Explicit construction of a dynamic Bessel bridge of dimension 3

Transcription

1 Explici consrucion of a dynamic Bessel bridge of dimension 3 Luciano Campi Umu Çein Albina Danilova February 25, 23 Absrac Given a deerminisically ime-changed Brownian moion Z saring from, whose imechange V ) saisfies V ) > for all >, we perform an explici consrucion of a process X which is Brownian moion in is own filraion and ha his zero for he firs ime a V τ), where τ := inf{ > : Z = }. We also provide he semimaringale decomposiion of X under he filraion joinly generaed by X and Z. Our consrucion relies on a combinaion of enlargemen of filraion and filering echniques. The resuling process X may be viewed as he analogue of a 3-dimensional Bessel bridge saring from a ime and ending a a he random ime V τ). We call his a dynamic Bessel bridge since V τ) is no known in advance. Our sudy is moivaed by insider rading models wih defaul risk, where he insider observes he firm s value coninuously on ime. The financial applicaion, which uses resuls proved in he presen paper, has been developed in he companion paper [6]. Inroducion In his paper, we are ineresed in consrucing a Brownian moion saring from a ime = and condiioned o hi he level for he firs ime a a given random ime. More precisely, le Z be he deerminisically ime-changed Brownian moion Z = + σs)dw s and le B be anoher sandard Brownian moion independen of W. We denoe V ) he associaed ime-change, i.e. V ) = σ2 s)ds for. Consider he firs hiing ime of Z of he level, denoed by τ. Our aim is o build explicily a process X of he form dx = db + α d, X =, where α is an inegrable and adaped process for he filraion joinly generaed by he pair Z, B) and saisfying he following wo properies:. X his level for he firs ime a ime V τ); 2. X is a Brownian moion in is own filraion. This resuling process X can be viewed as an analogue of 3-dimensional Bessel bridge wih a random erminal ime. Indeed, he wo properies above characerising X can be reformulaed as follows: X is a Brownian moion condiioned o hi for he firs ime a he random ime V τ). In order o emphasise he disinc propery ha V τ) is no known a ime, we call his process a dynamic Bessel bridge of dimension 3. The reason ha X his a V τ) raher han τ is simply due o he LAGA, Universiy Paris 3, and CREST, campi@mah.univ-paris3.fr. Deparmen of Saisics, London School of Economics, u.cein@lse.ac.uk. Deparmen of Mahemaics, London School of Economics, a.danilova@lse.ac.uk.

2 relaionship beween he firs hiing imes of by Z and a sandard Brownian moion saring a. The soluion o he above problem consiss of wo pars wih varying difficulies. The easy par is he consrucion of his process afer ime τ. Since V is a deerminisic funcion, he firs hiing ime of is revealed a ime τ. Thus, one can use he well-known relaionship beween he 3-dimensional Bessel bridge and Brownian moion condiioned on is firs hiing ime o wrie for τ, V τ)) { } X dx = db + d. X V τ) The difficul par is he consrucion of X unil ime τ. Thus, he challenge is o consruc a Brownian moion which is condiioned o say sricly posiive unil ime τ using a drif erm adaped o he filraion generaed by B and Z. Our sudy is moivaed by he equilibrium model wih insider rading and defaul as in [5], where a Kyle-Back ype model wih defaul is considered. In such a model, hree agens ac in he marke of a defaulable bond issued by a firm, whose value process is modelled as a Brownian moion and whose defaul ime is se o be he firs ime ha he firm s value his a given consan defaul barrier. I has been shown in [5] ha he equilibrium oal demand for such a bond, afer an appropriae ranslaion, is a process X which is a 3-dimensional Bessel bridge in insider s enlarged) filraion bu is a Brownian moion in is own filraion. These wo properies can be rephrased as follows: X is a Brownian moion condiioned o hi for he firs ime a he defaul ime τ. However, he assumpion ha he insider knows he defaul ime from he beginning may seem oo srong from he modelling viewpoin. To approach he realiy, one migh consider a more realisic siuaion when he insider doesn know he defaul ime bu however she can observe he evoluion hrough ime of he firm s value. Equilibrium consideraions, akin o he ones employed in [5], lead one o sudy he exisence of processes which we called dynamic Bessel bridges of dimension 3 a he beginning of his inroducion. The financial applicaion announced here has been performed in he companion paper [6], where he ools developed in he presen paper are used o solve expliciely he equilibrium model wih defaul risk and dynamic insider informaion, as oulined above. We refer o ha paper for furher deails. We will observe in he nex secion ha in order o make such a consrucion possible, one has o assume ha Z evolves faser han is underlying Brownian moion W, i.e. V ) for all. I can be proved see nex Secion 2) ha V ) canno be equal o in any inerval a, b) of R +. We will neverheless impose a sronger assumpion ha V ) > for all > in order o avoid unnecessary echnicaliies. In he conex of he financial marke described above his assumpions amouns o insider s informaion being more precise han ha of he marke maker see [] for a discussion of his assumpion). Moreover, an addiional assumpion on he behaviour of he ime change V ) in a neighbourhood of will be needed. Apar from he financial applicaion, which is our firs moivaion, such a problem is ineresing from a probabilisic poin of view as well. We have observed above ha he difficul par in obaining he dynamic Bessel bridge is he consrucion of a Brownian moion which is condiioned o say sricly posiive unil ime τ using a drif erm adaped o he filraion generaed by B and Z. Such a consrucion is relaed o he condiioning of a Markov process, which has been he opic of various works in he lieraure. The canonical example of his phenomenon is he 3-dimensional Bessel process which is obained when one condiions a sandard Brownian moion o say posiive. Chaumon [8] sudies he analogous problem for Lévy process whereas Beroin and 2

3 Doney [2] are concerned wih he siuaion for random walks and he convergence of heir respecive probabiliy laws. Beroin e al. [3] consrucs a Brownian pah over a fixed ime inerval wih a given minimum by performing ransformaions on a Brownian bridge. More recenly, Chaumon and Doney [9] revisis he Lévy processes condiioned o say posiive and shows a Williams ype pah decomposiion resul a he minimum of such processes. However, none of hese approaches can be adoped o perform he consrucion ha we are afer since i) he ime inerval in which we condiion he Brownian moion o be posiive is random and no known in advance; and ii) we are no allowed o use ransformaions ha are no adaped o he filraion generaed by B and Z. The paper is srucured as follows. In Secion 2, we formulae precisely our main resul Theorem 2.) and provide a parial jusificaion for is assumpions. Secion 3 conains he proof of Theorem 2., ha uses, in paricular, a echnical resul on he densiy of he signal process Z, whose proof is given in Secion 4. Finally, several echnical resuls used along our proofs have been relegaed in he Appendix for reader s convenience. 2 Formulaion of he main resul Le Ω, H, H = H ), P) be a filered probabiliy space saisfying he usual condiions. We suppose ha H conains only he P-null ses and here exis wo independen H-Brownian moions, B and W. We inroduce he process Z := + for some σ whose properies are given in he assumpion below. σs)dw s, 2.) Assumpion 2. There exis a measurable funcion σ : R +, ) such ha:. V ) := σ2 s)ds, ) for every > ; 2. There exiss some ε > such ha ε V ) ) 2 d <. Noice ha under his assumpions, Z and W generae he same minimal filraion saisfying he usual condiions. Consider he following firs hiing ime of Z: τ := inf{ > : Z = }, 2.2) where inf = by convenion. One can characerize he disribuion of τ using he well-known disribuions of firs hiing imes of a sandard Brownian moion. To his end le for a > where H, a) := P [T a > ] = lu, a) du, 2.3) T a := inf{ > : B = a}, and ) a l, a) := exp a2. 2π 3 2 3

