On Integral Equations Arising in the First-Passage Problem for Brownian Motion

Transcription

1 J. Inegral Equaions Al. Vol. 4, No. 4, 22, ( ) Research Reor No. 42, 2, De. Theore. Sais. Aarhus On Inegral Equaions Arising in he Firs-Passage Problem for Brownian Moion GORAN PESKIR 3 Le (B ) be a sandard Brownian moion sared a zero, le g : (; )! IR be a coninuous funcion saisfying g(+), le inf f > j B g() g be he firs-assage ime of B over g, and le F denoe he disribuion funcion of. Then he following sysem of inegral equaions is saisfied: n2 H n g() (s) n2 H n g()g(s) s for > and n ; ;..., where H n (x) R x H n(z) dz for n and H(x) '(x) ( 2) e x2 2 is he sandard normal densiy. These equaions are derived from a single maser equaion which may be viewed as a Chaman-Kolmogorov equaion of Volerra ye. The iniial idea in he derivaion of he maser equaion goes back o Schrödinger [23].. Inroducion Le (B ) be a sandard Brownian moion sared a zero, le g : (; )! IR be a coninuous funcion saisfying g(+), le (.) inf f > j B g() g be he firs-assage ime of B over g, and le F denoe he disribuion funcion of. The firs-assage roblem seeks o deermine F when g is given. The inverse firs-assage roblem seeks o deermine g when F is given. Boh he rocess B and he boundary g in hese formulaions may be more general, and our choice of Brownian moion is rimarily moivaed by he racabiliy of he exosiion. The facs o be resened below can be exended o more general Markov rocesses and boundaries (such as wo-sided ones) and he ime may also be discree. The firs-assage roblem has a long hisory and a large number of alicaions. Ye exlici soluions o he firs-assage roblem (for Brownian moion) are known only in a limied number of secial cases including linear or quadraic g. The law of is also known for a square-roo boundary g bu only in he form of a Lalace ransform (which aears inracable o inversion). The inverse roblem seems even harder. For examle, i is no known if here exiss a boundary g for which is exonenially disribued (cf. [2]). 3 Cenre for Mahemaical Physics and Sochasics (suored by he Danish Naional Research Foundaion). Mahemaics Subjec Classificaion 2. Primary 6J65, 45D5, 6J6. Secondary 45G5, 45G, 45Q5, 45K5. Key words and hrases: The (inverse) firs-assage roblem, Brownian moion, a curved (non-linear) boundary, a firs-assage ime, Markov rocess, he Chaman-Kolmogorov equaion, Volerra inegral equaions (of he firs and second kind), a sysem of nonlinear inegral equaions. goran@imf.au.dk

2 One way o ackle he roblem is o derive an equaion which links g and F. Moivaed by his fac many auhors have sudied inegral equaions in connecion wih he firs-assage roblem (see e.g. [23, 26,, 24, 9, 9, 22, 6]) under various hyoheses and levels of rigor. The main aim of his aer is o resen a unifying aroach o he inegral equaions arising in he firs-assage roblem ha is done in a rigorous fashion and wih minimal ools. The aroach naurally leads o a sysem of inegral equaions for g and F (Secion 6) in which he firs wo equaions conain he reviously known ones (Secion 5). These equaions are derived from a single maser equaion (Secion 3) ha can be viewed as a Chaman-Kolmogorov equaion of Volerra ye (Secion 2). The iniial idea in he derivaion of he maser equaion goes back o Schrödinger [23]. The maser equaion canno be reduced o a arial differenial equaion of forward or backward ye (cf. [4]). A key echnical deail needed o connec he second equaion of he sysem o known mehods leads o a simle roof of he fac ha F has a coninuous densiy when g is coninuously differeniable (Secion 4). The roblem of finding F when g is given is ackled using classic heory of linear inegral equaions (Secion 7). The inverse roblem is reduced o solving a sysem of nonlinear Volerra inegral equaions of he second kind (Secion 8). General heory of such sysems seems far from being comlee a resen. 2. Chaman-Kolmogorov equaions of Volerra ye I will be convenien o divide our discussion ino wo ars deending on if he ime se T of he Markov rocess (X ) 2T is eiher discree (finie or counable) or coninuous (uncounable). The sae sace S of he rocess may be assumed o be a subse of IR.. Discree ime (and sace). Recall ha (X n ) n is a (ime-homogeneous) Markov rocess if he following condiion is saisfied: (2.) E x (Y k j F k ) E Xk (Y ) for all (bounded) measurable Y and all k and x. ( Recall ha X x under P x, and ha X n k X n+k : ) Then he Chaman-Kolmogorov equaion (cf. [4, 4]) holds: (2.2) P x (X n z) X y P y (X nk z) P x (X k y) for x ; z in S and < k < n given and fixed, which is seen as follows: X (2.3) P x (X n z) X y X y X y y P x (X n z ; X k y) E x I(X k y) E x I(X nk z) k j F k I(X nk z) E x I(X k y) E Xk P x (X k y) P y (X nk z) uon using (2.) wih Y I (X nk z). 2

