Numerical Approximation of Partial Differential Equations Arising in Financial Option Pricing

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1 Numerical Approximaion of Parial Differenial Equaions Arising in Financial Opion Pricing Fernando Gonçalves Docor of Philosophy Universiy of Edinburgh 27 (revised version)

2 To Sílvia, my wife.

3 Declaraion I declare ha his hesis was composed by myself and ha he work conained herein is my own, excep where explicily saed oherwise in he ex. (Fernando Gonçalves)

4 Absrac We consider he Cauchy problem for a second-order parabolic PDE in half spaces, arising from he sochasic modelling of a mulidimensional European financial opion. To improve generaliy, he asse price drif and volailiy in he underlying sochasic model are aken ime and space-dependen and he payoff funcion is no specified. The numerical mehods and possible approximaion resuls are srongly linked o he heory on he solvabiliy of he PDE. We make use of wo heories: he heory of linear PDE in Hölder spaces and he heory of linear PDE in Sobolev spaces. Firs, insead of he problem in half spaces, we consider he corresponding problem in domains. This localized PDE problem is solvable in Hölder spaces. The soluion is numerically approximaed, using finie differences (wih boh he explici and implici schemes) and he rae of convergence of he ime-space finie differences scheme is esimaed. Finally, we esimae he localizaion error. Then, using he L 2 heory of solvabiliy in Sobolev spaces and in weighed Sobolev spaces, he soluion of he PDE problem is approximaed in space, also using finie differences. The approximaion in ime is considered in absrac spaces for evoluion equaions (making use of boh he explici and implici schemes) and hen specified o he second-order parabolic PDE problem. The raes of convergence are esimaed for he approximaion in space and in ime.

5 Acknowledgemens I express graiude o my supervisor, Professor Isván Gyöngy, for his mos valuable eachings and direcions for my PhD research.

6 Table of Conens Chaper 1 Inroducion 2 Chaper 2 European financial opions Sochasic processes background European opion sochasic modelling European opion pricing and parabolic PDE Chaper 3 Parabolic PDE in Hölder spaces: space and ime discreizaion Classical resuls Numerical approximaion Localizaion error esimae Chaper 4 Parabolic PDE in Sobolev and weighed Sobolev spaces: space discreizaion Classical resuls Numerical approximaion: bounded daa case Numerical approximaion: unbounded daa case Chaper 5 Evoluion equaions in absrac spaces: ime discreizaion Numerical approximaion under a general framework An example: he second-order parabolic PDE problem in weighed Sobolev spaces Chaper 6 Conclusion and furher research 13 Appendix A Noaion 15 Appendix B Useful resuls 11 Bibliography 111 1

7 Chaper 1 Inroducion Financial opions or derivaives are coningen financial claims and heir modelling is made in a sochasic framework (according wih he Financial Mahemaics heory iniiaed by he works of Fisher Black and Myron Scholes (1973) and Rober Meron (1973)). We are ineresed, in paricular, in one basic ype of financial opion: he European opion, in is general mulidimensional version (he opion on a baske of asses). The European opion modelling lies on he sochasic equaion describing he dynamic of he underlying asse prices. I is well known ha pricing an opion can be reduced, wih he use of Feynman-Kac formula, o solving he Cauchy problem wih a final condiion for a second-order parabolic PDE in half spaces, where he parabolic operaor s coefficiens associaed wih he firs and second-order parial derivaives are unbounded. The opic of his research is he numerical approximaion of he PDE arising from he sochasic financial problem, in his general mulidimensional version. In he available numerical analysis lieraure, several numerical schemes can be found for he European opion price approximaion. However, we could no find a sysemaic approach o he subjec, namely considering he PDE problem in is general form (wih ime and space dependen coefficiens and non specified independen erm and final condiion) and simulaneously producing he raes of convergence for he corresponding approximaion schemes. The aim of he presen sudy is o conribue o his sysemaic approach. We make some commens on he choice of he European opion (in he general mulidimensional form) as he derivaive ype moivaing his research. This choice seemed o be appropriae as is general modelling can be applied or be adaped, more or less easily, o he oher several ypes of opions wih no early exercise. A he same ime, he pariculariies of he sudy of each of he muliple differen ypes of opions are avoided in his firs sage. We expec ha our numerical approximaion sudy can be used beyond he paricular derivaive ype 2

8 moivaing i. Finally, we menion ha, in his research, we will no pu he emphasis in he numerical mehods sophisicaion: he basic finie differences explici and implici schemes will be used. We summarize he chapers conen. Chaper 2 - European financial opions Afer briefly reviewing he sochasic background for he European opion modelling, we consider he simple unidimensional Black-Scholes model and a few of is immediae generalizaions. Then, we ouline he way he parabolic PDE Cauchy problem arises from he sochasic problem. Chaper 3 - Parabolic PDE in Hölder spaces: space and ime discreizaion In his chaper, we follow he approach by N. V. Krylov (in Krylov [29]). We approximae he parabolic PDE Cauchy problem in Hölder spaces (imposing ha he operaor is non-degenerae ellipic in space and is coefficiens are bounded). We localize he problem on a bounded domain and sudy he approximaion for his localized problem, using boh he implici and he explici schemes. Then we esimae he localizaion error, i.e. he error due o considering he Cauchy problem on a bounded domain insead of he whole space. The main conen of he chaper is: Exisence and uniqueness resul for he soluion of he discree problem corresponding o he coninuous iniial-boundary value problem - his is a resul saed in Krylov [29], bu proved only for an ellipic problem. Esimae for he convergence rae of he discree problem soluion o he corresponding coninuous problem soluion - his resul is also saed in Krylov [29], bu proved only for an ellipic problem. We also esimae he rae of convergence for a case where weaker condiions are imposed over he iniial daa. Consrucion of discree operaors approximaing he corresponding coninuous operaor, using he explici and implici schemes - hese operaors are considered in Krylov [29], bu for a more specific example of he equaion. Sochasic represenaion of he soluions of he Cauchy and he iniialboundary value problems for a parabolic PDE, under milder condiions and capuring wider siuaions han we could find in he lieraure. Esimae of he localizaion error. These resuls are obained for he cases where srong and weak soluions of he corresponding sochasic equaion are considered. 3

