obacle problem. An alernaive weak Sobolev oluion approach i deal wih in Crépey Maoui [17]. We poulae from now on ha Aumpion 4.1 i All he, x, i coefficien of he generaor G are coninuou funcion; ii The funcion δ and f are locally Lipchiz coninuou w.r.., x, uniformly in y, i; iii τ i defined a he minimum of T and of he fir exi ime by X of a given domain D I, ha i: where for every i I : τ = inf{ ; X / D} T 20 D {i} = {ψ i > 0} for ome ψ i C 2 wih ψ i > 0 on {ψ i = 0}. 21 Noe ha Aumpion 4.1iii correpond o he anding example for τ in [16], in which cae, all he relaed aumpion are aified on mild condiion on he problem daa. Le D = [0, T ] D, and le In p E = [0, T I, p E := E \ In p E In p D = [0, T D, p D := E \ In p D 22 and for he parabolic inerior and he parabolic boundary of E and D, repecively he definiion of a hick boundary p D i made neceary by he preence of he jump in X. Given locally bounded funcion φ and ϕ on E wih ϕ of cla C 1,2 around, x, i for ome, x, i E, we define cf. 2 3: Gφ, ϕ i, x = ϕ i, x T r[ a i, xhϕ i, x ] + ϕ i, xβ i, x + Iφ i, x 23 wih Iφ i, x := φ i, x + δ i, x, y φ i, x f i, x, ymdy. 24

10 10 Abou he Pricing Equaion in Finance Le alo Gϕ and for Gϕ, ϕ. So in paricular cf. 2: Gϕ i, x + λ i,j, x ϕ j, x ϕ i, x = Gϕ i, x. 25 j I In he equel we denoe by V2 he following variaional inequaliy wih double obacle problem: max min Gu i, x g i, x, u, x, uσ i, x, Iu i, x, u i, x l i, x, u i, x h i, x = 0 on In p E, upplemened by he erminal condiion Ψ our erminal co funcion on p E. We alo conider he problem V1 obained by formally replacing h by + in V2, ha i min Gu i, x g i, x, u, x, uσ i, x, Iu i, x, u i, x l i, x = 0, on In p D, upplemened by a coninuou boundary condiion Φ prolongaing Ψ on p D. Remark 4.2 i We deliberaely wrie on p E raher han a T for V2 in order o make he formal analogy beween he Cauchy problem V2 and he Cauchy Dirichle problem V1 a complee a poible. ii The boundary including erminal condiion Φ in he Cauchy Dirichle problem V1 will be pecified laer in he paper a our value funcion u, characerized a he unique vicoiy oluion of cla P of V2. The following coninuiy propery of he inegral erm I of G cf. 24 play a crucial role in he heory of vicoiy oluion of nonlinear inegro-differenial equaion ee for inance Alvarez Tourin [1, p. 297]. Lemma 4.1 The funcion, x, i Iψ i, x i coninuou on E, for any coninuou funcion ψ on E. Proof. We decompoe Iψ i n, x n Iψ i, x = + where ψ i n, x n f i n, x n, y ψ i, xf i, x, y mdy ψ i n, x n + δ i n, x n, yf i n, x n, y ψ i, x + δ i, x, yf i, x, y mdy, ψ i n, x n + δ i n, x n, yf i n, x n, y ψ i, x + δ i, x, yf i, x, y mdy 26 = ψ i n, x n + δ i n, x n, y ψ i, x + δ i, x, y f i n, x n, ymdy + ψ i, x + δ i, x, y f i n, x n, y f i, x, y mdy goe o 0 a E n, x n, x, by Aumpion 4.1ii, and likewie for ψ i n, x n f i n, x n, y ψ i, xf i, x, y mdy

11 S. Crépey 11 which i of he ame form a 26 bu wihou δ i. The following definiion are obained by pecifying o problem V1 and V2 he general definiion of vicoiy oluion for nonlinear PDE ee [15, 20], adaping furher he reuling definiion o finie aciviy jump and yem of PIDE a in [1, 26, 4, 14]. Definiion 4.3 ai A locally bounded upper emiconinuou funcion u on E i called a vicoiy uboluion of V2 a, x, i In p E, if and only if for any ϕ C 1,2 E uch ha u i ϕ i reache a global maximum a, x, hen max min Gu, ϕ i, x g i, x, u, x, ϕσ i, x, Iu i, x, u i, x l i, x, u i, x h i, x 0. Equivalenly, u i a vicoiy uboluion of V2 a, x, i if and only if u i, x h i, x, and u i, x > l i, x, implie Gu, ϕ i, x g i, x, u, x, ϕσ i, x, Iu i, x 0, 27 or inequaliy 27 wih Gu, ϕ and Iu replaced by Gϕ and Iϕ, for any ϕ C 1,2 E uch ha u i ϕ i reache a global null maximum a, x, or, in urn, global null maximum