Convergence of option rewards for multivariate price processes

Transcription

1 Mathematcal Statstcs Stockholm Unversty Convergence of opton rewards for multvarate prce processes Robn Lundgren Dmtr Slvestrov Research Report 2009:10 ISSN

2 Postal address: Mathematcal Statstcs Dept. of Mathematcs Stockholm Unversty SE Stockholm Sweden Internet:

3 Convergence of opton rewards for multvarate prce processes Robn Lundgren Dmtr Slvestrov December 12, 2009 Abstract Amercan type optons wth general payoff functons possessng polynomal rate of growth are consdered for multvarate Markov prce processes. Convergence results are obtaned for optmal reward functonals of Amercan type optons for perturbed multvarate Markov processes. These results are appled to approxmaton tree type algorthms for Amercan type optons for exponental dffuson type prce processes. Applcaton to mean-reverse prce processes used to model stochastc dynamcs of energy prces are presented. Also applcaton to resellng of European optons are gven. Keywords: Amercan opton; convergence of opton rewards; bnomaltrnomal tree approxmaton; optmal stoppng; skeleton approxmaton; multvarate Markov prce process. Postal address: Stockholm Unversty, Department of Mathematcs, SE Stockholm E-mal:

4

5 Convergence of opton rewards for multvarate prce processes Robn Lundgren Dmtr Slvestrov December 12, 2009 Abstract Amercan type optons wth general payoff functons possessng polynomal rate of growth are consdered for multvarate Markov prce processes. Convergence results are obtaned for optmal reward functonals of Amercan type optons for perturbed multvarate Markov processes. These results are appled to approxmaton tree type algorthms for Amercan type optons for exponental dffuson type prce processes. Applcaton to mean-reverse prce processes used to model stochastc dynamcs of energy prces are presented. Also applcaton to resellng of European optons are gven. Keywords: Amercan opton; convergence of opton rewards; bnomal-trnomal tree approxmaton; optmal stoppng; skeleton approxmaton; multvarate Markov prce process. 1 Introducton An Amercan opton gves the holder the rght to at any moment before some future tme T, to exercse the opton to receve a payoff determned by a functon g (t, s). We study the model wth payoff functons admttng power type upper bounds and multvarate exponental Markov prce processes S (t). Optmal stoppng problems for Amercan type optons have been also studed n Jacka (1991), Km (1990), Peskr and Shryaev (2006), for models wth stochastc Mälardalen Unversty, School of Educaton, Culture and Communcaton, Appled Mathematcs Västerås Stockholm Unversty, Department of Mathematcs, Mathematcal Statstcs, Stockholm E-mal addresses:robn.lundgren@mdh.se (R. Lundgren) slvestrov@math.su.se (D. Slvestrov) 1

6 volatlty by Zhang and Lm (2006), for Amercan barrer optons n Gau, Huang and Subrahmanyam (2000), and for generalzed Amercan knock out opton n Lundgren (2007, 2010). Convergence of Amercan opton rewards have been studes n Amn and Khanna (1994), Coquet and Toldo (2007), Dupus and Wang (2005), Jönsson (2001, 2005), Jönsson, Kukush and Slvestrov (2004, 2005), Kukush and Slvestrov (2000, 2001, 2004), Prgent (2003), Slvestrov, Galochkn and Sbrtsev (1999). For recent results concernng optmal stoppng we refer to the works Dayank and Karatzas (2003), Ekström, Lndberg, Tysk and Wanntorp (2009), Henderson and Hobson (2008). We specally would lke to menton the papers Slvestrov, Jönsson and Stenberg (2006, 2007, 2009) on convergence of optmal reward functonals for Amercan type optons for one-dmensonal Markov prce processes modulated by stochastc ndces. We consder the convergence of reward functonals for Amercan type optons n a mult-asset settng. In the frst part of the paper, we generalze the results obtaned n the papers mentoned above to the model of multvarate Markov prce processes. Despte the man steps for analyss n multvarate case are smlar wth those n the unvarate case, the multvarate aspects complcate the correspondng proofs and do requre a separate consderaton. In the second part of the paper, we apply the general convergence results to the multvarate tree type approxmaton algorthms for Amercan type optons wth general payoff functons. We present the correspondng approxmatons for multvarate geometrc Brownan prce processes and mean-reverse dffuson prce processes. The latter model was used n Schwartz (1997) for descrpton of stochastc dynamcs for energy prces. Smlar model was also studed n Cortazar Gravet and Urzua (2008), where Amercan optons where studed usng Monte Carlo smulaton. Further applcatons presented n the paper relate to the approxmaton tree type algorthms for the model of optmal resellng of European type optons. We refer to the recent paper Lundgren, Slvestrov and Kukush (2008), where one can fnd addtonal detals n analyss of the resellng problem. The paper contans of 8 sectons. In Secton 2, we ntroduce the model of multvarate prce processes and Amercan type optons wth general payoff functons. In Secton 3, we present skeleton type approxmatons of reward functonals for contnuous tme prce processes by a smlar functonals for smpler mbedded dscrete tme models. In Secton 4, the results concerned condtons for convergence of reward functonals n dscrete tme models are gven. Secton 5 presents general results on convergence of reward functonals for Amercan type optons for contnuous tme prce processes. In Sectons 6, we llustrate our general convergence results by applyng them to multvarate exponental prce processes wth ndependent ncrements. In Sectons 7 and 8, we apply our general convergence results to approxmaton tree type algorthms for Amercan type optons for exponental dffuson type prce processes mentoned above. 2

7 2 Reward functonal for general Amercan type optons In ths secton we ntroduce reward functonals for Amercan type optons wth general payoff functons and formulate condtons whch guarantee that the functonals are well defned,.e take fnte values. The correspondng nequaltes are also used n followng proofs of our convergence results. For every ε 0, let Y (t) = (Y 1 (t),..., Y k (t)), t 0 be a càdlàg Markov process wth the phase space R k and transton probabltes P (t, y, t + s, A). We nterpret Y (t) as a vector log-prce process. Now, we defne a vector prce process S (t) = (S 1 (t),..., S k (t)), t 0 wth the phase space R k + = R + R +, where R + = (0, ), by the relatons S (t) = e Y (t), = 1,..., k, t 0. (2.1) Due to the one-to-one mappng and contnuty propertes of exponental functon, S (t) s also a càdlàg Markov process. For every ε 0, let g (t, s), (t, s) [0, ) R k + be a payoff functon. We assume g (t, s) to be a real-valued Borel measurable functon. Note that we do not assume payoff functons to be non-negatve. Let F t = σ( Y (s), s t), t 0 be the natural fltraton of σ-felds, assocated wth the vector log-prce process Y (t), t 0. It s useful to note that ths fltraton concdes wth the natural fltraton generated by the prce process S (t), t 0. We consder Markov moments τ wth respect to the fltraton F t, t 0. It means that τ s a random varable whch takes values n [0, ] and wth the property {ω : τ (ω) t} F t, t 0. Let M max,t be the class of all Markov moments τ T, where T > 0, and consder a class of Markov moments M T M max,t. Below we mpose condtons on prce processes and payoff functons whch guarantee that, for all ε small enough, sup E g(τ, S (τ ) <. (2.2) τ M max,t The man object of our studes s the reward functonal, that s, the maxmal expected payoff over dfferent classes of Markov moments, M T, Φ(M T ) = sup Eg (τ, S (τ )). (2.3) τ M T We are nterested n condtons of convergence for reward functonals for dfferent classes of stoppng tmes. In partcular, we formulate condtons mplyng the 3