9 Chaper 4 - Parabolic PDE in Sobolev and weighed Sobolev spaces: space discreizaion We consider he Cauchy problem in Sobolev spaces (assuming ha he operaor is non-degenerae ellipic in space bu imposing less regulariy from he daa) and sudy is space-discreized version in discree Sobolev spaces. Nex, in order o consider PDE wih unbounded coefficiens, we ake he problem in weighed Sobolev spaces and sudy is space-discreizaion in discree weighed Sobolev spaces. The main resuls we obain are: Exisence and uniqueness of he discreized problem soluion in discree Sobolev spaces. Esimae for he discree problem soluion rae of convergence o he corresponding coninuous problem soluion in Sobolev spaces. Sronger esimae for he paricular unidimensional (in space) case. Exisence and uniqueness resul for he discree problem soluion in discree weighed Sobolev spaces. Esimae for he discree problem soluion rae of convergence o he coninuous problem soluion in weighed Sobolev spaces. Chaper 5 - Evoluion equaions in absrac spaces: ime discreizaion We consider he approximaion in ime in absrac spaces for evoluion equaions, using boh he implici and he explici schemes. The paricular secondorder parabolic PDE problem approximaion is given as an example. We prove he following main resuls for each of he approximaion schemes: Exisence and uniqueness resul for he soluion of he discree problem. Esimae for he soluion of he discree problem. Esimae for he discree problem soluion rae of convergence o he corresponding coninuous problem soluion. Chaper 6 - Conclusion and furher research We discuss some of he resuls obained in he previous chapers and ouline furher research direcions. Appendix A - Noaion The noaion is mosly inroduced in he ex. For he convenience of he reader, we lis he basic noaion symbols used. Appendix B - Useful resuls We lis some basic inequaliies and convergence heorems we use. 4

10 Chaper 2 European financial opions We will inroduce he European financial opion. Basically, his derivaive is a conrac giving is owner he righ (and no he obligaion) o rade (eiher o buy or o sell) a sock (or a commodiy, an index or a currency) for a fixed price a a fixed fuure dae. We will skech he sochasic model for he pricing of a European opion and he way his problem can be reduced o solve he Cauchy problem for a secondorder parabolic PDE. Finally, we will discuss he poenialiy of he modelling for applicaion o oher ypes of opions. 2.1 Sochasic processes background In his secion we summarize he basic sochasic processes conceps and resuls (see e.g. Lamberon e all [34], pp , Friedman [18], ch. 5). Sochasic processes. Definiion A coninuous-ime sochasic process in a space E endowed wih a σ algebra E is a family (X ) R + of random variables defined on a probabiliy space (Ω, A, P) wih values in a measurable space (E, E). We inroduce he concep of filraion, which represens he informaion available a ime. Definiion Le (Ω, A, P) be a probabiliy space. A filraion (F ) is an increasing family of σ algebras included in A. A process (X ) is said o be adaped o he filraion (F ) if, for any, X is F measurable. We say ha he filraion F = σ(x s, s ) is generaed by he process (X ). We will work wih filraions which conain all he P null ses of A. The compleion of (F ) is he filraion generaed by boh 5

11 σ(x s, s ) and N (he σ algebra generaed by all he P null ses of A) and is called he naural filraion of he process (X ). A sopping ime is a random ime ha depends on he process (X ) in a non-anicipaive way. Definiion τ is a sopping ime wih respec o he filraion(f ) if τ is a mapping Ω [, + ] such ha, for any, {τ } F. The σ algebra associaed wih τ is F τ = {A A : for any, A {τ } F }, and represens he informaion available unil he random ime τ. Nex we sae some sopping ime properies(see Lamberon e all [34], p. 31). Proposiion The following hold 1. If S is a sopping ime hen S is F S measurable; 2. If S is a sopping ime, finie almos surely, and (X ) is a coninuous adaped process hen X S is F S measurable; 3. If S and T are wo sopping imes such ha S T P a.s. hen F S F T ; 4. If S and T are wo sopping imes hen S T = inf(s, T ) is a sopping ime. In paricular, if S is a sopping ime and is a deerminisic ime hen S is a sopping ime. Brownian moion. An imporan example of sochasic process is he Brownian moion (or Wiener process). This process is cenral in he financial opion modelling. Definiion A Brownian moion is a real-valued, coninuous sochasic process (X ), wih independen and saionary incremens. Tha is 1. Coninuiy: P a.s. he map s X s (ω) is coninuous; 2. Independen incremens: If s hen X X s is independen of F s = σ(x u, u s); 3. Saionary incremens: If s hen X X s and X s X have he same probabiliy law. We sae he Gaussian propery of a Brownian moion (see Lamberon e all [34], p. 31). 6

12 Theorem If (X ) is a Brownian moion hen X X is a normal random variable wih mean r and variance σ 2, where r and σ are consan real numbers. Definiion A Brownian moion is sandard if 1. X = P a.s.; 2. E(X ) = ; 3. E(X 2 ) =. In he sequel ex, if we do no sae differenly, a Brownian moion is assumed o be sandard. A sronger resul for he Gaussian propery holds (see Lamberon e all [34], p. 32). Theorem If (X ) is a Brownian moion and if (X 1,..., X d ) is a Gaussian vecor. 1 < < d hen We define he Brownian moion wih respec o a filraion. Definiion Areal-valued coninuous sochasic process is a(f ) Brownian moion if i saisfies 1. For any, X is F measurable; 2. If s hen X X s is independen of he σ algebra F s ; 3. If s hen X X s and X s X have he same probabiliy law. Maringales. The financial noion of arbirage, o be inroduced in he nex secion, is explained wih he concep of maringale. Definiion Le (Ω, A, P) be a probabiliy space and (F ) a filraion on his space. An adaped family (M ) of inegrable random variables, i.e. E( M ) < for any, is a maringale if, for any s, E(M F s ) = M s. We give some examples of maringales (see Lamberon e all [34], p. 32). Proposiion If (X ) is a sandard F Brownian moion hen 1. X is a F maringale; 2. X 2 is a F maringale; 3. exp(σx (σ 2 /2)) is a F maringale. 7

13 The maringale propery E(M F s ) = M s sill holds when and s are bounded sopping imes (see Lamberon e all [34], p. 34). Theorem (Opional sampling Theorem). If (M ) is a coninuous maringale wih respec o he filraion (F ), and if τ 1 and τ 2 are wo sopping imes such ha τ 1 τ 2 K, where K is a finie real number, hen M τ2 is inegrable and E(M τ2 F τ1 ) = M τ1 P a.s. We sae a propery of he hiing ime of a poin a by a Brownian moion (see Lamberon e all [34], p. 34). If a is a real number, we define T a := inf{s, X s = a} or + if ha se is empy. Proposiion Le (X ) be an F Brownian moion and a a real number. Then T a is a sopping ime, finie almos surely, and is disribuion is characerized by is Laplace ransform E(e λta ) = e 2λ a. Nex resul gives an esimae for he second-order momen of sup T M, where M is a square inegrable maringale (see Lamberon e all [34], p. 35). Theorem (Doob inequaliy). If (M ) T hen E ( sup T M 2) 4E( M T 2 ). is a coninuous maringale Sochasic inegral. In he financial opion modelling, we will deal wih expressions of he ype ( H sdw s ) T, where (W ) is a F Brownian moion and (H ) T is a F adaped process. As Brownian moion pahs are, almos surely, no differeniable a any poin, his inegral wih respec o a Brownian moion (he sochasic inegral ) needs o be defined. Le (W ) be a sandard F Brownian moion defined on a filered probabiliy space (Ω, A, (F ), P). Take T a sricly posiive, finie real number. We will begin by considering a se of processes called simple processes. Definiion (H ) T is a simple process if i can be wrien as (H )(ω) = p φ i (ω)1 ] i 1, i ](), i=1 where = < 1 < < p = T and φ i is F i 1 measurable and bounded. By definiion, he sochasic inegral of a simple process is he coninuous process (I(H) ) T defined for any ] k, k+1 ] as I(H) = φ i (W i W i 1 ) + φ k+1 (W W k ). 1 i k 8

