- Email: [email protected]
Besov regularity of stochastic measures
ARTICLE IN PRESS
Statistics & Probability Letters 77 (2007) 822–825 www.elsevier.com/locate/stapro
Besov regularity of stochastic measures$ Vadym M. Radchenko Department of Mathematical Analysis, Kyiv Taras Shevchenko University, Kyiv 01033, Ukraine Received 18 November 2004; received in revised form 1 August 2006; accepted 19 December 2006 Available online 20 January 2007
Abstract Let m be a random s-additive in probability set function defined on Borel subsets of ½a; b. We prove that if the process mð½a; tÞ; aptpb, has continuous paths, then they belong a.s. to the Besov space Bapp ð½a; bÞ for all pX2; 0oao1=p. r 2007 Elsevier B.V. All rights reserved. MSC: 60G17; 60G57 Keywords: Stochastic measures; Besov spaces
The Besov regularity of trajectories has been studied for some special classes of stochastic processes. Brownian motion trajectories have been studied in Roynette (1993). Other Gaussian processes were considered in Ciesielski et al. (1993). The Besov regularity of indefinite Skorohod integral w.r.t. fractional Brownian motion was studied in Lakhel et al. (2002), Nualart and Ouknine (2003). In the given note we consider the class of continuous stochastic processes generated by values of stochastic measures and prove the Besov regularity of their paths. Let L0 ¼ L0 ðO; F; PÞ be a set of all real-valued random variables defined on the probability space ðO; F; PÞ (more precisely, the set of equivalence classes). Convergence in L0 means the convergence in probability. Let X be an arbitrary set and B be a s-algebra of subsets of X. Definition. The s-additive mapping m : B ! L0 is called a stochastic measure. In other words, m is a vector measure with values in L0 . We do not assume positivity or moment existence for stochastic measures. In Kwapien´ and Woyczin´ski (1992) such m is called a general stochastic measure. Examples of stochastic measures are the following. Let XR¼ ½a; b Rþ , B be the s-algebra of Borel subsets of ½a; b, and W ðtÞ be the Brownian motion. Then b mðAÞ ¼ a I A ðtÞ dW ðtÞ is a stochastic measure on B. Moreover, by the same way any continuous square integrable martingale X ðtÞ; aptpb defines a stochastic measure m on B so that mððs; tÞ ¼ X ðtÞ X ðsÞ. If W H ðtÞ is a fractional RBrownian motion with Hurst index H412 and f : ½0; T ! R is a bounded measurable T function then mðAÞ ¼ 0 f ðtÞI A ðtÞ dW H ðtÞ is a stochastic measure on B too (this fact follows from Theorem 1.1 of Memin et al., 2001). Other examples may be found in Section 7.2 of Kwapien´ and Woyczin´ski (1992). $
This work was supported by Alexander von Humboldt Foundation, Grant 1074615. E-mail address: [email protected].
0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.12.001
ARTICLE IN PRESS V.M. Radchenko / Statistics & Probability Letters 77 (2007) 822–825
823
Let us consider an arbitrary stochastic process X ðtÞ; aptpb. Put mððs; tÞ ¼ X ðtÞ X ðsÞ and extend m to algebra B0 of all finite unions [lk¼1 ðak ; bk ða; b by additivity. Then m can be extended to a stochastic measure on the Borel s-algebra iff both the following conditions holds: P
(i) mðAn Þ ! 0 for any An 2 B0 ; An # ;; (ii) the set of random variables fmðAn Þ; nX1g is bounded in probability for any disjoint An 2 B0 (Theorem 1 of Radchenko, 1991). It is known that for any X; B the set of values of any stochastic measure is bounded in probability, i.e. lim sup PðjmðAÞj4cÞ ¼ 0
c!1 A2B
(see Talagrand, 1981). Furthermore, Theorem 7.1.2 of Kwapien´ and Woyczin´ski (1992) establishes that the set ( ) n X ck mðAk Þ; Ak1 \ Ak2 ¼ ; for k1 ak2 ; nX1; jck jp1 (1) k¼1
is bounded in probability. We consider the Besov space Bapq ð½a; bÞ; ½a; b R. Recall that the norm in this classical space for 1pp; qo1 and 0oao1 may be introduced by Z ba 1=q kf kap;q ¼ kf kLp ð½a;bÞ þ ðwp ðt; f ÞÞq taq1 dt , 0
where Z
p
wp ðt; f Þ ¼ sup jhjpt
1=p
jf ðx hÞ f ðxÞj dx
,
Ih
I h ¼ fx 2 ½a; b : x h 2 ½a; bg. The norm in the Besov space Bapp ð½a; bÞ is equivalent to the norm in the Slobodeckij space W ap . The main result of the paper will be based on Corollary 3.3 of Kamont (1997) and the following property of stochastic measures. P Lemma. Let m be a stochastic measure and an ; nX1, be a sequence of positive numbers such that 1 n¼1 an o1. Let Dkn 2 B; nX1; 1pkpl n , be such that for each n and k1 ak2 Dk1 n \ Dk2 n ¼ ;. Then 1 X
a2n
n¼1
ln X
m2 ðDkn Þo1
a:s.