8 followng convergence relaton: Φ(M max,t ) Φ(M(0) max,t ) as ε 0. (2.4) The frst condton assumes the absolute contnuty of payoff functons and mposes power type upper bounds on ther partal dervatves: A 1 : There exsts ε 0 > 0 such that for every 0 ε ε 0 : (a) functon g (t, s) s absolutely contnuous n t wth respect to the Lebesgue measure on [0, T ] for every fxed s R k + and n s wth respect to the Lebesgue measure on R k + for every fxed t [0, T ]; (b) for every s R k +, the partal dervatve g (t, s) K t 1 + K k 2 j=1 sγ 0 j for almost all t [0, T ] wth respect to the Lebesgue measure on [0, T ], where 0 K 1, K 2 < and γ 0 0; (c) for every t [0, T ], the partal dervatve g (t, s) s m K 3 + K 4 k j=1 sγm j for almost all s R k + wth respect to the Lebesgue measure on R k +, where 0 K 3, K 4 < and γ 1,..., γ k 0, m = 1,..., k. (d) for every t [0, T ] and for s R k +, the upper lmt lm s 0 g (t, s) K 5, where 0 K 5 <. Note that condton A 1 mples that the functon g (t, s) s contnuous n (t, s) [0, T ] R k +. Denote e y = (e y 1,..., e y k ). Then condton A1 can be re-wrtten n the equvalent form n terms of the functon g (t, e y ): A 1: There exsts ε 0 > 0 such that for every 0 ε ε 0 : (a) functon g (t, e y ) s absolutely contnuous n t wth respect to the Lebesgue measure on [0, T ] for every fxed y R k and n y wth respect to the Lebesgue measure on R k for every fxed t [0, T ]; (b) for every y R k, the partal dervatve n t s bounded as g (t,e y ) t K 1 + K k 2 j=1 eγ 0y j for almost all t [0, T ] wth respect to the Lebesgue measure on [0, T ], where 0 K 1, K 2 < and γ 0 0; (c) for every t [0, T ], the partal dervatve n y m s bounded as g (t,e y ) y m (K3 + K 4 k j=1 eγ my j )e y m for almost all y R k wth respect to the Lebesgue measure on R k, where 0 K 3, K 4 < and γ 1,..., γ k 0, m = 1,..., k. 4

9 (d) for every t [0, T ], the upper lmt lm y,=1,k g (t, e y ) K 5, where 0 K 5 <. We use the notatons E y,t and P y,t for expectaton and probablty calculated under condton Y (t) = y. For β, c, T > 0, = 1,..., k, defne the exponental moment modulus of compactness for the càdlàg process Y (t), t 0, β (Y ( ), c, T ) = sup 0 t t+u t+c T sup E y,t (e β Y y R k (t+u) Y (t) 1). We use the followng condton for exponental moment modulus of compactness for log-prce processes: C 1 : lm c 0 lm ε 0 β (Y ( ), c, T ) = 0, = 1,..., k for some β > γ, where γ = max(γ 0, γ 1 +1,..., γ k + 1) and γ 0, γ 1,..., γ k are the parameters ntroduced n condton A 1, and also the followng condton: C 2 : lm ε 0 Ee (0) <, = 1,..., k, where β s the parameter ntroduced n condton C 1. β Y The followng lemmas gves upper bounds for moments of the maxmum of prce processes whch are asymptotcally unform, wth respect to perturbaton parameter. Lemma 2.1. Let condtons C 1 and C 2 hold. Then, there exst 0 < ε 1 ε 0 and a constant L 1 < such that for every ε ε 1, and = 1,..., k, E exp{β sup Y (u) } L 1. (2.5) 0 u T Proof. The followng equalty holds for each 0 t T and = 1,..., k, βs (t) = exp{β sup Y (u) } = sup exp{β Y (u) }. (2.6) 0 u t 0 u t Note also that, by the defnton, the random varable, Let us also ntroduce random varables, βs (0) = exp{β Y (0) }. (2.7) βw [t, t ] = sup exp{β Y (t) Y (t ) }, 0 t t T. t t t 5

10 Defne a partton Π m = {0 = v (m) 0 <... < v m (m) = T } on the nterval [0, T ] by ponts v n (m) = nt/m, n = 0,..., m. Usng equalty (2.6) we can get the followng nequaltes for n = 1,... m and = 1,..., k, βs (v n (m) ) β S (v n 1) (m) + β S (v n 1) (m) + exp{β Y β S (v n 1)( (m) β W [v (m) sup exp{β Y (u) } (2.8) v (m) n 1 u v(m) n (v (m) n 1, v (m) n 1) } β W n ] + 1). [v n 1, (m) v n (m) ] Condton C 1 mples that for any constant e β < L 5 < 1 one can choose c = c(l 5 ) > 0 and then ε 1 = ε 1 (c) ε 0 such that for ε ε 1, and = 1,..., k, β (Y ( ), c, T ) + 1 e β L 5. (2.9) Moreover, condton C 2 mples that ε 1 can be chosen n such a way that, for some constant L 6 = L 6 (ε 1 ) <, the followng nequalty holds for ε ε 1 and = 1,..., k, E exp{β Y (0) } L 6. (2.10) where We need to show that the followng nequalty holds for ε ε 1 and = 1,..., k, sup 0 t t t +c T sup E y,t βw [t, t ] L 7, (2.11) y R k L 7 = eβ (e β 1)L 5 1 L 5 <. (2.12) Usng condton C 2, relatons (2.8), (2.10) (2.12), and Markov property of the process Y (t) we get, for ε ε 1, = 1,..., k and m = [T/c] + 1, and n = 1,..., m, E ( βs (v n (m) ) ) E ( βs (v n 1)E( (m) β W [v n 1, (m) v n (m) ] + 1/ Y (v (m) E β S (v n 1)(L (m) 7 + 1) E β S (0)(L 7 + 1) n L 6 (L 7 + 1) n. Fnally, for ε ε 1 and = 1,..., k, the followng nequalty holds E β S (v m (m) ) = E exp{β sup 0 u T n 1)) ) Y (u) } L 6 (L 7 + 1) m. (2.13) Relaton (2.13) mples that nequalty (2.5) holds, for ε ε 1, wth the constant, L 1 = L 6 (L 7 + 1) m. (2.14) 6

11 To show that relatons (2.11) and (2.12) holds, we have from relaton (2.9) that for every ε ε 1 and = 1,..., k, sup 0 t t t t +c T sup P y,t { Y (t ) Y (t) 1} (2.15) y R k sup 0 t t t t +c T E y,t exp{β Y sup y R k β(y ( ), c, T ) + 1 e β L 5 < 1. (t ) Y (t) } The process Y (t) s not a Markov process. Despte ths, an analogue of Kolmogorov nequalty can be obtaned by slght modfcaton of ts standard proof for Markov processes, see, for example Gkhman and Skorokhod (1971). We formulate t n the form of a lemma. Lemma 2.2. Let a, b > 0 and for the process Y (t) assume that the followng condton hold sup y R k P y,t { Y (t ) Y (t) a} L < 1, t t t. Then, for any vector y R k, P y,t { sup Y t t t e β (t) Y (t ) a + b} (2.16) 1 1 L P y,t { Y (t ) Y (t ) b}. We refer to the report Slvestrov, Jönsson and Stenberg (2006) for the proof. To shorten notatons denote the random varables. Note that W + = sup Y t t t (t) Y (t ), W = Y (t ) Y (t ). e βw + = β W [t, t ]. Usng (2.15) and Lemma 2.2, we get for every ε ε 1, = 1,..., k, 0 t t t + c T, y R k, and b > 0, P y,t {W b} 1 1 L 5 P y,t {W b}. (2.17) 7

12 Relatons (2.9) and (2.17) mply that for every ε ε 1, = 1,..., k, 0 t t t + c T and y R k, E y,t e βw + = 1 + β 1 + β = e β + β e β + βeβ 1 L 5 e βb P y,t {W + e βb db + β 1 b}db e β(1+b) P y,t {W + = e β + βeβ E y,t e βw 1 1 L 5 β 0 e βb P y,t {W b}db e βb P y,t {W b}db b}db = eβ 1 L 5 (E y,t e βw L 5 ) e β 1 L 5 ( β (Y ( ), c, T ) + 1 L 5 ) eβ (e β 1)L 5 1 L 5 = L 7. (2.18) Snce nequalty (2.18) holds for every ε ε 1 and 0 t t t + c T, y R k, t mply relaton (2.11). The proof s complete. Lemma 2.3. Let condtons A 1, C 1, and C 2 hold. Then, there exsts a constant L 2 < such that for every ε ε 1, sup E g(τ, S (τ ) τ M max,t E ( sup g (u, S (u)) ) β γ L 2. 0 u T (2.19) Proof. Consder the vectors s = (s 1,..., s k ), s = (s 1,..., s k ) Rk + and construct the vectors s = (s 1,..., s, s +1,..., s k ), s (v) = (s 1,..., s 1, v, s +1,..., s k ), = 1,..., k. By the defnton, s k = s and s 0 = s. Defne also s + = s s, s = s s, = 1,..., k. Usng condton A 1 we get the followng estmates, for ε ε 0 and s, s R k +, g (u, s ) g (u, s ) g (u, s ) g (u, s 1 ) =1 8