14 We wrie H sdw s = I(H). We nex sae some fundamenal properies of he sochasic inegral of a simple process (see Lamberon e all [34], p. 36). Proposiion If (H ) T is a simple process hen ( 1. H sdw s is a coninuous F maringale; ) T ( ( ) ) 2 ( ) 2. E H sdw s = E H2 s ds ; ( ) 3. E sup H 2 ( ) T T sdw s 4E H2 s ds. We exend he concep of sochasic inegral o a larger class of adaped processes H { ( H = (H ) T, (F ) adaped process : E We define he exension (see Lamberon e all [34], p. 38). ) } Hs 2 ds < +. Proposiion Le (W ) be an F Brownian moion. There exiss a unique linear mapping J from H o he space of he coninuous F maringales defined on [, T ], such ha 1. If (H ) T is a simple process hen P a.s. for any T, J(H) = I(H) ; 2. If T hen E(J(H) 2 ) = E ( ) H2 s ds. This linear mapping is unique in he sense ha if boh J and J saisfy he previous properies hen P a.s. T, J(H) = J (H). We denoe, for H H, H sdw s = J(H). We noe ha he condiion E( H2 s ds) < + in he definiion of H is saisfied if and only if E(sup T ( H sdw s) 2 ) < +. The following properies hold (see Lamberon e all [34], p. 38). Proposiion If (H ) T belongs o H hen ( ) 1. E sup H 2 ( ) T T sdw s 4E H2 s ds ; 2. If τ is a F sopping ime hen P a.s. τ H sdw s = 1 {s τ}h s dw s. 9

15 We exend he sochasic inegral o a class of processes saisfying a weaker inegrabiliy condiion. Le { H = (H s ) s T, (F ) adaped process : We define he exension o H (see Lamberon e all [34], p. 4). } Hs 2 ds < + P a.s.. Proposiion There exiss a unique linear mapping J from H ino he vecor space of coninuous processes defined on [, T ], such ha 1. If (H ) T is a simple process hen P a.s. T, J(H) = I(H) ; 2. If (H i ) i is a sequence of processes in H such ha (Hi s) 2 ds converges o in probabiliy hen sup T J(H i ) converges o in probabiliy. We wrie, for H H, H sdw s = J(H). In his case he inegral is no necessarily a maringale. We inroduce nex some basic conceps of Iô calculus. Le us define a Iô process. Definiion Le (Ω, F, (F ), P) be a filered probabiliy space and (W ) an F Brownian moion. (X ) T is an R valued Iô process if i can be wrien as P a.s. T, X = X + K s ds + H s dw s, where X is F measurable, (K ) T and (H ) T are F adaped processes, K s ds < + P a.s. and H s 2 ds < + P a.s. The previous decomposiion is unique (see Lamberon e all [34], p. 43). Proposiion If (M ) T is a coninuous maringale such ha M = K sds, wih P a.s. K s ds < + hen P a.s. T, M =. This implies ha 1. An Iô process decomposiion is unique. Tha means ha if X = X + K s ds + H s dw s = X + K sds + H sdw s hen X = X dp a.s. H s = H s ds dp a.e. K s = K s ds dp a.e.; 2. If (X ) T is a maringale of he form X + K sds + H sdw s hen K = d dp a.e. 1

16 In nex resul he sochasic inegral is defined in he inerval [, τ ], wih τ a sopping ime (he sochasic inegral is inerpreed as a random variable) (see Friedman [18], p. 72). Theorem Le f a process such ha E τ f() 2 d < and τ a sopping ime wih respec o F, τ T. Then he process τ f(s)dw (s), T, is a maringale and E τ f(s)dw (s) =. We sae Iô formula (see Lamberon e all [34], p. 44). Theorem (Iô formula). Le (X ) T be an Iô process X = X + K s ds + H s dw s, and f be a wice coninuously differeniable funcion. Then f(x ) = f(x ) + f (X s )dx s f (X s )d X, X s, where X, X := H2 s ds and f (X s )dx s := f (X s )K s ds+ f (X s )H s dw s. Also, if f is a funcion wice differeniable wih respec o x and once differeniable wih respec o, wih coninuous parial derivaives in (, x), hen f(, X ) = f(, X ) + + f s(s, X s )ds f x(s, X s )dx s f xx(s, X s )d X, X s. We give he inegraion by pars formula (see Lamberon e all [34], p. 46). Proposiion (Inegraion by pars formula). Le (X ) and Y be wo Iô processes, X = X + K sds + H sdw s and Y = Y + K sds + H sdw s. Then X Y = X Y + wih X, Y := H sh sds. X s dy s + Y s dx s + X, Y, We have a mulidimensional version of Iô formula o be applied when f is a funcion of several Iô processes, each of hem funcion of several Brownian moions. Definiion A p dimensional F Brownian moion is an R p valued F adaped process (W = (W 1,..., W p )), where all he (W i ) are independen sandard F Brownian moions. 11

17 We define he mulidimensional Iô process. Definiion (X ) T is a (mulidimensional) Iô process if X = X + K s ds + p i=1 H i sdw i s, where K and all he processes (H) i are adaped o F, K s ds < + P a.s. and (Hi s) 2 ds < + P a.s. We sae he mulidimensional Iô formula (see Lamberon e all [34], p. 48). Theorem (Mulidimensional Iô formula). processes X i = X i + K i sds + p j=1 Le (X 1,..., X d ) be d Iô H i,j s dw j s, and f a funcion wice differeniable wih respec o x and once differeniable wih respec o, wih coninuous parial derivaives in (, x). Then f(, X 1,..., X d ) = f(, X 1,..., X d ) d i=1 d i,j=1 f x i (s, X1 s,..., X d s )dx i s f s (s, X1 s,..., X d s )ds 2 f x i x j (s, X1 s,..., X d s )d X i, X j s, wih dx i s = K i sds + p j=1 Hij s dw j s and d X i, X j s = p m=1 Him s Hs jm ds. Sochasic differenial equaions. We begin by considering a ype of process ha, as i will be menioned laer, models he behaviour of cerain financial asses. Le S = x + S s (µds + σdw s ), (2.1) where σ and µ are real numbers and (W ) is a Brownian moion. We show nex ha he process S = x exp((µ σ 2 /2) + σw ) solves (2.1). Le f(, x) = x exp((µ σ 2 /2) + σx) so ha we can wrie S = f(, W ). As (W ) is an Iô process (idenifying K s = and H s = 1) we apply Iô formula and obain S = f(, W ) = f(, W ) f s(s, W s )ds + f xx(s, W s )d W, W s. 12 f x(s, W s )dw s