replaced by global null ric maximum a, x herein. ii A locally bounded lower emiconinuou funcion u on E i called a vicoiy uperoluion of V2 a, x, i In p E if and only if for any ϕ C 1,2 E uch ha u i ϕ i reache a global minimum a, x, hen max min Gu, ϕ i, x g i, x, u, x, ϕσ i, x, Iu i, x, u i, x l i, x, u i, x h i, x 0. Equivalenly, u i a vicoiy uperoluion of V2 a, x, i if and only if u i, x l i, x, and u i, x < h i, x implie Gu, ϕ i, x g i, x, u, x, ϕσ i, x, Iu i, x 0, 28 or inequaliy 27 wih Gu, ϕ and Iu replaced by Gϕ and Iϕ, for any ϕ C 1,2 E uch ha u i ϕ i reache a global null minimum a, x, or, in urn, global null maximum replaced by global null ric maximum a, x herein. iii A coninuou funcion u on E u i called a vicoiy oluion of V2 a, x, i In p E, if and only if i i boh a vicoiy uboluion and a vicoiy uperoluion of V2 a, x, i. bi By a P vicoiy uboluion u of V2 on E for he boundary condiion Ψ, we mean a funcion of cla P on E, which i a vicoiy uboluion of V2 on In p E, and uch ha u Ψ poinwie on p E. ii By a P vicoiy uperoluion u of V2 on E for he boundary condiion Ψ, we mean a funcion of cla P on E, which i a vicoiy uperoluion of V2 on In p E, and uch ha u Ψ poinwie on p E. iii By a P vicoiy oluion u of V2 on E, we mean a funcion ha i boh a P-uboluion and a P-uperoluion of V2 on E hence u = Ψ on p E. c The noion of rep. P vicoiy uboluion, uperoluion and oluion of V1 a, x, i In p D rep. on E, given a coninuou boundary condiion Φ prolongaing Ψ on p D are hen defined by immediae adapaion of par a and b above, ubiuing V1 o V2, + o h, In p D o In p E, C 0 E C 1,2 D o C 1,2 E, p D o p E and Φ o Ψ herein.

12 12 Abou he Pricing Equaion in Finance Remark 4.4 i A claic oluion if any of V1, rep. V2, i necearily a vicoiy oluion of V1, rep. V2. ii A vicoiy oluion u of V2 necearily aifie l u h. However a vicoiy uboluion rep. uperoluion u of V2 doe no need o verify u l rep. u h. Likewie a vicoiy oluion v of V1 necearily aifie l u, however a vicoiy uboluion v of V1 doe no need o verify u l. iii The fac ha Gu, ϕ and Iu may be replaced by Gϕ and Iϕ in 27 or 28 or he analog inequaliie regarding V1 can be hown by an immediae adapaion o he preen conex of Barle e al. [4, Lemma 3.3 p. 66] ee alo Alvarez Tourin [1, ee Definiion 2 Equivalen p. 300], uing he monooniciy aumpion M.1.iii on g. Since we only conider emioluion in he vicoiy ene in hi paper, rep. P uboluion, uperoluion and oluion are o be underood henceforh a rep. P vicoiy uboluion, uperoluion and oluion. 5 Exience Theorem 5.1 i The value funcion u cf. Definiion 3.4bi i a P-oluion of V2 on E for he erminal condiion Ψ on p E. ii The value funcion v cf. Definiion 3.4bii i a P-oluion of V1 on E for he boundary condiion u on p D. Proof. i u i a coninuou funcion of cla P on E, by definiion. Moreover by definiion of u and Y we have ha, T referring o he iniial condiion T, x, i : u i T, x = YT T = Ψi x l i, x Y = u i, x h i, x. So u = Ψ poinwie a T and l u h on E. Le u how ha u i a uboluion of V2 on In p E. We le he reader check likewie ha u i a uperoluion of V2 on In p E. Le hen, x, i In p E and ϕ C 1,2 E be uch ha u i ϕ i reache i maximum a, x. Given ha u h, i uffice o prove ha Gϕ i, x g i, x, u, x, ϕσ i, x, Iϕ i, x 0, 29 auming furher ha u i, x > l i, x and u i, x = ϕ i, x cf. Definiion 4.3i. Suppoe by conradicion ha 29 doe no hold. Then by coninuiy uing in paricular Lemma 4.1: ψ, y := Gϕ i, y + g i, y, u, y, ϕ i σ i, y, Iϕ i, y < 0 30 for any, y uch ha [, + α] and y x α, for ome mall enough α > 0 wih + α < T. Le θ = inf { ; X x α, N i, Y = l i, X } + α 31 Ŷ, Ẑ, V, V, K = 1 <θ Y + 1 θ u i θ, Xθ, 1 θ Z, 1 θ V, 1 θ V, K θ 32 Ỹ, Z, Ṽ = ϕ i X θ, 1 θ ϕσ i, X, 33 [ 1 θ ϕ i, X + δ i, X, y ϕ i, X ]. y