3 g z x k n Figure. A symbolic drawing of he Chaman-Kolmogorov equaion (2.2). The arrows indicae a ime evoluion of he samle ahs of he rocess. The verical line a k reresens he sae sace of he rocess. The equaions (2.) have a similar inerreaion. A geomeric inerreaion of he Chaman-Kolmogorov equaion (2.2) is shown in Figure (noe ha he verical line assing hrough k is given and fixed). Alhough for (2.2) we only considered he ime-homogeneous Markov roery (2.) for simliciy, i should be noed ha a more general Markov rocess creaes essenially he same icure. Imagine now on Figure ha he verical line assing hrough k begins o move coninuously and evenually ransforms ino a new curve sill searaing x from z as shown in Figure 2. The quesion hen arises naurally how he Chaman-Kolmogorov equaion (2.2) exends o his case. An eviden answer o his quesion is saed in Theorem 2.. This fac is hen exended o he case of coninuous ime and sace in Theorem 2.2 below. Theorem 2. Le (X n ) n be a Markov rocess (aking values in a counable se S ), le x and z be given and fixed in S, le g : IN! S be a funcion searaing x and z relaive o X (i.e. if X x and X n z for some n, hen here exiss k n such ha X k g(k) ), and le (2.4) inf f k j X k g(k) g be he firs-assage ime of X (2.5) P x (X n z) k nx PX n z j X k g(k) P x ( k). over g. Then he following sum equaion holds: 3

4 z g x k n Figure 2. A symbolic drawing of he inegral equaion (2.5)-(2.6). The arrows indicae a ime evoluion of he samle ahs of he rocess. The verical line a k has been ransformed ino a ime-deenden boundary g. The equaions (2.6)-(2.7) have a similar inerreaion. Moreover, if he Markov rocess X (2.6) P x (X n X z) n P g(k)(x nk z) P x ( k). is ime-homogeneous, hen (2.5) reads as follows: k Proof. Since g searaes x and z relaive o X, we have: (2.7) P x (X n X z) n P x (X n z ; k). k On he oher hand, by he Markov roery: (2.8) P x (X n z j F k ) P Xk (X n z) and he fac ha f kg 2 F k, we easily find: (2.9) P x (X n z ; k) P X n z j X k g(k) P x ( k). Insering his ino (2.7) we obain (2.5). The ime-homogeneous simlificaion (2.6) follows hen immediaely, and he roof is comlee. The equaions (2.5) and (2.6) exend o he case when he sae sace S is uncounable. In his case he relaion z in (2.5) and (2.6) can be relaced by he relaion 2 G where G is any measurable se ha is searaed from he iniial oin x relaive o X in he sense described above. The exensions of (2.5) and (2.6) obained in his way will be omied. 4