18 As d W, W = d, S = x + = x + S s (µ σ 2 /2)ds + S s µds + S s σdw s. S s σdw s S s σ 2 ds The uniqueness of his soluion can obained using Proposiion We have he following heorem (see Lamberon e all [34], p. 47): Theorem Le σ, µ be wo real numbers, T a sricly posiive consan and (W ) a Brownian moion. There exiss a unique Iô process (S ) T which saisfies, for any T, equaion (2.1). This process is given by S = x exp((µ σ 2 /2) + σw ). We consider now he equaion X = Z + b(s, X s )ds + σ(s, X s )dw s, (2.2) a more general version of equaion (2.1). Equaion (2.2) is also wrien: dx = b(, X )d + σ(, X )dw, X = Z. Equaions of his ype are called sochasic differenial equaions and heir soluions are called diffusions. equaions. We define he soluion of equaion (2.2). Mos financial asses are modelled using hese Definiion Le (Ω, A, P) be a probabiliy space equipped wih a filraion (F ). Le b and σ be funcions such ha b : R + R R, σ : R + R R, Z a F measurable random variable and (W ) a F Brownian moion. A soluion o he equaion (2.2) is an F adaped sochasic process (X ) ha 1. For any, he inegrals b(s, X s)ds and σ(s, X s)dw s exis, i.e. such b(s, X s ) ds < + and 2. (X ) saisfies (2.2), i.e. σ(s, X s ) 2 ds < + P a.s.; P a.s. X = Z + b(s, X s )ds + σ(s, X s )dw s. We sae he exisence and uniqueness of he soluion of equaion (2.2) (see Lamberon e all [34], pp. 49-5). 13

19 Theorem If b and σ are coninuous funcions and if here exis consans K, L < + such ha 1. b(, x) b(, y) + σ(, x) σ(, y) K x y, 2. b(, x) + σ(, x) L(1 + x ), 3. E( Z 2 ) < +, for all R +, for all x, y R. Then here exiss a unique soluion of (2.2) in [, T ], T. Moreover, his soluion saisfies E(sup T X 2 ) < +. The uniqueness means ha if (X ) T and (Y ) T are wo soluions of (2.2) hen P a.s. T, X = Y. We exend he sochasic differenial equaion analysis o he mulidimensional case. Le W = (W 1,..., W p ) an R p valued F Brownian moion; b : R + R d R d, b(s, x) = (b 1 (s, x),..., b d (s, x)); σ : R + R d R d p, σ(s, x) = (σ ij (s, x)) 1 i d,1 j p ; Z = (Z 1,..., Z d ) an F measurable random variable in R d. Consider he mulidimensional equaion X i = Z i + b i (s, X s )ds + p j=1 which can be wrien X = Z + b(s, X s )ds + σ ij (s, X s )dw j s, for i = 1,..., d, σ(s, X s )dw s. (2.3) We sae he exisence and uniqueness of a soluion of (2.3) (see Lamberon e all [34], p. 53). If x R d, denoe by x he Euclidean norm of x and if σ R d p denoe σ 2 = 1 i d, 1 j p (σij ) 2. Theorem Assume ha b and σ are coninuous funcions and ha here exis consans K, L < + such ha 1. b(, x) b(, y) + σ(, x) σ(, y) K x y, 2. b(, x) + σ(, x) L(1 + x ), 3. E( Z 2 ) < +, for all R +, for all x, y R d. Then here exiss a unique soluion of (2.3) in [, T ], T. Moreover, his soluion saisfies E(sup T X 2 ) < +. 14

20 (2.2). We nex sae he flow and Markov properies for he soluion of equaion We say ha an F adaped process (X ) saisfies he Markov propery if, for any bounded Borel funcion f and for any s and such ha s, E(f(X ) F s ) = E(f(X ) X s ). Inuiively, his means ha he fuure behaviour of (X ) depends only on he value X and no on any oher previous informaion. We will see ha his propery will play an imporan role in he financial opion pricing. Le us denoe by Xs,x, for s, he soluion of equaion (2.2) saring from x a ime. For s, Xs,x saisfies X,x s = x + s b(u, X,x u )du + s σ(u, X,x u )dw u. We sae he flow propery of X (see Lamberon e all [34], p. 54). Lemma Under he assumpions of Theorem 2.1.3, if s hen X,x s = X,Xx s P a.s. For he Markov propery of X, we have he following resul (see Lamberon e all [34], p. 55): Theorem Le (X ) be a soluion of (2.2). Then (X ) is a Markov process wih respec o he filraion (F ). Furhermore, for any bounded Borel funcion f, we have P a.s. E(f(X ) F s ) = φ(x s ), wih φ(x) = E(f(X s,x )). We sae an exension of Theorem , resul useful when ineres rae models are considered (see Lamberon e all [34], p. 55). Theorem Le (X ) be a soluion of (2.2) and r(s, x) be a non-negaive measurable funcion. Then, for > s, P a.s. E (e ) s r(u,xu)du f(x ) F s = φ(x s ), wih φ(x) = E (e s r(u,xs,x u )du f(x s,x ) ). I is also wrien as E (e ) s r(u,xu)du f(x ) F s = E (e s r(u,xs,x u )du f(x s,x ) ) x=xs. 15

21 2.2 European opion sochasic modelling In his secion we briefly presen he Black-Scholes model (see e.g. Lamberon e all [34], pp ). Saemen of he problem. An European opion on a sock S is a conrac giving is owner he righ o rade he sock (o buy i in he case of a call opion or o sell i in he case of a pu opion) for a fixed price K (he srike price) a a fuure dae T (he opion mauriy or expiry). If, a ime T, he opion s owner ops o rade he sock he opion is said o be exercised. In he mos simple case, he payoff of an opion is C T = (S T K) + = max(s T K, ), for a call opion and P T = (K S T ) + = max(k S T, ), for a pu opion. The model we will ouline enables us o deermine he price for his ype of securiy, ha is, wha is he value a ime of an opion worh C T (for a call) or P T (for a pu) a ime T. As consequence of a model s assumpion (he absence of arbirage opporuniy o be menioned laer), we have he pu-call pariy equaion C P = S Ke r(t ), which holds for all < T. Then i suffices o consider one of he wo cases: we will approach he call opion case. Remark In he model we are presening we assume, for simplificaion, ha he sock does no pay dividends unil he expiraion dae T. Remark We have defined a European opion on a sock. I can be defined in he same way on a commodiy, an index or a currency. Behaviour of prices. We will consider a model wih wo asses: a riskless asse S and a risky asse S. Their price behaviour is described as follows. For S we have he ordinary differenial equaion ds = rs d, 16