13 S. Crépey 13 Noe ha θ >, P almo urely. Thu, uing alo he coninuiy of u i : Ŷ = Y = u i, x = ϕ i, x = Ỹ, P -a.. 34 Moreover, by uing he R2BSDE equaion for Y in which K ;+ = 0 on [, θ] by he relaed minimaliy condiion, given ha l i, X < Y on [, θ, we have for r in [, θ : Ŷ r = u i r, Xr + g i ζ, Xζ, uζ, Xζ, Ẑ ζ, r ζdζ r r ẐζdB ζ V ζ y χ dy, dζ, K ; r K ; 35 which alo hold rue for = r = θ by definiion of Ŷ in 32. Furhermore for < r = θ eiher χ, whence X, do no jump a θ, and 35 hold again a r = θ by paage o he limi a r θ in 35, or cf. Definiion 3.2i N doe no jump a θ, in which cae he R2BSDE equaion for Y inegraed beween and θ direcly give 35 for r = θ. In concluion 35 hold for r in [, θ]. Beide, by applicaion of he Iô formula 6 o he funcion ϕ defined by ϕ j = ϕ i for all j I, we ge for any [, θ] : dϕ i, X = G ϕ, Xd + ϕσ, XdB + ϕ i, X + δ, X, y ϕ i, X χ dy, d = G ϕ, Xd + ϕσ, XdB + ϕ i, X + δ, X, y ϕ i, X χ dy, d = Gϕ i, Xd + ϕσ i, XdB + ϕ i, X + δ i, X, y ϕ i, X χ dy, d, where he econd equaliy ue 25 applied o ϕ and he hird one exploi he fac ha N canno jump before θ and ha χ canno jump a θ if N doe. Hence cf. 33: Ỹ = ϕ i θ, Xθ = ϕ i θ, X θ Z rdb r Gϕ i r, Xrdr Z rdb r Ṽ r y χ dy, dr ψr, X r g i r, X r, ur, X r, ϕσ i r, X r, Iϕ i r, X r Ṽ r y χ dy, dr, by definiion 30 of ψ. In concluion we have for [, θ] : Ŷ = u i θ, X θ + Ỹ = ϕ i θ, X θ g i ζ, Xζ, uζ, Xζ, Ẑ ζ, r ζdζ ; K θ Ẑ ζdb ζ K ; ψζ, Xζ g i ζ, Xζ, uζ, Xζ, Z ζ, Iϕ i ζ, Xζ dζ Z ζdb ζ dr V ζ y χ dy, dζ 36 Ṽ ζ y χ dy, dζ 37

14 14 Abou he Pricing Equaion in Finance wih by definiion 9 of r ζ = r ζ V ζ, 24 of I and 33 of Ṽ : g i ζ, X ζ, uζ, X ζ, Ẑ ζ, r ζ dζ = g i ζ, X ζ, uζ, X ζ, Z ζ, Iϕ i ζ, X ζ g i ζ, Xζ, uζ, Xζ, Ẑ ζ, dζ = g i ζ, Xζ, uζ, Xζ, Z ζ, Vζ yf i ζ, Xζ, ymdy dζ Ṽ ζ yf i ζ, Xζ, ymdy dζ. Oherwie aid: Ŷ, Ẑ, V olve he opped BSDE on [, + α] wih driver cf. Definiion 2.2d and Remark 2.3i g i, X, u, X, z, vyf i, X ;, ymdy d d K and erminal condiion u i θ, Xθ a θ; Ỹ, Z, Ṽ olve he opped BSDE on [, + α] wih driver g i, X, u, X, z, vyf i, X, ymdy d ψ, Xd and erminal condiion ϕ i θ, Xθ a θ. Seing δy = Ŷ Ỹ, we deduce by andard compuaion ee for inance he proof of [18, Comparion Principle]: Γ δy = E [ Γ θδy θ + Γ da ] 38 where: δyθ = Ŷ θ Ỹ θ = ui θ, Xθ ϕi θ, Xθ 0, by making = θ in and ince ui ϕ i ; da = ψr, Xd ; d K, o ha A i ricly decreaing on [, θ], by 30; Γ i a ricly poiive proce, he o-called adjoin of δy ee, for inance, [18]. Since furhermore θ > P -a.., we deduce ha Γ da < 0 P -a.., whence δy < 0, by 38. Bu hi conradic 34. ii v i a coninuou funcion on E, by definiion. Moreover by definiion of u, v, Y and Ȳ including he definiion 20 of τ, we have, for, x, i p D : and for any, x, i E : v i, x = Ȳ = Y = u i, x, l i, x Ȳ = v i, x. So v = u on p D and l v on E. We now how ha v i a uboluion of V1 on In p D. We le he reader check likewie ha v i a uperoluion of V1 on In p D. Le hen, x, i In p D and ϕ C 0 E C 1,2 D be uch ha v i ϕ i reache i maximum a, x. We need o prove ha Gϕ i, x g i, x, v, x, ϕσ i, x, Iϕ i, x 0, 39 auming furher v i, x > l i, x and v i, x = ϕ i, x cf. Definiion 4.3i. Suppoe by conradicion ha 39 doe no hold. Then by coninuiy, y, i In p D and ψ, y := Gϕ i, y + g i, y, v, y, ϕσ i, y, Iϕ i, y < 0 40