k¼1
Proof. Suppose that " # ln 1 X X 2 2 P an m ðDkn Þ ¼ þ1 ¼ e0 40. n¼1
k¼1
We find that for any c40 there exists j such that " # j ln X X a2n m2 ðDkn ÞXc Xe0 =2. P n¼1
(2)
k¼1
For j from (2), let us consider the set ( ) j ln X X 2 2 O1 ¼ o 2 O : an m ðDkn ÞXc , n¼1
k¼1
and the set of independent symmetric Bernoulli random variables ekn ; 1pnpj; 1pkpl n ; defined on other probability space ðO0 ; F0 ; P0 Þ, P0 ðekn ¼ 1Þ ¼ P0 ðekn ¼ 1Þ ¼ 12. We have the following consequence of
ARTICLE IN PRESS V.M. Radchenko / Statistics & Probability Letters 77 (2007) 822–825
824
Paley–Zigmund inequality 2 !2 X 04 P lkn ekn Xð1=4Þ 1pnpj;1pkpl n
X
3 l2kn 5X1=8;
lkn 2 R
1pnpj;1pkpl n
(see, for example, Lemma V.4.3(a) of Vakhania et al., 1987 or Lemma 0.2.1 of Kwapien´ and Woyczin´ski, 1992 for l ¼ 14). By applying this inequality taking lkn ¼ an mðDkn ; oÞ for a fixed o 2 O1 we get 2 3 !2 j ln X X P0 4o0 : an ekn ðo0 ÞmðDkn ; oÞ Xc=45X1=8. n¼1
k¼1
Integrating the above inequality with respect to measure P over O1 we obtain 2 3 !2 j ln X X P P0 4ðo; o0 Þ : an ekn ðo0 ÞmðDkn ; oÞ Xc=45Xe0 =16. n¼1
k¼1
Hence by using Fubini’s theorem, we have that there exists o00 2 O0 such that 2 3 !2 j ln X X P4o : an ekn ðo00 ÞmðDkn ; oÞ Xc=45Xe0 =16. n¼1
k¼1
Recalling that each ekn ðo00 Þ ¼ 1 or ekn ðo00 Þ ¼ 1 we obtain sets Bn ; C n 2 B such that " # j pffiffiffi X a ðmðBn Þ mðC n ÞÞX c=2 Xe0 =16. P n¼1 n
(3)
We have
j X X 1 max an ð1Bn ðxÞ 1C n ðxÞÞp a. n¼1 n x2X n¼1
(4)
Recall that e0 40 is fixed and c40 is arbitrary. Therefore, (3) and (4) contradict the boundedness in probability of the sums (1). This completes the proof of the Lemma. & The main result of the paper is the following. Theorem. Let X ¼ ½a; b R, B be the Borel s-algebra, m be a stochastic measure on B and the process mðtÞ ¼ mð½a; tÞ; aptpb, have continuous paths. Then for any pX2; 0oao1=p, the path of mðtÞ with probability 1 belongs to the Besov space Bapp ð½a; bÞ. Proof. When f : ½a; b ! R is a continuous function, Corollary 3.3 of Kamont (1997) shows that the convergence of the series 1 X n¼1
2
nðap1Þ
2n X
jf ða þ k2n ðb aÞÞ f ða þ ðk 1Þ2n ðb aÞÞjp
k¼1
implies that f 2 Bapp ð½a; bÞ. Obviously, it is sufficient to prove the convergence for the second power of differences of function values. By taking an ¼ 2nðap1Þ=2 ;
Dkn ¼ ða þ ðk 1Þ2n ðb aÞ; a þ k2n ðb aÞ; 1pkp2n
in the Lemma, we see that continuous paths of mðtÞ a.s. satisfy the mentioned condition. & In particular the statement of the Theorem may by applied to the paths of any continuous square integrable martingale.
