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1 hs arcle appeared n a journal publshed by Elsever. he aached copy s furnshed o he auhor for nernal non-commercal research and educaon use, ncludng for nsrucon a he auhors nsuon and sharng wh colleagues. Oher uses, ncludng reproducon and dsrbuon, or sellng or lcensng copes, or posng o personal, nsuonal or hrd pary webses are prohbed. In mos cases auhors are permed o pos her verson of he arcle (e.g. n Word or ex form o her personal webse or nsuonal reposory. Auhors requrng furher nformaon regardng Elsever s archvng and manuscrp polces are encouraged o vs: hp://

2 Auhor's personal copy Sascs and Probably Leers 78 ( Conens lss avalable a ScenceDrec Sascs and Probably Leers journal homepage: On exendng classcal flerng equaons Mchael A. Kourzn a, Hongwe Long b, a Deparmen of Mahemacal and Sascal Scences, Unversy of Albera, Edmonon, Canada 6G 2G1 b Deparmen of Mahemacal Scences, Florda Alanc Unversy, Boca Raon, FL 33431, USA a r c l e n f o a b s r a c Arcle hsory: Receved 12 March 27 Receved n revsed form 19 February 28 Acceped 2 June 28 Avalable onlne 17 June 28 MSC: prmary 6G35 secondary 6H15 In hs paper, we gve a drec dervaon of he Duncan Morensen Zaka flerng equaon, whou assumng rgh connuy of he sgnal, nor s flraon, and whou he usual fne energy condon. As a consequence, he Fujsak Kallanpur Kuna equaon s also derved. Our resuls can be appled o flerng problems n whch he sgnal process has α-sable (α > 1 componens, and he sensor funcon s lnear. 28 Elsever B.V. All rghs reserved. 1. Inroducon Classcal connuous-me flerng heory requres ha he sgnal process sasfes he fne energy condon gven below n (1. However, some sgnals lke he α-sable processes, or sochasc processes sasfyng a large class of sochasc dfferenal equaons (SDEs drven by α-sable processes, wll no longer sasfy he momen condon (e.g. he second momen of an α-sable random varable does no exs when 1 < α < 2. In hs paper, we generalze he classcal flerng equaons, by weakenng he fne energy condon. Our new resuls can be appled o parameer esmaon, and flerng for he geophyscal models dscussed n e.g. Dlevsen (1999a,b. In hese models, clmae change s relaed o an SDE drven by an α-sable process (wh α 1.75, and nference s done hrough massve ce core samples corruped by such hngs as dang error, ce shfng and melng and refreezng. Le {X, be a measurable Markov process, akng values n a complee separable merc space S, and lvng on a complee probably space (Ω, F, P. he classcal flerng problem s o descrbe he condonal dsrbuon of sgnal X, gven he collecon {Y s, s of dsored, corruped, paral observaons Y h(x s ds + W, where h : S R d s measurable and W s a sandard Brownan moon ndependen of X. A common goal s o derve a sochasc dfferenal equaon (SDE for he condonal dsrbuon. Under he fne energy condon E h(x 2 d <, wh denong he Eucldean dsance, Fujsak e al. (1972 obaned a SDE for he condonal expecaons Ef (X Y s, s wh f belongng o he doman of he generaor of X. We shall refer o hs equaon as he FKK equaon, bu s equally well known as he Kushner-Sraonovch equaon. An equvalen, ye smpler, equaon s he (1 Correspondng auhor. E-mal addresses: mkourz@mah.ualbera.ca (M.A. Kourzn, hlong@fau.edu (H. Long /$ see fron maer 28 Elsever B.V. All rghs reserved. do:1.116/j.spl