13 =1 =1 =1 s + s s + s g (u, s (v)) dv v 1 (K 3 + K 4 ( j=1 (s j) γ + (v ) γ + j=+1 (K 3 (s + s ) + K 4( (s + j )γ (s + s ))) j=1 (K 3 + K 4 (s + j )γ )(s + s ). =1 j=1 (s j ) γ ))dv (2.20) By lettng s = s = (s 1,..., s k ) R k + and s j 0 n (2.20) we get the followng nequalty, for ε ε 0 and s R k +, g (u, s) K 5 + K 3 s + K 4 (s j ) γ s K 5 + K 3 K 5 + K 3 =1 =1 j=1 (1 s ) γ + K 4 =1 = L 8 + L 9 (s ) γ, =1 (1 s j s ) γ j=1 (1 + (s ) γ ) + K 4 =1 =1 j=1 =1 (1 + (s j ) γ + (s ) γ ) (2.21) where L 8 = K 5 + kk 3 + k 2 K 4, L 9 = K 3 + 2kK 4. Let us denote for the moment S = sup 0 u T Usng nequalty (2.21) we get S (u), Y = sup Y (u), = 1,..., k. 0 u T sup 0 u T g (u, S (u)) L 8 + L 9 S γ. (2.22) 9 =1

14 Usng relaton (2.6) from the proof of Lemma 2.1 and nequalty (2.21) we get ( sup g (u, S (u)) ) β γ 0 u T (k + 1) β γ 1 β γ (L 8 + L β γ 9 (1 + S β ) =1 = L 10 + L 11 S β L 10 + L 11 e βy, =1 =1 (2.23) where L 10 = (k + 1) β γ 1 β γ (L β γ 8 + kl9 ), L 11 = (k + 1) β γ 1 L Thus, by takng expectaton n (2.23) and usng Lemma 2.1, we get for ε ε 1, β γ 9. E ( sup g (u, S (u)) ) β γ 0 u T L 10 + L 11 Ee βy L 10 + kl 11 L 1 = L 2 <. =1 (2.24) The proof s complete. Inequalty (2.24) mples that for ε ε 1, Φ(M max,t ) sup E g(τ, S (τ ) τ M max,t E sup 0 u T g (u, S γ β (u)) L2 <. (2.25) Thus, the reward functonal Φ(M max,t ) s well defned for ε ε 1. 3 Skeleton approxmatons for reward functonals In ths secton we gve explct and unform wth respect to perturbaton parameter estmates n so-called skeleton approxmatons where reward functonals for contnuous tme prce processes are approxmated by the correspondng reward functonals for dscrete tme prced processes. These approxmatons plays the key role n the proofs on the correspondng convergence results. They also have ther own value. For example, such approxmatons can be used for justfcaton of Monte Carlo type algorthms, where contnuous type prce processes should be approxmated by the correspondng dscrete tme processes. 10

15 Let Π = {0 = t 0 < t 1 <... t N = T } be a partton on the nterval [0, T ] and ˆM Π,T d(π) = max 1 N (t t 1 ). Consder the class of all Markov moments from M max,t, whch only take the values t 0, t 1,... t N, and the class M Π,T of all Markov moments τ from ˆM Π,T such that the event {ω : τ (ω) = t j } σ( Y (t 0 ),..., Y (t j )) for j = 0,... N. By defnton, M Π,T ˆM Π,T M max,t. (3.1) Relatons (3.1) mply that, under condtons of Lemma 2.2, for ε ε 1, < Φ(M Π,T ) Φ( ˆM Π,T ) Φ(M max,t ) <. (3.2) The reward functonals Φ(M max,t ), Φ( ˆM Π,T ), and Φ(M Π,T ) correspond to Amercan type opton n contnuous tme, Bermudan type opton n contnuous tme, and Amercan type opton n dscrete tme, respectvely. In the frst two cases, the underlyng prce process s a contnuous tme Markov type prce process, whle n the thrd case the correspondng prce process s a dscrete tme Markov type process. The random varables Y (t 0 ), Y (t 1 ),..., Y (t N ) are connected n a dscrete tme nhomogeneous Markov chan wth the phase space R k, transton probabltes P (t n, y, t n+1, A), and ntal dstrbuton P (A). Note that we have slghtly modfed the standard defnton of a dscrete tme Markov chan by consderng moments t 0,..., t N as the moments of jumps for the Markov chan Y (t n ) nstead of the moments 0,..., N. Ths s done n order to synchronze the dscrete and contnuous tme models. Thus, the optmzaton problem (2.3) for the class M Π,T s really a problem of optmal expected reward for Amercan type optons n dscrete tme. The followng lemma establshes useful equalty between reward functonals Φ(M Π,T ) and Φ( ˆM Π,T ). Lemma 3.1. Let condtons A 1, C 1 and C 2 hold. Then, for any partton Π = {0 = t 0 < t 1 <... < t N = T } on the nterval [0, T ] and ε ε 1 where ε 1 s defned n (2.9) and (2.10), Φ(M Π,T ) = Φ( ˆM Π,T ). (3.3) Proof. A smlar result was gven n Kukush and Slvestrov (2000, 2004) and we shortly present the modfed verson of the correspondng proof. The optmzaton problem (2.3) for the class ˆM Π,T can be consdered as a problem of optmal expected reward for Amercan type optons wth dscrete tme. To see ths let us add to the random varables Y tn addtonal components Y n = 11

16 { Y (t), t n 1 < t t n } wth the correspondng phase space Y endowed by the correspondng σ-feld. As Y 0 we can take an arbtrary pont n Y. Consder the extended Markov chan Y n = ( Y (t n ), Y n ) wth the phase space Y = Y Y. As above, we slghtly modfy the standard defnton and count moments t 0,..., t N as moments of jumps for the ths Markov chan nstead of moments 0,..., N. Ths s done n order to synchronze the dscrete and contnuous tme models. Denote by M Π,T the class of all Markov moments τ t N for the dscrete tme Markov chan Y n and consder the reward functonal, Φ( M Π,T ) = τ sup M Π,T Eg (τ, S (τ )). (3.4) It s readly seen that the optmzaton problem (2.3) for the class equvalent to the optmzaton problem (3.4),.e., ˆM Π,T s Φ( ˆM Π,T ) = Φ( M Π,T ). (3.5) It s well known, (See, for example Peskr and Shryaev (2006)) that the optmal stoppng moment τ exsts n any dscrete tme Markov model, and the optmal decson {τ = t n } depends only on the value Y n. Moreover the optmal Markov moment has a frst httng tme structure,.e., t s the frst moment the process enters some set D, that s τ = mn(t n : Y n D n ), where D n, n = 0,..., N are some measurable subsets of the phase space Y. The optmal stoppng domans are determned by the transton probabltes of the extended Markov chan Y n. However, the extended Markov chan Y n has transton probabltes dependng only on values of the frst component Y (t n ). As was shown n Kukush and Slvestrov (2004), the optmal Markov moment has n ths case the frst httng tme structure of the form τ = mn(t n : Y (t n ) D n ), where D n, n = 0,..., N are some measurable subsets of the phase space of the frst component Y. Therefore, for the optmal stoppng moment τ the decson {τ = t n } depends only on the value Y (t n ), and τ M Π,T. Hence, Φ(M Π,T ) Eg (τ, S (τ )) = Φ( ˆM Π,T ). (3.6) Inequaltes (3.2) and (3.6) mply the equalty (3.3) and the proof s complete. The followng theorem gves a skeleton approxmaton for the reward functonals Φ(M max,t ) whch s asymptotcally unform wth respect to the perturbaton parameter ε. Theorem 3.2. Let condtons A 1, C 1, and C 2 hold. Let also ε ε 1 and d(π) c, where ε 1 and c are defned n relatons (2.9) and (2.10). Then there exst constants 12