4 Recall ha P[T a > H s ] = [Ta>s]H s, a B s ), s <. Thus, since V is deerminisic and sricly increasing, Z V )) is a sandard Brownian moion in is own filraion saring a, and consequenly Hence, P[τ > H s ] = [τ>s] HV ) V s), Z s ). 2.4) P[V τ) > ] = H, ), for every, i.e. V τ) = T in disribuion. Here we would like o give anoher formulaion for he funcion H in erms of he ransiion densiy of a Brownian moion killed a. Recall ha his ransiion densiy is given by q, x, y) := 2π exp ) )) x y)2 x + y)2 exp, 2.5) 2 2 for x > and y > see Exercise.5), Chaper III in [7]). Then one has he ideniy H, a) = q, a, y) dy. 2.6) In he sequel, for any process Y, F Y is going o denoe he minimal filraion saisfying he usual condiions and wih respec o which Y is adaped. The following is he main resul of his paper. Theorem 2. There exiss a unique srong soluion o Moreover, τ q x V s) s, X s, Z s ) V τ) X = + B + qv s) s, X s, Z s ) ds + l a V τ) s, X s ) ds. 2.7) τ lv τ) s, X s ) i) Le F X = N σx s ; s ), where N is he se of P-null ses. Then, X is a sandard Brownian moion wih respec o F X := F X ) ; ii) V τ) = inf{ > : X = }. The proof of his resul is posponed o he nex secion. We conclude his secion by providing a jusificaion for our assumpion V ) > for all >. Firs, observe ha we necessarily have V ) for any. This follows from he fac ha if he consrucion in Theorem 2. is possible, hen V τ) is an F B,Z -sopping ime since i is an exi ime from he posiive real line of he process X. Indeed, if V ) < for some > so ha V ) >, hen [V τ) < ] canno belong o F B,Z since [V τ) < ] [τ > ] = [τ < V )] [τ > ] / F Z, and ha τ is no F -measurable. B We will nex see ha when V ) consrucion of a dynamic Bessel bridge is no possible. Similar argumens will also show ha V ) canno be equal o in an inerval. We are going o adap o our seing he argumens used in [], Proposiion 5.. 4

5 To his end consider any process X = +B + α sds for some H-adaped and inegrable process α. Assume ha X is a Brownian moion in is own filraion an ha τ = inf{ : X = } a.s. and fix an arbirary ime. The wo processes M Z s := P[τ > F Z s ] and M X s := P[τ > F X s ], for s, are uniformly inegrable coninuous maringales, he former for he filraion F Z,B and he laer for he filraion F X. In his case, Doob s opional sampling heorem can be applied o any pair of finie sopping imes, e.g. τ s and τ, o ge he following: M X τ s = E[M X τ F X τ s] = E[ τ> F X τ s] = E[M Z τ F X τ s] = E[M Z τ s F X τ s], where he las equaliy is jus an applicaion of he ower propery of condiional expecaions and he fac ha M Z is maringale for he filraion F Z,B which is bigger han F X. We also obain E[M X τ s M Z τ s) 2 ] = E[M X τ s) 2 ] + E[M Z τ s) 2 ] 2E[M X τ sm Z τ s]. Noice ha, since he pairs X, τ) and Z, τ) have he same law by assumpion, he random variables M X τ s and M Z τ s have he same law oo. This implies On he oher hand we can obain E[M X τ s M Z τ s) 2 ] = 2E[M X τ s) 2 ] 2E[M X τ sm Z τ s]. E[M X τ sm Z τ s] = E[M X τ se[m Z τ s F X τ s]] = E[M X τ s) 2 ], which implies ha M X τ s = M Z τ s for all s. Using he fac ha M Z s = τ>s H s, Z s ), M X s = τ>s H s, X s ), s <, one has H s, X s ) = H s, Z s ) on [τ > s]. Since he funcion a Hu, a) is sricly monoone in a whenever u >, he las equaliy above implies ha X s = Z s for all s < on he se [τ > s]. being arbirary, we have ha ha X τ s = Z τ s for all s. We have jus proved ha, before τ, X and Z coincide, which conradics he fac ha B and Z are independen, so ha he consrucion of a Brownian moion condiioned o hi for he firs ime a τ is impossible. A possible way ou is o assume ha he signal process Z evolves faser han is underlying Brownian moion W, i.e. V ), ) for all as in our assumpions on σ. We prove our main resul in he following secion. 3 Proof of he main resul Noe firs ha in order o show he exisence and he uniqueness of he srong soluion o he SDE in 2.7) i suffices o show hese properies for he following SDE τ q x V s) s, Y s, Z s ) Y = y + B + ds, y >, 3.8) qv s) s, Y s, Z s ) 5

6 and ha Y τ >. Indeed, he drif erm afer τ is he same as ha of a 3-dimensional Bessel bridge from X τ o over he inerval [τ, V τ)]. Noe ha V τ) = T in disribuion implies ha τ has he same law as V T ) which is finie since T is finie and he funcion V ) is increasing o infiniy as ends o infiniy. Thus τ is a.s. finie. By Corollary in [4] he exisence and uniqueness of he srong soluion of 3.8) is equivalen o he exisence of a weak soluion and pahwise uniqueness of srong soluion when he laer exiss. More precisely, afer proving pahwise uniqueness for he SDE 3.8), and hus esablishing he uniqueness of he sysem of 2.) and 3.8), in Lemma 3., we will consruc a weak soluion, Y, Z), o his sysem. The weak exisence and pahwise uniqueness will hen imply Y, Z) = h, y, β, W ) for some measurable h and some Brownian moion β in view of Corollary in [4]. Moreover, he second par of Corollary in [4] will finally give us h, y, B, W ) as he srong soluion of he sysem described by 2.) and 3.8). In he sequel we will ofen work wih a pair of SDEs defining A, Z) where A is a semimaringale given by an SDE whose drif coefficien depends on Z. In order o simplify he saemens of he following resuls, we will shorly wrie exisence and/or uniqueness of he SDE for A, when we acually mean he corresponding propery for he whole sysem. We sar wih demonsraing he pahwise uniqueness propery. Lemma 3. Pahwise uniqueness holds for he SDE in 3.8). Proof. I follows from direc calculaions ha q x, x, z) q, x, z) = z x + exp 2xz ) exp 2xz ) 2z. 3.9) Moreover, qx,x,z) q,x,z) Y and Y 2. Then Y τ Y 2 τ ) 2 = 2 is decreasing in x for fixed z and. Now, suppose here exis wo srong soluions, τ { Ys Ys 2 qx V s) s, Ys, Z s ) ) qv s) s, Ys, Z s ) q xv s) s, Ys 2 }, Z s ) qv s) s, Ys 2 ds., Z s ) The exisence of a weak soluion will be obained in several seps. Firs we show he exisence of a weak soluion o he SDE in he following proposiion and hen conclude via Girsanov s heorem. Proposiion 3. There exiss a unique srong soluion o Y = y + B + τ where f, x, z) := exp 2xz ) exp 2xz ) 2z. Moreover, P[Y τ > and Y τ >, > ] =. fv s) s, Y s, Z s ) ds y >, 3.) Proof. Pahwise uniqueness can be shown as in Lemma 3.; hus, is proof is omied. Observe ha if Y is a soluion o 3.), hen dy 2 = 2Y db + 2 [τ>] Y fv ), Y, Z ) + ) d. 6

