Jae Gil Choi and Young Seo Park
|
|
- Cecilia Fisher
- 5 years ago
- Views:
Transcription
1 Kangweon-Kyungki Math. Jour. 11 (23), No. 1, pp TRANSLATION THEOREM ON FUNCTION SPACE Jae Gil Choi and Young Seo Park Abstract. In this paper, we use a generalized Brownian motion process to define a translation theorem. First we establish the translation theorem for function space integrals. We then obtain the general translation theorem for functionals on function space. 1. Introduction. In [1], Cameron and Martin introduced the transformation of Wiener integrals under the translation. In [4], Chang, Skoug and Park studied translation theorems for Fourier-Feynman transforms and conditional Fourier-Feynman transforms. In this paper, we study a translation theorem for functionals on function space but with x in a very general function space C a,b [, T ] rather than in the Wiener space. The Wiener process used in [1,4] is free of drift and is stationary in time while the stochastic processes used in this paper is nonstationary in time and is subject to a drift a(t). In Section 2 of this paper, we give the basic concepts and notations. In Section 3, we study a translation theorem for function space integrals. Finally, in Section 4, we establish the general translation theorem for functionals on function space. 2. Definitions and preliminaries. Let D [, T ] and let (Ω, B, P ) be a probability measure space. A real valued stochastic process Y on (Ω, B, P ) and D is called a generalized Brownian motion process if Y (, ω) almost everywhere and Received January 3, Mathematics Subject Classification: 6J65,28C2. Key words and phrases: Generalized Brownian motion process, function space, function space integrals, translation theorem.
2 18 Jae Gil Choi and Young Seo Park for t < t 1 < < t n T, the n-dimensional random vector (Y (t 1, ω),, Y (t n, ω)) is normally distributed with density function (2.1) K( t, η) ( (2π) n exp 1 2 n () ) 1/2 ((η j a(t j )) (η j 1 a(t j 1 ))) 2 where η (η 1,, η n ), η, t (t 1,, t n ), a(t) is an absolutely continuous real-valued function on [, T ] with a(), a (t) L 2 [, T ], and b(t) is a strictly increasing, continuously differentiable real-valued function with b() and b (t) > for each t [, T ]. As explained in [8, p.18-2], Y induces a probability measure µ on the measurable space (R D, B D ) where R D is the space of all real valued functions x(t), t D, and B D is the smallest σ-algebra of subsets of R D with respect to which all the coordinate evaluation maps e t (x) x(t) defined on R D are measurable. The triple (R D, B D, µ) is a probability measure space. This measure space is called the function space induced by the generalized Brownian motion process Y determined by a( ) and b( ). We note that the generalized Brownian motion process Y determined by a( ) and b( ) is a Gaussian process with mean function a(t) and covariance function r(s, t) minb(s), b(t). By Theorem 14.2 [8, p.187], the probability measure µ induced by Y, taking a separable version, is supported by C a,b [, T ] (which is equivalent to the Banach space of continuous functions x on [, T ] with x() under the sup norm). Hence (C a,b [, T ], B(C a,b [, T ]), µ) is the function space induced by Y where B(C a,b [, T ]) is the Borel σ-algebra of C a,b [, T ]. Let L 2 a,b [, T ] be the Hilbert space of functions on [, T ] which are Lebesgue measurable and square integrable with respect to the Lebesgue Stieltjes measures on [, T ] induced by a( ) and b( ): i.e., (2.2) L 2 a,b[, T ] v : v 2 (s)db(s) < and v 2 (s)d a (s) < where a (t) denotes the total variation of the function a on the interval [, t].
3 Translation theorem on function space 19 For convenience, let BV [, T ] be the space of bounded variation functions on [, T ]. We denote the function space integral of a B(C a,b [, T ])-measurable functional F by F (x)dµ(x) whenever the integral exists. C a,b [,T ] 3. Translation theorem for function space integrals. Let (C a,b [, T ], B(C a,b [, T ]), µ) be the function space induced by the generalized Brownian motion process defined in Section 1. In this section we will obtain a translation theorem for function space integrals over (C a,b [, T ], B(C a,b [, T ]), µ). For a partition τ t 1,, t n of [, T ] with t < t 1 < < t n T, define a function X τ : C a,b [, T ] R n by X τ (x) (x(t 1 ),, x(t n )). For x C a,b [, T ], define the function [X τ (x)] [x] n : [, T ] R by (3.1) [x] n (t) x(t j 1 ) + b(t) b(t j 1) (x(t j) x(t j 1 )) for each t [t j 1, t j ], j 1, 2,, n. In case, [x] n is called the polygonalized form of x. Similarly, for ξ (ξ 1, ξ 2,, ξ n ) R n, define the function [ ξ] n : [, T ] R by (3.2) [ ξ] n (t) ξ j 1 + b(t) b(t j 1) (ξ j ξ j 1 ) for each t [t j 1, t j ], j 1, 2,, n, and ξ. For any positive integer n, define the point t j by t j T n j where j 1, 2,, n.