15 S. Crépey 15 for any, y uch ha [, + α] and y x α, for ome mall enough α > 0. Le θ = inf{ ; X x α, N i, Ȳ = l i, X} + α τ 41 Ŷ, Ẑ, V, V, K = 1 <θ Ȳ + 1 θ v i θ, Xθ, 1 θ Z, 1 θ V, 1 θ V, K θ 42 Ỹ, Z, Ṽ = ϕ i X θ, 1 θ ϕσ i, X, 43 [ 1 θ ϕ i, X + δ i, X, y ϕ i, X ]. y Uing in paricular he fac ha D i open in 20, we have ha θ >, P almo urely. Thu, uing alo he coninuiy of v i : Ŷ = Ȳ = v i, x = ϕ i, x = Ỹ, P -a.. 44 Noe ha by he minimaliy condiion in he opped RBSDE for Ȳ, we have ha K = 0 on [, θ], ince l i, X < Ȳ on [, θ and θ τ. By uing he opped RBSDE equaion for Ȳ, we hu have for r in [, θ : Ŷ r = v i r, Xr + g i ζ, Xζ, vζ, Xζ, Ẑ ζ, r ζdζ r r ẐζdB ζ V ζ y χ dy, dζ, 45 which alo hold rue for = r = θ by definiion of Ŷ in 42. Furhermore for < r = θ eiher χ, whence X, do no jump a θ, and 45 hold again a r = θ by paage o he limi a r θ in 35, or N doe no jump a θ, in which cae he opped RBSDE equaion for Ȳ wrien beween and θ direcly give 45 for r = θ. In concluion we have for [, θ] : wih cf. 18: g i ζ, X ζ, vζ, X ζ, Ẑ ζ, r ζ Ŷ = v i θ, X θ + dζ = Ẑ ζdb ζ g i ζ, X ζ, vζ, X ζ, Ẑ ζ, r ζdζ V ζ y χ dy, dζ 46 g i ζ, Xζ, vζ, Xζ, Ẑ ζ, Vζ yf i ζ, Xζ, ymdy dζ. Oherwie aid Ŷ, Ẑ, V olve he opped BSDE on [, + α] wih driver cf. Remark 2.3i g i, X, v, X, z, vyf i, X, ymdy d where vy refer o a generic funcion v M, B, mdy; R, no o be confued wih he value funcion v = v i, x in vζ, Xζ and erminal condiion vi θ, Xθ a θ. Beide one can how a in par i above ha Ỹ, Z, Ṽ olve he opped BSDE on [, + α] wih driver g i, X, v, X, z, vyf i, X, ymdy d ψ, Xd and erminal condiion ϕ i θ, Xθ a θ. We conclude a in par i.

16 16 Abou he Pricing Equaion in Finance 6 Uniquene In hi Secion we conider he iue of uniquene of a oluion o V2 and V1, repecively. We prove a diconinuou emioluion comparion principle or maximum principle for our problem, which implie in paricular uniquene of a P-oluion o hee problem. For relaed comparion and/or uniquene reul we refer he reader o Alvarez Tourin [1], Barle e al. [4] and Barle Imber [5], Pardoux e al. [26], Harraj e al. [21], Amadori [2, 3], Ma Cvianic [25], among oher. Aumpion 6.1 i The funcion b, σ and δ are locally Lipchiz coninuou in, x, i, uniformly in y regarding δ; ii There exi, for every R > 0, a nonnegaive funcion η R coninuou and null a 0 modulu of coninuiy uch ha g i, x, u, z, r g i, x, u, z, r η R x x 1 + z for any [0, T ], i I, z, r R and x, x, u R k wih x, x, u R ; iii The funcion g i i non-decreaing w.r.. u j, for any i, j I 2 wih i j. Noe ha by Aumpion 6.1i, we have in paricular b σ δ C1 + x on E. 47 Theorem 6.1 We have µ ν on E, for any P-uboluion µ and P-uperoluion ν of V2 on E wih erminal condiion Ψ on p E, repecively of V1 on E wih boundary condiion u on p D. Given Theorem 5.1, hi immediaely implie he following reul. Propoiion 6.2 The value funcion u, rep. v, i he unique P-oluion, he maximal P-uboluion, and he minimal P-uperoluion, of V2 on E wih erminal condiion Ψ on p E, repecively of V1 on E wih boundary condiion u on p D. Remark 6.2 If Theorem 6.1 hold in he pecial cae when g i i non-decreaing w.r.. u j for any i, j I 2 o g non-increaing w.r.. u a a whole, raher han g i non-increaing w.r.. u j for any i, j I 2 wih i j in Aumpion 6.1iii, hen Theorem 6.1 alo hold in general, by applicaion of hi pecial cae o he ranformed funcion e R µ i, x and e R ν i, x for large enough R. Indeed e R µ and e R ν are repecively P-uboluion and P-uperoluion of he ranformed problem for V2 max min Gϕ i, x e R g i, x, e R ϕ, x, e R ϕσ i, x, e R Iϕ i, x Rϕ i, x, ϕ i, x e R l i, x, ϕ i, x e R h i, x = 0 on In p E, upplemened by he erminal condiion ϕ = e R Ψ on p E and likewie wih h = + for V1 on In p D, upplemened by he boundary condiion ϕ = e R Φ on p D. Now, for R large enough, Aumpion 6.1iii and he Lipchiz coninuiy propery of g w.r.. he la variable imply ha g, x, e R u, e R z, e R r + Ru i non-decreaing w.r.. u. Given hee obervaion, we may and do reduce aenion, in order o prove Theorem 6.1, o he cae where he funcion g i non-decreaing w.r.. u. Theorem 6.1 in cae of V2 i hen an immediae conequence of he following Lemma leing α go o 0 in par iii. The proof for V1 i idenical, ubiuing V1 o V2, + o h, In p D o In p E and C 0 E C 1,2 D o C 1,2 E in Lemma 6.3 below and i proof.