5 2. Coninuous ime (and sace). A assage from he discree o he coninuous case inroduces some echnical comlicaions (regular condiional robabiliies) which we se aside in he sequel. Recall ha (X ) is a Markov rocess if he following condiion is saisfied: (2.) P (X 2 G j F s ) P (X 2 G j X s ) for all measurable G (2.) P ( ; x; 3 ; G) and all s <. Then he Chaman-Kolmogorov equaion (cf. [4, 4]) holds: S P ( 2 ; y; 3 ; G) P ( ; x; 2 ; dy) where P ( i ; x; j ; G) P (X j 2 G j X i x) and < 2 < 3 are given and fixed. Kolmogorov [4] calls (2.) he fundamenal equaion, noes ha (under a desired Markovian inerreaion) i is saisfied if he sae sace S is finie or counable ( he oal robabiliy law ), and in he case when S is uncounable akes i as a new axiom. If X j under X i x has a densiy funcion f saisfying (2.2) P ( i ; x; j ; G) for all measurable (2.3) f ( ; x; 3 ; z) f ( i ; x; j ; z) dz G G, hen he equaions (2.) reduce o: f ( ; x; 2 ; y) f ( 2 ; y; 3 ; z) dy S for x and z in S and < 2 < 3 given and fixed. Kolmogorov [5] saes ha his inegral equaion was sudied by Smoluchowski [25], recalls ha he roved in [4] ha under some addiional condiions f saisfies cerain differenial equaions of arabolic ye (he forward and he backward equaion), and in a foonoe acknowledges ha hese differenial equaions were inroduced by Fokker [] and Planck [2] indeendenly of he Smoluchowski inegral equaion. (The Smoluchowski inegral equaion [25] is a ime homogeneous version of (2.3). The Bachelier-Einsein equaion (cf. [2, 8]): (2.4) f ( +s; z) f (; z x) f (s; x) dx S is a sace-ime homogeneous version of he Smoluchowski equaion.) Wihou going ino furher deails on hese facs, we will only noe ha he inerreaion of he Chaman-Kolmogorov equaion (2.2) described above by means of Figure carries over o he general case of he equaion (2.), and he same is rue for he quesion raised above by means of Figure 2. The following heorem exends he resul of Theorem 2. on his maer. Theorem 2.2 (cf. Schrödinger [23] and Fore [,. 27]) Le (X ) be a srong Markov rocess wih coninuous samle ahs sared a x, le be a coninuous funcion saisfying g(+) x, le g : (; )! IR (2.5) inf f > j X g() g be he firs-assage ime of X over g, and le F Fx denoe he disribuion funcion of. 5

6 Then he following inegral equaion holds: (2.6) P x (X 2 G) P X 2 G j X s g(s) for each measurable se G conained in [g(); ). Moreover, if he Markov rocess X is ime-homogeneous, hen (2.6) reads as follows: (2.7) P x (X 2 G) P g(s) Xs 2 G. for each measurable se G conained in [g(); ). Proof. The key argumen in he roof is o aly a srong Markov roery a ime. This can be done informally (wih G [g(); ) given and fixed) as follows: (2.8) P x (X 2 G) P x (X 2 G ; ) E x I ( ) E x E x I(X 2 G) j s which is (2.6). In he las ideniy above we used ha for s (2.9) E x I (X 2 G) j s P X 2 G j X s g(s) I (X 2 G) j P X 2 G j X s g(s) we have: which formally requires a recise argumen. This is wha we do in he res of he roof. For his, recall ha ( ) is a srong Markov rocess if he following condiion is saisfied: (2.2) E z (Y j F ) E (Y ) for all (bounded) measurable Y and all soing imes. I urns ou, however, ha he rocess has o be chosen carefully. This is due o he fac ha he ime on he lef-hand side of (2.9) is deerminisic. For examle, by aking Y f (X ) on f g we fail o achieve Y f (X ), since X X + whenever. We hus choose (; X ) and define: (2.2) inf f > j 2 C g (2.22) inf f > j 2 C [ D g where C f (s; y) j < s < ; y < g(s) g and D f (s; y) j < s < ; y g(s) g, so ha C [ D f (s; y) j < s < g. Thus under P (;x) i.e. P x, and moreover + since boh and are hiing imes of he rocess o closed (oen) ses, he second se being conained in he firs one, so ha. Seing F (s; y) G (y) and Y F ( ), we hus see ha Y F ( ) F ( + ) F ( ) Y, which by means of (2.2) imlies ha (2.23) E z (F ( ) j F ) E (F ( )). 6