22 where S is he price of he asse a ime and r is a non-negaive consan represening he riskless rae of ineres. Assuming an iniial condiion S = 1, we have S = e r,. For S we have he sochasic differenial equaion ds = S (µd + σdb ), (2.4) where µ and σ > are consans represening he expeced reurn or average growh rae of he asse (drif rae) and he sandard deviaion of reurns (volailiy), respecively, and (B ) is a sandard Brownian moion. The model is valid on [, T ]. As we saw in he previous secion (Theorem ), a closed-form unique soluion for he sochasic differenial equaion can be deermined S = S exp((µ σ 2 /2) + σb ), where S is he sock price observed a ime. The process (log(s )) is a (non necessarily sandard) Brownian moion. We hen have he following properies for he process (S ): 1. Coninuiy of he sample pahs; 2. Independen of he relaive incremens: If u hen (S S u )/S u is independen of σ(s v, v u); 3. Saionariy of he relaive incremens: If u hen (S S u )/S u and (S u S )/S have he same probabiliy law. These properies characerize he sock price behaviour assumed in Black- Scholes model. Sraegies. A sraegy is defined as a process φ = (φ) T = ((H, H )), wih values in R 2, adaped o he naural filraion (F ) of he Brownian moion. The componens H and H of he porfolio (H, H ) are he quaniies of riskless asse and risky asse, respecively, held a ime. The value of he porfolio a ime is V (φ) = H S + H S. 17

23 We define sraegies in which he decisions made on he composiion of he porfolio do no affec is value, ha is, changes in he porfolio value would only be brough by price moves. Definiion A self-financing sraegy is a pair φ of adaped processes (H ) T and (H ) T saisfying 1. H d + (H ) 2 d < + a.s.; 2. H S + H S = H S + H S + H uds u + H uds u a.s., for all [, T ]. Denoe he discouned price of he risky asse by S = e r S. We have he following resul (see Lamberon e all [34], p. 65): Proposiion Le φ = (φ) T = ((H, H )) be an adaped process wih values in R 2, saisfying H d + (H ) 2 d < + a.s. Le V (φ) = H S + H S and Ṽ(φ) = e r V (φ). Then φ defines a self-financing sraegy if and only if for all [, T ]. Ṽ (φ) = V (φ) + H u d S u a.s., Remark The model we are presening assumes ha he(coninuous) changes in he porfolio composiion are made wih no cos (he model is called wih no ransacion coss). Girsanov s Theorem. Maringale represenaion. In order o price an opion, we will consruc self-financing sraegies replicaing he opion. We need firs o consider an equivalen probabiliy measure under which discouned prices of asses are maringales. We define equivalen probabiliies (see Lamberon e all [34], p. 66). Definiion Le (Ω, A, P) be a probabiliy space. A probabiliy measure Q on (Ω, A) is absoluely coninuous wih respec o P if A A P(A) = Q(A) =. Theorem Q is absoluely coninuous relaive o P if and only if here exiss a non-negaive random variable Z on (Ω, A) such ha A A Q(A) = Z(ω)dP(ω). Z is called densiy of Q relaive o P and denoed dq/dp. A Definiion Le Q and P be wo probabiliy measures on (Ω, A). P and Q are equivalen if each one is absoluely coninuous relaive o he oher. Wih nex resul, a probabiliy measure Q equivalen o a given probabiliy measure P is consruced (see Lamberon e all [34], p. 66). 18

24 Theorem (Girsanov s Theorem). Le (Ω, F, (F ) T, P) be a probabiliy space wih (F ) T he naural filraion of he sandard Brownian moion (B ) T. Le (θ ) T be an adaped process saisfying ( θ2 sds < + a.s. and such ha he process (L ) T defined by L = exp θ ) sdb s 1 2 θ2 sds is a maringale. Then, under probabiliy P (L) wih densiy L T relaive o P, he process (W ) T defined by W = B + θ sds is a sandard Brownian moion. The sochasic inegral is invarian by change of equivalen probabiliy (see Lamberon e all [34], p. 79). Proposiion Assume ha he hypohesis of Theorem are saisfied. Le (H ) T be an adaped process such ha H2 s ds < P a.s. Le he processes and X = Y = H s db s + H s θ s ds, under P H s dw s, under P (L), wih W = B + θ sds and P (L) he probabiliy measure defined in Theorem Then X = Y. We sae nex a resul on he represenaion of a Brownian maringale in erms of a sochasic inegral (see Lamberon e all [34], p. 67). Theorem Le (B ) T be a sandard Brownian moion on a probabiliy space (Ω, F, P) and le (F ) T be is naural filraion. Le (M ) T be a square-inegrable maringale, wih respec o (F ) T. There exiss an adaped process (H ) T such ha E( H2 s ds) < + and [, T ] M = M + H s db s a.s. Opion pricing. We consider now he problem of deermining he price of an opion. Firs, we show ha here exiss a probabiliy P equivalen o P under which he discouned risky asse price S = e r S is a maringale. From equaion (2.4), we have d S = re r S d + e r ds = S ((µ r)d + σdb ). (2.5) Seing W = B + (µ r)/σ, we obain d S = S σdw. (2.6) 19

25 Owing o Theorem 2.2.9, wih θ = (µ r)/σ, here exiss a probabiliy measure P equivalen o P under which (W ) T is a sandard Brownian moion. As, from Proposiion 2.2.1, he sochasic inegral is invarian by change of equivalen probabiliy, under P we have S = S exp(σw σ 2 /2), and, by Proposiion , S is a maringale. Remark The erm (µ r) in (2.5) is called he risk premium. Remark If we apply he ransformaion W insead of o S, from ds = S (µd + σdb ) = B + (µ r)/σ o S we obain ds = S (rd + σdw ), and, for he same reasons, (W ) T is a sandard Brownian moion under he equivalen probabiliy measure P. Noe ha he drif µ is replaced by he riskless ineres rae r, so ha, under P he risk premium for S is null. This is why he probabiliy measure P is someimes called risk-neural. We will resric he sudy o he class of admissible sraegies. Definiion A sraegy φ = ((H, H )) T is admissible if i is selffinancing and if he discouned value Ṽ(φ) = H + H S of he corresponding porfolio is, for all, non-negaive and such ha sup [,T ] Ṽ is square inegrable under P. For a self-financing sraegy φ, from Proposiion and equaion (2.6) we have Ṽ = V + H u σ S u dw u. If, addiionally, φ is admissible, from Proposiion we have ha (Ṽ) is a square-inegrable maringale under P. Then, under P, for any admissible sraegy φ, Ṽ (φ) = ṼT (φ) = P a.s. This expresses he no arbirage opporuniy hypohesis of he model. We define a call opion by a non-negaive, F T measurable, random variable h (he opion payoff). Definiion An opion is replicable if here is an admissible sraegy φ = ((H, H )) T such ha a ime T is value equals he opion payoff V T (φ) = h. 2