17 S. Crépey 17 Lemma 6.3 Le Λ 1 = kλ where Λ i he Lipchiz conan of g cf. Aumpion M.1.ii. Given a P-uboluion µ and a P-uperoluion ν of V2 on E, le q 1 be an ineger greaer han q 2 uch ha µ, ν P q2, where q 2 i provided by our aumpion ha µ, ν P in Theorem 6.1. Then, auming g non-decreaing w.r.. u : i ω = µ ν i a P-uboluion of min w, Gω Λ 1 max j I ωj + + ωσ + Iω + = 0 on E wih null boundary condiion on p E, in he ene ha, fir, ω 0 on p E, and econd, ω i, x > 0 implie Gϕ i, x Λ 1 max j I ωj, x + + ϕ i, xσ i, x + Iϕ i, x for any, x, i In p E and ϕ C 1,2 E uch ha ω i ϕ i reache a global null maximum a, x. ii There exi C 1 > 0 uch ha he regular funcion i a ric P-uperoluion of min on E, in he ene ha χ > 0 and χ i, x = 1 + x q1 C1T e χ, Gχ Λ 1 χ + χσ + Iχ + = 0 on E. iii max i I ω i + αχ on [0, T ], for any α > 0. Gχ Λ 1 χ + χσ + Iχ + > 0 49 Thi Lemma i an adapaion o our e-up of he analog reul in [4] ee alo [26] and [21]. Here are he main difference our aumpion are in fac fied o financial applicaion, ee Secion 8: i We conider a model wih jump in X and regime repreened by N, wherea Barle e al. [4] or Harraj e al. [21] only conider jump in X, and Pardoux e al. [26] only conider regime; ii We work wih finie jump meaure fm d, jump ize δ wih linear growh in x, and diconinuou emioluion wih polynomial growh in x, wherea Barle e al. [4] or Harraj e al. conider general Levy meaure, bounded jump, and coninuou emioluion wih ub-exponenial ricly including polynomial growh in x; iii Barle e al. [4] deal wih claic BSDE wihou barrier; iv We conider ime-dependen coefficien b, σ, δ wherea Barle e al. [4] conider homogenou dynamic. Becaue of hee difference we provide a deailed proof, which i deferred o he Appendix. 7 Approximaion An imporan feaure of diconinuou vicoiy emioluion comparion principle like Theorem 6.1 above i ha hey enure he abiliy of he relaed PIDE problem, providing in paricular generic condiion enuring he convergence of a wide family of deerminiic approximaion cheme able, monoone and conien approximaion cheme originally inroduced for PDE by Barle and Souganidi [6], and furher exended o PIDE by Briani, La Chioma and Naalini [14]. The

18 18 Abou he Pricing Equaion in Finance following reul hu exend o model wih regime hu yem of PIDE he reul of [6, 14], among oher. I i worh reing ha o he be of our knowledge, hee reul are he fir convergence reul in he lieraure regarding approximaion cheme of vicoiy oluion o yem of PIDE. The following Lemma i andard and elemenary, and hu aed wihou proof. Lemma 7.1 Le E h h>0 denoe a family of ube of E, uch ha for any, x, i E, here exi equence h n, n, x n verifying h n 0 and E hn n, x n, i, x, i a n. Le u h h>0 be a family of uniformly locally bounded real funcion wih u h defined on he e E h, for any h > 0. i For any, x, i E, he e of limi of he following kind: lim n + ui h n n, x n wih h n 0 and E hn n, x n, i, x, i a n, 50 i non empy and compac in R. I admi a uch a malle and a greae elemen: u i, x u i, x in R. ii The funcion u, repecively u, defined in hi way, i locally bounded and lower emi-coninuou on E, repecively locally bounded and upper emi-coninuou on E. We call i he lower limi, repecively upper limi, of u h h>0 a, x, i a h 0 +. We ay ha u h converge o l a, x, i E a h 0, and we denoe : lim u i h h, x h = l, h 0 + E h h,x h,i,x,i if and only if u i, x = u i, x = l, or, equivalenly: lim n + ui h n n, x n = l for any h n 0 e E hn n, x n, i, x, i. iii If u h converge poinwie everywhere o a coninuou funcion u on E, hen hi convergence i locally uniform: max u h u 0 E h C a h 0 +, for any compac ube C of E. Definiion 7.1 Le u be given familie of operaor G h = G h u i h, x h, δ h = δ h u i h, x h, I h = I h u i h, x h devoed o approximae Gu i h, x h, u i h, x h and Iu i h, x h on E h for real-valued funcion u on E, repecively. For L =, I or G : i we ay ha he relaed dicreiaion L h = δ h, I h or G h i monoone, if L h u i 1 h, x h L h u i 2 h, x h 51 for any funcion u 1 u 2 on E h wih u i 1 h, x h = u i 2 h, x h ; ii we ay ha he relaed dicreiaion cheme L h h>0 i conien wih L if and only if for any coninuou funcion ϕ on E of cla C 1,2 around, x, i, we have: L h ϕ + ξ h i h, x h Lϕ i, x 52 whenever h 0 +, E h h, x h, i, x, i E and R ξ h 0.