7 In he secial case z (; x) his reads: (2.24) E (;x) I (X 2 G) j F E (;g()) I (X 2 G) where F on he lef-hand side can be relaced by since he righ-hand side defines a measurable funcion of. I follows hen immediaely from such modified (2.24) ha (2.25) E (;x) I (X 2 G) j s E (s;g(s)) I (X 2 G) and since ^ we see ha (2.25) imlies (2.9) for s. Thus he final se in (2.8) is jusified and herefore (2.6) is roved as well. The ime-homogeneous simlificaion (2.7) is a direc consequence of (2.6), and he roof of he heorem is comlee. The roof of Theorem 2.2 jus resened is no he only ossible one. The roof of Theorem 3.2 given below can easily be ransformed ino a roof of Theorem 2.2. Ye anoher quick roof can be given by alying he srong Markov roery of he rocess (; X ) o esablish (2.24) (mulilied by I( ) on boh sides) wih ^ on he lef-hand side and on he righ-hand side. The righ-hand side hen easily ransforms o he righ-hand side of (2.6) hus roving he laer. In order o examine he scoe of he equaions (2.6) in a clearer manner, we will leave he realm of a general Markov rocess in he sequel, and consider he case of a sandard Brownian moion insead. The facs and mehodology resened below exend o he case of more general Markov rocesses (or boundaries) alhough some of he formulas may be less exlici. 3. The maser equaion The following noaion will be used hroughou: (3.) '(x) 2 ex2 2, 8(x) x '(z) dz, 9(x) 8(x) for x 2 IR. We begin his secion by recalling he resul of Theorem 2.2. Thus, le g : (; )! IR be a coninuous funcion saisfying g(+), and le F denoe he disribuion funcion of from (2.5). If secialized o he case of sandard Brownain moion (B ) sared a zero, he equaion (2.7) wih G [g(); ) reads as follows: (3.2) 9 g() 9 g()g(s) s where he scaling roery B B of B is used, as well as ha (z +B ) defines a sandard Brownian moion sared a z whenever z 2 IR.. Derivaion. I urns ou ha he equaion (3.2) is jus one in he sequence of equaions ha can be derived from a single maser equaion. This maser equaion can be obained by aking G [z; ) in (2.7) wih z g(). We now resen ye anoher roof of his derivaion. 7

8 Theorem 3. (The Maser Equaion) Le (B ) be a sandard Brownian moion sared a zero, le g : (; )! IR be a coninuous funcion saisfying g(+), le (3.3) inf f > j B g() g be he firs-assage ime of B over g, and le F denoe he disribuion funcion of. Then he following inegral equaion holds: (3.4) 9 z for all z g() where >. 9 zg(s) s Proof. We will make use of he srong Markov roery of he rocess (; B ) a ime. This makes he resen argumen close o he argumen used in he roof of Theorem 2.2. For each > le z() from [g(); ) be given and fixed. Seing f (; x) I(x z()) R and Y e s f ( s ) ds by he srong Markov roery ( of he rocess ) given in (2.2) wih, and he scaling roery of B, we find: (3.5) e P B z() d E E E E E e f ( ) d j F E e E Y j F e f ( ) d E e E (Y ) F (d) e ( +s) f ( +s ) ds j F e E (;g()) e s f ( s ) ds e e s P g()+b s z(+s) ds F (d) e e 9 z(+s)g() s ds F (d) s e e 9 z(r)g() r (r) dr F (d) r r e 9 z(r)g() F (d) dr r for all >. By he uniqueness heorem for Lalace ransform i follows ha (3.6) P B z() 9 z()g(s) s which is seen equivalen o (3.4) by he scaling roery of B. The roof is comlee. 8

9 2. Consan and linear boundaries. I will be shown in Secion 4 ha when g is C on (; ) hen here exiss a coninuous densiy f F of. The equaion (3.2) hen becomes: (3.7) 9 g() 9 g()g(s) s f (s) ds for >. This is a linear Volerra inegral equaion of he firs kind in f if g is known (i is a nonlinear equaion in g if f is known). Is kernel (3.8) K(; s) 9 g()g(s) s is nonsingular in he sense ha he maing (s; ) 7! K(; s) for s < is bounded. If g() c wih c 2 IR, hen (3.2) or (3.7) reads as follows: (3.9) P ( ) 2P (B c) and his is he reflecion rincile of André [], Bachelier [2, 964,. 64] and Lévy [6,. 293]. If g() + wih 2 IR and >, hen (3.7) reads as follows: (3.) 9 g() 9 s f (s) ds where we see ha he kernel K(; s) is a funcion of he difference s and hus of a convoluion ye. Sandard Lalace ransform echniques herefore can be alied o solve he equaion (3.) yielding he following exlici formula: (3.) f () 32 ' + which is he well-known resul of Doob [5,. 397] and Malmquis [7,. 526]. Closed form exressions for f in he case of more general boundaries g will be reaed using classic heory of inegral equaions in Secion 7 below. 3. Numerical calculaion. The fac ha he kernel (3.8) of he equaion (3.7) is nonsingular in he sense exlained above makes his equaion esecially aracive o numerical calculaions of f if g is given. This can be done using he simle idea of Volerra (daing back o 896). Seing j jh for j ; ;... ; n where h n and n is given and fixed, we see ha he following aroximaion of he equaion (3.7) is valid (when g is C for insance): (3.2) nx j K(; j ) f ( j ) h b() where we se b() 9(g() ). In aricular, alying his o each i (3.3) ix j for i ; 2;... ; n. Seing: K( i ; j ) f ( j ) h b( i ) yields: 9