17 L 3, L 4 < such that the followng skeleton approxmaton nequalty holds, for ε ε 1, Φ(M max,t ) Φ(M L 3 d(π) + L 4 Π,T ) =1 β (Y ( ), d(π), T ) β γ β. (3.7) Proof. Let us assume that ε ε 1 and d(π) c. For any Markov moment τ M max,t and a partton Π = {0 = t 0 < t 1 <... < t N = T } one can defne the dscretzaton of ths moment, { τ tk, f t [Π] = k 1 τ < t k, k = 1,... N, T, f τ = T. Let τ be -optmal Markov moment n the class M max,t,.e., Eg (τ, S (τ )) Φ(M max,t ). (3.8) Such -optmal Markov moment always exsts for any > 0, by the defnton of the reward functonal Φ(M max,t ). By the defnton, the Markov moment τ [Π] (3.3) gven n Lemma 3.1 mples that By the defnton, Eg (τ [Π], S (τ [Π])) Φ( τ ˆM Π,T. Ths fact and relaton ˆM Π,T ) = Φ(M Π,T ) Φ(M max,t ). (3.9) τ [Π] τ + d(π). (3.10) Now nequaltes (3.8) and (3.9) mply the followng skeleton approxmaton nequalty, 0 Φ(M max,t ) Φ(M Π,T ) + Eg (τ, S (τ )) Eg (τ [Π], S (τ [Π])) + E g (τ, S (τ )) g (τ [Π], S (τ [Π])). (3.11) Now, we need to mprove relaton (2.20). We use the same notatons and let also 0 u, u T and u + = u u, u = u u. 13

18 Usng condton A 1 we get the followng estmates, for ε ε 0 and s, s R k +, 0 u, u T, g (u, s ) g (u, s ) g (u, s ) g (u, s ) + g (u, s ) g (u, s ) g (u, s ) g (u, s ) + g (u, s ) g (u, s 1 ) u + g (u, s ) du + u u =1 (K 1 + K 2 (s j) γ 0 )du u u + + =1 s + s (K 1 + K 2 + j=1 1 (K 3 + K 4 ( j=1 =1 s + s (s + j )γ 0 ) u u j=1 g (u, s (v)) dv v (s j) γ + (v ) γ + (K 3 + K 4 (s + j )γ ) s s. =1 j=1 j=+1 (s j ) γ ))dv (3.12) In the case, where vectors s = e y, s = e y,.e., s = e y, s = e y, = 1,..., k, and, therefore, s ± = e y±, = 1,..., k, where y + = y y, y = y y, = 1,..., k, nequalty (3.12) can be transformed, usng nequalty e y y+ 1 y + y = y y, to the followng form, g (u, e y ) g (u, e y ) (K 1 + K 2 e γ 0y + j ) u u + =1 j=1 (K 3 + K 4 e γ y + j e y + ) y y. j=1 To shorten notatons denote the random varables τ = τ, τ = τ [Π], Y = Y (τ ), Y = Y (τ ), = 1,..., k. and random vectors Y = (Y 1,..., Y k), Y = (Y 1,..., Y k ). 14 (3.13)

19 Also defne random varables Y + = Y Y, = 1,..., k. Then, Y + Y = sup Y (u), = 1,..., k. (3.14) 0 u T Thus by substtutng random varables ntroduced above n the nequalty (3.13) and then usng (3.14) we readly get that the followng estmate holds under condton A 1, for ε ε 0, g (τ Y, e ) g (τ Y, e ) (K 1 + K 2 e γ 0Y + )(τ τ ) + =1 =1 (K 3 + K 4 e γ Y + j e Y + ) Y Y (K 1 + K 2 + =1 j=1 e γ 0Y )(τ τ ) =1 (K 3 + K 4 e γ Y j e Y ) Y Y j=1 (3.15) Now, applyng Hölder nequalty (wth parameters p = γ β γ and q = and then β β wth parameters p = γ 1 and q = 1 ) to the correspondng products of random γ γ varables on the rght hand sde n (3.15) and by usng nequalty (2.5) gven n Lemma 2.1 and relaton (3.10), we can wrte down the followng estmate for the expectaton on the rght hand sde n (3.11), for ε ε 1, E g (τ, S (τ )) g (τ [Π], S (τ [Π])) = E g (τ Y, e ) g (τ Y, e ) (K 1 + K 2 Ee γ 0Y )d(π) + K 3 + K 4 =1 (K 1 + K 2 + K 4 =1 =1 j=1 =1 j=1 Ee γ Y j e Y Y Y (Ee γ 0 β =1 E Y Y γ Y ) γ β )d(π) + K3 (E Y Y =1 (Ee γ β γ Y j e β γ Y ) γ β (E Y Y β β γ β γ ) β β β γ β γ ) β (3.16) 15

20 where (K 1 + K 2 + K 4 =1 (Ee βy ) γ β )d(π) + K3 =1 ((Ee γ β j=1 γ 1 Y j ) γ 1 γ (K 1 + kk 2 (L 1 ) γ β )d(π) + K3 + K 4 =1 j=1 (K 1 + kk 2 (L 1 ) γ β )d(π) =1 (E Y Y β β γ β γ ) β (Ee βy ) 1 γ γ ) β (E Y Y β β γ β γ ) β =1 (E Y Y β β γ β γ ) β ((L 1 ) γ 1 γ (L1 ) 1 γ γ ) β (E Y Y β β γ β γ ) β + (K 3 + kk 4 (L 1 ) γ β ) (E Y Y =1 β β γ β γ ) β The last step n the proof s to show that, for ε ε 1 and = 1,..., k, where E Y Y β γ β L 12 = sup y 0 L 12 β (Y ( ), d(π), T ), (3.17) y β β γ e βy 1 Indeed, n ths case, (3.16) take the form E g (τ <. (3.18), S (τ )) g (τ [Π], S (τ [Π])) L 3 d(π) + L 4 =1 (3.19) β (Y ( ), d(π), T ) β β γ, L 3 = K 1 + kk k (L 1 ) γ β, L4 = (K 3 + kk 4 (L 1 ) γ β )(L12 ) β β γ. (3.20) Note that the quantty on the rght hand sde n (3.19) does not depend on. Thus, we can substtute t n (3.11) and then pass to zero n ths relaton that wll result n nequalty (3.7) gven n Theorem 3.2. To get nequalty (3.17), we employ the method from Slvestrov (1974) for estmaton of moments for ncrements of stochastc processes stopped at Markov type stoppng moments. Introduce the functon f Π (t) = t j+1 t for t j t < t j+1, j = 0,..., N 1 and 0 for t = t (N) N. The functon f Π(t) s contnuous from the rght on the nterval [0, T ] and 0 f Π (t) d(π). 16

21 and It follows from the defnton of functon f Π (t) that Y (τ ) Y (τ τ = τ + f Π (τ ), [Π]) = Y (τ ) Y (τ + f Π (τ )). Use agan partton Π m of nterval [0, T ] by ponts v (m) n = nt/m, n = 0,..., m, and consder the random varables, τ [ Π m ] = { v (m) j, f v (m) j 1 τ < v (m) j, j = 1,... N, T, f τ = T. Thus we have τ τ [ Π m ] τ +T/m, and the random varables τ [ Π m ] a.s. τ as m. Snce the Y (t) s a vector of càdlàg processes, we also get the followng relaton, for each = 1..., k Q m, = Y a.s. Q (τ [ Π m ]) Y (τ [ Π m ] + f Π (τ [ Π m ])) β = Y β γ (3.21) (τ ) Y (τ + f Π (τ )) β β γ as m. Note also that Q m, are non-negatve random varables and the followng estmate holds for any m = 1,... and = 1,..., k, Q m, ( Y 2 β (τ [ Π m ]) + Y (τ [ Π m ] + f Π (τ [ Π m ])) ) β β γ 1 ( Y 2 β β γ ( sup 0 u T (τ [ Π m ]) β β γ Y (u) ) β β γ 2 β β γ L12 exp{β sup 0 u T β γ + Y (τ [ Π m ] + f Π (τ [ Π m ])) β β γ ) Y (u) }. (3.22) Inequalty (2.5) gven n Lemma 2.1, mples that the random varable on the rght hand sde n (3.22) has a fnte expectaton, and relaton (3.21), we get by Lebesgue theorem that, for ε ε 1 and = 1..., k, EQ m, EQ as m. (3.23) Let us now fnd an estmate of EQ m. To reduce the notaton let us denote for the moment Y n+1, = Y (v n+1) (m) and Y n+1, = Y (v n+1)+f (m) Π (v n+1)). (m) Snce τ s a Markov moment for the Markov process Y (t), the random varables χ(v n (m) τ < v n+1) (m) and Y n+1, Y n+1, β β γ are condtonally ndependent wth respect to random vector 17