4 2 Jae Gil Choi and Young Seo Park Lemma 3.1. Let [x] n be the polygonalized form of x as in (3.1) and let F be a bounded and continuous functional on C a,b [, T ]. Then (3.3) lim F ([x] n )dµ(x) F (x)dµ(x). n C a,b [,T ] C a,b [,T ] Proof. Let ε > be given. For any x C a,b [, T ], there exists an integer n n (ε) such that for all t t 1/n, we have (3.4) x(t ) x(t ) < ε 2. By using (3.1) and (3.4), we have for each n n and t j 1 t t j, (3.5) x(t) [x] n (t) x(t) x(t j 1) b(t) b(t j 1) (x(t j) x(t j 1 )) x(t) x(t j 1 ) + x(t j) x(t j 1 ) Hence ε 2 + ε 2 ε. (3.6) lim n [x] n(t) x(t) uniformly on [, T ]. Since F is continuous on C a,b [, T ], (3.7) lim n F ([x] n) F (x). Let F n (x) F ([x] n ). Then for all n N, F n F. So, by using the Bounded Convergence Theorem, we have the desired result. Lemma 3.2. Let ϕ(t) be of bounded variation function on [, T ] and let x (t) t ϕ(s)db(s). Then x C a,b [, T ] and we have (3.8) lim n (x (t j ) x (t j 1 )) 2 ϕ 2 (s)db(s)
5 and (3.9) lim n Translation theorem on function space 21 ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 )))(x (t j ) x (t j 1 )) ϕ(s)dx(s) ϕ(s)da(s). Proof. Since ϕ BV [, T ], x (t) t ϕ(s)db(s) is absolutely continuous on [, T ]. Observe that (x (t j ) x (t j 1 )) 2 ( ) 2 x (t j ) x (t j 1 ) (b(t j ) b(t j 1)) By using the Cauchy s Mean Value Theorem in the above equation, we have (3.1) (x (t j ) x (t j 1 )) 2 ϕ 2 (ξ j )() where ξ j [t j 1, t j ] for each j 1, 2,, n. Similarly, we have (3.11) ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 )))(x (t j ) x (t j 1 )) ( ) x (t j ) x (t j 1 ) (x(t j ) x(t j 1 )) b(t j ) b(t j 1 ) ( ) x (t j ) x (t j 1 ) (a(t j ) a(t j 1 )) ϕ(ξ j )(x(t j ) x(t j 1 )) ϕ(ξ j )(a(t j ) a(t j 1 )).
6 22 Jae Gil Choi and Young Seo Park Hence equations (3.1) and (3.11) converge to the followings ϕ 2 (s)db(s) and ϕ(s)dx(s) ϕ(s)da(s) as n, respectively. Thus we have the desired results. Theorem 3.3. Let ϕ and x be given as in Lemma 3.2 and let F be a B(C a,b [, T ])-measurable functional. Then F (x+x ) is B(C a,b [, T ])- measurable and (3.12) F (y)dµ(y) F (x + x )J(x, x )dµ(x) C a,b [,T ] where (3.13) J(x, x ) exp 1 2 ϕ 2 (s)db(s) + C a,b [,T ] ϕ(s)da(s) ϕ(s)dx(s). Proof. It suffices to show the case in which the functional F is bounded on C a,b [, T ]. Let us first consider the case F is bounded, continuous, and F (y) for any y x C a,b [, T ] : x > M where M >. Then F (x + x ) is measurable and there exists a positive real number K such that F (x) K for all x C a,b [, T ]. Let τ : t < t 1 < < t n T be a partition of [, T ]. Define a function G on R n by G( ξ) F ([ ξ] n ) for each ξ R n. Then G is bounded and continuous. Hence we see that (3.14) F ([y] n ) G(y(t 1 ),, y(t n )). By using (3.14) and the Change of Variables Theorem, we have (3.15) F ([y] n )dµ(y) C a,b [,T ] C a,b [,T ] ( (2π) n R n G(y(t 1 ),, y(t n ))dµ(y) n () ) 1/2 G(v1,, v n ) exp 1 2 ((v j a(t j )) (v j 1 a(t j 1 ))) 2 d v.