19 S. Crépey 19 Aumpion 7.2 The funcion R u, p, r g,, u, pσ, r R E 53 i non-decreaing, in he ene ha for any u, p, r u, p, r coordinae by coordinae in R, we have g i, x, u, pσ i, x, r g i, x, u, p σ i, x, r for any, x, i E. Remark 7.3 i Of coure, he previou monooniciy and coniency aumpion are abrac condiion which need o be verified carefully on a cae-o-cae bai for any concree approximaion cheme under conideraion. The mo ringen le andard condiion eem o be he one regarding he monooniciy of g w.r.. p in 53. Noe however ha hi condiion i obviouly aified in every of he following hree cae: he funcion g = g i, x, u, z, r doe no depend on he argumen z which i for inance aified in he cae of rik-neural pricing problem in finance, cf. Secion 8; σ i null model baed on pure jump procee; noe however ha he coninuiy of he funcion τ of, x, i cf. Remark 3.6 will ypically fail o be aified in hi cae for domain a imple a D = { x < R} I, τ being defined a in Aumpion 4.1iii; he dimenion d i equal o one calar Lévy-Poion componen X of X and i dicreized by finie difference decenered forward, whence an upwind dicreizaion cheme for ϕσ, by nonnegaiviy of σ in he calar cae. ii Under he weaker aumpion ha g i, x, u, pσ i, x, r i non-decreaing w.r.. p, r and cf. Aumpion 6.1iii non-decreaing w.r.. u j for j i, hen he mapping u i, x ũ i, x := e R u i, x for R large enough ranform he problem ino one in which Aumpion 7.2 hold ee Remark 6.2. Suiable approximaion cheme may hen be applied o he ranformed problem, and a convergen approximaion o he oluion of he original problem i recovered by eing u i h, x := er ũ i h, x. Lemma 7.2 Le u be given monoone and conien approximaion cheme G h h>0, δ h h>0 and I h h>0 for G, and I repecively, g aifying he monooniciy Aumpion 7.2. a Le u h h>0 be uniformly polynomially bounded u h bounded by C1 + x p for ome C and p independen of h and aify max min G h u i h h, x h g i h, x h, u h h, x h, δ h u h σ i h, x h, I h u i h h, x h, 54 u i h h, x h l i h, x h, u i h h, x h h i h, x h = 0 55 on In p E E h and u h = Ψ on p E E h for any h > 0. Then: i he upper and lower limi u and u of u h a h 0, are repecively vicoiy uboluion and uperoluion of V2 on In p E; ii we have u Ψ u poinwie a T. b Le v h h>0 be uniformly polynomially bounded and aify min G h vh i h, x h g i h, x h, v h h, x h, δ h v h σ i h, x h, I h vh i h, x h, 56 vh i h, x h l i h, x h = 0 57 on In p D E h and v h = u on p D E h for any h > 0. Then: i he upper and lower limi v and v of v h a h 0, are repecively vicoiy uboluion and uperoluion of V1 on In p D; ii we have v u= Ψ v poinwie a T.

20 20 Abou he Pricing Equaion in Finance Proof. We only prove a, ince he proof of b i imilar cf. commen preceding Lemma 6.3. Noe ha we only have v u v a T in b, and no necearily v u v on p D, which would be he formal analogue of u Ψ u a T in a; ee commen in ii below. i We prove ha u i a vicoiy uboluion of V2 on In p E. The fac ha u i a vicoiy uperoluion of V2 on In p E can be hown likewie. Fir noe ha u h, by 54 on In p E E h, inequaliy Ψ h on p E cf. M.2.ii and coninuiy of h and Ψ. Le hen, x, i In p E be uch ha u i, x > l i, x and, x maximize ricly u i ϕ i a zero for ome funcion ϕ C 1,2 E. We need o how ha cf. 27: Gϕ i, x g i, x, u, x, ϕσ i, x, Iϕ i, x By a claical argumen in he heory of vicoiy oluion ee e.g. Barle and Souganidi [6], here exi, for any h > 0, a poin, x in [0, T ] B R, where B R i a ball wih large radiu R around x, uch ha we omi he dependence of, x in h for noaional impliciy: u i h ϕ i + u h ϕ i, x 59 wih equaliy a, x, and ξ h := u h ϕ i, x goe o 0 = u ϕ i, x, whence u i h, x goe o u i, x, a h 0 cf. an analog aemen and i juificaion in he econd par of he proof of par ii below. Therefore u i, x > l i, x implie ha u i h, x > li, x for h mall enough, whence by 54: G h u i h, x g i, x, u h, x, δ h u h σ i, x, I h u i h, x Given 59, we hu have by monooniciy of he cheme and of g Aumpion 7.2: G h ϕ + ξ h i, x g i, x, u h, x, δ h ϕ + ξ h σ i, x, I h ϕ + ξ h i, x g i, x, u, x, ϕσ i, x, Iϕ i, x + η + η R x x 1 + ϕσ i, x + kλ max j I uj h, x uj, x + + Λ δ h ϕ + ξ h σ i, x ϕσ i, x + ΛI h ϕ + ξ h i, x Iϕ i, x +, where in he la inequaliy cf. proof of Lemma 6.3i in he Appendix: η i a modulu of coninuiy of g i on a large compac e around, x, u, x, ϕσ i, x, Iϕ i, x ; η R i he modulu of coninuiy anding in Aumpion 6.1ii; he hree la erm come from he Lipchiz coninuiy and monooniciy properie of g. Inequaliy 58 follow by ending h o zero in he previou inequaliy, uing he coniency 52 of he cheme. ii Le u how furher ha u and u aify he boundary condiion in he o-called weak vicoiy ene on p E, namely in he cae of u he relaed aemen and proof are imilar in he cae of u: Inequaliy 58 hold for any, x, i p E and ϕ C 1,2 E uch ha u i, x > l i, x Ψ i, x 61 and, x maximize globally and ricly u i ϕ i a zero. A in par i, here exi, for any h > 0, a poin, x in [0, T ] B R we omi he dependence of, x in h for noaional impliciy, where B R i a ball wih large radiu R around x, uch ha inequaliy 59 hold wih equaliy a, x, and ξ h = u h ϕ i, x, whence u i h, x ui, x, goe o zero a h 0. Therefore inequaliy 61 implie ha u i h, x > l i, x Ψ i, x