22 Y (v n+1). (m) Usng ths fact and nequalty f Π (v n+1) (m) d(π), we get, for ε ε 1, and = 1..., k, EQ m, = E Y n=0 m 1 (τ [ Π m ]) Y (τ [ Π m ] + f Π (τ [ Π m ])) β m 1 = E Y n+1, Y n+1, β β γ χ(v (m) n τ < v n+1) (m) = n=0 E{χ(v (m) n m 1 sup E (m) y,v n=0 y R k n+1 m 1 n=0 m 1 n=0 β γ τ < v (m) n+1)e{ Y n+1, Y n+1, β β γ / Y (v (m) n+1)}} Y n+1, Y n+1, β β γ P{v (m) n τ < v n+1} (m) L 12 sup E (m) y,v exp{β Y n+1, Y n+1, }P{v n (m) τ < v n+1} (m) y R k n+1 L 12 β (Y ( ), d(π), T )P{v n (m) τ < v n+1} (m) L 12 β (Y ( ), d(π), T ). Relatons (3.23) and (3.24) mply that, for ε ε 1 and = 1..., k, EQ (3.24) = E Y (τ ) Y (τ + f Π (τ )) β β γ (3.25) L 12 β (Y ( ), d(π), T ). Ths nequalty s equvalent to nequalty (3.19). The proof s complete. 4 Convergence of rewards for dscrete tme prce processes In ths secton we present convergence results for reward functonals for Amercan type optons for dscrete tme prce processes. These results have ther own value and are also essentally used n the proofs of the correspondng convergence results for contnuous tme prce processes. Let us now formulate condtons of convergence for dscrete tme reward functonals Φ(M Π,T ) for a gven partton Π = {0 = t 0 < t 1... < t N = T } on the nterval [0, T ]. In ths case t s natural to use condtons based on the transton probabltes between the sequental moments of ths partton and values of the payoff functons at the moments of ths partton. Condton A 1 s replaced by a smpler condton: 18

23 A 2 : There exsts ε 2 > 0 such that, for every 0 ε ε 2, functon g (t n, s) K 6 +K k 7 =1 sγ, for n = 0,..., N and s Rk + for some γ 1 and constants K 6, K 7 <. Condton A 2 can be re-wrtten n terms of functons g (t, e y ): A 2: There exsts ε 2 > 0 such that, for every 0 ε ε 2, functon g (t n, e y ) K 6 + K k 7 =1 eγy, for n = 0,..., N and y R k for some γ 1 and constants K 6, K 7 <. We also need an assumpton about convergence of the payoff functon. In dscrete tme case, the dervatves of the payoff functons are not nvolved. In ths case, the payoff functons can be dscontnuous. Ths s compensated by a stronger assumpton concerned the convergence of the payoff functons. We requre local unform convergence for the payoff functon on some sets, whch later wll be assumed to have the value 1 for the correspondng lmt transton probabltes and the lmt ntal dstrbuton: A 3 : There exsts a measurable set S t n R k + for every n = 0,..., N, such that g (t n, s ) g (0) (t n, s) as ε 0, for any s s S t n and n = 0,..., N. Condton A 3 can be re-wrtten n terms of functons g (t, e y ): A 3: There exsts measurable sets Y t n R k, n = 0,..., N, such that the payoff functon g (t n, e y ) g (0) (t n, e y ) as ε 0 for any y y Y t n and n = 0,..., N. The sets S t n and Y t n n condtons A 3 and A 3 are connected by the followng relaton S t n = { s = e y : y Y t n }, n = 0,..., N. Let us now formulate condtons assumed for the transton probabltes and ntal dstrbutons of the process Y (t). The frst condton assumes weak convergence (denoted by the symbol ) of the transton probabltes that should be locally unform wth respect to ntal states from some sets, and also that the correspondng lmt measures are concentrated on these sets: B 1 : There exst measurable sets Y t n R k, n = 0,..., N such that: (a) P (t n, y, t n+1, ) P (0) (t n, y, t n+1, ) as ε 0, for any y y Y t n as ε 0 and n = 0,..., N 1; (b) P (0) (t n, y, t n+1, Y t n+1 Y t n+1 ) = 1 for every y Y t n and n = 0,..., N 1. 19

24 A typcal example s when the sets Ȳ t n and Ȳ t n are at most fnte or countable sets. Then the assumpton that the measures P (0) (t n, y, t n+1, A), A B R k have no atom mples that condton B 1 (b) holds. The second condton assumes weak convergence of the ntal dstrbutons to some dstrbuton that s assumed to be concentrated on the sets of convergence for the correspondng transton probabltes: B 2 : (a) P ( ) P (0) ( ) as ε 0; (b) P (0) (Y 0 Y 0) = 1, where Y 0 and Y 0 are the sets ntroduced n condtons A 3 and B 1. A typcal example s when the sets Ȳ t n and Ȳ t n are at most fnte or countable sets. Then the assumpton that the measures P (0) (A) have no atom mples that condton B 2 (b) holds. We also weaken condton C 1 by replacng t by a smpler condton, whch s mpled by condton C 1 : C 3 : lm ε 0 sup y R k E y,tn (e (t n ) 1) <, n = 0,..., N 1, = 1,..., k, for some β > γ, where γ s the parameter ntroduced n condton A 2. β Y (t n+1 ) Y Condton C 2 does not change and takes the followng form: C 4 : lm ε 0 Ee (t 0 ) <, = 1,..., k, where β s the parameter ntroduced n condton C 3. β Y Condton C 3 mples that there exsts a constant L 13 < and ε 3 ε 2 such that for n = 0,..., N 1, ε ε 3, and = 1,... k, sup E y,tn (e β Y y R k (t n+1 ) Y (t n ) 1) L 13. (4.1) Condton C 4 mples that ε 3 can be chosen n such a way that, for some constant L 14 <, the followng nequalty holds for ε ε 3 and = 1,..., k, β Y Ee (0) L 14. (4.2) A dscrete analogue to nequalty (2.25) can be wrtten as Φ(M Π,T ) sup E g(τ, S (τ ) τ M Π,T E max 0 N g (t, S γ β (t )) c1 <. (4.3) The followng theorem gves condtons of convergence for reward functonals ) for a gven partton Π. Φ(M Π,T 20

25 Theorem 4.1. Let condtons A 2, A 3, B 1, B 2, C 3, and C 4 hold. Then, the followng asymptotc relaton holds for the partton Π = {0 = t 0 < t 1 < t N = T } on the nterval [0, T ], Φ(M Π,T ) Φ(M(0) Π,T ) as ε 0. (4.4) Proof. Snce ε 0 n (4.4), we can assume that ε ε 2. The reward functons are defned by the followng recursve relatons, and, for n = 0,..., N 1, w (t N, y) = g (t N, e y ), y R k, w (t n, y) = max(g (t n, e y ), E y,tn w (t n+1, Y (t n+1 )), y R k. As follows from general results on optmal stoppng for dscrete tme Markov processes, see Chow Robbns and Segmund (1971), Shryaev (1976), the reward functonal, Φ(M Π,T ) = Ew (t 0, Y (0)). (4.5) Condton A 2 drectly mples that the followng power type upper bound for the reward functon w (t N, y) holds, for ε ε 3 and y R k, w (t N, y) L 1,N + L 2,N e γ y, (4.6) =1 where L 1,N = K 6, L 2,N = K 7 <. (4.7) Also, accordng condtons A 3, for an arbtrary y y (0) as ε 0, where y (0) Ỹ t N Y t N, w (t N, y ) w (0) (t N, y (0) ) as ε 0. (4.8) Let us prove that relatons smlar wth (4.6), (4.7), and (4.8) hold for the reward functons w (t N 1, y). 21