10 (3.4) a ij 2K( i ; j ), x j f ( j ), b i 2b( i )h we see ha he sysem (3.3) reads as follows: (3.5) ix a ij x j b i ( i ; 2;... ; n ) j he simliciy of which is obvious (cf. [9]). We conjecure ha his sysem consiues an efficien mehod for numerical comuaion of f when g is given. Some examles of his comuaion are resened in [28] where references o oher numerical mehods can be found as well. 4. Remarks. I follows from (3.) ha for in (3.3) wih g() + we have: (3.6) P ( < ) e 2 whenever and >. This shows ha F in (3.4) does no have o be a roer disribuion funcion bu generally saisfies F (+) 2 (; ]. On he oher hand, recall ha Blumenhal s - law imlies ha P ( ) is eiher or for in (3.3) and a coninuous funcion g : (; )! IR. If P ( ) hen g is said o be an uer funcion for B, and if P ( ) hen g is said o be a lower funcion for B. Kolmogorov s es (see e.g. [3, ]) gives sufficien condiions on g o be an uer or lower funcion. I follows by Kolmogorov s es ha 2 log log is a lower funcion for B, and (2+") log log is an uer funcion for B for every " >. 4. The exisence of a coninuous firs-assage densiy The equaion (3.7) is a Volerra inegral equaion of he firs kind. These equaions are generally known o be difficul o deal wih direcly, and here are wo sandard ways of reducing hem o Volerra inegral equaion of he second kind. The firs mehod consiss of differeniaing boh sides in (3.7) wih resec o, and he second mehod (Theorem 7.) makes use of an inegraion by ars in (3.7) (see e.g. [2,. 4-4]). Our focus in his secion is on he firs mehod. Being led by his objecive we now resen a simle roof of he fac ha F is C when g is C (comare he argumens given below wih hose given in [27,. 323] or [9,. 322]). Theorem 4. Le (B ) be a sandard Brownian moion sared a zero, le g : (; )! IR be an uer funcion for B, and le in (3.3) be he firs-assage ime of B over g. If g is coninuously differeniable on (; ) hen has a coninuous densiy f. Moreover, he following ideniy is saisfied: (4.) for all 9 2 f () 9 f (s) ds s Proof.. Seing G() 9(g() ) and K(; s) 9((g()g(s)) s) for s <

11 we see ha (3.2) ( i.e. (3.4) wih z g() ) reads as follows: (4.2) G() K(; s) for all >. Noe ha K(; ) () 2 for every > since (g()g(s)) s! as s " for g ha is C on (; ). Noe also K(; s) s 2 g()g(s) g () s ' g()g(s) s for < s <. Hence we see ha (@K@)(; ) is no finie ( whenever g () 6 ), and we hus roceed as follows. 2. Using (4.2) we find by Fubini s heorem ha (4.4) lim "# 2 2" lim K( 2 ; s) "# G( 2 )G( ) 2 F (2 )F ( ) K(; s) d K( ; s) 2" " K(s+"; s) for < 2 <. On he oher hand, we see from (4.3) ha (4.5) for all 2 [ ; 2 ] " K(; s) s and " >, while by Fubini s heorem i is easily verified ha (4.6) s d <. We may hus by he dominaed convergence heorem (alied wice) inerchange he firs limi and he firs inegral in (4.4) yielding: (4.7) 2 a leas for hose 2 [ ; 2 ] for K(; s) d G( 2 )G( ) 2 s <. F (2 )F ( ) I follows from (4.6) ha he se of all > for which (4.8) fails is of Lebesgue measure zero. 3. To verify (4.8) for all > we may noe ha a sandard rule on he differeniaion under an inegral sign can be alied in (3.4), and his yields he following equaion: (4.9) ' z s ' zg(s) s