3 Auhor's personal copy 3196 M.A. Kourzn, H. Long / Sascs and Probably Leers 78 ( Duncan Morensen Zaka (DMZ equaon for he unnormalzed condonal expecaon (see Zaka (1969. here are wo mehods o derve he DMZ equaon. he frs one s o use he Kallanpur Srebel formula, he FKK equaon and Io s rule (e.g., Szprglas (1978 and Davs and Marcus (198. he second mehod s o derve he DMZ equaon drecly under (1 as done by Ocone (1984 (see also references heren. he fne energy condon (1 has prevously been mposed n he dervaon of he FKK and DMZ equaons. Ye, hese equaons (see (6 and (7 n Secon 2 are well defned only under he condon E h(x s ds <. Indeed, follows from Secon 3, ha a marngale formulaon of he flerng problem holds under he condon h(x s 2 ds < a.s. herefore, s naural o ponder (see Remark 3 of Kurz and Ocone (1988, wheher soluons o hese flerng equaons exs under condons weaker han (1. In hs paper, we answer hs queson n he affrmave. Heren, we derve he DMZ equaon whou assumng rgh connuy or he sandard fne energy condon. In Secon 2, we provde our noaon and man resuls. In Secon 3, we consder a marngale problem relaed o he unnormalzed fler and prove heorem 1. In Secon 4, we derve he DMZ equaon and FKK equaon (heorem 2. he Appendx conans some echncal resuls used n prevous Secons. 2. Noaon and man resuls We le X and Y be as n he nroducon, and defne F X σ {X s, s, F X σ {F X, N, F X + ε> F X +ε, F W σ {W s, s, F Y σ {σ {Y s, s, N, where N s he collecon of P-null ses. Le B(S be he class of R-valued bounded measurable funcon on S and P(, x, Γ (, x S, Γ B(S be he ranson funcon for X. Is ranson semgroup, defned by f (x f (yp(, x, dy for f B(S, s generally only measurable. We le S J {f B(S : bp lm f f, where bp-lm sands for bounded ponwse lm, and assume ha ( f (x s a jonly measurable funcon of (, x for all f J. hen, J would conan he connuous bounded funcons f X were rgh connuous, whch we do no assume. We defne { D f J : here exss g f J such ha ( f (x f (x + ( s g f (xds, x S g f s unquely deermned so we le Lf g f for f D. L s called he weak generaor of {X, and D s s doman (e.g. Dynkn (1965. I follows from Kallanpur and Karandkar (1985 ha D s measure-deermnng class f J s bp-dense n B(S. o calculae he condonal expecaons π (f Ef (X F Y, f B(S, we fx > and use he condons on h h(x 2 d < a.s. and (2 E h(x d <. Compared wh he fne energy condon (1, our new condons (2 and (3 are more general, allowng for nsance h(x x and X o be an α-sable process wh α > 1. We se F X,Y. σ {F X, F Y, F+ X,Y ε> F+ε X,Y, Y σ {F Y ε> +ε, F X, and M (f f (X f (X Lf (X s ds, f D. (4 I follows by Lemma 6 