21 S. Crépey 21 for h mall enough, whence, x, i In p E and by 54: G h u i h, x g i, x, u h, x, δ h u h σ i, x, I h u i h, x Inequaliy 58 follow like in par i above. Now here he argumen only work a T and canno be adaped o he cae of problem V1 on he whole of p D, cf. commen a he beginning of he proof, by a claical argumen in he heory of vicoiy oluion ee Alvarez Tourin [1, boom of p. 303] or Amadori [2, 3], any vicoiy uboluion or uperoluion of V2 on In p E aifying he boundary condiion in he weak vicoiy ene on p E, aifie i poinwie a T. So, in our cae, uppoe for inance by conradicion ha for ome x wih T, x p E. Le u hen inroduce he funcion u i T, x > Ψ i T, x 63 in which ϕ i ε, x = u i, x x x 2 C ε T 64 ε C ε > up,x [ η,t ] B 1x y x G 2 i 2y x, x + g i i σ y x 2 i, x, u, x,, x, I, x ε ε ε 65 goe o a ɛ 0, where B 1 x denoe he cloed uni ball cenered a x in. There exi, for any ε > 0, a poin, x in [0, T ] B R we omi he dependence of, x in ε for noaional impliciy, where B R i a ball wih large radiu R around x, uch ha: for any ε > 0 he relaed poin, x maximize ϕ i ε over [0, T ] B R,, x T, x and u i, x u i T, x a ε 0. To juify he la poin, noe ha by he maximizing propery of, x we have ha whence in paricular cf. 64 ϕ i εt, x ϕ i ε, x o 0 x x 2 ε + C ε T u i, x u i T, x 66 u i T, x u i, x. 67 Since u i locally bounded, 66 implie ha, x T, x a ε 0, which, join o he upper emi-coninuiy of u and o 67, implie ha u i, x u i T, x a ε 0. Now we have l Ψ poinwie a T, herefore 63 join o he fac ha lim ε 0 u i, x = u i T, x imply ha u i, x > l i, x, for ε mall enough. In virue of he reul already eablihed a hi poin of he proof, he funcion, y x y 2 ε + C ε T hu aifie he relaed vicoiy uboluion inequaliy a, x, i, o C ε G y x 2 i 2y x, x g i i σ y x 2 i, x, u, x,, x, I, x 0, ε ε ε which for ε mall enough conradic 65.

22 22 Abou he Pricing Equaion in Finance Propoiion 7.3 Le u h h>0, rep. v h>0, denoe a able, monoone and conien approximaion cheme in he ene ha all condiion in Lemma 7.2a, rep. b are aified for he value funcion u, rep. v. a u h u locally uniformly on E a h 0. b v h v locally uniformly on E a h 0, provided v h v= u on p D { < T }. Proof. a By Lemma 7.2a, he upper and lower limi u and u are P-uboluion and P- uperoluion of V2 on E. So u u, by Theorem 6.1. Moreover u u by conrucion cf. Lemma 7.1i. Thu finally u = u, which implie ha u h u locally uniformly on E a h 0, by Lemma 7.1iii. b By Lemma 7.2bi, v and v are repecively vicoiy uboluion and uperoluion of V1 on In p D. Moreover, hey aify v u v a T, by Lemma 7.2bii. If, in addiion, v h v= u on p D { < T }, hen finally v u v on p D, and v and v are P-uboluion and P-uperoluion of V1 on E. We conclude like in par a. 8 Applicaion o Finance We refer he reader o [13, 11, 16, 18] for veraile applicaion in finance of he model conidered in hi paper. Noe ha he aumpion of he preen paper and [16] are epecially fied o financial applicaion, for which cf. in paricular he dicuion following Lemma 6.3: i Jump are moivaed by he empirical evidence of he volailiy mile on financial derivaive marke, and regime can be hough of a he imple, whence mo robu, way o implemen ochaic volailiy; ii Finie jump meaure are generally enough in pracice for pricing applicaion, jump ize are ypically proporional o x, pricing funcion ypically have polynomial growh, and diconinuou vicoiy emioluion comparion principle are required o enure convergence of deerminiic approximaion cheme cf. Secion 7; iii Obacle are par of he problem in he cae of American and Game opion; iv Many model ue ime-dependen coefficien, for he ake of calibrabiliy o marke daa. Again ee Remark 3.3ii, we exclude mall jump in he model rericing our aenion o finie jump meaure mdy in X for impliciy of preenaion only, ye our approach could be exended o general Lévy jump meaure wihou major change if wihed. Ye he preenaion of he vicoiy argumen would be much le uer friendly, accouning in paricular for he recen finding of Barle and Imber [5]. Conidering ha mall jump may be approximaed by diffuive erm in X, we are no aware of any applicaion in pricing for which reoring o mall jump would be a real pracical neceiy he iuaion i differen in aiical finance, for which mall jump are peraining o heavy ail modeling. So we preferred no o inroduce mall jump in hi paper, in order no o make he preenaion even more cumberome. In rik-neural pricing problem in finance ee [16, 11], he funcion g i ypically of he form: g i, x, u, z, r = g1, i x g2, i xu i + u j u i λ i,j, x, 68 j I\{i} for dividend and inere-rae relaed funcion g 1 and g 2 or dividend and inere-rae adjued for credi pread, in a more general conex of defaulable coningen claim. In reference o Remark 7.3i, noe ha he funcion g in 68 doe in fac no depend on z, o ha he mo ringen condiion in Aumpion 7.2 ha regarding he monooniciy of g w.r.. p i rivially verified in hi cae.