12 for all z > g() wih > uon differeniaing in (3.4) wih resec o z. By Faou s lemma hence we ge: (4.) s ' g()g(s) s lim inf z#g() lim inf z#g() s ' zg(s) s s ' zg(s) s ' g() < for all >. Now for s < close o we know ha '((g()g(s)) s) in (4.) is close o 2 >, and his easily esablishes (4.8) for all >. R 4. Reurning back o (4.7) i is easily seen using (4.3) ha 7! (@K@)(; s) is righ-coninuous a 2 ( ; 2 ) if we have: n (4.) n s! for n # as n!. To check (4.) we firs noe ha by assing o he limi for z # g() in (4.9), and using (4.8) wih he dominaed convergence heorem, we obain (5.) below for all >. Noing ha (s; ) 7! '((g()g(s)) s) aains is sricly osiive minimum c > over < 2 and s <, we may wrie: n (4.2) n n s c n s ' g(n c n ' g(n ) n )g(s) n s n s ' g(n )g(s) n s where he final exression ends o zero as n! by means of (5.) below R and using (4.8) wih he dominaed convergence heorem. Thus (4.) holds and herefore 7! (@K@)(; s) is righ-coninuous. I can be similarly verified ha his maing is lef-coninuous a each 2 ( ; 2 ) and hus coninuous on (; ). 5. Dividing finally by 2 in (4.7) and hen leing 2!, we obain: (4.3) F () 2 K(; s) for all >. Since he righ-hand side of (4.3) defines a coninuous funcion of >, i follows ha f F is coninuous on (; ), and he roof is comlee. 5. Derivaion of known equaions In he revious roof we saw ha he maser equaion (3.4) can be once differeniaed wih resec o z imlying he equaion (4.9), and ha in (4.9) one can ass o he limi for z # g() obaining he following equaion: (5.) for all >. ' g() g()g(s) s ' s F(ds) 2

13 The urose of his secion is o show how he equaions (4.) and (5.) yield some known equaions sudied reviously by a number of auhors.. We assume hroughou ha he hyoheses of Theorem 4. are fulfilled (and ha > is given and fixed). Rewriing (4.) more exlicily by comuing derivaives on boh sides gives: (5.2) g() 2 g () 32 ' g() 2 f () + Recognizing now he ideniy (5.) mulilied by g () ar of he ideniy (5.2) by 2, we ge: (5.3) g() g() 32 ' f () + g()g(s) 2 g()g(s) (s) 32 (s) 32 g () s ' g()g(s) s f (s) ds. wihin (5.2), and mulilying he remaining g()g(s) ' f (s) ds. s This equaion has been derived and sudied by Ricciardi e al. [22] using oher means. Moreover, he same argumen shows ha he facor 2 can be removed from (5.2) yielding: g() (5.4) g () ' g() g()g(s) 32 f () + g () ' g()g(s) (s) 32 f (s) ds. s s This equaion has been derived indeendenly by Ferebee [9] and Durbin [6]. Ferebee s derivaion is, se aside echnical oins, he same as he one resened here. Williams [7] resens ye anoher derivaion of his equaion (assuming ha f exiss). [Mulilying boh sides of (3.7) by 2r() and boh sides of (5.) by 2(k()+g ()), and adding he resuling wo equaions o he equaion (5.3), we obain he equaion (2.9)+(3.4) in Buonocore e al. [3] derived by oher means.] 2. Wih a view o he inverse roblem (of finding g if f is given) i is of ineres o roduce as many non-equivalen equaions linking g o f as ossible. (Recall ha (3.7) is a nonlinar equaion in g if f is known, and nonlinear equaions are marked by a non-uniqueness of soluions.) For his reason i is eming o derive addiional equaions o he one given in (5.) saring wih he maser equaion (3.4) and roceeding similarly o (4.9) above. A sandard rule on he differeniaion under an inegral sign can be inducively alied o (3.4), and his gives he following equaions: (5.5) n2 '(n) z zg(s) (s) n2 '(n) s for all z > g() and all n where >. Recall ha (5.6) ' (n) (x) () n h n (x) '(x) for x 2 IR and n where h n is a Hermie olynomial of degree n for n. Noing ha ' (x) x'(x) and recalling (5.3) we see ha a assage o he limi for z # g() in (5.5) is no sraighforward when n 2 bu comlicaed. For his reason we will no ursue i in furher deail here. 3. The Chaman-Kolmogorov equaion (2.) is known o admi a reducion o he forward and backward equaion (cf. [4]) which are arial differenial equaions of arabolic ye. No such 3