26 We get usng relaton (4.6), for ε ε 3 and y R k, E y,tn 1 g (t N, e Y (t N ) ) L 1,N + L 2,N E y,tn 1 L 1,N + L 2,N E y,tn 1 L 1,N + L 2,N L 1,N + L 2,N =1 e γ y E y,tn 1 e =1 e γ y E y,tn 1 e =1 L 1,N + L 2,N (L ) e γ y γ Y e (t N ) y e γ y. =1 =1 γ Y (t N ) Y (t N 1 ) β Y (t N ) Y (t N 1 ) γ Y e (t N ) (4.9) where Relaton (4.9) mples that, for ε ε 3 and y R k, w (t N 1, y) max( g (t N 1, e y ), E y,t w (t N, Y (t N )) ) K 6 + L 1,N + (K 7 + L 2,N (L )) e γ y = L 1,N 1 + L 2,N 1 e γ y, =1 =1 (4.10) L 1,N 1 = K 6 + L 1,N, L 2,N 1 = K 7 + L 2,N (L ) <. (4.11) Let us ntroduce, for every n = 0,..., N 1 and y R k, a random vector Y n ( y) such that P{ Y n ( y) A} = P (t n, y, t n+1, A), A B R k. Let us prove that, for any y y (0) Ỹ t N 1 Y t N 1 as ε 0, the followng relaton takes place, w (t N, Y N 1 ( y )) w (0) (t N, Y (0) N 1 ( y(0) )) as ε 0. (4.12) Take an arbtrary sequence ε r ε 0 = 0 as r. From condton B 1 we have for an arbtrary y (ε r) y (ε 0) Ỹ t N 1 Y t N 1 as r, Y (ε r) N 1 ( y(εr) ) Y (ε 0) N 1 ( y(ε 0) ) as r, (4.13) and P{ Y (ε 0) N 1 ( y(ε 0) ) Ỹ t N Y t N } = 1. (4.14) 22

27 Now, accordng to Skorokhod representaton theorem, one can construct random varables Y (ε r) N 1 ( y(ε r) ), r = 0, 1,... on some probablty space (Ω, F, P) such that, for every r = 0, 1,..., and A B R k, and Denote P{ Y (ε r) N 1 ( y(εr) ) A} = P{ Y (ε r) N 1 ( y(εr) ) A}, (4.15) Y (ε r) N 1 ( y(εr) ) a.s. Y (ε 0) N 1 ( y(ε 0) ) as r. (4.16) A N 1 = {ω Ω : Y (εr) N 1 ( y(ε r), ω) Y (ε 0) N 1 ( y(ε 0), ω) as r } and B N 1 = {ω Ω : Y (ε 0) N 1 ( y(ε 0), ω) Ỹ t N Y t N }. Relaton (4.16) mples that P(A N 1 ) = 1. Also relatons (4.14) and (4.15) mply that P(B N 1 ) = 1. Thus, P(A N 1 B N 1 ) = 1. By relaton (4.8) and the defnton of the sets A N 1 and B N 1, the sequence w (εr) (t N, Y (ε r) N 1 ( y(εr), ω)) w (ε0) (t N, Y (ε 0) N 1 ( y(ε 0), ω)) as r, for every ω A N 1 B N 1,.e., the random vectors w (ε r) (t N, Y (εr) N 1 ( y(ε r) )) a.s. w (ε 0) (t N, Y (ε 0) N 1 ( y(ε 0) )) as r. (4.17) Relaton (4.15) mples that, for every r = 0, 1,..., and A B R k, P{w (εr) (t N, Y (ε r) N 1 ( y(εr) )) A} = P{w (εr) (t N, Y (ε r) N 1 ( y(εr) )) A}. (4.18) Relatons (4.17) and (4.18) mply that the random varables w (εr) (t N, Y (ε r) N 1 ( y(εr) )) w (ε 0) (t N, Y (ε 0) N 1 ( y(ε 0) )) as r. (4.19) Snce an arbtrary choce of a sequence ε r ε 0, relaton (4.19) mples relaton of weak convergence (4.12). Usng nequaltes (4.1) and (4.10), and condton C 3 we get for any sequence y y (0) Ỹ t N 1 Y t N 1 as ε 0, and for ε ε 3, E w (t N, Y N 1 ( y )) β γ = E yε,t N 1 w (t N, Y (t N )) β γ γ Y E y, t N 1 (L 1,N + L 2,N e (t N ) ) β γ (k + 1) β γ 1 ((L 1,N ) β γ + (L 2,N ) β γ E y,t N 1 =1 =1 e β y e β Y (t N ) y ) (4.20) 23

28 and, therefore, (k + 1) β γ 1 ((L 1,N ) β γ + (L2,N ) β γ (L13 + 1) =1 e β y ) lm ε 0 E w (t N, Y N 1 ( y )) β γ <. (4.21) Relatons (4.12) and (4.21) mply that for any sequence y ε y 0 Ỹk t N 1 Y k t N 1 as ε 0, E yε,t N 1 w (t N, Y (t N )) E y0,t N 1 w (0) (t N, Y (0) (t N )) as ε 0. (4.22) Relatons (4.22), (2.1) and condton A 3 mply that for any sequence y ε y 0 Ỹ k t N 1 Y k t N 1 as ε 0, w (t N 1, y ) = g (t N 1, e y ) E y,t N 1 w (t N, Y (t N )) w (0) (t N 1, y (0) ) = g (0) (t N 1, e y(0) ) E y (0),t N 1 w (0) (t N, Y (0) (t N )) as ε 0. (4.23) Relatons (4.10), (4.11), and (4.23) are analogues of relatons (4.6), (4.7), and (4.8). By repeatng the recursve procedure descrbed above, we fnally get that, under condtons A 2, A 3, B 1, and C 3, for ε ε 3, n = 0, 1,..., N, and y R k, where constants, w (t n, y) L 1,n + L 2,n e γ y, (4.24) =1 L 1,n, L 2,n <, (4.25) and that, for an arbtrary y n y n (0) as ε 0, where y n (0) Ỹ t n Y t n, and for every n = 0, 1,..., N, w (t n, y n ) w (0) (t n, y n (0) ) as ε 0. (4.26) Let us take an arbtrary sequence ε r ε 0 = 0 as r. By condton B 2, the random vectors Y (ε r) (0) Y (ε 0) (0) as r, (4.27) and P{ Y (ε 0) (0) Ỹ t 0 Y t 0 } = 1. (4.28) Accordng to Skorokhod representaton theorem, one can construct random varables 24

29 Y (εr) (0), r = 0, 1,... on some probablty space (Ω, F, P) such that for every r = 0, 1,..., and A B R k, P{ Y (ε r) (0) A} = P{ Y (ε r) (0) A}, (4.29) and Let us denote Y (ε r) (0) a.s. Y (ε 0) (0) as r. (4.30) and A = {ω Ω : Y (ε r) (0, ω) Y (ε 0) (0, ω) as r } B = { ω Ω : Y (ε 0) (0, ω) Y t 0 Y t 0 }. Relaton (4.30) mples that P(A) = 1. Relatons (4.28) and (4.29) mply that P(B) = 1. Thus, P(A B) = 1. By relaton (4.26) and the defnton of sets A and B, the non-random sequence w (ε r) (t 0, Y (ε r) (0, ω)) w (ε 0) (t 0, Y (ε 0) (0, ω)) as j, for every ω A B,.e., the random vectors w (ε r) (t 0, Y (ε r) (0)) a.s. w (ε 0) (t 0, Y (ε 0) (0)) as r. (4.31) Relaton (4.29) mples that for every r = 0, 1,..., and A B R k, P{w (εr) (t 0, Y (εr) (0)) A} = P{w (εr) (t 0, Y (εr) (0)) A}, (4.32) Relatons (4.31) and (4.32) mply that the random vectors w (εr) (t N, Y (εr) (0)) w (ε 0) (t N, Y (ε 0) (0)) as r. (4.33) Snce an arbtrary choce of a sequence ε r ε 0, relaton (4.33) mples that w (t 0, S (0)) w (0) (t 0, S (0) (0)) as ε 0. (4.34) Usng nequaltes (4.2) and (4.24), and condton C 4 we get for ε ε 3, E w (t 0, Y (0)) β γ E(L1,0 + L 2,0 e (k + 1) β γ 1 ((L 1,0 ) β γ =1 + (L2,0 ) β γ E =1 (k + 1) β γ 1 ((L 1,0 ) β γ + (L2,0 ) β γ L14 ), γ Y (0) ) β γ β Y e (0) ) (4.35) 25