n he Appendx ha Lf (X sds F X and {M (f, s an {F X -marngale under P. We defne he nnovaon process ν Y π s(hds. hen, by Lemma 2.2 and Remark 2.1 of Fujsak e al. (1972, we know ha (ν, F Y, P s a d-dmensonal marngale wh a connuous verson under he condon (3. Now, we defne { A exp h(x s, dy s 1 h(x s 2 ds (5 2 and fnd ha {A 1,, s a { Y -marngale and { (W, Y, s a d-dmensonal Brownan moon on (Ω, F, P. hese resuls are sandard n flerng heory, and can be easly derved by usng he ndependence of W and X. Now, we defne a new probably measure va Grsanov s heorem (3 dp Y dp Y A 1 E A 1 Y. hen, P (X, Y 1 P (X, W 1, {(A, Y,, s a marngale under P, and {Y s a sandard Brownan moon ndependen of {X under P. Moreover, one has he Kallanpur Srebel formula π (f Ef (X F Y Ef (X A F Y py (f EA F Y p Y (1,

4 Auhor's personal copy M.A. Kourzn, H. Long / Sascs and Probably Leers 78 ( where p Y (f. Ef (X A F Y for all f D. Hence, he condonal sascs of X gven F Y, n erms of P, can be calculaed from hose for he new measure P. Now, we sae our man resuls. heorem 1. Suppose ha h sasfes (2. hen, p Y (f py (f py Y s (Lf ds s an {F -marngale under P for each f D. heorem 2. Suppose h sasfes (2 and (3. hen, he DMZ and FKK equaons p Y (f py (f + p Y s (Lf ds + p Y s (hf, dy s a.s. (6 π (f π (f + π s (Lf ds + hold for all,, f D. π s (hf π s (hπ s (f, dν s a.s. (7 Remark 1. In Eqs. (6 and (7 as well as her proofs we ake { p Y (hf, and {π (hf, o be he oponal projecons of {Ēf (X h(x A F Y, and {Ef (X h(x F Y,, respecvely. Remark 2. When Kurz and Ocone (1988 consdered he unqueness of soluons o (6 and (7 (n he case S s locally compac, hey used he condon (3 and π s(h 2 ds < a.s. whch s smlar o our condon (2. In heorem 2, we provde exsence of soluons o (6 and (7 under condons (2 and (3, whch complemens he resuls n Kurz and Ocone ( Proof of heorem 1 We gve some prelmnary resuls before provng heorem 1. Noe ha he augmened flraon {F X,Y need no be rgh connuous. By Lemma 7 n he Appendx, {A, s ndsngushable from an almos surely connuous {F X,Y - progressvely measurable process. In he sequel, {A, s aken o be hs progressvely measurable process. Noe ha M (f f (X f (X Lf (X sds s an {F X -marngale bu no necessarly o be cadlag, and {F X s no necessarly rgh connuous. We defne M + (f lm M s (f, s whch makes sense by upcrossng nequaly. hen, M + (f s acually a cadlag {F X + -marngale (see Meyer (1966. Le Z (f f (X + Lf (X sds + M + (f. We frs prove he followng resul: Lemma 3. p Y (f Ef (X A F Y EZ (f A F Y. Proof. We need only show EM + (f A F Y E(M +s (f M (f A F Y EM (f A F Y. However, one has ha EE(M +s (f M (f A F X,Y F Y EA E(M +s (f M (f F X,Y F Y. Usng domnaed convergence heorem and leng s, one has EM + (f A F Y { Lemma 4. A M + (f s an F X,Y + -marngale under P. Proof. For τ <, we have ha E A M + (f Fτ+ X,Y E E A Y τ M + (f Fτ+ X,Y A τ E M + (f Fτ+ X,Y We defne M (f. Z (f A f (X A τ M τ + (f a.s. Lemma 5. {M (f, s an {F X,Y + -marngale under P. EM (f A F Y. A s Lf (X s ds, f D. (8