23 S. Crépey A relevan Exenion Given he preence of dicree coupon a dicree coupon dae in many produc payoff like converible bond, ee [11, 9, 12], i i imporan in pracice o be able o deal wih he cae where he funcion g in 68 ee [10, 12], and more generally o, he running co funcion g in hi paper, i no coninuou, bu may be preen iolaed lef-diconinuiie in ime a cerain dicree coupon dae. Thi moivae he following exenion o he reul of hi paper ee alo [16]. Le u be given 0 = T 0 < T 1 < < T m 1 < T m = T m repreening dicree coupon dae in he financial inerpreaion. Le u e, for l = 1,, m cf. 22 E l = E {T l 1 T l }, In p E l = In p E {T l 1 < T l }, p E l = E l \ In p E l D l = D {T l 1 T l }, In p D l = In p D {T l 1 < T l }, p D l = E l \ In p D l, and le u be given, on every e E l, a funcion g l aifying he aumpion of he preen paper. We now conider a running co funcion g in our problem he funcion obained by concaenaion of he g l on each E {T l 1 < T l }. Noe ha he e In p E l, rep. In p D l, pariion In p E, rep. In p D. Definiion 8.1 i A locally bounded upper emiconinuou, rep. lower emiconinuou, rep. rep. coninuou, funcion u on E, i called a vicoiy uboluion, rep. uperoluion, rep. rep. oluion, of V2 a, x, i In p E, if and only if he rericion of u o E l wih, x, i In p E l i a vicoiy uboluion, rep. uperoluion, rep. rep. oluion, of V2 a, x, i, relaive o E l cf. Definiion 4.3. ii A P vicoiy uboluion, rep. uperoluion, rep. rep. oluion u o V2 on E for he boundary condiion Ψ a T i hen formally defined a in Definiion 4.3, wih he embedded noion of vicoiy uboluion, rep. uperoluion, rep. rep. oluion, of V2 on a any, x, i in In p E defined a in i above. iii The noion of rep. P vicoiy uboluion, uperoluion and oluion of V1 a, x, i In p D rep. on E, given a coninuou boundary condiion Φ on p D uch ha Φ = Ψ a T are defined imilarly cf. Definiion 4.3c. Propoiion 8.1 Uing Definiion 8.1 for he involved noion of rep. P vicoiy emi- oluion, all he reul of hi paper are ill rue under he currenly relaxed aumpion on g. Proof. Fir noe ha he reul of [16] cf. Remark 3.6 alo hold rue under hi relaxed aumpion on g hi wa he concluding obervaion in [16]. A for he reul of hi paper properly aid, Theorem 5.1 ill hold rue, by immediae inpecion of i proof. Moreover, under he l by l verion of Aumpion 6.1ii on he g l, Lemma 6.3 and Theorem 6.1 whence Propoiion 6.2 can now be proven ogeher ieraively on l a follow. Le µ and ν denoe a P-uboluion and a P-uperoluion v of V2 on E he proof i analog for V1. Lemma 6.3 relaive o E m i proven in exacly he ame way a before. We hu have cf. Theorem 6.1 µ ν on E m. We can hen eablih likewie he verion of Lemma 6.3 relaive o E m 1 for he erminal condiion given by he rericion of µ ν o p E l which i non-poiive, by he fir ep of he proof. So µ ν on E m 1, and o on unil l = 1, which finally prove Lemma 6.3 and Theorem 6.1 a a whole. Lemma 7.1 i of coure no affeced by he relaxaion of he aumpion on g. Finally, given he new definiion 8.1, Lemma 7.2ai can be proven exacly a before, on each In p E l, and he proof of Lemma 7.2aii doe no change a all. Lemma 7.2a i hu ill rue, and o i likewie Lemma 7.2b, whence Propoiion 7.3 a before.