7 Inspired by his formulaion we consider he following SDE: du = 2 U db + 2 [τ>] U fv ), ) U, Z ) + d, 3.) wih U = y 2. In Lemma 3.2 i is shown ha here exiss a weak soluion o his SDE which is sricly posiive in he inerval [, τ]. This yields in paricular ha he absolue values can be removed from he SDE 3.) considered over he inerval [, τ]. Thus, i follows from an applicaion of Iô s formula ha U is a weak, herefore srong, soluion o 3.) in [, τ] due o pahwise uniqueness and Corollary in [4]. The global soluion can now be easily consruced by he addiion of B B τ afer τ. This furher implies ha Y is sricly posiive in [, τ] since U is clearly sricly posiive. Lemma 3.2 There exiss a weak soluion o du = 2 U db + 2 U fv ), ) U, Z ) + d, 3.2) wih U = y 2 upo and including τ. Moreover, he soluion is sricly posiive in [, τ]. Proof. Consider he measurable funcion g : R + R 2 [, ] defined by x f, x, z), for, x, z), ) R R+ g, x, z) =, for, x, z), ) R, ),, for, x, z) {} R 2 and he following SDE: dũ = 2 Ũ db + ) 2gV ), Ũ, Z ) + d. 3.3) Observe ha if we can show he exisence of a posiive weak soluion o 3.3), hen U = Ũ τ ) is a posiive weak soluion o 3.2) upo ime τ. I follows from Corollary..2 and Theorem 6..7 in [9] ha he maringale problem defined by he sochasic differenial equaions for Ũ, Z) wih he sae space R2 is well-posed upo an explosion ime, i.e. here exiss a weak soluion o 3.3), along wih 2.), valid upo he explosion ime by Theorem 5.4. in [4]. Fix one of hese soluions and call i Ũ, Z). Then, since he range of g is [, 3], i follows from Lemma A. ha Ũ is nonnegaive and here is no explosion. Nex i remains o show he sric posiiviy of U in [, τ]. Firs, le a and b be sricly posiive numbers such ha As xe x e x ime ae a e a = 3 4 and be b e b = 2. is sricly decreasing for posiive values of x, one has < a < b. Now define he sopping I := inf{ < τ : U Z V ) a}, 2 where inf = τ by convenion. As U τ Z τ =, U Z = y 2, and V ) > for >, we have ha < I < τ, ν y -a.s. by coninuiy of U, Z) and V, where ν y is he probabiliy measure associaed o he fixed weak soluion. Moreover, U > on he se [ I ]. 7

8 Noe ha C := 2 U Z V ) is coninuous on, ) and C I = a. Thus, τ := inf{ > I : C = } > I. Consider he following sequence of sopping imes: J n := inf{i n τ : C /, b)} I n+ := inf{j n τ : C = a} for n N {}, where inf = τ by convenion. Our aim is o show ha τ = τ = lim n J n, a.s.. We sar wih esablishing he second equaliy. As J n s are increasing and bounded by τ, he limi exiss and is bounded by τ. Suppose ha J := lim n J n < τ wih posiive probabiliy. Noe ha by consrucion we have I n J n I n+ and, herefore, lim n I n = J. Since C is coninuous, one has lim n C In = lim n C Jn. However, as on he se [J < τ] we have C In = a and C Jn = b for all n, we arrive a a conradicion. Therefore, τ = J. Nex, we will demonsrae ha τ = τ. Observe ha since τ is finie, a.s., and U does no explode unil τ, one has ha C τ =. Therefore, τ τ and hus C τ =. Suppose ha τ < τ wih posiive probabiliy. Then, we claim ha on his se C Jn = b for all n, which will lead o a conradicion since hen b = lim n C Jn = C τ =. We will show our claim by inducion.. For n =, recall ha I < τ by consrucion. Also noe ha on I, J ] he drif erm in 3.2) is greaer han 2 as xe x is sricly decreasing for posiive values of x and due o he e x choice of a and b. Therefore he soluion o 3.2) is sricly posiive in I, J ] in view of Lemma A.2 since a 2-dimensional Bessel process is always sricly posiive. Thus, C J = b. 2. Suppose we have C Jn = b. Then, due o coninuiy of C, I n < τ. For he same reasons as before, he soluion o 3.2) is sricly posiive in I n, J n ]. Thus, C Jn = b. Thus, we have shown ha for all >, U τ >, a.s.. In order o show ha U τ > consider he sopping ime I := sup{i n : I n < τ}. Then, we mus have ha I < τ a.s. since oherwise a = C I = C τ =, anoher conradicion. Similar o he earlier cases he drif erm in I, τ] is larger han 2, hus, U τ > as well. Proposiion 3.2 There exiss a unique srong soluion o 3.8) which is sricly posiive on [, τ]. Proof. Due o Proposiion 3. here exiss a unique srong soluion, Y, of 3.). Define L ) by L = and Y Z dl = [τ>] L V ) db. Observe ha here exiss a soluion o he above equaion since ) Ys Z 2 s [τ>s] ds <, a.s.. V s) s Indeed, since Y and Z are well-defined and coninuous upo τ, we have sup s τ Y s Z s <, a.s. and hus he above expression is finie in view of Assumpion If L ) is a rue maringale, hen for any T >, Q T on H T defined by dq T dp T = L T, 8

9 where P T is he resricion of P o H T, is a probabiliy measure on H T equivalen o P T. Then, by Girsanov Theorem see, e.g., Theorem 3.5. in [4]) under Q T τ Y = y + β T q x V s) s, Y s, Z s ) + qv s) s, Y s, Z s ) ds, for T where β T is a Q T -Brownian moion. Thus, Y is a weak soluion o 3.8) on [, T ]. Therefore, due o Lemma 3. and Corollary in [4], here exiss a unique srong soluion o 3.8) on [, T ], and i is sricly posiive on [, τ] since Y has his propery. Since T is arbirary, his yields a unique srong soluion on [, ) which is sricly posiive on [, τ]. Thus, i remains o show ha L is a rue maringale. Fix T > and for some n < n T consider [ n τ ) Y Z 2 E exp d)]. 3.4) 2 n τ V ) As boh Y and Z are posiive unil τ, Y Z ) 2 Y 2 + Z 2 R + Z 2 by comparison where R saisfies R = y Rs db s + 3. Therefore, since R and Z are independen, he expression in 3.4) is bounded by [ )] [ n )] n E exp E exp 2 2 E [ exp 2 R T n R υ)d n n υ)d )] E [ exp n Z 2 υ)d 2 Z T ) 2 n n υ)d )], 3.5) where Y := sup s Y s for any càdlàg process Y and υ) := 2. V ) ) Recall ha Z is only a ime-changed Brownian moion where he ime change is deerminisic and R is he square of he Euclidian norm of a 3-dimensional sandard Brownian moion wih iniial value y 2,, ). Thus, since V T ) > T, he above expression is going o be finie if )] E [exp y n 2 β V T ) )2 υ)d <, 3.6) n where β is a sandard Brownian moion and E x is he expecaion wih respec o he law of a sandard Brownian moion saring a x. Indeed, i is clear ha, by ime change, 3.6) implies ha he second expecaion in he RHS of 3.5) is finie. Moreover, since RT is he supremum over [, T ] of a 3-dimensional Bessel square process, i can be bounded above by he sum of hree supremums of squared Brownian moions over [, V T )] remember ha V T ) > T ), which gives ha 3.6) is an upper bound for he firs expecaion in he RHS of 3.5) as well. In view of he reflecion principle for sandard Brownian moion see, e.g. Proposiion 3.7 in Chap. 3 of [7]) he above expecaion is going o be finie if n n υ)d < V T ). 3.7) 9