ARTICLE IN PRESS V.M. Radchenko / Statistics & Probability Letters 77 (2007) 822–825
825
Embeddings of the Besov spaces (see, for example, Section 3.2.4 of Triebel, 1983) implies that a continuous path of mðtÞ a.s. belongs to Bapq ð½a; bÞ; qXpX2; 0oao1=p. Note that with probability 1 the trajectories of the Brownian motion do not belong to the Besov spaces Bapq ð½0; 1Þ for 12oao1; p; qX2 (Theorem 1 of Roynette, 1993). I am grateful to Prof. M. Za¨hle-Ziezold for fruitful discussions during the preparation of this paper. References Ciesielski, Z., Kerkyacharian, G., Roynette, B., 1993. Quelques espaces fonctionnels associe´s a` des processus gaussiens. Stud. Math. 107, 171–204. Kamont, A., 1997. A discrete characterization of Besov spaces. Approx. Theory Appl. (N.S.) 13, 63–77. Kwapien´, S., Woyczin´ski, W.A., 1992. Random Series and Stochastic Integrals: Single and Multiple. Birkha¨user, Boston. Lakhel, H., Ouknine, Y., Tudor, C.A., 2002. Besov regularity for the indefinite Skorohod integral with respect to the fractional Brownian motion: the singular case. Stochastics Stoch. Reports 74, 597–615. Memin, J., Mishura, Y., Valkeila, E., 2001. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51, 197–206. Nualart, D., Ouknine, Y., 2003. Besov regularity of stochastic integrals with respect to the fractional Brownian motion with parameter H41=2. J. Theoret. Probab. 16, 451–470. Radchenko, V.N., 1991. Integrals with respect to random measures and random linear functionals. Teor. Veroyatnost. i Primenen. 36, 594–596 (Russian), translation in Theory Probab. Appl. 1992, 36, 621–623. Roynette, B., 1993. Mouvement Brownien et espaces de Besov. Stochastics Stoch. Reports 43, 221–260. Talagrand, M., 1981. Les mesures vectorielles a valeurs dans L0 sont borne´es. Ann. Sci. E´cole Norm. Sup. 14, 445–452. Triebel, H., 1983. Theory of Function Spaces. Geest & Portig K.-G., Leipzig and Birkha¨user, Basel. Vakhania, N.N., Tarieladze, V.I., Chobanian, S.A., 1987. Probability Distributions on Banach Spaces. D. Reidel Publishing Co, Dordrecht.
Statistics & Probability Letters 77 (2007) 822–825 www.elsevier.com/locate/stapro
Besov regularity of stochastic measures$ Vadym M. Radchenko Department of Mathematical Analysis, Kyiv Taras Shevchenko University, Kyiv 01033, Ukraine Received 18 November 2004; received in revised form 1 August 2006; accepted 19 December 2006 Available online 20 January 2007
Abstract Let m be a random s-additive in probability set function defined on Borel subsets of ½a; b. We prove that if the process mð½a; tÞ; aptpb, has continuous paths, then they belong a.s. to the Besov space Bapp ð½a; bÞ for all pX2; 0oao1=p. r 2007 Elsevier B.V. All rights reserved. MSC: 60G17; 60G57 Keywords: Stochastic measures; Besov spaces
The Besov regularity of trajectories has been studied for some special classes of stochastic processes. Brownian motion trajectories have been studied in Roynette (1993). Other Gaussian processes were considered in Ciesielski et al. (1993). The Besov regularity of indefinite Skorohod integral w.r.t. fractional Brownian motion was studied in Lakhel et al. (2002), Nualart and Ouknine (2003). In the given note we consider the class of continuous stochastic processes generated by values of stochastic measures and prove the Besov regularity of their paths. Let L0 ¼ L0 ðO; F; PÞ be a set of all real-valued random variables defined on the probability space ðO; F; PÞ (more precisely, the set of equivalence classes). Convergence in L0 means the convergence in probability. Let X be an arbitrary set and B be a s-algebra of subsets of X. Definition. The s-additive mapping m : B ! L0 is called a stochastic measure. In other words, m is a vector measure with values in L0 . We do not assume positivity or moment existence for stochastic measures. In Kwapien´ and Woyczin´ski (1992) such m is called a general stochastic measure. Examples of stochastic measures are the following. Let XR¼ ½a; b Rþ , B be the s-algebra of Borel subsets of ½a; b, and W ðtÞ be the Brownian motion. Then b mðAÞ ¼ a I A ðtÞ dW ðtÞ is a stochastic measure on B. Moreover, by the same way any continuous square integrable martingale X ðtÞ; aptpb defines a stochastic measure m on B so that mððs; tÞ ¼ X ðtÞ X ðsÞ. If W H ðtÞ is a fractional RBrownian motion with Hurst index H412 and f : ½0; T ! R is a bounded measurable T function then mðAÞ ¼ 0 f ðtÞI A ðtÞ dW H ðtÞ is a stochastic measure on B too (this fact follows from Theorem 1.1 of Memin et al., 2001). Other examples may be found in Section 7.2 of Kwapien´ and Woyczin´ski (1992). $
This work was supported by Alexander von Humboldt Foundation, Grant 1074615. E-mail address: [email protected].