5 Auhor's personal copy 3198 M.A. Kourzn, H. Long / Sascs and Probably Leers 78 ( Proof. By negraon by pars, we have Z (f A M + (f A + f (X A + M + (f A + f (X A + Lf (X s ds A A s Lf (X s ds + u Lf (X s dsda u. hs mples ha M (f s an {F+ X,Y -local marngale. Moreover, by (8, we fnd ( M (f f (X + Lf (X s ds + M + (f A + f (X + A s Lf (X s ds ( sup f (x + sup Lf (x A + M + (f A + sup f (x + sup Lf (x x S x S x S x S So, M (f s of class DL, hence an {F+ X,Y -marngale, by he fac ha A and M + (f A are boh {F+ X,Y -marngales, hence of class DL. Proof of heorem 1. We ake condonal expecaons on (8, and use Fubn s heorem as well as ndependen ncremens o fnd EZ (f A F Y Ef (X + where M (f. E M (f F Y zero mean {F Y -marngale. 4. Proof of heorem 2 s an { F Y EA s Lf (X s F Y s ds + M (f a.s., A s ds. -marngale by Lemma 5. Hence, by Lemma 3, p Y (f py (f py s (Lf ds s a o derve he DMZ equaon, we denfy M (f as he desred sochasc negral. By marngale represenaon (e.g., Problem of Karazas and Shreve (1988, we know ha M (f s connuous, and here exss R d -valued {F Y - progressvely measurable process {α f, such ha M (f αf s, dy s and αf 2 d < a.s. Also, M (f s he unque F Y -measurable random varable wh E M (f ξ E M (f ξ for all bounded ξ F Y. Whou loss of generaly, we can ake ξ E ξ F Y wh ξ F { Y. Snce F Y s connuous, we have ha E ξ F Y has a connuous modfcaon (see II.2.9 of Revuz and Yor (1991, and by almos sure monooncy of condonal expecaon, we can make hs modfcaon bounded. hus, M (f αf s, dy s s he F Y -measurable random varable wh E M (f ξ E M (f ξ for all connuous {F Y -marngales ξ Φ s, dy s wh Φ progressvely measurable and. c ξ sup ξ (ω <. ω, We can also ake Φ s bounded by Lemma 8 n he Appendx. Now, n order o calculae E M (f ξ, we defne he soppng mes. σ N nf { > : α f, s dy s > N. hen, E α f s 2 1 s σn ds < so ha α f s 1 s σn L 2 (P almos everywhere. Proof of heorem 2. Snce he FKK equaon can be easly derved by usng Io s formula, negraon by pars and he DMZ equaon, we jus derve he DMZ equaon here. From he proof of heorem 1, we know ha p Y (f py (f p Y s (Lf ds M (f, where M (f. E M (f F Y ha α f p Y (hf a.s.. We se ξ m A A τ m A τ m 1 M σn (f E E M (f F Y I follows ha αf s, dy s { s an F Y -marngale. o derve he DMZ equaon, suffces o prove be a refnng paron of, and defne operaor Φ s, dy s, le {τ m, τ m,..., τ m 1 m. By Doob s oponal samplng heorem, we have F Y σ N E M (f F σ Y N a.s. EM σn (f ξ σn E M (f ξ σn. (9 (1

6 Auhor's personal copy M.A. Kourzn, H. Long / Sascs and Probably Leers 78 ( By marngale dfference echnque, he {Y -marngale propery of ξ σn and Lemma 5, we fnd ha E M (f ξ σn E m (M (f ξ σn 1 E m M (f ξ σn τ m + EM 1 τ m (f m 1 ξ σn E m M (f m ξ σn E m M (f m ξ σn. ( hen, usng (8, we have for some consan C 1 > E m M (f m ξ σn E m (Z (f A m ξ σn E C 1 E τ m A s Lf (X s m ξ σn ds A s ds max m j ξ σn, (12 j m whch ends o zero when m by domnaed convergence