26 Noe ha for an opion o be replicable h has o be square inegrable under P. This necessary condiion is saisfied when h is wrien as h = g(s T ), wih g(x) = (x K) +. We saw above ha, for an admissible sraegy φ, (Ṽ) is a square-inegrable maringale under P. If φ replicaes he opion, from Ṽ = E (ṼT F ), we have V = E ( ) e r(t ) h F. I could also be shown ha if h is square inegrable under P hen here is an admissible sraegy replicaing he opion. We have he following main resul which defines he opion price (see Lamberon e all [34], p. 69): Theorem In he Black-Scholes model, any opion defined by a nonnegaive F T measurable random variable h, which is square-inegrable under he probabiliy P, is replicable and he value a ime of any replicaing porfolio is given by V = E ( ) e r(t ) h F. (2.7) The expression E ( ) e r(t ) h F defines he opion value a ime. Remark If he opion value is wrien h = g(s T ), under srong hypohesis over g i would be possible o deermine explicily he replicaing porfolio, ha is he composiion of he porfolio (H, H ) saisfying (2.7). We make a final commen. Recall ha in he modelling we assumed ha here were no dividend paymens and no ransacion coss. The inclusion of coninuously payed dividends in he model is immediae. Unforunaely, his is no consisen wih he discree (usual annual) dividend paymen in finance world. This poins o he need o combine he coninuous modelling we have presened wih discree modelling for he dividend paymen. The same idea applies o he inclusion of ransacion coss: he changes in he porfolio composiion should raher be considered discree. Several models for hese purposes are available in he Financial Mahemaics lieraure (see e.g. Wilmo [47]) 2.3 European opion pricing and parabolic PDE We will show he way he problem of pricing an European opion is relaed o a parabolic PDE Cauchy problem (see e.g. Lamberon e all [34], pp ). We will consider a more general version of he problem we have presened. 21

27 Le (X ) be a diffusion in R, soluion of he sochasic differenial equaion dx = b(, X )d + σ(, X )dw, (2.8) where b and σ saisfy he assumpions of Theorem Le also r(, x) be a bounded coninuous real-valued funcion defined on R + R, modelling he riskless ineres rae. We wrie he payoff funcion h as h = g(x T ). We wan o compue V = E ( e r(s,x s)ds g(x T ) F ). As a consequence of Theorem , V can be wrien where G(, x) = E ( e saring from x a ime. r(s,x,x s V = G(, X ), )ds g(x,x T )), and Xs,x denoes he soluion of (2.8) Firs we sae some resuls relaing he infiniesimal generaor of a diffusion. Infiniesimal generaor of a diffusion. Le b and σ saisfy he assumpions of Theorem We sae he following resul (see Lamberon e all [34], p. 98): Proposiion For any ime le A (x) be he differenial operaor ha maps a C 2 funcion v from R o R o a funcion A v such ha (A v)(x) = σ2 (, x) 2 v (x) + b(, x) v 2 x2 x (x). Le u(, x) be a C 1,2 real-valued funcion defined on R + R wih bounded derivaives in x. Le X be a soluion of (2.8). Then he process is a maringale. M = u(, X ) ( As u + u ) (s, Xs )ds The differenial operaor A is called he infiniesimal generaor of he diffusion (X ). We sae a more general resul where discouned prices are considered (see Lamberon e all [34], p. 98). Proposiion Le he assumpions of Proposiion be saisfied. Le r(, x) be a bounded coninuous real-valued funcion defined on R + R. Then he process M = e r(s,x s)ds u(, X ) is a maringale. e s ( r(v,x v)dv A s u ru + u ) (s, Xs )ds 22

28 This resul sill holds in he mulidimensional case. Le W = (W 1,..., W p ) an R p valued F Brownian moion; b : R + R d R d, b(s, x) = (b 1 (s, x),..., b d (s, x)); σ : R + R d R d p, σ(s, x) = (σ ij (s, x)) 1 i d,1 j p. Consider he mulidimensional sochasic differenial equaion dx i = b i (, X )d + which can be wrien p j=1 σ ij (, X )dw j, for i = 1,..., d, dx = b(, X )d + σ(, X )dw. (2.9) We assume ha he assumpions of Theorem are saisfied. For any ime we define he differenial operaor A which maps a C 2 funcion v from R d o R o he funcion (A v)(x) = 1 2 v 2 aij (, x) x i x (x) + j bi (, x) v (x), (2.1) xi where (a ij (, x)) is he marix wih componens a ij (, x) = p σ ik (, x)σ jk (, x). We have he following resul (see Lamberon e all [34], p. 99): k=1 Proposiion Le u(, x) be a C 1,2 real-valued funcion defined on R + R d wih bounded derivaives in x and (X ) a soluion of sysem (2.9). Le r(, x) be a bounded coninuous real-valued funcion defined on R + R d. Then he process M = e ( r(s,xs)ds u(, X ) e s r(v,xv)dv A s u ru + u ) (s, X s )ds is a maringale. Opion pricing and solving a PDE. We will now esablish he connecion beween pricing an opion and solving a parabolic PDE problem. We consider he mulidimensional sochasic differenial equaion (2.9). Le (X ) be he soluion of (2.9), g(x) a funcion from R d o R and r(, x) a bounded coninuous real-valued funcion defined on R + R d. We wan o compue V = E (e r(s,x s)ds g(x T ) F ). 23