30 and, therefore, Relatons (4.34) and (4.36) mply that, lm ε 0 E w (t 0, Y (0)) β γ <. (4.36) Ew (t 0, Y (0)) Ew (0) (t 0, Y (0)) as ε 0. (4.37) Formula (4.5) and relaton (4.37) mply relaton (4.4) gven n Theorem Convergence of rewards for contnuous tme prce processes We are now ready to formulate and to prove our man convergence result for rewards of Amercan type optons for contnuous tme processes. Now we formulate condtons of convergence for dscrete tme reward functonals ) for contnuous tme model. As was mentoned above, n the dscrete tme case, the payoff functons can be dscontnuous. In the contnuous tme case, the dervatves of the payoff functons are nvolved n condton A 1. The correspondng assumptons mply contnuty of the payoff functons. Ths gve us possblty to weaken the assumpton concernng the convergence of the payoff functons and just to requre ther pontwse convergence: Φ(M max,t A 4 : g (t, s) g (0) (t, s) as ε 0, for every (t, s) [0, T ] R k +. Condton A 4 can be re-wrtten n terms of functon g (t, e y ): A 4: g (t, e y ) g (0) (t, e y ) as ε 0, for every (t, y) [0, T ] R k. Remark 1. See for example the paper Kukush and Slvestrov (2004) for an example where a payoff functon are approxmated by a perturbed dscrete payoff functon. Let us now formulate condtons assumed for the transton probabltes and the ntal dstrbutons of process Y (t). The frst condton assumes weak convergence of the transton probabltes that should be locally unform wth respect to ntal states from some sets, and also that the correspondng lmt measures are concentrated on these sets: B 3 : There exst measurable sets Y t R k, t [0, T ] such that: (a) P (t, y, t + u, ) P (0) (t, y, t + u, ) as ε 0, for any y y Y t as ε 0 and 0 t < t + u T ; 26

31 (b) P (0) (t, y, t + u, Y t+u ) = 1 for every y Y t and 0 t < t + u T. The second condton assumes weak convergence of the ntal dstrbutons to some dstrbuton that s assumed to be concentrated on the sets of convergence for the correspondng transton probabltes: B 4 : (a) P ( ) P (0) ( ) as ε 0; (b) P (0) (Y 0 ) = 1, where Y 0 s the set ntroduced n condton B 3. The followng theorem presents our man convergence result. It gves condtons of convergence for reward functonals Φ(M max,t ). Theorem 5.1. Let condtons A 1, A 4, B 3, B 4, C 1, and C 2 hold. Then Φ(M max,t ) Φ(M(0) max,t ) as ε 0. (5.1) Proof. Let Π N = {0 = t 0 < t 1 <... t N = T } be a sequence of parttons on the nterval [0, T ]. Snce we study the case when ε 0, we can assume that d(π n ) c and ε ε 1 where c and ε 1 are defned n relatons (2.9) and (2.10) ths ensures that the correspondng reward functonals are fnte. The followng lemmas play the key role n the proof. Lemma 5.2. Let condtons A 1, C 1, and C 2 hold, the followng relaton holds for any sequence of parttons Π N such that d(π N ) 0 as N, lm lm N ε 0 (Φ(M max,t ) Φ(M Π N,T )) = 0. (5.2) Proof. Ths Lemma s a drect corollary of Theorem 3.2, whch mples that, under condtons A 1, C 1, and C 2, there exst constants L 3, L 4 < such that the followng skeleton approxmaton nequalty holds for ε ε 1 and N such that d(π N ) c, Φ(M max,t ) Φ(M Π N,T ) L 3 d(π N ) + L 4 ( =1 Ths estmate drectly mples relaton (5.2). (5.3) β (Y ( ), d(π N ), T )) β γ β. Lemma 5.3. Let condtons A 1, A 4, B 3, B 4, C 1, and C 2 hold. Then, condtons of Theorem 4.1 hold for any partton Π N = {0 = t 0 < t 1 < t N = T } on the nterval [0, T ] and, therefore, the followng asymptotc relaton holds, Φ(M Π N,T ) Φ(M(0) Π N,T ) as ε 0. (5.4) 27

32 Proof. Take an arbtrary partton Π N = {0 = t 0 < t 1 < t N = T } on the nterval [0, T ]. Inequalty (2.21) gven n the proof of Lemma 2.3, mples that, for every n = 0,..., N, s R k +, and ε ε 0, g (t n, s) L 8 + L 9 (s ) γ, (5.5) where γ = max(γ 0, γ 1 + 1,..., γ k + 1). Thus condton A 2 holds for partton Π N wth the constants K 6 = L 8, K 7 = L 9 and the parameter γ gven above. Also, nequalty (2.20) gven n the proof of Lemma 3, mples that for every n = 0,..., N, s, s R k +, and ε ε 0, =1 g (t n, s ) g (t n, s ) (K 3 + K 4 (s + j )γ )(s + s =1 j=1 ). (5.6) and, therefore, for any 0 < u < u + <, = 1,..., k sup u s,s <u+, s s c,=1,...,k g (t n, s ) g (t n, s ) L 15 c, (5.7) where L 15 = (K 3 + K 4 (u + j )γ ). =1 Condton A 4 and nequaltes (5.5) and (5.7) mply that for every n = 0,..., N, and 0 < u < u <, = 1,..., k, the condtons of Ascol-Arzelá theorem holds for payoff functons g (t n, s), u s < u +, = 1,..., k, for every n = 0, 1,..., N. Thus, these functons converge unformly,.e., j=1 sup u s <u +,=1,...,k g (t n, s) g (0) (t n, s) 0 as ε 0. (5.8) Relaton (5.8) mples that condton of locally unform convergence A 4 holds for partton Π N wth the correspondng sets S t n = R k +, for n = 0, 1,..., N. It reman to show that condton C 1 mples that condton C 3 holds. Condton C 1 mples that for any constant L 16 < one can choose c = c(l 16 ) > 0 and then ε 4 = ε 4 (c) < ε 2 such that for ε ε 4 and = 1,..., k, β (Y ( ), c, T ) L 16. (5.9) 28

33 Take an arbtrary nteger 0 n N and consder the unform partton t n = u (m) 0 <... < u (m) m = t n+1 on the nterval [t n, t n+1 ] by ponts u (m) j = (t n+1 t n )j. m Relaton (5.9) and the Markov property of the processes Y (t) mply that for ε ε 4, m = [ (t n+1 t n ) ] + 1 (n ths case (t n+1 t n ) c), y R k, j = 1,..., m and c m = 1,..., k, E y,tn (e β Y (u (m) j ) Y (u (m) E y,tn e = E y,tn ((e + e = E y,tn {(e 0 ) 1) β Y (u (m) j 1 ) Y (u (m) 0 ) β Y e (u (m) j ) Y (u (m) β Y (u (m) j 1 ) Y (u (m) β Y (u (m) j ) Y (u (m) E{(e + E y,tn {E{(e E y,tn (e 0 ) 1)e j 1 ) 1) β Y (u (m) j 1 ) Y (u (m) β Y (u (m) j ) Y (u (m) 0 ) 1) β Y (u (m) j ) Y (u (m) β Y (u (m) j 1 ) Y (u (m) j 1 ) 1 β Y (u (m) j ) Y (u (m) j 1 ) j 1 ) 1 + 1) Y (u (m) j 1 )}} j 1 ) 1) Y (u (m) 0 ) 1)(L ) + L 16 j 1 )}} (5.10) Fnally, we get, for ε ε 4, y R k, and = 1,..., k, E y,tn (e β Y (t n+1 ) Y = E y,tn (e (t n ) 1) β Y (u (m) m ) Y (u (m) 0 ) 1) m 1 ((L ) m + L 12 (L ) j ) = (2(L ) m 1) <. j=0 (5.11) Thus condton C 3 holds wth parameter β gven n condton A 1. Fnally, condton C 2 s equvalent to condton C 4. The proof of Lemma 5.3 s complete. Lemmas 5.2 and 5.3 let us make the fnal ffth step n the proof of Theorem 5.1. We employ the followng nequalty that can be wrtten down for any partton Π N, Φ(M max,t ) Φ(M(0) max,t ) Φ(M max,t ) Φ(M Π N,T ) + Φ(M (0) max,t ) Φ(M(0) Π N,T ) + Φ(M Π N,T ) Φ(M(0) Π N,T ). (5.12) Usng ths nequalty and relaton (5.4) gven n Lemma 5.3 we get for any partton Π N and for d(π N ) < c, lm ε 0 Φ(M max,t ) Φ(M(0) max,t ) lm Φ(M ε 0 max,t ) Φ(M Π N,T ) + Φ(M(0) max,t ) Φ(M(0) Π N,T ). (5.13) 29