0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.12.001
ARTICLE IN PRESS V.M. Radchenko / Statistics & Probability Letters 77 (2007) 822–825
823
Let us consider an arbitrary stochastic process X ðtÞ; aptpb. Put mððs; tÞ ¼ X ðtÞ X ðsÞ and extend m to algebra B0 of all finite unions [lk¼1 ðak ; bk ða; b by additivity. Then m can be extended to a stochastic measure on the Borel s-algebra iff both the following conditions holds: P
(i) mðAn Þ ! 0 for any An 2 B0 ; An # ;; (ii) the set of random variables fmðAn Þ; nX1g is bounded in probability for any disjoint An 2 B0 (Theorem 1 of Radchenko, 1991). It is known that for any X; B the set of values of any stochastic measure is bounded in probability, i.e. lim sup PðjmðAÞj4cÞ ¼ 0
c!1 A2B
(see Talagrand, 1981). Furthermore, Theorem 7.1.2 of Kwapien´ and Woyczin´ski (1992) establishes that the set ( ) n X ck mðAk Þ; Ak1 \ Ak2 ¼ ; for k1 ak2 ; nX1; jck jp1 (1) k¼1
is bounded in probability. We consider the Besov space Bapq ð½a; bÞ; ½a; b R. Recall that the norm in this classical space for 1pp; qo1 and 0oao1 may be introduced by Z ba 1=q kf kap;q ¼ kf kLp ð½a;bÞ þ ðwp ðt; f ÞÞq taq1 dt , 0
where Z
p
wp ðt; f Þ ¼ sup jhjpt
1=p
jf ðx hÞ f ðxÞj dx
,
Ih
I h ¼ fx 2 ½a; b : x h 2 ½a; bg. The norm in the Besov space Bapp ð½a; bÞ is equivalent to the norm in the Slobodeckij space W ap . The main result of the paper will be based on Corollary 3.3 of Kamont (1997) and the following property of stochastic measures. P Lemma. Let m be a stochastic measure and an ; nX1, be a sequence of positive numbers such that 1 n¼1 an o1. Let Dkn 2 B; nX1; 1pkpl n , be such that for each n and k1 ak2 Dk1 n \ Dk2 n ¼ ;. Then 1 X
a2n
n¼1
ln X
m2 ðDkn Þo1
a:s.
k¼1
Proof. Suppose that " # ln 1 X X 2 2 P an m ðDkn Þ ¼ þ1 ¼ e0 40. n¼1
k¼1
We find that for any c40 there exists j such that " # j ln X X a2n m2 ðDkn ÞXc Xe0 =2. P n¼1
(2)
k¼1
For j from (2), let us consider the set ( ) j ln X X 2 2 O1 ¼ o 2 O : an m ðDkn ÞXc , n¼1
k¼1
and the set of independent symmetric Bernoulli random variables ekn ; 1pnpj; 1pkpl n ; defined on other probability space ðO0 ; F0 ; P0 Þ, P0 ðekn ¼ 1Þ ¼ P0 ðekn ¼ 1Þ ¼ 12. We have the following consequence of
ARTICLE IN PRESS V.M. Radchenko / Statistics & Probability Letters 77 (2007) 822–825
824
Paley–Zigmund inequality 2 !2 X 04 P lkn ekn Xð1=4Þ 1pnpj;1pkpl n
X
3 l2kn 5X1=8;
lkn 2 R
1pnpj;1pkpl n
(see, for example, Lemma V.4.3(a) of Vakhania et al., 1987 or Lemma 0.2.1 of Kwapien´ and Woyczin´ski, 1992 for l ¼ 14). By applying this inequality taking lkn ¼ an mðDkn ; oÞ for a fixed o 2 O1 we get 2 3 !2 j ln X X P0 4o0 : an ekn ðo0 ÞmðDkn ; oÞ Xc=45X1=8. n¼1
k¼1
Integrating the above inequality with respect to measure P over O1 we obtain 2 3 !2 j ln X X P P0 4ðo; o0 Þ : an ekn ðo0 ÞmðDkn ; oÞ Xc=45Xe0 =16. n¼1
k¼1
Hence by using Fubini’s theorem, we have that there exists o00 2 O0 such that 2 3 !2 j ln X X P4o : an ekn ðo00 ÞmðDkn ; oÞ Xc=45Xe0 =16. n¼1
k¼1
Recalling that each ekn ðo00 Þ ¼ 1 or ekn ðo00 Þ ¼ 1 we obtain sets Bn ; C n 2 B such that " # j pffiffiffi X a ðmðBn Þ mðC n ÞÞX c=2 Xe0 =16. P n¼1 n
(3)
We have
j X X 1 max an ð1Bn ðxÞ 1C n ðxÞÞp a. n¼1 n x2X n¼1