heorem. herefore, E M (f ξ σn s he lm of 1 E m A m ξ σn Z τ m(f + 1 τ m 1 EA τ m 1 m ξ σn m Z (f (13 as m. By he { Y -marngale propery of ξ σn, follows ha EH m ξ σn, H L 1 ( Ω, Y τ m 1, P so he second erm n (13 s zero. For he frs erm n (13, one uses negraon by pars o fnd ha τ m m A m ξ σn 1 s σn h(x s A s, Φ s ds τ m 1 τ m σ N τ m 1 σ N A s Φ s, dy s + τ m τ m 1 h(x s A s ξ s σn, dy s (14 and noes ha he rgh hand sde of (14 s a {Y -local marngale dfference by Proposon of Karazas and Shreve (1988 (h(x. s Y -progressve. Moreover, here s some consan C 2 > such ha τ m ( m A m ξ σn 1 s σn h(x s A s, Φ s ds C 2 m A + A s h(x s ds. hus, f we defne. ρ k nf { > : τ m 1 A s Φ s, dy s h(x s A s ξ s σn, dy s > k, hen { m A ρk m ξ ρk σ N ρ k τ m ρ k τ m 1 s σn A s h(x s, Φ s ds s unformly negrable by condon (3 and he fac ha 1 k1 {A, s a {Y -marngale under P, hence of class DL. herefore, by (14, Lemma 9 n he Appendx and he fac ha Z τ m(f s bounded and Y τ 1-measurable, m we have τ m E m A m ξ σn Z τ m(f τ m 1 lm k E lm k E lm k E. m A ρk m ξ ρk σ N Z τ m(f ( τ m σ N ρ k Z τ m(f Z τ m(f E E1 s σn A s h(x s, Φ s Z τ m(f ds ρk τ m ρ k τ m 1 A s Φ s, dy s + τ m 1 σ N ρ k ( τ m σ N ρ k A s Φ s, dy s + τ m 1 σ N ρ k 1 s σn A s h(x s, Φ s Z τ m(f ds τ m ρ k τ m 1 ρ k τ m ρ k τ m 1 ρ k h(x s A s ξ s σn, dy s h(x s A s ξ s σn, dy s Y τ m 1 (15

7 Auhor's personal copy 32 M.A. Kourzn, H. Long / Sascs and Probably Leers 78 ( By condon (3 and he fac ha 1 s σn A s h(x s, Φ s σ {F X, s F Y, we fnd ha E 1 s σn A s h(x s, Φ s (M + τ m (f M s (f E 1 s σn A s h(x s, Φ s (M + τ m (f M + s (f + E 1 s σn A s h(x s, Φ s (M + s (f M s(f For any s τ m, 1 τ m, we have by (4 Z τ m(f f (X + s Lf (X u du + τ m f (X s + M + τ m (f M s (f + s a.e. (16 Lf (X u du + M + τ m (f τ m s Lf (X u du. Consequenly, by (16, he boundedness of Lf, Φ as well as condon (3, we have τ m { τ m lm E1 s σn A s h(x s, Φ s Z τ m(f ds lm E 1 s σn A m s h(x s, Φ s f (X s m 1 + E τ m 1 1 s σn A s h(x s, Φ s (M + τ m (f M s (f + τ m s 1 τ m 1 E1 s σn A s h(x s, Φ s Lf (X u du ds E1 s σn A s h(x s, Φ s f (X s ds. herefore, by (1 (17, follows ha (17 E M σn (f ξ σn E1 s σn A s h(x s, Φ s f (X s ds. (18 Now, by Io s somery propery, we noe ha E M σn (f ξ σn E1 s σn α f, Φ s s ds. (19 Combnng (18 and (19, we ge d d E M σn (f ξ σn E 1 σn α f, Φ E 1 σn h(x f (X A, Φ. By (3, follows ha Ēh(X f (X A F Y α f lm N 1 σ N α f exss for a.e. (, and lm E 1 σn h(x f (X A F Y N lm 1 σ N E h(x f (X A F Y N E h(x f (X A F Y, (2 a.s. for a.e. (, snce 1 σn s F Y -measurable. hus, we fnd ha α f p Y (hf a.s. for a.e. (,. Hence, by Fubn s heorem, we have ha α f p Y (hf 2 d a.s. for all > and p Y (f py (f + p Y s (Lf ds + p Y s (hf, dy s hs complees he proof. Acknowledgemens a.s. he auhors are graeful o he referee for very helpful commens. he frs auhor was suppored by an NSERC Dscovery Gran. he second auhor was suppored by FAU Sar-up fundng a he C. E. Schmd College of Scence.

