Modelli Clamfim info sul corso Spazi Euclidei 16 settembre 2015

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1 CLAMFIM Bologna Modell Clamfm nfo sul corso Spaz Euclde 16 settembre 2015 professor Danele Rtell danele.rtell@unbo.t 1/26?

2 rogramma roflo scentfco del docente Rtell Rtell sept15.pdf 2/26?

3 Our startng pont wll therefore be the BS equaton whch s taken for granted: V t σ2 S 2 2 V V + rs S2 S rv = 0, S 0, t [0, T ], (1) where V (S, t) s the value of the opton, S the prce of the underlyng, t the tme, T the expraton date, σ the volatlty of the underlyng and r the rsk-free nterest rate. The BS Eq. (1) s a lnear parabolc equaton of the form v t = 2 v x + a v 2 x + bv, (2) 3/26?

4 Syllabus ) Funzon d due o pù varabl: dervate parzal, massm e mnm lber e vncolat 4/26?

5 Syllabus ) Funzon d due o pù varabl: dervate parzal, massm e mnm lber e vncolat ) Equazon dfferenzal ordnare del prmo ordne 4/26?

6 Syllabus ) Funzon d due o pù varabl: dervate parzal, massm e mnm lber e vncolat ) Equazon dfferenzal ordnare del prmo ordne ) Msura e ntegrale d Lebesgue 4/26?

7 Syllabus ) Funzon d due o pù varabl: dervate parzal, massm e mnm lber e vncolat ) Equazon dfferenzal ordnare del prmo ordne ) Msura e ntegrale d Lebesgue v) Trasformata d Fourer ed Equazon alle dervate parzal per la fnanza 4/26?

8 Syllabus ) Funzon d due o pù varabl: dervate parzal, massm e mnm lber e vncolat ) Equazon dfferenzal ordnare del prmo ordne ) Msura e ntegrale d Lebesgue v) Trasformata d Fourer ed Equazon alle dervate parzal per la fnanza v) Introduzone alla statstca matematca 4/26?

9 Bblografa Sldes e altro materale da ams campus. 5/26?

10 rove d esame a) 28 ottobre 2015 ore 14 Aula 21 6/26?

11 rove d esame a) 28 ottobre 2015 ore 14 Aula 21 b) 8 gennao 2016 ore 14 Aula 21 6/26?

12 rove d esame a) 28 ottobre 2015 ore 14 Aula 21 b) 8 gennao 2016 ore 14 Aula 21 Modaltà d esame prova scrtta teorco pratca a lbr chus 6/26?

13 Student n debto d Rcerca Operatva ossono segure e fare l esame: non c sono stat sostanzal cambament d programma Student n debto corso prof. lazz aa 2010/2011 Devono sostenere l esame con programma e modaltà attual 7/26?

14 Rcevmento prevo appuntamento va mal non s tratta d un metodo d dssuasone ma dell ottmzzazone del tempo d tutt 8/26?

15 Rcevmento prevo appuntamento va mal non s tratta d un metodo d dssuasone ma dell ottmzzazone del tempo d tutt Eserctazon gestte da Alessandro Gambn, sono parte ntegrante del corso 8/26?

16 Rcevmento prevo appuntamento va mal non s tratta d un metodo d dssuasone ma dell ottmzzazone del tempo d tutt Eserctazon gestte da Alessandro Gambn, sono parte ntegrante del corso Homework verranno assegnat esercz da svolgere, che se rconsegnat ne temp stablt saranno tenut n consderazone per la valutazone e consentranno d alleggerre la prova scrtta del 28 ottobre 8/26?

17 Calendaro ndcatvo delle consegne Homework lunedì 28 settembre lunedì 5 ottobre lunedì 12 ottobre lunedì 19 ottobre 9/26?

18 Eucldean Space For any n N let R n denote the n-fold Cartesan product of R wth tself R n := {(x 1, x 2,..., x n ) : x j R for j = 1, 2,..., n} 10/26?

19 Eucldean Space For any n N let R n denote the n-fold Cartesan product of R wth tself R n := {(x 1, x 2,..., x n ) : x j R for j = 1, 2,..., n} By Eucldean space we shall mean R n together wth the Eucldean nner product we are gong to ntroduce. The nteger n s called the dmenson of R n, elements x = (x 1, x 2,..., x n ) of R n are called ponts or vectors or ordered n-tuples, and the numbers x j are called coordnates, or components, of x 10/26?