7 Translation theorem on function space 23 Let β j x (t j ) and u j v j β j. Then we see that (3.16) F ([x + x ] n ) G(x(t 1 ) + x (t 1 ),, x(t n ) + x (t n )) G(x(t 1 ) + β 1,, x(t n ) + β n )) and (3.17) ((v j a(t j )) (v j 1 a(t j 1 ))) 2 ((u j a(t j )) (u j 1 a(t j 1 ))) 2 + (β j β j 1 ) 2 + 2(β j β j 1 )((u j a(t j )) (u j 1 a(t j 1 ))). By applying (3.16) and (3.17) above to the last equation of (3.15), we have (3.18) C a,b [,T ] F ([y] n )dµ(y) exp 1 (x (t j ) x (t j 1 )) 2 F ([x + x ] n ) 2 b(t j ) b(t j 1 ) C a,b [,T ] ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 ))) exp b(t j ) b(t j 1 ) (x (t j ) x (t j 1 )) dµ(x). Assume that x + x x C a,b [, T ] : y > M. Then we see that [x + x ] n x + x M and so we have (3.19) x(t j ) x (t j ) + x(t j ) + x (t j ) x + M
8 24 Jae Gil Choi and Young Seo Park for any x C a,b [, T ]. By using (3.11) and (3.19), we obtain that ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 )))(x (t j ) x (t j 1 )) b(t j ) b(t j 1 ) ϕ(ξ j )(x(t j ) x(t j 1 )) ϕ(ξ j )(a(t j ) a(t j 1 )) ϕ(ξ j )(x(t j ) x(t j 1 )) n + ϕ(ξ j )(a(t j ) a(t j 1 )) ϕ(ξ n)x(t n ) ϕ(ξ n ) x(t n ) + (ϕ(ξ j ) ϕ(ξ j 1 ))x(t j 1 ) + ϕ(ξ n)a(t n ) (ϕ(ξ j ) ϕ(ξ j 1 ))a(t j 1 ) ϕ(ξ j ) ϕ(ξ j 1 ) x(t j 1 ) + ϕ(ξ n ) a(t n ) + ( x + M + a ) ( ϕ + V T (ϕ)) ϕ(ξ j ) ϕ(ξ j 1 ) a(t j 1 ) where V T (ϕ) is the total variation of ϕ on [, T ]. So the integrand of (3.18) is bounded by the following K exp( x + M + a )( ϕ + V T (ϕ)). Thus, by using the Bounded Convergence Theorem, (3.8), and (3.9), the expression (3.17) converges to exp 1 ϕ 2 (s)db(s) 2 F (x + x ) exp C a,b [,T ] ϕ(s)dx(s) + ϕ(s)da(s) dµ(x). Hence by using Lemma 2.1, we obtain the equation (3.12) above.
9 Translation theorem on function space 25 Now, let F be a nonnegative, bounded and continuous functional on C a,b [, T ]. For each n N, define the function M n by 1 ( u n) M n (u) n + 1 u (n u n + 1) (n + 1 u). Then M n is a continuous real valued function. Let F n (x) F (x)m n ( x ). Then F n satisfies the hypothesis of the first case. So proceeding as in the proof above, we obtain C a,b [,T ] F n (y)dµ(y) exp 1 2 C a,b [,T ] ϕ 2 (s)db(s) + F n (x + x ) exp ϕ(s)da(s) ϕ(s)dx(s) dµ(x). Since F n is a monotone increasing sequence of functionals, F n F as n. Hence by using the Monotone Convergence Theorem, we have the desired result. Theorem 3.4. Let ϕ, x, and F be given as in Theorem 3.3. Then C a,b [,T ] F (x + x )dµ(x) exp 1 ϕ 2 (s)db(s) ϕ(s)da(s) 2 F (x) exp ϕ(s)dx(s) dµ(x). C a,b [,T ] Proof. Let G(x) F (x) exp ϕ(s)dx(s). Then by using equa-
10 26 Jae Gil Choi and Young Seo Park tion (2.12), we have C a,b [,T ] C a,b [,T ] C a,b [,T ] C a,b [,T ] F (x) exp ϕ(s)dx(s) dµ(x) G(x)dµ(x) G(x + x )J(x, x )dµ(x) F (x + x ) exp ϕ(s)d(x(s) + x (s)) exp 1 ϕ 2 (s)db(s) + ϕ(s)da(s) 2 1 T exp ϕ 2 (s)db(s) + ϕ(s)da(s) 2 C a,b [,T ] ϕ(s)dx(s) dµ(x) F (x + x )dµ(x). Hence we have the desired result. 4. The general translation theorem In this section we consider the general translation theorem for functionals on C a,b [, T ]. For u, v L 2 a,b [, T ], let (u, v) a,b u(t)v(t)d[b(t) + a (t)]. Then (, ) a,b is an inner product on L 2 a,b [, T ] and u a,b (u, u) a,b is a norm on L 2 a,b [, T ]. In particular note that u a,b if and only if u(t) a.e. on [, T ]. Furthermore (L 2 a,b [, T ], a,b) is a separable Hilbert space. Let e j be a complete orthogonal set of real-valued functions of bounded variation on [, T ] such that, j k (e j, e k ) a,b 1, j k,