10 However, Assumpion 2. yields ha T υ)d <. Therefore, we can find a finie sequence of real numbers = < <... < nt ) = T ha saisfy 3.7). Since T was arbirary, his means ha we can find a sequence n ) n wih lim n n = such ha 3.4) is finie for all n. Then, i follows from Corollary in [4] ha L is a maringale. The above proposiion esablishes as a lower bound o he soluion of 3.8) over he inerval [, τ], however, one can obain a igher bound. Indeed, observe ha qx q, x, z) is sricly increasing in z on [, ) for fixed, x) R Moreover, q x q x, x, ) := lim q z q, x, z) = x x. Therefore, qx q V ), Y, Z ) > qx q V ), Y, ) = Y Y V ) for, τ]. Alhough qx q, x, z) is no Lipschiz in x hus, sandard comparison resuls don apply), if Y < Z hen he comparison resul of Exercise in [4] can be applied o obain P[Y R ; < τ] = where R is given by3.8). However, his sric inequaliy may break down a = when Y Z, and, hus, rendering he resuls of Exercise is inapplicable. Neverheless, we will show in Proposiion 3.4 ha P[Y R ; < τ] = where R is he soluion of { } R s R = y + B + ds, y >. 3.8) R s V s) s Before proving he comparison resul we firs esablish ha here exiss a unique srong soluion o he SDE above and i equals in law o a scaled, ime-changed 3-dimensional Bessel process. We incidenally observe ha he exisence of a weak soluion o an SDE similar o ha in 3.8) is proved in Proposiion 5. in [7] along wih is disribuional properies. Unforunaely, our SDE 3.8) canno be reduced o heirs and moreover, in our seing, exisence of a weak soluion is no enough. Proposiion 3.3 There exiss a unique srong soluion o 3.8). Moreover, he law of R is equal o he law of R = R ), where R = λ ρ Λ where ρ is a 3-dimensional Bessel process saring a y and ) λ := exp V s) s ds, Λ := λ 2 s ds. Proof. Noe ha x x is decreasing in x and, hus, pahwise uniqueness holds for 3.8). Thus, i suffices o find a weak soluion for he exisence and he uniqueness of srong soluion. Consider he 3-dimensional Bessel process ρ which is he unique srong soluion see Proposiion 3.3 in Chap. VI in [7]) o ρ = y + B + ρ s ds. Therefore, ρ Λ = y + B Λ + Λ ρ s ds. Now, M = B Λ is a maringale wih respec o he imechanged filraion H Λ ) wih quadraic variaion given by Λ. By inegraion by pars we see ha { dλ ρ Λ ) = λ dm + λ } ρ Λ d. λ ρ Λ V )

11 Since λ ρ Λ = y and λ2 sd[m, M] s =, we see ha λ ρ Λ is a weak soluion o 3.8). This obviously implies he equivalence in law. Proposiion 3.4 Le R be he unique srong soluion o 3.8). Then, P[Y R ; < τ] = where Y is he unique srong soluion of 3.8). Proof. Noe ha R Y = { qx q V s) s, R s, ) q } x q V s) s, Y s, Z s ) ds, so ha by Tanaka s formula see Theorem.2 in Chap. VI of [7]) { R Y ) + qx = [Rs>Ys] q V s) s, R s, ) q } x q V s) s, Y s, Z s ) ds { qx = [Rs>Ys] q V s) s, R s, ) q } x q V s) s, Y s, ) ds { qx + [Rs>Ys] q V s) s, Y s, ) q } x q V s) s, Y s, Z s ) ds { qx [Rs>Ys] q V s) s, R s, ) q } x q V s) s, Y s, ) ds, since he local ime of R Y a is idenically see Corollary.9 n Chap. VI of [7]). Le τ n := inf{ > : R Y = n }. Noe ha as R is sricly posiive and Y is sricly posiive on [, τ], lim n τ n > τ. Since for each q x q, x, ) q x, y, ) q + ) n 2 x y for all x, y [/n, ), we have R τn Y τn ) + R s τn Y s τn ) + V s) s + ) n 2 ds. Thus, by Gronwall s inequaliy see Exercise 4 in Chap. V of [8]), we have R τn Y τn ) + = since V s) s + ) n 2 ds < by Assumpion 2.. Thus, he claim follows from he coninuiy of Y and R and he fac ha lim n τ n > τ. Remark 3. Noe ha he above proof does no use he paricular SDE saisfied by Z. The resul of he above proposiion will remain valid as long as Z is nonnegaive and Y is he unique srong soluion of 3.8), sricly posiive on [, τ]. Since he soluion o 3.8) is sricly posiive on [, τ] and he drif erm in 2.7) afer τ is he same as ha of a 3-dimensional Bessel bridge from X τ o over [τ, V τ)], we have proved

12 Proposiion 3.5 There exiss a unique srong soluion o 2.7). Moreover, he soluion is sricly posiive in [, τ]. Using he well-known properies of a 3-dimensional Bessel bridge see, e.g., Secion 2..3, in paricular expression 2.9) in [2]), we also have he following Corollary 3. Le X be he unique srong soluion of 2.7). Then, V τ) = inf{ > : X = }. Thus, in order o finish he proof of Theorem 2. i remains o show ha X is a sandard Brownian moion in is own filraion. We will achieve his resul in several seps. Firs, we will obain he canonical decomposiion of X wih respec o he minimal filraion, G, saisfying he usual condiions such ha X is G-adaped and τ is a G-sopping ime. More precisely, G = G ) where G = u> Gu, wih G := N σ{x s, s }, τ ) and N being he se of P-null ses. Then, we will iniially enlarge his filraion wih τ o show ha he canonical decomposiion of X in his filraion is he same as ha of a Brownian moion saring a in is own filraion enlarged wih is firs hiing ime of. This observaion will allow us o conclude ha he law of X is he law of a Brownian moion. In order o carry ou his procedure we will use he following key resul, he proof of which is deferred unil he nex secion for he clariy of he exposiion. We recall ha H, a) = q, a, y)dy, where q, a, y) is he ransiion densiy of a Brownian moion killed a. Proposiion 3.6 Le X be he unique srong soluion of 2.7) and f : R + R be a bounded measurable funcion wih a compac suppor conained in, ). Then E[ [τ>] fz ) G ] = [τ>] fz) qv ), X, z) HV ), X ) dz. Using he above proposiion we can easily obain he G-canonical decomposiion of X. Corollary 3.2 Le X be he unique srong soluion of 2.7). Then, τ H x V s) s, X s ) V τ) M := X HV s) s, X s ) ds l a V τ) s, X s ) τ lv τ) s, X s ) ds is a sandard G-Brownian moion saring a. Proof. I follows from Theorem 8..5 in [3] and Lemma A.4 ha X [ ] q x V s) s, X s, Z s ) E [τ>s] qv s) s, X s, Z s ) G s V τ) l a V τ) s, X s ) ds τ lv τ) s, X s ) ds 2