14 derivaion or reducion is generally ossible in he enire-as deenden case of he equaion (2.6) or (2.7), and he same is rue for he maser equaion (3.4) in aricular. We showed above how he differeniaion wih resec o z in he maser equaion (3.4) leads o he densiy equaion (5.), which ogeher wih he disribuion equaion (3.2) yields known equaions (5.3) and (5.4). I was also indicaed above ha no furher derivaive wih resec o z can be aken in he maser equaion (3.4) so ha he assage o he limi for z # g() in he resuling equaion becomes sraighforward. 6. Derivaion of new equaions. Exanding on he revious facs a bi furher we now noe ha i is ossible o roceed in a reverse order and inegrae he maser equaion (3.4) wih resec o z as many imes as we lease. This yields he whole secrum of new non-equivalen equaions, which aken ogeher wih (3.2) and (5.), may lay a fundamenal role in he inverse roblem (Secion 8). Theorem 6. Le (B ) be a sandard Brownian moion sared a zero, le g : (; )! IR be a coninuous funcion saisfying g(+), le in (3.3) be he firs-assage ime of B over g, and le F denoe he disribuion funcion of. Then he following sysem of inegral equaions is saisfied: (6.) n2 H n g() for > and n ; ;..., where we se: (6.2) H n (x) x (s) n2 H n g()g(s) H n(z) dz wih H ' being he sandard normal densiy from (3.). s Remark. For n he equaion (6.) is he densiy equaion (5.). For n he equaion (6.) is he disribuion equaion (3.2). All equaions in (6.) for n 6 are nonsingular (in he sense ha heir kernels are bounded over he se of all (s; ) saisfying s < T ). Proof. Le > be given and fixed. Inegraing (3.4) we ge: z (6.3) 9 dz z dz z 9 z g(s) s for all z g() by means of Fubini s heorem. Subsiuing u z and v (z g(s)) s we can rewrie (6.3) as follows: (6.4) z 9(u) du which is he same as he following ideniy: (6.5) H z s (zg(s)) s s H zg(s) s 4 9(v) dv

15 for all z g() uon using ha H is defined by (6.2) above wih n. Inegraing (6.5) as (3.4) rior o (6.3) above, and roceeding similarly by inducion, we ge: z (6.6) n2 H n zg(s) (s) n2 H n s for all z g() and all n. (This equaion was also esablished earlier for n in (3.4) and for n in (4.9).) Seing z g() in (6.6) above we obain (6.) for all n. (Using ha 9(x) 2 '(x) for all x > i is easily verified by inducion ha all inegrals aearing in (6.)-(6.6) are finie.) As he equaion (6.) was also roved earlier for n in (3.2) and for n in (5.) above, we see ha he sysem (6.) holds for all n, and he roof of he heorem is comlee. 2. In view of our consideraions in Subsecion of Secion 5 above i is ineresing o esablish he analogues of he equaions (5.3) and (5.4) in he case of oher equaions in (6.). For his, fix n and > in he sequel, and noe ha aking a derivaive wih resec o in (6.) gives: n g() g() g (6.7) + n2 () H n g() 2 n2 H n n 2 (s)n2 H n + (s) n2 H n g()g(s) 2 32 s! g () s g()g(s) F s 2(s) 32 (ds). g()g(s) Recognizing now he ideniy (6.) ( wih n insead of n using ha H H n n ) mulilied by g () wihin (6.7), and mulilying he remaining ar of he ideniy (6.7) by 2, we ge:! g() (6.8) n2 nh n g() g() H n (s) n2 g()g(s) nh n g()g(s) s s Moreover, he same argumen shows ha he facor 2! n (6.9) n2 H g() n g() g () g() H 32 n (s) n2 n (s) H n g()g(s) s! g()g(s) H n. s can be removed from (6.7) yielding:! g()g(s) g () g()g(s) H (s) 32 n. s s 5

16 Each of he equaions (6.8) and (6.9) is conained in he sysem (6.). No equaion of he sysem (6.) is equivalen o anoher equaion from he same sysem bu iself. 7. A closed exression for he firs-assage disribuion In his secion we briefly ackle he roblem of finding F when g is given using classic heory of linear inegral equaions (see e.g. [2]). The key ool in his aroach is he fixed-oin heorem for conracive maings, which saes ha a maing T : X! X, where (X; d) is a comlee meric sace, saisfying: (7.) d(t (x); T (y)) d(x; y) for all x; y 2 X wih some 2 (; ) has a unique fixed oin in X, i.e. here exiss a unique oin x 2 X such ha T (x ) x. Using his rincile and some of is ramificaions develoed wihin he heory of inegral equaions, he aers [8] and [22] resen exlici exressions for F in erms of g in he case when X is aken o be a Hilber sace L 2. These resuls will here be comlemened by describing a narrow class of boundaries g ha allow X o be he Banach sace B(IR + ) of all bounded funcions h : IR +! IR equied wih he su-norm: (7.2) khk sujh()j. While examles from his class range from a consan o a square-roo boundary, he aroach iself is marked by simliciy of he argumen. Theorem 7. Le (B ) be a sandard Brownian moion sared a zero, le g : IR +! IR be a coninuous funcion saisfying g() >, le in (3.3) be he firs-assage ime of B over g, and le F denoe he disribuion funcion of. Assume, moreover, ha g is C on (; ), increasing, concave, and ha i saisfies: (7.3) g() g() + c for all wih some c >. Then we have: X (7.4) F () h() + n K n(; s) h(s) ds where he series converges uniformly over all, and we se: (7.5) h() 29 g() (7.6) K(; s) s! ' 2 g (s) g()g(s) g()g(s) (s) s 6