11 and for each v L 2 a,b [, T ], let Translation theorem on function space 27 v n (t) (v, e j ) a,b e j (t) for n 1, 2,. Then for each v L 2 a,b [, T ], the Paley-Wiener- Zygmund(PWZ) stochastic integral v, x is defined by the formula v, x lim n v n (t)dx(t) for all x C a,b [, T ] for which the limit exists. Remark 4.1. Following are some facts about the PWZ stochastic integral (i) For each v L 2 a,b [, T ], the PWZ integral v, x exists for µ-a.e. x C a,b [, T ]. (ii) The PWZ integral v, x is essentially independent of the complete orthonormal set e j. (iii) If v BV [, T ], then PWZ integral v, x equals the Riemann -Stieltjes integral v(s)dx(s) for s-a.e. x C a,b[, T ]. (iv) The PWZ integral has the expected linearity properties. Lemma 4.1. Let ϕ n C a,b [, T ] BV [, T ] for each n N and let ϕ n converge in the space L 2 a,b [, T ] as n. Then for any real number λ, expλ ϕ n(t)dx(t) converges in the space L 2 (C a,b [, T ]) as n. Proof. The proof given in [3] with the current hypotheses on a(t) and b(t) also works here. Now, we obtain the general translation theorem of a functional on C a,b [, T ]. Theorem 4.2. Let ϕ(t) L 2 a,b [, T ] and let x (t) t ϕ(s)db(s). If F be a B(C a,b [, T ])-measurable functional on C a,b [, T ], then F (x+x ) is B(C a,b [, T ])-measurable and F (y)dµ(y) exp 1 2 (ϕ2, b ) + (ϕ, a ) (4.1) C a,b [,T ] C a,b [,T ] F (x + x ) exp ϕ, x dµ(x)
12 28 Jae Gil Choi and Young Seo Park where and (ϕ 2, b ) (ϕ, a ) ϕ 2 (t)b (t)dt ϕ(t)a (t)dt ϕ(t)db(t) ϕ(t)da(t). Proof. It suffices to show the case in which the functional F is bounded and continuous on C a,b [, T ]. Let e j be complete orthonormal set in L 2 a,b [, T ] with e j C a,b [, T ] BV [, T ] for each j N. For each n N, let (4.2) ϕ n (t) (ϕ, e j ) a,b e j (t). Then ϕ n C a,b [, T ] BV [, T ] and (4.3) ϕ n ϕ a,b as n. For each n N, define (4.4) x,n (t) t ϕ n (s)db(s). Then, by using equation (3.12), we see that (4.5) C a,b [,T ] exp 1 2 F (y)dµ(y) ϕ 2 n(s)db(s) + C a,b [,T ] ϕ n (s)da(s) F (x + x,n ) exp ϕ n (s)dx(s) dµ(x).
13 Translation theorem on function space 29 By using Cauchy-Schwarz inequality, we have t x (t) x,n (t) (ϕ(s) ϕ n (s))db(s) (4.6) (ϕ(s) ϕ n (s))χ [,t] (s) d[b(s) + a (s)] ϕ ϕ n a,b b(t) + a (t) ϕ ϕ n a,b b(t ) + a (T ). Hence by using (4.6) and (4.3), we obtain that (4.7) x x,n. Thus for all x C a,b [, T ] (4.8) F (x + x,n ) F (x + x ). Since F is bounded, by applying the Bounded Convergence Theorem, we have (4.9) F (x + x,n ) F (x + x ) 2 dµ(x). C a,b [,T ] Note that ϕ n (t)dx(t) ϕ, x µ a.e.x C a,b [, T ]. So by using (4.3) and Lemma 4.1 with λ 1 exp ϕ n (t)dx(t) exp ϕ, x in the space L 2 (C a,b [, T ]). Further, by equation (4.9) above, we obtain (4.1) C a,b [,T ] F (x + x,n ) exp C a,b [,T ] ϕ n (t)dx(t) dµ(x) F (x + x ) exp ϕ, x dµ(x). Hence by using equations (4.5) and (4.1) we have the desired equation (4.1). We next use Theorem 4.2 to evaluate a translation theorem of functionals on C a,b [, T ].