13 is a G-Brownian moion. However, [ ] q x V s) s, X s, Z s ) E [τ>s] qv s) s, X s, Z s ) G s q x V s) s, X s, z) qv s) s, X s, z) = [τ>s] qv s) s, X s, z) HV s) s, X s ) dz = [τ>s] q x V s) s, X s, z) dz HV s) s, X s ) = [τ>s] qv s) s, x, z) dz HV s) s, X s ) x = [τ>s] H x V s) s, X s ) HV s) s, X s ). x=xs A naive way o show ha X as a soluion of 2.7) is a Brownian moion is o calculae he condiional disribuion of τ given he minimal filraion generaed by X saisfying he usual condiions. Alhough, as we will see laer, he condiional disribuion of V τ) given an observaion of X is defined by he funcion H as defined in 2.3), verificaion of his fac leads o a highly non-sandard filering problem. For his reason we use an alernaive approach which uilizes he well-known decomposiion of Brownian moion condiioned on is firs hiing ime as in [5]. We shall nex find he canonical decomposiion of X under G τ := G τ ) where G τ = G στ). Noe ha G τ = F X + στ). Therefore, he canonical decomposiion of X under G τ would be is canonical decomposiion wih respec o is own filraion iniially enlarged wih τ. As we shall see in he nex proposiion i will be he same as he canonical decomposiion of a Brownian moion in is own filraion iniially enlarged wih is firs hiing ime of. Proposiion 3.7 Le X be he unique srong soluion of 2.7). Then, is a sandard G τ -Brownian moion saring a. V τ) l a V τ) s, X s ) X lv τ) s, X s ) ds Proof. Firs, we will deermine he law of τ condiional on G for each. Le f be a es 3

14 funcion. Then E [ [ ] [τ>] fτ) G = E E [ ] ] [τ>] fτ) H G ] = E [ [τ>] fu)σ 2 u)lv u) V ), Z ) du G = [τ>] fu)σ 2 u) lv u) V ), z) qv ), X, z) dz du HV ), X ) = [τ>] fu)σ 2 u) H V u) V ), z) qv ), X, z) dz du HV ), X ) = [τ>] fu)σ 2 u) qs, z, y) dy qv ), X, z) s HV ), X ) dz du s=v u) V ) = [τ>] fu)σ 2 u) s = [τ>] fu)σ 2 u) s = [τ>] = [τ>] qv ), X, z) HV ), X ) qv ) + s, X, y) HV ), X ) fu)σ 2 u) H V u), X ) HV ), X ) du fu)σ 2 u) lv u), X ) HV ), X ) du. Thus, P[τ du, τ > G ] = [τ>] σ 2 lv u),x) u) HV ),X du. ) Then, i follows from Theorem.6 in [6] ha τ la V τ) s, X s ) M lv τ) s, X s ) H ) xv s) s, X s ) HV s) s, X s ) qs, z, y) dz dy du s=v u) V ) dy du s=v u) V ) is a G τ -Brownian moion as in Example.6 in [6]. This complees he proof. Corollary 3.3 Le X be he unique srong soluion of 2.7). Then, X is a Brownian moion wih respec o F X. Proof. I follows from Proposiion 3.7 ha G τ - decomposiion of X is given by V τ) { } X s X = + µ + ds, X s V τ) s where µ is a sandard G τ -Brownian moion vanishing a. Thus, X is a 3-dimensional Bessel bridge from o of lengh V τ). As V τ) is he firs hiing ime of for X and V τ) = T in disribuion, he resul follows using he same argumen as in Theorem 3.6 in [5]. Nex secion is devoed o he proof of Proposiion Condiional densiy of Z Recall from Proposiion 3.6 ha we are ineresed in he condiional disribuion of Z on he se [τ > ]. To his end we inroduce he following change of measure on H. Le P be he resricion ds 4

15 of P o H and define P τ, on H by dp τ, dp = [τ>] P[τ > ]. Noe ha his measure change is equivalen o an h-ransform on he pahs of Z unil ime where he h-ransform is defined by he funcion HV ) V ), ) and H is he funcion defined in 2.3) see Par 2, Sec. VI.3 of [] for he definiion and properies of h-ransforms). Noe also ha [τ>s] HV ) V s), Z s )) s [,] is a P, H)-maringale as a consequence of 2.4). Therefore, an applicaion of Girsanov s heorem yields ha under P τ, X, Z) saisfy dz s = σs)dβs + σ 2 s) H xv ) V s), Z s ) ds HV ) V s), Z s ) 4.9) dx s = db s + q xv s) s, X s, Z s ) ds, qv s) s, X s, Z s ) 4.2) wih X = Z = and β being a P τ, -Brownian moion. Moreover, due o he propery of h- ransforms, ransiion densiy of Z under P τ, is given by P τ, HV ) V s), z) [Z s dz Z r = x] = qv s) V r), x, z) HV ) V r), x). 4.2) Thus, P τ, [Z s dz Z r = x] = pv ); V r), V s), x, z) where H s, z) p; r, s, x, z) = qs r, x, z) H r, x). 4.22) Noe ha p is he ransiion densiy of he Brownian moion killed a afer he analogous h- ransform where he h-funcion is given by H s, x). Lemma 4. Le Fs τ,,x = σx r ; r s) N τ, where X is he process defined by 4.2) wih X =, and N τ, is he collecion of P τ, -null ses. Then he filraion Fs τ,,x ) s [,] is righ-coninuous. The proof of he above lemma is rivial once we observe ha F τ,,x τ n s ) s [,], where τ n := inf{s > : X s = n }, is righ coninuous. This follows from he observaion ha Xτn is a Brownian moion under an equivalen probabiliy measure, which can be shown using he argumens of Proposiion 3.2 along wih he ideniy 3.9) and he fac ha X is bounded upo τ n. Thus, for each n one has F τ,,x τ n Fu τ,,x = Fτ τ,,x n u = Fτ τ,,x n s s>u = s>u F τ,,x τ n Fs τ,,x ) = Fs τ,,x s>u ) F τ,,x τ n Indeed, since n F τ,,x τ n = Fτ τ,,x, leing n end o infiniy yields he conclusion. The reason for he inroducion of he probabiliy measure P τ, and he filraion Fs τ,,x ) s [,] is ha P τ,, Fs τ,,x ) s [,] )-condiional disribuion of Z can be characerised by a Kushner-Sraonovich equaion which is well-defined. Moreover, i gives us P, G)-condiional disribuion of Z. Indeed, observe ha P τ, [τ > ] = and for any se E G, [τ>] E = [τ>] F for some se F F τ,,x 5 see