29 As in he unidimensional case, i can be proved ha V = G(, X ), wih G(, x) = E (e ) T r(s,xs,x )ds g(x,x T ), where X,x s denoes he soluion of (2.9) saring from x a ime. The following main resul is obained owing o Proposiion and characerizes he funcion G as a soluion of a parabolic parial differenial equaion (see Lamberon e all [34], p. 99). Theorem Le u(, x) be a C 1,2 real-valued funcion defined on [, T ] R d wih bounded derivaives in x and (X ) a soluion of sysem (2.9). Le A be he operaor defined by (2.1) and r(, x) a bounded coninuous real-valued funcion defined on R + R d. If u saisfies ( A u ru + u ) (, x) = (, x) [, T ] R d, u(t, x) = g(x) x R d hen (, x) [, T ] R d u(, x) = G(, x) = E (e ) r(s,xs,x )ds g(x,x T ). This resul offers a mehod o deermine he price of an European opion which consiss in solving he corresponding PDE problem. To compue G(, x) = E (e ) T r(s,xs,x )ds g(x,x T ), we have o solve A u ru + u = in [, T ] Rd, u(t, x) = g(x) for x R d. (2.11) Equaion (2.11) characerizes a parabolic PDE problem wih a final condiion. We need o consider he proper funcion spaces for his problem o be well defined. We noe ha o have u = G, he soluion u of (2.11) has o saisfy he smoohness assumpions in Theorem In general, some regulariy assumpions have o be made on he coefficiens b and σ and he operaor A have o saisfy he ellipiciy condiion λ >, (, x) [, T ] R d, ξ R d d d a ij (, x)ξ i ξ j λ ξ i 2. i,j=1 i=1 Le us exemplify he mehod for he simple unidimensional Black-Scholes model (see Lamberon e all [34], pp. 1-11). 24

30 We consider he sochasic differenial equaion ds = S (µd + σdb ) where µ and σ > are consans and (B ) is a sandard F Brownian moion. We have ha (see Remark ), under he risk-neural probabiliy measure P, he asse price S saisfies ds = S (rd + σdw ), where r is a consan and (W ) is a sandard F Brownian moion. The operaor A is now independen of ime and is given by This operaor is no ellipic. A = σ2 2 x2 2 x + rx 2 x. We consider he diffusion X = log(s ). Since S = S e (r σ2 /2)+σW, we have ha (X ) is soluion of dx = (r σ 2 /2)d + σdw. The infiniesimal generaor of his diffusion A log = σ2 2 2 x + (r 2 σ2 /2) x has consan coefficiens and he ellipiciy condiion is saisfied. If we wan o compue he opion price G(, x), we hen have o find a soluion v C 1,2 (R + R), wih bounded derivaives in x, of he problem Finally, A log v rv + v = in [, T ] R, v(t, x) = g(ex ) for x R. G(, x) = v(, log(x)). The above example presened in Lamberon e all [34], can be generalized o he mulidimensional version of Black-Scholes model (also wih consan coefficiens and ineres rae). Le B = (B 1,..., B d ) an R d valued F Brownian moion; µ = (µ 1,..., µ d ) a consan vecor; σ = (σ ij ) 1 i,j d a consan marix. The sochasic differenial equaion modelling he asse prices is ( ) d ds i = S i µ i d + σ ij db j, for i = 1,..., d, j=1 25

31 and can be wrien ds = Ŝ(µd + σdb ), where Ŝ denoes de diagonal marix wih diagonal elemens S i, i = 1,..., d. We assume ha marix σ is posiive definie and define ρ := (r,..., r), wih r a consan. Owing o Theorem 2.2.9, wih θ = σ 1 (µ ρ), here exiss a probabiliy measure P equivalen o P under which W = B + σ 1 (µ ρ) is a R d valued sandard Brownian moion (see Ellio e all [14], p. 168). We obain The infiniesimal generaor of he diffusion S is and i is no ellipic. ds = Ŝ(ρd + σdw ), (2.12) A = 1 2 (σσ ) ij x i x j 2 x i x + j rxi x, i In he same way as for he unidimensional case, i could be checked ha he sochasic differenial equaion (2.12) has he unique soluion (( ) ) S i = S i exp r 1 d d (σ ij ) 2 + σ ij W j, for i = 1,..., d. 2 j=1 j=1 We use he logarihmic ransformaion X i = log(s i ), i = 1,..., d, and denoe i X = log(s ). We have ha (X ) is soluion of ( ) dx i = r 1 d d (σ ij ) 2 d + σ ij dw j, for i = 1,..., d, 2 j=1 j=1 and is infiniesimal generaor is ( A log = 1 2 (σσ ) ij 2 x i x + r 1 j 2 ) d (σ ij ) 2 x. i j=1 The coefficiens in A log are consan and, as σ is a posiive definie marix, he ellipiciy condiion is saisfied. To compue he opion price G(, x),we have o find a soluion v C 1,2 (R + R d ), wih bounded derivaives in x, of he problem A log v rv + v = in [, T ] Rd, v(t, x) = g(e x ) for x R d, and hen obain G(, x) = v(, log(x)), 26

32 where e x := (e x1,..., e xd ) and log(x) := (log(x 1 ),..., log(x d )). In hese wo simple examples, he drif µ and he volailiy σ were considered consan. Therefore we have a closed-form soluion for he sochasic differenial equaion modelling he asse prices and, wih he help of a logarihmic ransformaion, we could offse he linear growh of he equaion coefficiens and obain a differenial operaor A wih consan coefficiens. A more difficul siuaion occurs when µ and σ are no consan. In his case here does no exis in general a closed-form soluion for he sochasic equaion. We will approach his problem in Chaper 4, considering he appropriae funcion spaces in order o obain he (uniform) ellipiciy in space of he operaor A. We make a final commen on he applicaion poenialiy of he (mulidimensional) European opion modelling we have considered. We see ha i exends Black-Scholes model in several ways: The opion depends on several underlying asses; The payoff funcion is no specified; The coefficiens of he sochasic equaion modelling he sock prices are assumed o be ime and space-dependen. The model applies direcly o opions on a baske of asses (baske opions or rainbow opions). The higher dimensionaliy ogeher wih he non-specificaion of he payoff funcion allows he model o be adaped o oher ypes of opions wih no early exercise (ha is, for which he exercise can only occur a a fixed ime T ) (see e.g. Lamberon e all [34], Wilmo [47]). For insance, o: European opions on fuure conracs and foreign-exchange; Compound opions : his ype of opion is an opion on anoher opion; Exchange opions : in his case he opion gives he righ o exchange an asse for anoher; Some pah-dependen ypes of opions as Asian opions. The ime and space-dependency of he sochasic equaion s coefficiens confers flexibiliy o he model: he assumpion ha he coefficiens are consan would be resricive, mainly for opions wih disan expiraion daes. 27