20 Two vectors x, y are sad to be equal f and only f ther components are equal;.e., x j = y j for j = 1, 2,..., n. The zero vector s the vector whose components are all zero;.e., 0 := (0, 0,..., 0). When n = 2 (respectvely, n = 3), we usually denote the components of x by x, y (respectvely, by x, y, z). 11/26?

21 Defnton. Let x = (x 1, x 2,..., x n ), y = (y 1, y 2,..., y n ) R n be vectors and α R be a scalar. () The sum of x and y s the vector x + y := (x 1 + y 1, x 2 + y 2,..., x n + y n ) () The product of a scalar α and a vector x s the vector αx = (αx 1, αx 2,..., αx n ) () The (Eucldean) dot product (or scalar product or nner product) of x and y s the scalar x y := x 1 y 1 + x 2 y x n y n 12/26?

22 Observe that (v) The dfference of x and y s the vector x y := (x 1 y 1, x 2 y 2,..., x n y n ) 13/26?

23 We defne the standard bass of R n to be the collecton e 1,..., e n, where e j s the pont n R n whose j-th coordnate s 1, and all other coordnates are 0. By defnton each x = (x 1,..., x n ) R n can be wrtten as a lnear combnaton of the e j s: x = n x j e j = n x e j e j j=1 j=1 14/26?

24 We defne the standard bass of R n to be the collecton e 1,..., e n, where e j s the pont n R n whose j-th coordnate s 1, and all other coordnates are 0. By defnton each x = (x 1,..., x n ) R n can be wrtten as a lnear combnaton of the e j s: x = n x j e j = n x e j e j j=1 j=1 Notce that the usual bass {e j } conssts of parwse orthogonal vectors n the sense that e j e k = 0 when j k. In partcular, the usual bass s an orthogonal bass. 14/26?

25 Defnton. Let x R n. The (Eucldean) norm (or magntude) of x s the scalar ( n ) 1/2 x := k=1 x 2 k 15/26?

26 Defnton. Let x R n. The (Eucldean) norm (or magntude) of x s the scalar ( n ) 1/2 Remark. x := x 2 k k=1 x 2 = x x 15/26?

27 Defnton. Let x R n. The (Eucldean) norm (or magntude) of x s the scalar ( n ) 1/2 Remark. x := x 2 k k=1 x 2 = x x Theorem: Cauchy Schwarz nequalty x y x y 15/26?

28 Norm propertes. If x, y R n, then () x 0 wth equalty only when x = 0 () αx = α x for all scalars α () x + y x + y () are sad trangle nequaltes x y x y 16/26?

29 Topology Defnton () For each r > 0, the open ball centered at a of radus r s B r (a) := {x R n : x a < r} 17/26?

30 Topology Defnton () For each r > 0, the open ball centered at a of radus r s B r (a) := {x R n : x a < r} () For each r 0, the closed ball centered at a of radus r s B r (a) := {x R n : x a r} 17/26?

31 Topology Defnton () For each r > 0, the open ball centered at a of radus r s B r (a) := {x R n : x a < r} () For each r 0, the closed ball centered at a of radus r s B r (a) := {x R n : x a r} When n = 1, the open ball centered at a of radus r s the open nterval (a r, a + r), and the closed ball s the closed nterval [a r, a + r]. 17/26?

32 Defnton () A set V n R n s sad to be open f and only f for every a V there s an ε > 0 such that B ε (a) V 18/26?

33 Defnton () A set V n R n s sad to be open f and only f for every a V there s an ε > 0 such that B ε (a) V () A set E n R n s sad to be closed f E c := R n \ E s open 18/26?

34 Defnton () A set V n R n s sad to be open f and only f for every a V there s an ε > 0 such that B ε (a) V () A set E n R n s sad to be closed f E c := R n \ E s open Remark Every open ball s an open set 18/26?

35 Defnton () A set V n R n s sad to be open f and only f for every a V there s an ε > 0 such that B ε (a) V () A set E n R n s sad to be closed f E c := R n \ E s open Remark Every open ball s an open set Remark If a R n then R n \ {a} s open and that {a} s closed 18/26?