16 Lemma 5.. in [4] and he remarks ha follow). Then, i follows from he definiion of condiional expecaion ha E [ ] [ fz ) [τ>] G = [τ>] E τ, fz ) ] F τ,,x, P a.s ) Thus, i is enough o compue he condiional disribuion of Z under P τ, wih respec o Fs τ,,x ) s [,]. In order o achieve his goal we will use he characerizaion of he condiional disribuions obained by Kurz and Ocone [5]. We refer he reader o [5] for all unexplained deails and erminology. Le P be he se of probabiliy measures on he Borel ses of R + opologized by weak convergence. Given m P and m inegrable f we wrie mf := R fz)mdz). The nex resul is direc consequence of Lemma. and subsequen remarks in [5]: Lemma 4.2 There is a P-valued F τ,,x -opional process π ω, dx) such ha π sf = E τ, [fz s ) F τ,,x s ] for all bounded measurable f. Moreover, π s) s [,] has a càdlàg version. Le s recall he innovaion process I s = X s s π rκ r dr qxv r) r,xr,z) where κ r z) := qv r) r,x r,z). Alhough i is clear ha I depends on, we don emphasize i in he noaion for convenience. Due o Lemma A.4 πsκ s exiss for all s. In order o be able o use he resuls of [5] we firs need o esablish he Kushner-Sraonovich equaion saisfied by πs) s [,). To his end, le BA) denoe he se of bounded Borel measurable real valued funcions on A, where A will be alernaively a measurable subse of R 2 + or a measurable subse of R +. Consider he operaor A : B[, ] R + ) B[, ] R + ) defined by A φs, x) = φ s s, x) + 2 σ2 s) 2 φ x 2 s, x) + σ2 s) H x H V ) V s), x) φs, x), 4.24) x wih he domain DA ) = Cc [, ] R + ), where Cc is he class of infiniely differeniable funcions wih compac suppor. By Lemma A.3 he maringale problem for A is well-posed over he ime inerval [, ε] for any ε >. Therefore, i is well-posed on [, ) and is unique soluion is given by s, Z s ) s [,) where Z is defined by 4.9). Moreover, he Kushner-Sraonovich equaion for he condiional disribuion of Z is given by he following: π sf = π f + s π ra f)dr + s [ π r κ r f) π rκ r π rf ] di r, 4.25) for all f Cc R + )see Theorem in [3] and noe ha he condiion herein is saisfied due o Lemma A.4). Noe ha f can be easily made an elemen of DA ) by redefining i as fn where n Cc R + ) is such ha ns) = for all s [, ). Thus, he above expression is rigorous. The following heorem is a corollary o Theorem 4. in [5]. Theorem 4. Le m be an F τ,,x -adaped càdlàg P-valued process such ha m sf = π f + s m ra f)dr + s [ m r κ r f) m rκ r m rf ] di m r, 4.26) for all f C c R + ), where I m s = X s s m rκ r dr. Then, m s = π s for all s <, a.s.. 6

17 Proof. Proof follows along he same lines as he proof of Theorem 4. in [5], even hough, differenly from [5], we allow he drif of X o depend on s and X s, oo. This is due o he fac ha [5] used he assumpion ha he drif depends only on he signal process, Z, in order o ensure ha he join maringale problem X, Z) is well-posed, i.e. condiions of Proposiion 2.2 in [5] are saisfied. Noe ha he relevan maringale problem is well posed in our case by Proposiion A.. Now, we can sae and prove he following corollary. Corollary 4. Le f BR + ). Then, πsf = fz)pv ); s, V s), X s, z) dz, R + for s < where p is as defined in 4.22). Proof. Le ρ; s, x, z) := pv ); s, V s), x, z). Direc compuaions lead o ρ s + H xv ) s, x) HV ) s, x) ρ x + 2 ρ xx 4.27) ) = σ 2 Hx V ) V s), z) s) HV ) V s), z) ρ + 2 σ2 s)ρ zz. Define m P by m sf := R + fz)ρ; s, X s, z)dz. Then, using he above pde and Io s formula one can direcly verify ha m solves 4.26). Finally, Theorem 4. gives he saemen of he corollary. Now, we have all necessary resuls o prove Proposiion 3.6. Proof of Proposiion 3.6. Noe ha as X is coninuous, F τ,,x = s< F s τ,,x. Fix r < and le E Fr τ,,x. We will show ha for any f Cc R + ) [ E τ, [fz ) E ] = E τ, fz) qv ), X ], z) R + HV ), X ) dz E. Since Z is coninuous and f is bounded we have E τ, [fz ) E ] = lim s E τ, [fz s ) E ]. As s will evenually be larger han r, E Fs τ,,x for large enough s and, hen, Corollary 4. and anoher applicaion of he Dominaed Convergence Theorem will yield [ ] lim E τ, [fz s ) E ] = lim E τ, fz)pv ); V s) s, X s, z) dz E s s R + = E τ, [lim s z R + fz)pv ); V s) s, X s, z) dz E Since X is sricly posiive unil τ by Proposiion 3.5, min s X s >. This yields ha HV ) s,x s) is bounded ω-by-ω) for s. Moreover, qv s) s, X s, ) is bounded by 2πV s) s). Thus, in view of 4.22), pv ); V s) s, X s, z) Kω) V s) s HV ) V s), z), ]. 7

18 where K is a consan. Since V s) s) can be bounded when s is away from, H is bounded by, and f has a compac suppor, i follows from he Dominaed Convergence Theorem ha lim fz)pv ); V s) s, X s, z) dz = fz) qv ), X, z) s R + R + HV ), X ) dz, Pτ, a.s.. This in urn shows, E τ, [fz ) E ] = E τ, [lim s fz s ) E ] = E τ, [ fz) qv ), X ], z) R + HV ), X ) dz E. The claim now follows from 4.23). References [] Back, K., and H. Pedersen 998): Long-lived informaion and inraday paerns. Journal of Financial Marke,, [2] Beroin, J. and Doney, R. A. 994): On condiioning a random walk o say nonnegaive. Ann. Probab., 224), pp [3] Beroin, J., Piman, J. and Ruiz de Chavez, J. 999): Consrucions of a Brownian pah wih a given minimum. Elecron. Comm. Probab., 4, pp [4] Bielecki, T. and Rukowski, M. 22): Credi Risk: Modeling, Valuaion and Hedging, Springer-Verlag: New York. [5] Campi, L. and Çein, U. 27): Insider rading in an equilibrium model wih defaul: a passage from reduced-form o srucural modelling. Finance and Sochasics, 4), [6] Campi, L., Çein, U., and Danilova, A. 2): Equilibrium model wih defaul and insider s dynamic informaion. Finance and Sochasics, forhcoming. [7] Carr, P. and Linesky, V. 26): A jump o defaul exended CEV model: an applicaion of Bessel processes. Finance and Sochasics, 3), pp [8] Chaumon, L. 996): Condiionings and pah decomposiions for Lévy processes. Sochasic Process. Appl., 64), pp [9] Chaumon, L. and Doney, R. A. 25): On Lévy processes condiioned o say posiive. Elecron. J. Probab.,, pp [] Doob, J.L. 984): Classical poenial heory and is probabilisic counerpar, Springer. [] Föllmer, H., Wu, C.-T., and M. Yor 999): Canonical decomposiion of linear ransformaions of wo independen Brownian moions moivaed by models of insider rading. Sochasic Processes and heir Applicaions 84, [2] Kallenberg, O. 22): Foundaions of Modern Probabiliy 2nd Ediion), Springer-Verlag. 8