14 3 Jae Gil Choi and Young Seo Park Theorem 4.3. Let ϕ, x, and F be given as in Theorem 4.2. Then F (x + x )dµ(x) C a,b [,T ] exp 1 2 (ϕ2, b ) (ϕ, a) C a,b [,T ] F (x) exp ϕ, x dµ(x). Proof. Let G(x) F (x) exp ϕ(s)dx(s). Then, proceeding as in the proof of Theorem 3.4, we have the desired result. References [1] R.H. Cameron and W.T. Martin, Translations of Wiener integrals under translations, Annals of Math. 45(2), [2] R.H. Cameron and R.E. Graves, Additive functionals on a space of continuous functions I, Trans. Amer. Math. Soc. 7, [3] S.J. Chang and D.M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math , [4] S.J. Chang, C. Park and D. Skoug, Translation theorems for Fourier-Feynman transforms and conditional Fourier-Feynman transforms, Rocky Mountain J. of Math. 3(2), [5] G.W. Johnson and D. Skoug, Notes on the Feynman integral I, Pacific J. Math. 93, [6], Notes on the Feynman integral II, J. Func. Anal. 41, [7], Notes on the Feynman integral III : The Schrödinger equation, Pacific J. Math. 15, [8] J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, Department of Mathematics Dankook University Cheonan , Korea jgchoi@dankook.ac.kr Department of Mathematics Dankook University Cheonan , Korea yseo@dankook.ac.kr
Generalized Analytic Fourier-Feynman Transforms with Respect to Gaussian Processes on Function Space
ilomat 3:6 6), 65 64 DO.98/L6665C Published by aculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Generalized Analytic ourier-eynman Transforms
More informationSOME EXPRESSIONS FOR THE INVERSE INTEGRAL TRANSFORM VIA THE TRANSLATION THEOREM ON FUNCTION SPACE
J. Korean Math. Soc. 53 (2016), No. 6, pp. 1261 1273 http://dx.doi.org/10.4134/jkms.j150485 pissn: 0304-9914 / eissn: 2234-3008 SOME EXPRESSIONS FOR THE INVERSE INTEGRAL TRANSFORM VIA THE TRANSLATION THEOREM
More informationA Present Position-Dependent Conditional Fourier-Feynman Transform and Convolution Product over Continuous Paths
International Journal of Mathematical Analysis Vol. 9, 05, no. 48, 387-406 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.589 A Present Position-Dependent Conditional Fourier-Feynman Transform
More informationINTEGRATION FORMULAS INVOLVING FOURIER-FEYNMAN TRANSFORMS VIA A FUBINI THEOREM. Timothy Huffman, David Skoug, and David Storvick
J. Korean Math. Soc. 38 (1), No., pp. 41 435 INTEGRATION FORMULAS INVOLVING FOURIER-FEYNMAN TRANSFORMS VIA A FUBINI THEOREM Timothy Huffman, David Skoug, and David Storvick Abstract. In this paper we use
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More informationON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary
More informationChapter 4. The dominated convergence theorem and applications
Chapter 4. The dominated convergence theorem and applications The Monotone Covergence theorem is one of a number of key theorems alllowing one to exchange limits and [Lebesgue] integrals (or derivatives
More informationOn pathwise stochastic integration
On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic
More informationON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR TERM
Internat. J. Math. & Math. Sci. Vol. 22, No. 3 (999 587 595 S 6-72 9922587-2 Electronic Publishing House ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR
More informationThree hours THE UNIVERSITY OF MANCHESTER. 24th January
Three hours MATH41011 THE UNIVERSITY OF MANCHESTER FOURIER ANALYSIS AND LEBESGUE INTEGRATION 24th January 2013 9.45 12.45 Answer ALL SIX questions in Section A (25 marks in total). Answer THREE of the
More informationSolutions to Tutorial 11 (Week 12)
THE UIVERSITY OF SYDEY SCHOOL OF MATHEMATICS AD STATISTICS Solutions to Tutorial 11 (Week 12) MATH3969: Measure Theory and Fourier Analysis (Advanced) Semester 2, 2017 Web Page: http://sydney.edu.au/science/maths/u/ug/sm/math3969/
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationGAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM
GAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM STEVEN P. LALLEY 1. GAUSSIAN PROCESSES: DEFINITIONS AND EXAMPLES Definition 1.1. A standard (one-dimensional) Wiener process (also called Brownian motion)
More informationCitation Osaka Journal of Mathematics. 41(4)
TitleA non quasi-invariance of the Brown Authors Sadasue, Gaku Citation Osaka Journal of Mathematics. 414 Issue 4-1 Date Text Version publisher URL http://hdl.handle.net/1194/1174 DOI Rights Osaka University
More informationWeighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 401 412 (2013) http://campus.mst.edu/adsa Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationPart II Probability and Measure
Part II Probability and Measure Theorems Based on lectures by J. Miller Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationMeasure, Integration & Real Analysis
v Measure, Integration & Real Analysis preliminary edition 10 August 2018 Sheldon Axler Dedicated to Paul Halmos, Don Sarason, and Allen Shields, the three mathematicians who most helped me become a mathematician.
More informationFACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355. Analysis 4. Examiner: Professor S. W. Drury Date: Tuesday, April 18, 2006 INSTRUCTIONS
FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355 Analysis 4 Examiner: Professor S. W. Drury Date: Tuesday, April 18, 26 Associate Examiner: Professor K. N. GowriSankaran Time: 2: pm. 5: pm. INSTRUCTIONS
More informationGaussian Processes. 1. Basic Notions
Gaussian Processes 1. Basic Notions Let T be a set, and X : {X } T a stochastic process, defined on a suitable probability space (Ω P), that is indexed by T. Definition 1.1. We say that X is a Gaussian
More informationKarhunen-Loève decomposition of Gaussian measures on Banach spaces
Karhunen-Loève decomposition of Gaussian measures on Banach spaces Jean-Charles Croix GT APSSE - April 2017, the 13th joint work with Xavier Bay. 1 / 29 Sommaire 1 Preliminaries on Gaussian processes 2
More informationMi-Hwa Ko. t=1 Z t is true. j=0
Commun. Korean Math. Soc. 21 (2006), No. 4, pp. 779 786 FUNCTIONAL CENTRAL LIMIT THEOREMS FOR MULTIVARIATE LINEAR PROCESSES GENERATED BY DEPENDENT RANDOM VECTORS Mi-Hwa Ko Abstract. Let X t be an m-dimensional
More informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationBrownian Motion and Conditional Probability
Math 561: Theory of Probability (Spring 2018) Week 10 Brownian Motion and Conditional Probability 10.1 Standard Brownian Motion (SBM) Brownian motion is a stochastic process with both practical and theoretical
More informationLebesgue Integration: A non-rigorous introduction. What is wrong with Riemann integration?