33 Chaper 3 Parabolic PDE in Hölder spaces: space and ime discreizaion We have o consider he proper funcion spaces for he parabolic PDE problem we sudy o be well defined. In he presen chaper, we will consider he solvabiliy of he PDE in Hölder spaces, following he presenaion of Krylov [29]. To approximae he soluion of he Cauchy problem in half spaces, we firs sudy he approximaion of he soluion of he corresponding (localized) problem in domains. Then we esimae he error due o he problem localizaion. In he previous chaper, arising from he sochasic modelling of he sock price, we considered a parabolic problem (wih final condiion and null erm) Au + cu + u = in [, T ] R d, u(t, x) = g(x) in R d, where A(, x) = aij (, x) x i x + j bi (, x) x i and g is a given funcion. In his chaper and in he following chapers (excep for Secion 3.3 where he sochasic represenaion of he PDE problem is needed) we will consider he more sandard form of he PDE problem (wih iniial condiion) Lu u + f = in [, T ] R d, u(, x) = g(x) in R d, (3.1) where L(, x) = a ij 2 (, x) x i x + j bi (, x) + c (, x), xi and f and g are given funcions (wih f no necessarily null). Noe ha problem (3.1) (wih he iniial condiion u(, x) = g(x)), using he change of variable (, x) (T, x), is obviously equivalen o he problem wih final condiion u(t, x) = g(x) Lu + u + f = in [, T ] R d, u(t, x) = g(x) in R d. 28

34 3.1 Classical resuls We inroduce he Hölder spaces (see Krylov [29], pp and ). Le U be a domain in R d, meaning an open subse of R d. For k =, 1, 2,... we denoe C k loc (U) he se of all funcions u : U R whose derivaives Dα u for α k are coninuous in every bounded subse V of U. We define u ;U := [u] ;U := sup u, [u] k;u := max U α =k Dα u ;U. Definiion For k =, 1, 2,..., he space C k (U) is he Banach space of all funcions u Cloc k (U) for which he norm u k;u = k j= [u] j;u is finie. If < δ < 1, we call u Hölder coninuous wih exponen δ in U if he seminorm [u] δ;u = u(x) u(y) sup x,y U, x y x y δ is finie. The seminorm is called Hölder s consan of u of order δ. We define [u] k+δ;u := max α =k [Dα u] δ;u. Definiion For < δ < 1 and k =, 1, 2,..., he Hölder space C k+δ (U) is he Banach space of all funcions u C k (U) for which he norm is finie. u k+δ;u = u k;u + [u] k+δ;u Now denoe R d+1 = {(, x) : R, x R d }. In R d+1 define he parabolic disance beween he poins z 1 = ( 1, x 1 ), z 2 = ( 2, x 2 ) as ρ(z 1, z 2 ) := x 1 x /2. We fix a consan δ (, 1). If u is a real-valued funcion defined in Q R d+1, we denoe [u] δ/2,δ;q := sup z 1 z 2, z i Q u(z 1 ) u(z 2 ), u ρ δ δ/2,δ;q := u ;Q + [u] δ/2,δ;q. (z 1, z 2 ) 29

35 Definiion For < δ < 1, C δ/2,δ (Q) is he Banach space of all funcions u defined in Q for which u δ/2,δ;q <. We inroduce he parabolic Hölder spaces. Definiion For < δ < 1, he parabolic Hölder space C 1+δ/2,2+δ (Q) is he Banach space of all real-valued funcions u(z) defined in Q for which boh d 1. [u] 1+δ/2,2+δ;Q := [u ] δ/2,δ;q + [u x i x j] δ/2,δ;q i,j=1 d 2. u 1+δ/2,2+δ;Q := u ;Q + u x ;Q + u ;Q + u x i x j ;Q + [u] 1+δ/2,2+δ;Q are finie. We now summarize some classical resuls on solvabiliy of parabolic PDE in Hölder spaces. Consider he ellipic and parabolic operaors of order m. Definiion Le m 1 be an ineger and a α (x) be some real-valued funcions in R d, given for any muli-index α wih α m. The operaor L = α m aα (x)d α is called mh order (uniformly) ellipic if here exiss a consan λ > called he consan of ellipiciy, such ha a α (x)ξ α λ ξ m x, ξ R d. α m i,j=1 Definiion Le m 1 be an ineger and a α (, x) be some given realvalued funcions in R d+1, wih α m a muli-index. The operaor L /, wih L = α m aα (, x)d α is called mh order (uniformly) parabolic if here exiss a consan λ > such ha a α (, x)ξ α λ ξ m (, x) R d+1, ξ R d. α m Consider he second-order operaor (in he non-divergence form) L(, x) = a ij 2 (, x) x i x + j bi (, x) + c (, x), (3.2) xi wih real coefficiens. We assume ha, for some λ > and for each >, he operaor saisfies a ij (, x)ξ i ξ j λ ξ 2, for all x, ξ R d, so ha L is uniformly ellipic wih respec o he space variables, wih consan of ellipiciy λ. Then, 3

36 for each, he symmeric marix (a ij (x, )) is posiive definie for any x R d. We also assume ha here exiss a consan K such ha a δ/2,δ K, b δ/2,δ K, c δ/2,δ K, where δ (, 1) is fixed. We consider firs he Cauchy problem for second-order parabolic equaions in half spaces. Le T (, ), Q = [, T ] R d. The problem o be solved is Lu u + f = in Q, u(, x) = g(x) in R d, (3.3) where f and g are given funcions. Remark In he presenaion of Krylov [29], he parabolic equaion is defined for he ime variable aking values in (, T ), wih T (, ]. As, for any consan µ, he funcion v(, x) = u(, x)e µ saisfies Lv µv v + fe µ = if and only if u saisfies Lu u + f =, we se c wihou loss of generaliy. We have he following exisence and uniqueness resul for he soluion of (3.3) (see Krylov [29], p. 14). Theorem Assume ha c µ for a consan µ >. Le g C 2+δ (R d ) and f C δ/2,δ (Q). Then here exiss a unique funcion u C 1+δ/2,2+δ (Q) such ha i saisfies (3.3). Moreover, here is a consan N depending only on d, λ, δ, K and µ such ha u 1+δ/2,2+δ;Q N( f δ/2,δ;q + g 2+δ ). We consider now he iniial-boundary value problem in Q = [, T ] U, wih U R d a bounded domain. For his, we give a preliminary definiion (see Krylov [29], p. 78). Denoe B R (x ) R d he open ball in R d wih cener x and radius R. For any U R d, denoe U he boundary of U. Denoe also R d + = {(x, x d ) : x = (x 1,..., x d 1 ) R d 1, x d > }. Definiion Le r > and U be a bounded domain in R d. We wrie U C r (or U C r ) and say ha he domain U is of class C r if here are numbers ρ, K > such ha for any poin x U here exiss a one-o-one mapping ψ of B ρ (x ) ono a domain D R d such ha 1. D + := ψ(b ρ (x ) U) R d + and ψ(x ) = ; 2. ψ(b ρ (x ) U) = D {y R d : y d = }; 3. [ψ] s;bρ (x ) + [ψ 1 ] s;d K for any s [, r], and ψ 1 (y 1 ) ψ 1 (y 2 ) K y 1 y 2 for any y i D. We say ha he diffeomorphism ψ sraighens he boundary near x. 31