17 (7.7) K n+ (; s) for s < and n. Moreover, inroducing he funcion: (7.8) R(; s) s X n K (; r)k n (r; s) dr K n (; s) for s <, he following reresenaion is valid: (7.9) F () h() + for all >. R(; s) h(s) ds (g()g(s)) s and v F (s) Proof. Seing u 9 using he inegraion by ars formula, we obain: g() (7.) 9 2 F 9 s in he inegral equaion (3.2) and F (s) ds for each > ha is given and fixed in he sequel. Using he noaion of (7.5) and (7.6) above we can rewrie (7.) as follows: (7.) F () K (; s) F (s) ds h(). Inroduce a maing T on B(IR + ) by seing: (7.2) (T (G))() h() + for G 2 B(IR + ). Then (7.) reads as follows: (7.3) T (F ) F K (; s) G(s) ds and he roblem reduces o solve (7.3) for F in B(IR + ). In view of he fixed-oin heorem quoed above, we need o verify ha T from B(IR + ) ino iself wih resec o he su-norm (7.2). For his, noe: (7.4) k T (G )T (G 2 ) k su su (T (G G 2 ))() K (; s) G (s)g 2 (s) ds su is a conracion K (; s) ds kg G 2 k. Since s! g(s) is concave and increasing, i is easily verified ha s 7! (g()g(s)) s is decreasing and hus s 7! 9(g()g(s)) s is increasing on (; ). I imlies ha (7.5) : su K (; s) ds 9 g()g(s) s ds 7

18 su 9 ds su 2 g()g() s (c) < using he hyohesis (7.3). This shows ha T is a conracion from he Banach sace B(IR + ) ino iself, and hus by he fixed-oin heorem here exiss a unique F in B(IR + ) saisfying (7.3). Since he disribuion funcion F of belongs o B(IR + ) and saisfies (7.3), i follows ha F mus be equal o F. Moreover, he reresenaion (7.4) follows from (7.) and he well-known formula for he resolven of he inegral oeraor K T h associaed wih he kernel K : (7.6) I K P n K n uon using Fubini s heorem o jusify ha K n+ in (7.7) is he kernel of he inegral oeraor K n+ for n. Likewise, he final claim abou (7.8) and (7.9) follows by he Fubini-Tonelli heorem since all kernels in (7.6) and (7.7) are non-negaive, and so are all mas s 7! K n (; s) h(s) in (7.4) as well. This comlees he roof. Leaving aside he quesion on usefulness of he mulile-inegral series reresenaion (7.4), i is an ineresing mahemaical quesion o find a similar exression for F in erms of g ha would no require addiional hyoheses on g such as (7.3) for insance. In his regard esecially hose g saisfying g(+) seem roblemaic as hey lead o singular (or weakly singular) kernels generaing he inegral oeraors ha urn ou o be non-conracive. 8. The inverse roblem In his secion we will reformulae he inverse roblem of finding g when F is given using he resul of Theorem 6.. Recall from here ha g and F solve: (8.) n2 H n g() (s) n2 H n g()g(s) s R for > and n where H n (x) H x n(z) dz wih H '. Then he inverse roblem reduces o answer he following hree quesions: Quesion 8.. Does here exis a (coninuous) soluion 7! g() of he sysem (8.)? Quesion 8.2. Is his soluion unique? Quesion 8.3. Does he (unique) soluion 7! g() solve he inverse firs-assage roblem i.e. is he disribuion funcion of from (3.3) equal o F? I may be noed ha each equaion in g of he sysem (8.) is a nonlinear Volerra inegral equaion of he second kind. Nonlinear equaions are known o lead o non-unique soluions, so i is hoed ha he oaliy of counably many equaions could counerbalance his deficiency. Perhas he main examle one should have in mind is when F has a coninuous densiy f. Noe ha in his case f (+) can be sricly osiive (and finie). Some informaion on ossible behaviour of g a zero for such f can be found in [2]. 8

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