36 Defnton () A set V n R n s sad to be open f and only f for every a V there s an ε > 0 such that B ε (a) V () A set E n R n s sad to be closed f E c := R n \ E s open Remark Every open ball s an open set Remark If a R n then R n \ {a} s open and that {a} s closed Remark For each n N, the empty set and the whole space R n are both open and closed. 18/26?

37 Defnton Let n N and a R n, let V be an open set whch contans a, and suppose that f : V \ {a} R. Then f(x) s sad to converge to l, as x approaches a, f and only f for every ε > 0 there s a δ > 0 (δ n general depends on ε, f, V, and a) such that 0 < x a < δ = f(x) l < ε 19/26?

38 Defnton Let n N and a R n, let V be an open set whch contans a, and suppose that f : V \ {a} R. Then f(x) s sad to converge to l, as x approaches a, f and only f for every ε > 0 there s a δ > 0 (δ n general depends on ε, f, V, and a) such that In ths case we wrte 0 < x a < δ = f(x) l < ε lm x a f(x) = l and call l the lmt of f(x) as x approaches a. 19/26?

39 Defnton Let E R n and let f : E R. () f s sad to be contnuous at a E f and only f for every ε > 0 there s a δ > 0 (whch n general depends on ε, f and a) such that x a < δ and x E = f(x) f(a) < ε 20/26?

40 Defnton Let E R n and let f : E R. () f s sad to be contnuous at a E f and only f for every ε > 0 there s a δ > 0 (whch n general depends on ε, f and a) such that x a < δ and x E = f(x) f(a) < ε () f s sad to be contnuous on E f and only f f s contnuous at every x E. 20/26?

41 A subset B R n s bounded f there exsts M > 0 such that x M for any x B 21/26?

42 A subset B R n s bounded f there exsts M > 0 such that x M for any x B It s fundamental that f a set s both closed and bounded (we call t compact), then so s ts mage under any contnuous functon. 21/26?

43 A subset B R n s bounded f there exsts M > 0 such that x M for any x B It s fundamental that f a set s both closed and bounded (we call t compact), then so s ts mage under any contnuous functon. Theorem. Let n, m N. If H s compact n R n and f : H R m s contnuous on H, then f(h) s compact n R m. 21/26?

44 Theorem. [K. Weerstrass]. Suppose that H s a nonempty subset of R n and f : H R. If H s compact, and f s contnuous on H, then M := sup{f(x) : x H} and m := nf{f(x) : x H} are fnte real numbers. Moreover, there exst ponts x M, x m H such that M = f(x M ) and m = f(x m ). 22/26?

45 artal dervatves The most natural way to defne dervatves of functons of several varables s to allow one varable to move at a tme. If f : R n R s a real functon of n varables the partal dervatve f xj exsts exsts at a pont a f and only f the lmt f f(a + he j ) f(a) (a) := lm x j h 0 h 23/26?

46 Hgher-order partal dervatves are defned by teraton. For example, the second-order partal dervatve of f wth respect to x j and x k s defned by f xj x k := 2 f := ( ) f x k x j x k x j when t exsts. Second-order partal dervatves are called mxed when j k 24/26?

47 Defnton. Let V be a nonempty, open subset of R n, let f : V R, and let p N. () f s sad to be C p on V f and only f each partal dervatve of f of order k p exsts and s contnuous on V. 25/26?

48 Defnton. Let V be a nonempty, open subset of R n, let f : V R, and let p N. () f s sad to be C p on V f and only f each partal dervatve of f of order k p exsts and s contnuous on V. () f s sad to be C on V f and only f f s C p for all p N 25/26?

49 Defnton. Let V be a nonempty, open subset of R n, let f : V R, and let p N. () f s sad to be C p on V f and only f each partal dervatve of f of order k p exsts and s contnuous on V. () f s sad to be C on V f and only f f s C p for all p N C p (V ) denotes the set of functons that are C p on an open set V 25/26?

50 Theorem. [Claraut-Schwarz] Suppose that V s open n R 2, that (a, b) V, and that f : V R. If f s C 2 on V 2 f y x (a, b) = 2 f (a, b) x y 26/26?

51 Theorem. [Claraut-Schwarz] Suppose that V s open n R 2, that (a, b) V, and that f : V R. If f s C 2 on V 2 f y x (a, b) = 2 f (a, b) x y Theorem. If f s C 2 on an open subset V of R n, f a V, and f j k, then 2 f x j x k (a) = 2 f x k x j (a) 26/26?