8 Auhor's personal copy M.A. Kourzn, H. Long / Sascs and Probably Leers 78 ( Appendx In hs Appendx, we provde some echncal resuls whch have been used prevously. Lemma 6. Lf (X sds F X and {M (f, s an {F X -marngale.. Proof. I follows as n Remark of Karazas and Shreve (1988 ha Lf (X sds s ndsngushable from an {F X - progressvely measurable process and consequenly we can redefne Lf (X sds such ha Lf (X sds F X for all. hen, follows from Eher and Kurz (1986 p. 162 ha E (M τ (f M (f F X he lemma follows by nong G. a.s., τ. { G F X : E (M τ (f M (f 1 G s a monoone class. Lemma 7. {A, s ndsngushable from an almos surely connuous {F X,Y -progressvely measurable process. Proof. here exs R d -valued {F X -smple processes {U n wh non-random mes { n j sasfyng n < n 1 <, lm j n j such ha sup, Y n Y j n j 1 h(x s, dy s j: n j U n n j 1 n probably (e.g. Proposon , Remark , he developmen on p. 146, and he developmen of Proposon of Karazas and Shreve (1988. Now, we defne he {Y -cadlag marngales A n exp U n j 1, n Y n Y j n 1 j 1 2 U n 2 n ( n j n j 1 j 1. j: n j By Doob s nequaly, we have for any gven ε > ( P sup A n A ε E A n A /ε. Now, by Lemma 9 (o follow, A n A n probably and E A n E A 1 mply ha E A n A as n. herefore, ( lm P sup A n A ε, ε >. n hus, here s a subsequence n m such ha sup A n m A a.s. Moreover, {A n m, s {F X,Y -progressvely measurable. Now, we redefne A lm sup m A n m, {A, s connuous a.s. and {F X,Y -progressvely measurable. Le L Y M { connuous {F -marngales M on, wh sup,ω M (ω < and {. ( L M β s, dy s : βs {F Y -progressvely measurable and sup β (ω,ω β s, dy s (ω <. Lemma 8. L M s L2 -dense n L M. Proof. We le M β s, dy s be n L M and defne. βn s β s 1 βs n. By domnaed convergence, we fnd 2 E β s β n, s dy s E β s β n s 2 ds. Now, we defne λ n,m. nf { > : βn, s dy s > m and fnd ha λ n,m as m by connuy. hen, M n,m. λ n,m β n s, dy s L M and here s a sequence M n,m n ha converges o M n L 2.

9 Auhor's personal copy 322 M.A. Kourzn, H. Long / Sascs and Probably Leers 78 ( For he reader s convenence, we sae he followng basc resul (see heorem of Chung (1974 ha we have reled upon heavly. Lemma 9. Le < r <, V n L r and V n V n probably. hen he followng hree properes are equvalen: ( { V n r s unformly negrable; ( V n V n L r ; ( E( V n r E( V r. References Chung, K.L., A Course n Probably heory, 2nd ed. Academc Press, New York. Davs, M.H.A., Marcus, S.I., 198. An Inroducon o nonlnear flerng. In: Hazewnkel, M., Wllems, J.C. (Eds., Sochasc Sysem: he Mahemacs of Flerng and Idenfcaon and Applcaon. D. Redel Publshng Company, pp Dlevsen, P.D., 1999a. Observaon of α-sable nose nduced mllennal clmae changes from an ce-core record. Geophys. Res. Le. 26, Dlevsen, P.D., 1999b. Anomalous jumpng n a double-well poenal. Physcal Revew E 6, Dynkn, E.B., Markov Processes, vol. 1. Sprnger-Verlag, Berln. Eher, S.N., Kurz,.G., Markov Processes, Characerzaon and Convergence. Wley, New York. Fujsak, M., Kallanpur, G., Kuna, H, Sochasc dfferenal equaons for he nonlnear flerng problem. Osaka J. Mah. 9, Kallanpur, G., Karandkar, R.L., Whe nose calculus and nonlnear flerng heory. Ann. Probab. 13, Karazas, I., Shreve, S.E., Brownan Moon and Sochasc Calculus. Sprnger-Verlag, Berln. Kurz,.G., Ocone, D.L., Unque characerzaon of condonal dsrbuons n nonlnear flerng. Ann. Probab. 16, Meyer, P.A., Probably and Poenals. Blasdell, Walham, MA. Ocone, D., Remarks on he fne energy condon n addve whe nose flerng. Sysems Conrol Le. 5, Revuz, D., Yor, M., Connuous Marngales and Brownan Moon. Sprnger-Verlag, Berln. Szprglas, J., Sur l équvalence d équaons dfférenelle sochasques à valeurs mesures nervenan dans le flrage Markoven non lnéare. Ann. Ins. H. Poncaré Sec. B 14, Zaka, M., On he opmal flerng of dffuson processes. Z. Wahrsch. Verw. Gebee 11,