Lebesgue Integration: A non-rigorous introduction What is wrong with Riemann integration? xample. Let f(x) = { 0 for x Q 1 for x / Q. The upper integral is 1, while the lower integral is 0. Yet, the function
More informationIntroduction to Empirical Processes and Semiparametric Inference Lecture 09: Stochastic Convergence, Continued
Introduction to Empirical Processes and Semiparametric Inference Lecture 09: Stochastic Convergence, Continued Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationShift Invariant Spaces and Shift Generated Dual Frames for Local Fields
Communications in Mathematics and Applications Volume 3 (2012), Number 3, pp. 205 214 RGN Publications http://www.rgnpublications.com Shift Invariant Spaces and Shift Generated Dual Frames for Local Fields
More informationPREDICTABLE REPRESENTATION PROPERTY OF SOME HILBERTIAN MARTINGALES. 1. Introduction.
Acta Math. Univ. Comenianae Vol. LXXVII, 1(28), pp. 123 128 123 PREDICTABLE REPRESENTATION PROPERTY OF SOME HILBERTIAN MARTINGALES M. EL KADIRI Abstract. We prove as for the real case that a martingale
More informationA COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE
More informationi. Bonic R. and Frampton J., Differentiable functions on certain Banach spaces, Bull. Amer. Math. Soc. 71(1965),
References i. Bonic R. and Frampton J., Differentiable functions on certain Banach spaces, Bull. Amer. Math. Soc. 71(1965), 393-395. 2. Cameron R. H. and Graves R., Additive functionals on a space of continuous
More informationChapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.
Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space
More information6.2 Fubini s Theorem. (µ ν)(c) = f C (x) dµ(x). (6.2) Proof. Note that (X Y, A B, µ ν) must be σ-finite as well, so that.
6.2 Fubini s Theorem Theorem 6.2.1. (Fubini s theorem - first form) Let (, A, µ) and (, B, ν) be complete σ-finite measure spaces. Let C = A B. Then for each µ ν- measurable set C C the section x C is
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More information9 Brownian Motion: Construction
9 Brownian Motion: Construction 9.1 Definition and Heuristics The central limit theorem states that the standard Gaussian distribution arises as the weak limit of the rescaled partial sums S n / p n of
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationAPPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS
APPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS J.M.A.M. VAN NEERVEN Abstract. Let µ be a tight Borel measure on a metric space, let X be a Banach space, and let f : (, µ) X be Bochner integrable. We show
More informationA New Wavelet-based Expansion of a Random Process
Columbia International Publishing Journal of Applied Mathematics and Statistics doi:10.7726/jams.2016.1011 esearch Article A New Wavelet-based Expansion of a andom Process Ievgen Turchyn 1* eceived: 28
More informationThe Codimension of the Zeros of a Stable Process in Random Scenery
The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar
More informationThe Nemytskii operator on bounded p-variation in the mean spaces
Vol. XIX, N o 1, Junio (211) Matemáticas: 31 41 Matemáticas: Enseñanza Universitaria c Escuela Regional de Matemáticas Universidad del Valle - Colombia The Nemytskii oerator on bounded -variation in the
More informationSINGULAR MEASURES WITH ABSOLUTELY CONTINUOUS CONVOLUTION SQUARES ON LOCALLY COMPACT GROUPS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 10, Pages 2865 2869 S 0002-9939(99)04827-3 Article electronically published on April 23, 1999 SINGULAR MEASURES WITH ABSOLUTELY CONTINUOUS
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationFourier Series. ,..., e ixn ). Conversely, each 2π-periodic function φ : R n C induces a unique φ : T n C for which φ(e ix 1
Fourier Series Let {e j : 1 j n} be the standard basis in R n. We say f : R n C is π-periodic in each variable if f(x + πe j ) = f(x) x R n, 1 j n. We can identify π-periodic functions with functions on
More informationHomework 11. Solutions
Homework 11. Solutions Problem 2.3.2. Let f n : R R be 1/n times the characteristic function of the interval (0, n). Show that f n 0 uniformly and f n µ L = 1. Why isn t it a counterexample to the Lebesgue
More informationBEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction
Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of
More informationA class of non-convex polytopes that admit no orthonormal basis of exponentials
A class of non-convex polytopes that admit no orthonormal basis of exponentials Mihail N. Kolountzakis and Michael Papadimitrakis 1 Abstract A conjecture of Fuglede states that a bounded measurable set
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationFourier Series. 1. Review of Linear Algebra
Fourier Series In this section we give a short introduction to Fourier Analysis. If you are interested in Fourier analysis and would like to know more detail, I highly recommend the following book: Fourier
More informationSome Properties of NSFDEs
Chenggui Yuan (Swansea University) Some Properties of NSFDEs 1 / 41 Some Properties of NSFDEs Chenggui Yuan Swansea University Chenggui Yuan (Swansea University) Some Properties of NSFDEs 2 / 41 Outline
More informationSCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE
SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE BETSY STOVALL Abstract. This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid
More informationChapter 6. Integration. 1. Integrals of Nonnegative Functions. a j µ(e j ) (ca j )µ(e j ) = c X. and ψ =
Chapter 6. Integration 1. Integrals of Nonnegative Functions Let (, S, µ) be a measure space. We denote by L + the set of all measurable functions from to [0, ]. Let φ be a simple function in L +. Suppose
More information1 Weak Convergence in R k
1 Weak Convergence in R k Byeong U. Park 1 Let X and X n, n 1, be random vectors taking values in R k. These random vectors are allowed to be defined on different probability spaces. Below, for the simplicity
More informationFACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355. Analysis 4. Examiner: Professor S. W. Drury Date: Wednesday, April 18, 2007 INSTRUCTIONS
FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355 Analysis 4 Examiner: Professor S. W. Drury Date: Wednesday, April 18, 27 Associate Examiner: Professor K. N. GowriSankaran Time: 2: pm. 5: pm.
More informationThe Cameron-Martin-Girsanov (CMG) Theorem
The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version,
More informationOn an uniqueness theorem for characteristic functions
ISSN 392-53 Nonlinear Analysis: Modelling and Control, 207, Vol. 22, No. 3, 42 420 https://doi.org/0.5388/na.207.3.9 On an uniqueness theorem for characteristic functions Saulius Norvidas Institute of
More informationMATH & MATH FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING Scientia Imperii Decus et Tutamen 1
MATH 5310.001 & MATH 5320.001 FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING 2016 Scientia Imperii Decus et Tutamen 1 Robert R. Kallman University of North Texas Department of Mathematics 1155
More informationLecture 22 Girsanov s Theorem
Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n
More informationASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying
More informationON CHARACTERISTIC 0 AND WEAKLY ALMOST PERIODIC FLOWS. Hyungsoo Song
Kangweon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 161 167 ON CHARACTERISTIC 0 AND WEAKLY ALMOST PERIODIC FLOWS Hyungsoo Song Abstract. The purpose of this paper is to study and characterize the notions
More informationSUMMARY OF RESULTS ON PATH SPACES AND CONVERGENCE IN DISTRIBUTION FOR STOCHASTIC PROCESSES
SUMMARY OF RESULTS ON PATH SPACES AND CONVERGENCE IN DISTRIBUTION FOR STOCHASTIC PROCESSES RUTH J. WILLIAMS October 2, 2017 Department of Mathematics, University of California, San Diego, 9500 Gilman Drive,
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationSome basic elements of Probability Theory
Chapter I Some basic elements of Probability Theory 1 Terminology (and elementary observations Probability theory and the material covered in a basic Real Variables course have much in common. However
More informationHardy-Stein identity and Square functions
Hardy-Stein identity and Square functions Daesung Kim (joint work with Rodrigo Bañuelos) Department of Mathematics Purdue University March 28, 217 Daesung Kim (Purdue) Hardy-Stein identity UIUC 217 1 /
More informationTools from Lebesgue integration
Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given
More informationHarnack Inequalities and Applications for Stochastic Equations
p. 1/32 Harnack Inequalities and Applications for Stochastic Equations PhD Thesis Defense Shun-Xiang Ouyang Under the Supervision of Prof. Michael Röckner & Prof. Feng-Yu Wang March 6, 29 p. 2/32 Outline
More informationAffine and Quasi-Affine Frames on Positive Half Line
Journal of Mathematical Extension Vol. 10, No. 3, (2016), 47-61 ISSN: 1735-8299 URL: http://www.ijmex.com Affine and Quasi-Affine Frames on Positive Half Line Abdullah Zakir Husain Delhi College-Delhi
More informationLEBESGUE MEASURE AND L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L 2 Space 4 Acknowledgments 9 References 9
LBSGU MASUR AND L2 SPAC. ANNI WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue
More informationWELL-POSEDNESS FOR HYPERBOLIC PROBLEMS (0.2)
WELL-POSEDNESS FOR HYPERBOLIC PROBLEMS We will use the familiar Hilbert spaces H = L 2 (Ω) and V = H 1 (Ω). We consider the Cauchy problem (.1) c u = ( 2 t c )u = f L 2 ((, T ) Ω) on [, T ] Ω u() = u H
More informationMATH 6605: SUMMARY LECTURE NOTES
MATH 6605: SUMMARY LECTURE NOTES These notes summarize the lectures on weak convergence of stochastic processes. If you see any typos, please let me know. 1. Construction of Stochastic rocesses A stochastic
More informationThe Hilbert Transform and Fine Continuity
Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval
More informationBrownian Motion and Stochastic Calculus
ETHZ, Spring 17 D-MATH Prof Dr Martin Larsson Coordinator A Sepúlveda Brownian Motion and Stochastic Calculus Exercise sheet 6 Please hand in your solutions during exercise class or in your assistant s
More informationANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.
ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n
More informationGaussian Random Fields: Geometric Properties and Extremes
Gaussian Random Fields: Geometric Properties and Extremes Yimin Xiao Michigan State University Outline Lecture 1: Gaussian random fields and their regularity Lecture 2: Hausdorff dimension results and
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationMeasure and Integration: Solutions of CW2
Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost
More informationAdmissible vector fields and quasi-invariant measures
Admissible vector fields and quasi-invariant measures Denis Bell Department of Mathematics, University of North Florida 4567 St. Johns Bluff Road South,Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu
More informationMan Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No. 4, December 26 GIRSANOV THEOREM FOR GAUSSIAN PROCESS WITH INDEPENDENT INCREMENTS Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim *** Abstract.
More informationWeak convergence and Brownian Motion. (telegram style notes) P.J.C. Spreij
Weak convergence and Brownian Motion (telegram style notes) P.J.C. Spreij this version: December 8, 2006 1 The space C[0, ) In this section we summarize some facts concerning the space C[0, ) of real
More informationGENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C ALGEBRA AND APPROXIMATE ALGEBRA HOMOMORPHISMS. Chun Gil Park
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 3 (003), 183 193 GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C ALGEBRA AND APPROXIMATE ALGEBRA HOMOMORPHISMS Chun Gil Park (Received March
More informationSang-baek Lee*, Jae-hyeong Bae**, and Won-gil Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 6, No. 4, November 013 http://d.doi.org/10.14403/jcms.013.6.4.671 ON THE HYERS-ULAM STABILITY OF AN ADDITIVE FUNCTIONAL INEQUALITY Sang-baek Lee*,
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationAN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 95, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu AN EXTENSION OF THE
More informationMATH 202B - Problem Set 5
MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationReminder Notes for the Course on Measures on Topological Spaces
Reminder Notes for the Course on Measures on Topological Spaces T. C. Dorlas Dublin Institute for Advanced Studies School of Theoretical Physics 10 Burlington Road, Dublin 4, Ireland. Email: dorlas@stp.dias.ie
More informationON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS
1 2 3 ON MATRIX VALUED SQUARE INTERABLE POSITIVE DEFINITE FUNCTIONS HONYU HE Abstract. In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important
More information1.3.1 Definition and Basic Properties of Convolution
1.3 Convolution 15 1.3 Convolution Since L 1 (R) is a Banach space, we know that it has many useful properties. In particular the operations of addition and scalar multiplication are continuous. However,
More information(B(t i+1 ) B(t i )) 2
ltcc5.tex Week 5 29 October 213 Ch. V. ITÔ (STOCHASTIC) CALCULUS. WEAK CONVERGENCE. 1. Quadratic Variation. A partition π n of [, t] is a finite set of points t ni such that = t n < t n1
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More informationRiesz Representation Theorems
Chapter 6 Riesz Representation Theorems 6.1 Dual Spaces Definition 6.1.1. Let V and W be vector spaces over R. We let L(V, W ) = {T : V W T is linear}. The space L(V, R) is denoted by V and elements of
More informationFUNCTIONAL ANALYSIS HAHN-BANACH THEOREM. F (m 2 ) + α m 2 + x 0
FUNCTIONAL ANALYSIS HAHN-BANACH THEOREM If M is a linear subspace of a normal linear space X and if F is a bounded linear functional on M then F can be extended to M + [x 0 ] without changing its norm.
More informationOutline of Fourier Series: Math 201B
Outline of Fourier Series: Math 201B February 24, 2011 1 Functions and convolutions 1.1 Periodic functions Periodic functions. Let = R/(2πZ) denote the circle, or onedimensional torus. A function f : C
More informationS. Lototsky and B.L. Rozovskii Center for Applied Mathematical Sciences University of Southern California, Los Angeles, CA
RECURSIVE MULTIPLE WIENER INTEGRAL EXPANSION FOR NONLINEAR FILTERING OF DIFFUSION PROCESSES Published in: J. A. Goldstein, N. E. Gretsky, and J. J. Uhl (editors), Stochastic Processes and Functional Analysis,
More information