19 [3] Kallianpur, G. 98): Sochasic Filering Theory, Springer-Verlag. [4] Karazas, I., and S. E. Shreve 99): Brownian Moion and Sochasic Calculus 2nd Ediion), Springer. [5] Kurz, T. G., and D. L. Ocone 988): Unique characerizaion of condiional disribuions in nonlinear filering. The Annals of Probabiliy, 8), pp [6] Mansuy, R. and M. Yor 26): Random Times and Enlargemens of Filraions in a Brownian Seing, Springer-Verlag. [7] Revuz, D., and M. Yor 999): Coninuous Maringales and Brownian Moion 3rd Revised Ediion), Springer-Verlag. [8] Proer, P. 25): Sochasic inegraion and differenial equaions Second ediion, Version 2., Correced 3rd prining), Springer-Verlag. [9] Sroock, D.W. and Varadhan, S.R.S. 997): Mulidimensional Diffusions Processes, Springer. [2] Yor, M. 997): Some Aspecs of Brownian Moion: Some recen Maringale problems. Vol. 2, Lecures in Mahemaics ETH Zrich. Birkhauser Verlag, Basel. A Auxiliary resuls and heir proofs A. Comparison resuls Lemma A. Suppose ha d : R + R 2 + [, M] for some consan M > is a measurable funcion and Y is a srong soluion o Y = y + 2 Ys db s + ds, Y s, Z s )ds for some y upo an explosion ime τ. Then, P[τ = ] = and P[ Y Y M, ] =, where Y M = y + 2 Y M s db s + Proof. Le τ n := inf{ > : Y n}. By Tanaka s formula, Y τn Y M τ n ) + = 2 Y τ n = 2 τn τn τn τn Y s Y M s Mds. ) [Ys>Y s M ] db s M ds, Y s, Z s )) [Ys>Y M s ] ds + L Y Y M ) τn, Ys [Ys<]dB s ds, Y s, Z s ) [Ys<]ds + L Y ) τn 9

20 where L Y Y M ) and L Y ) are he local imes of Y Y M and Y a, respecively. We will firs show ha Y is nonnegaive upo τ n. Since τn Y s [< Ys<] Y s ds and x dx =, i follows from Lemma 3.3 in Chap. IX of [7] ha L Y τn ) = for all. Thus, E [ Y τ ] τn τn ] n = E [ 2 Ys [Ys<]dB s ds, Y s, Z s ) [Ys<]ds, since he sochasic inegral is a maringale having a bounded inegrand. Thus, Y τn, a.s. for every and any n. Similarly, τn [<Ys Y M s s ) 2 s Y s Y M <] Y s Y M ds = τn [<Ys Y M s s ) 2 Y s Y M <] Y s Y M s ds, where he firs equaliy is due o he fac ha Y M implies Y s on he se [Y s Ys M > ], and he second inequaliy follows from he elemenary fac ha x y x y. Thus i follows from Lemma 3.3 in Chap. IX of [7] ha L Y τn Y τ M n ) = for all and E [ Y τn Y M τ n ) +] = 2E E [ τn [ τn Y s Y M s ] ) [Ys>Y s M ]db s M ds, Y s, Z s )) [Ys>Y s M ] ds ], since he sochasic inegral τ n Y s Ys M ) [Ys>Y s M ]db s ) is a maringale having a bounded inegrand. Thus, Y τn Y τ M n, a.s. for every and any n. Since Y and Y M are coninuous upo ime τ n, we have P[ Y τn Y τ M n, ] =. By aking he limi as n, we obain P[ Y τ Y M τ, ] =. Since Y M is non-explosive, his implies ha τ =, a.s.. In view of he above lemma, he hypohesis of he nex lemma is no vacuous. Lemma A.2 Suppose ha d : R + R 2 + [, M] for some consan M > is a measurable funcion and Y is he nonnegaive srong soluion o Y = y + 2 Ys db s + ds, Y s, Z s )ds, for some y. Moreover, suppose ha here exiss wo sopping imes S T such ha d S) T, Y S) T, Z S) T ) [a, b] [, M] for some consans a and b. Then, P[Y T a Y T Y T b, ] =, where S { Y a = Y S + 2 } Ys a db s + ads S S { } Y b = Y S + 2 Ys b db s + bds. S 2

21 Proof. Observe ha using he similar argumens as in he previous lemma, one obains ha L Y Y a ) = L Y Y b ) =. Thus, by Tanaka s formula, Y T Y b T ) + = 2 Y a T Y T ) + = 2 T S T S T S T S Y s Ys b ) [Ys>Y s b] db s b ds, Y s, Z s )) [Ys>Y b s ] ds Y a s Y s ) [Y a s >Y s]db s ds, Y s, Z s ) a) [Y a s >Y s]ds. Observe ha he sochasic inegrals above are nonnegaive local maringales, herefore hey are supermaringales. Thus, by aking he expecaions we obain [ E Y T Y T b ) +] E [ Y a T Y T ) +]. Hence, he conclusion follows. A.2 Maringale problems and some L 2 esimaes In he nex lemma we show ha he maringale problem relaed o Z as defined in 4.9) is well posed. Recall ha A is he associaed infiniesimal generaor defined in 4.24). We will denoe he resricion of A o B[, ε] R + ) by A ε. Lemma A.3 Fix ε > and le µ P. Then, he maringale problem A ε, µ) is well-posed. Moreover, he SDE 4.9) has a unique weak soluion for any nonnegaive iniial condiion and he soluion is sricly posiive on s, ε] for any s [, ε]. Proof. Le s [, ε] and z R +. Then, direc calculaions yield { } dz r = σr)dβ r + σ 2 r) Z r η r, Z r ) dr, for r [s, ε], Z r A.28) wih Z s = z, where η r, y) := V ) V r) V ) V r) exp 2πu 5 exp 2πu 3 y2 2u y2 2u ) du ), A.29) du hus, η r, y) [, V ) V ε) ] for any r [, ε] and y R +. Firs, we show he uniqueness of he soluions o he maringale problem. Suppose here exiss a weak soluion aking values in R + o he SDE above. Thus, here exiss Z, β) on some filered probabiliy space Ω, F, F r ) r [, ε], P ) such ha { } d Z r = σr)d β r + σ 2 r) Zr Z r η r, Z r ) dr, for r [s, ε], 2

22 wih Z s = z. Consider R which solves d R r = σr)d β r + σ 2 r) Rr dr, A.3) wih R s = z. Noe ha his equaion is he SDE for a ime-changed 3-dimensional Bessel process wih a deerminisic ime change and he iniial condiion R s = z. Therefore, i has a unique srong soluion which is sricly posiive on s, ε] see in Chap. XI of [7]). Then, from Tanaka s formula see Theorem.2 in Chap. VI of [7]), since he local ime of R Z a is idenically see Corollary.9 in Chap. VI of [7]), we have { } Z R ) + = [ Zr> R r] σ2 r) Zr Z r η r, Z r ) Rr dr, where he las inequaliy is due o η, and a < b whenever a > b >. Thus, Zr R r for r [s, ε]. Define L r ) r [, ε] by L = and dl r = L r Zr η r, Z r ) d β r. If L r ) r [, ε] is a rue maringale, hen Q on F ε defined by dq d P = L ε, is a probabiliy measure on F ε equivalen o P. Then, by Girsanov Theorem see, e.g., Theorem 3.5. in [4]) under Q d Z r = σr)d β Q r + σ 2 r) Zr dr, for r [s, ε], wih Z s = z, where β Q is a Q-Brownian moion. This shows ha Z, β Q ) is a weak soluion o A.3). As A.3) has a unique srong soluion which is sricly posiive on s, ε], any weak soluion o 4.9) is sricly posiive on s, ε]. Thus, due o Theorem in [9], he maringale problem for δ z, A ε ) has a unique soluion. Noe ha alhough he drif coefficien is no bounded, Theorem in [9] is sill applicable when L is a maringale. Thus, i remains o show ha L is a rue maringale when Z is a posiive soluion o A.28). For some n < n ε consider [ )] n E exp 2 Z r η r, Z r )) 2 dr. A.3) n The expression in A.3) is bounded by [ n ) 2 E exp R2 2 r dr)] n V ) V ε) [ )] E exp 2 R r) 2 n n V ) V ε)) 2 where Y := sup s Y s for any càdlàg process Y. Recall ha R is only a ime-changed Bessel process where he ime change is deerminisic and, herefore, R2 r is he square of he Euclidian 22