(4)
Recall that e0 40 is fixed and c40 is arbitrary. Therefore, (3) and (4) contradict the boundedness in probability of the sums (1). This completes the proof of the Lemma. & The main result of the paper is the following. Theorem. Let X ¼ ½a; b R, B be the Borel s-algebra, m be a stochastic measure on B and the process mðtÞ ¼ mð½a; tÞ; aptpb, have continuous paths. Then for any pX2; 0oao1=p, the path of mðtÞ with probability 1 belongs to the Besov space Bapp ð½a; bÞ. Proof. When f : ½a; b ! R is a continuous function, Corollary 3.3 of Kamont (1997) shows that the convergence of the series 1 X n¼1
2
nðap1Þ
2n X
jf ða þ k2n ðb aÞÞ f ða þ ðk 1Þ2n ðb aÞÞjp
k¼1
implies that f 2 Bapp ð½a; bÞ. Obviously, it is sufficient to prove the convergence for the second power of differences of function values. By taking an ¼ 2nðap1Þ=2 ;
Dkn ¼ ða þ ðk 1Þ2n ðb aÞ; a þ k2n ðb aÞ; 1pkp2n
in the Lemma, we see that continuous paths of mðtÞ a.s. satisfy the mentioned condition. & In particular the statement of the Theorem may by applied to the paths of any continuous square integrable martingale.
ARTICLE IN PRESS V.M. Radchenko / Statistics & Probability Letters 77 (2007) 822–825
825
Embeddings of the Besov spaces (see, for example, Section 3.2.4 of Triebel, 1983) implies that a continuous path of mðtÞ a.s. belongs to Bapq ð½a; bÞ; qXpX2; 0oao1=p. Note that with probability 1 the trajectories of the Brownian motion do not belong to the Besov spaces Bapq ð½0; 1Þ for 12oao1; p; qX2 (Theorem 1 of Roynette, 1993). I am grateful to Prof. M. Za¨hle-Ziezold for fruitful discussions during the preparation of this paper. References Ciesielski, Z., Kerkyacharian, G., Roynette, B., 1993. Quelques espaces fonctionnels associe´s a` des processus gaussiens. Stud. Math. 107, 171–204. Kamont, A., 1997. A discrete characterization of Besov spaces. Approx. Theory Appl. (N.S.) 13, 63–77. Kwapien´, S., Woyczin´ski, W.A., 1992. Random Series and Stochastic Integrals: Single and Multiple. Birkha¨user, Boston. Lakhel, H., Ouknine, Y., Tudor, C.A., 2002. Besov regularity for the indefinite Skorohod integral with respect to the fractional Brownian motion: the singular case. Stochastics Stoch. Reports 74, 597–615. Memin, J., Mishura, Y., Valkeila, E., 2001. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51, 197–206. Nualart, D., Ouknine, Y., 2003. Besov regularity of stochastic integrals with respect to the fractional Brownian motion with parameter H41=2. J. Theoret. Probab. 16, 451–470. Radchenko, V.N., 1991. Integrals with respect to random measures and random linear functionals. Teor. Veroyatnost. i Primenen. 36, 594–596 (Russian), translation in Theory Probab. Appl. 1992, 36, 621–623. Roynette, B., 1993. Mouvement Brownien et espaces de Besov. Stochastics Stoch. Reports 43, 221–260. Talagrand, M., 1981. Les mesures vectorielles a valeurs dans L0 sont borne´es. Ann. Sci. E´cole Norm. Sup. 14, 445–452. Triebel, H., 1983. Theory of Function Spaces. Geest & Portig K.-G., Leipzig and Birkha¨user, Basel. Vakhania, N.N., Tarieladze, V.I., Chobanian, S.A., 1987. Probability Distributions on Banach Spaces. D. Reidel Publishing Co, Dordrecht.