34 Fnally, relaton (5.2) gven n Lemma 5.2 mples (note that relaton ε 0 admt also the case where ε = 0) that the expresson on the rght hand sde n (5.13) can be forced to take a value less then any ε > 0 by choosng the partton Π N wth the dameter d(π N ) small enough. Ths proves the asymptotc relaton (5.1) and thus the proof of Theorem 5.1 s complete. In concluson of ths secton, let us formulate some useful suffcent condtons for an mportant condton of moment compactness C 1. Let us ntroduce the modulus of J-compactness, for h, c > 0, = 1,..., k, (Y ( ), h, c, T ) = sup 0 t t+u t+c T sup P y,t { Y (t + u) Y (t) h}. y R k The followng condton of J-compactness plays the key role n functonal lmt theorems for Markov type càdlàg processes: C 5 : lm c 0 lm ε 0 (Y ( ), h, c, T ) = 0, h > 0, = 1,..., k. Introduce also the quantty, whch represents the maxmum of moment generatng functons for ncrements of the log-prce processes Y (t), = 1,..., k, Ξ β (Y ( ), T ) = sup 0 t t+u T sup E y,t e β(y y R k (t+u) Y (t)), β R 1. The followng condton formulated n terms of these moment generatng functons can be effectvely verfed n many cases: C 6 : lm ε 0 Ξ ±β (Y ( ), T ) <, = 1,..., k, for some β > β, where β s the parameter ntroduced n condton C 1. Lemma 5.4. Condtons C 5 and C 6 mply condton C 1. Proof. Usng Hölder nequalty we get the followng estmates, for every 0 t t + u T, y R k + and = 1,..., k, E y,t e β Y (t+u) Y (t) 1 (e βh 1) + E y,t e β Y (t+u) Y (t) χ( Y (t + u) Y (t) h) (e βh 1) + (E y,t e β Y (t+u) Y (t) ) β β P y,t { Y (t + u) Y (t) h}. (5.14) 30

35 The followng nequalty, whch connect the exponental moment modulus of compactness wth the modulus of J-compactness, follows from (5.14), β (Y ( ), c, T ) (e βh 1) + ( β (Y ( ), c, T ) + 1) β β (Y ( ), h, c, T ). (5.15) Also, the followng estmate takes place, for every 0 t t + u T, y R k+, and = 1,..., k, E y,t e β Y (t+u) Y (t) = E y,t e β (Y + E y,t e β (Y E y,t e β (Y (t+u) Y (t)) χ(y (t+u) Y (t)) χ(y (t+u) Y (t)) + E y,t e β (Y (t + u) Y (t)) (t + u) < Y (t)) (t+u) Y (t)). (5.16) Ths estmate mples the followng nequalty, β (Y ( ), c, T ) + 1 Ξ β (Y ( ), T ) + Ξ β (Y ( ), T ). (5.17) Relatons (5.15) and (5.17) mply the statement of Lemma Multvarate exponental prce processes wth ndependent ncrements In ths secton we show general convergence results for the mportant case of exponental prce processes wth ndependent ncrements. Let us consder the model where the log-prce process Y (t), t 0 s a càdlàg process wth ndependent ncrements. We also assume for smplcty that the ntal state of process Y (0) = y =, = 1,..., k) s a constant. The process Y (t) s a càdlàg Markov process wth transton probabltes whch are connected wth the dstrbutons of ncrements for ths process P (t, t + u, A) by the followng relaton, (y P (t, y, t + u, A) = P (t, t + u, A y) = P{ y + Y (t + u) Y (t) A}. (6.1) Let us assume the followng standard condton of weak convergence for dstrbutons of ncrements for log-prce processes: 31

36 D 1 : P (t, t + u, ) P (0) (t, t + u, ) as ε 0, 0 t t + u T. Representaton (6.1) mples n an obvous way that condton B 3 holds wth the sets Y t = R k, t [0, T ],.e., dstrbutons of ncrements for the processes Y (t) locally unformly weakly converge, f condton D 1 holds. Thus, n the case of processes wth ndependent ncrements, the condton B 3 wth the sets Y t = R k s, n fact, equvalent to the standard condton of weak convergence for such processes. In ths case the J-compactness modulus (Y ( ), h, c, T ) takes the followng form: (Y ( ), h, c, T ) = sup 0 t t+u t+c T P{ Y (t + u) Y (t) h}. Thus, condton C 5 s reduced to the standard J-compactness condton for the log-prce processes wth ndependent ncrements: D 2 : lm c 0 lm ε 0 (Y (t), h, c, T ) = 0, h > 0, = 1,..., k. Note that condtons D 1 and D 2 mply J-convergence of processes Y (t), t [0, T ] to process Y (0) (t), t [0, T ] as ε 0 and stochastc contnuty of the lmt process. Also, the quanttes Ξ β (Y ( ), T ), = 1,..., k take a smplfed form, Ξ β(y ( ), T ) = sup Ee 0 t t+u T β(y (t+u) Y Therefore, condton C 6 takes the followng form: (t)), β R 1. D 3 : lm ε 0 Ξ ±β ( ), T ) <, = 1,..., k, for some β > β, where β s the parameter ntroduced n condton C 1. (Y Accordng to Lemma 5.4, condtons D 1 and D 2 mply condton C 1. Condton B 4 s reduced n ths case to the followng condton: D 4 : lm ε 0 y = y 0. Note that y 0 can be any vector wth real-valued components snce the set Y 0 = R k. Note that, condton D 4 mples also condton C 4. The followng theorem summarzes the remarks above. Theorem 6.1. Let condtons A 1, A 4, and D 1 D 4 hold for the exponental prce processes wth ndependents S (t). Then Φ(M max,t ) Φ(M(0) max,t ) < as ε 0. (6.2) 32

37 The skeleton approxmatons Y (t) = Y (0) ([t/ε]), t 0 for a stochastcally contnuous càdlàg log-prce process Y (0) (t), t 0 wth ndependent ncrements gve a good example of the model ntroduced above. In ths case, condtons D 1 and D 2 automatcally hold. Condton D 3 s mpled by the followng condton: D 5 : Ξ ±β ( ), T ) <, = 1,..., k, for some β > β, where β s the parameter ntroduced n condton C 1. (Y (0) Thus, f we let condtons A 1, A 4, and D 5 hold, then the statement of Theorem 6.1 holds for the exponental prce processes S (t) = e Y (0) ([t/ε]), t [0, T ]. Note that, n ths case, Theorem 3.2 yelds a stronger result n the form of explct estmates for the accuracy of skeleton approxmatons for reward functons. Note also that the optmal expected rewards for the skeleton prce processes S (t) = e Y (0) ([t/ε]) can be estmated wth the use of Monte Carlo smulaton. 7 Bnomal tree opton reward approxmatons for multvarate Brownan moton In order to llustrate the results presented above, let us consder the model where k = 2 and the bvarate geometrc Brownan prce process S (0) Y (t) = e (0)(t), t 0, where the log-prce process Y (0) (t) = (Y (0) 1 (t), Y (0) 2 (t)), t 0 s a bvarate Brownan moton wth components Y (0) (t) = y (0) + µ t + σ W (t), t 0, = 1, 2, whch are correlated,.e., EW 1 (t)w 2 (t) = ρt, t 0. We approxmate the process Y (0) (t), t 0 wth a bvarate bnomal sum-process Y (t) = (Y 1 (t), Y 2 (t)), t 0 wth components Y (t) = y (0) + 1 k [t/ε] Y k,, t 0, = 1, 2. Here Y n = (Y n,1, Y n,2 ), n = 1, 2,... are, for every ε > 0,..d. random vectors whch have the followng structure, (Y n,1, Y n,2 ) = (+u 1, +u 2 ) p ++, (+u 1, u 2 ) p +, wth prob. ( u 1, +u 2 ) p +, ( u 1, u 2 ) p. (7.1) Snce we assumed that the ntal state Y (0) = y (0) = (y (0) 1, y (0) 2 ) s a constant whch does not depend of ε, condtons B 4 and C 4 automatcally hold. In order to ft the bvarate bnomal sum-processes defned above to the lmt bvarate Brownan moton, we should ft expectatons, varances, and covarance 33