23 norm a ime V r) of a 3-dimensional sandard Brownian moion, saring a z,, ) a ime V s). Thus, by using he same argumens as in Proposiion 3.2, we ge ha he above expression is going o be finie if [ )] EV z n n s) exp 2 β V ε) )2 V ) V ε)) 2 <, where β is a sandard Brownian moion and Es x is he expecaion wih respec o he law of a sandard Brownian moion saring a x a ime s. In view of he reflecion principle for sandard Brownian moion see, e.g. Proposiion 3.7 in Chap. 3 of [7]) he above expecaion is going o be finie if n n V ) V ε)) 2 < V ε). Clearly, we can find a finie sequence of real numbers = < <... < nt ) = T ha saisfy above. Now, i follows from Corollary in [4] ha L is a maringale. In order o show he exisence of a nonnegaive soluion, consider he soluion, R, o A.3), which is a ime-changed 3-dimensional Bessel process, hus, nonnegaive. Then, define L r by L = and dl r = L r R r η r, R r ) d β r. ) r [, ε] Applying he same esimaion o L as we did for L yields ha L is a rue maringale. Then, Q on F ε defined by dq = L ε d P, is a probabiliy measure on F ε under which R solves { } d Z r = σr)d β r Q + σ 2 r) Zr Z r η r, Z r ) dr, for r [s, ε], wih Z s = z and β Q is a Q-Brownian moion. This means ha he nonnegaive process R is a weak soluion of A.28). Therefore, he maringale problem δ z, A ε ) has a soluion by Proposiion 5.4. and Corollary in [4] since σ is locally bounded. Thus, he maringale problem δ z, A ε ) is well-posed for any z R +. The well-posedness of he maringale problem for µ, A ε ) follows from Theorem 2. in [2] since P z is he unique soluion of he maringale problem for δ z, A ε ) for any z R +. We are now ready o show ha he join maringale problem for X, Z) defined by he operaor A : B[, ) R 2 +) B[, ) R 2 +) which is given by Aφs, x, z) = φ s s, x, z) + 2 φ 2 x 2 s, x, z) + 2 σ2 s) 2 φ s, x, z) z2 + q x V ) V s), x, z) φ q wih he domain DA) = C c [, ) R 2 +). x s, x, z) + σ2 s) H z H V ) V s), z) φs, x, z), z Proposiion A. Le µ P 2 where P 2 is he se of probabiliy measures on he Borel ses of R 2 + opologized by weak convergence. Then, he maringale problem µ, A) is well-posed. 23

24 Proof. Clearly, if µ, A ε ) is well-posed for any ε >, where A ε is he resricion of A o B[, ε], R + ), hen µ, A) is well-posed. As in he proof of Lemma A.3, he problem of wellposedness of µ, A ε ) can be reduced o ha of δ x,z, A ε ) for any fixed x, z) R 2 + due o Theorem 2. in [2] and Proposiion.6 in Chap. III of [7]. To his end, in view of Proposiion 5.4. and Corollary in [4], i suffices o show he exisence and he uniqueness of weak soluions o he sysem of SDEs defined by 4.9) and 4.2) wih he iniial condiion ha X s = x and Z s = z for a fixed s [, ε]. We will consider he following hree cases o finish he proof. Case : x >, z >. In Lemma A.3 we have proved he exisence and he uniqueness of a weak soluion o he SDE 4.9) for any iniial condiion Z s = z for s [, ε] and z. Thus, here exiss Z, β) on some filered probabiliy space Ω, F, F r ) r [, ε], P ) such ha Z, β) solves he SDE 4.9) wih he iniial condiion Z s = z. Wihou loss of generaliy we can assume ha he space Ω, F, F r ) r [, ε], P ) suppors anoher Brownian moion, B, independen of β. Then, Proposiion 3.2 yields ha here exiss a unique srong and sricly posiive soluion o 4.2) on Ω, F, F r ) r [, ε], P ). Indeed, he proof of Proposiion 3.2 would remain he same as long as he iniial condiion for Z is sricly posiive and one observes ha alhough Z is no a Brownian moion, he finieness of 3.5) sill follows from 3.6) since Z is sricly posiive and bounded from above by a ime-changed 3-dimensional Bessel process and he ime change is given by V ). This demonsraes ha here exiss a weak soluion o he sysem of SDEs. Moreover, he soluion is unique in law since X is pahwise uniquely deermined by Z, which is unique in law. Case 2: x =, z. We can use he same argumens as in he previous case once we esablish Lemma 3.2 over he ime inerval [s, ε]. Noe ha we only need o show he sric posiiviy of he soluion as he exisence of a nonnegaive weak soluion follows along he same lines. Consider he sequence of sopping imes τ n ) n τ n := inf{r [s, ε] : U r = n }, where inf = ε. On τ n, ε] he soluion exiss and is sricly posiive as in Case since Z τn > and U τn = n when τ n < ε. Consider τ := inf n τ n. If τ = s, we are done. Suppose τ > s wih some posiive probabiliy. Then, on his se U = for τ. However, his conradics he fac ha U solves 3.2) on [s, ε]. Case 3: x >, z =. As in he previous case i only remains o esablish he sric posiiviy of he soluion of 3.2), which exiss by he same argumens. Again consider he following sequence of sopping imes: τ n := inf{r [s, ε] : Z r = n }, where inf = ε. Tha he weak soluion o 3.2) is sricly posiive on τ n, ε] follows from Case if X τn >, and from Case 2 if X τn =. Since inf n τ n = s by Lemma A.3, we have he sric posiiviy on [s, ε]. Lemma A.4 Le Z, X) be he unique srong soluions o 2.) and 2.7). Then hey solve he maringale problem on he inerval [, ) defined by 4.9) and 4.2) wih he iniial condiion X = Z =. Moreover, under Assumpion 2. we have 24

25 [ ) ] 2 i) E qxq [τ>s] V s) s, X s, Z s ) ds <. [ ] ii) E τ, 2 qxq V s) s, X s, Z s )) ds <. iii) E τ, [ ε σ 2 s) H x H V ) V s), Z s) ds] 2 <, for any ε >. Proof. Recall ha dpτ, dp = [τ>] P[τ>] and ha E τ, denoes he expecaion operaor wih respec o P τ,. Hence, under P τ,, Z, X) saisfy 4.9) and 4.2) wih he iniial condiion X = Z =, which implies ha hey solve he corresponding maringale problem. i) & ii) Noe ha = E P[τ > ] E τ, [ E [ [τ>] [ ) 2 qx q V s) s, X s, Z s ) ds] qx q V s) s, X s, Z s ) [τ>s] qx q V s) s, X s, Z s ) ) 2 ds] ) 2 ds]. Thus, i suffices o prove he firs asserion since P[τ > ] > for all. Recall from 3.9) ha q x, x, z) q, x, z) = z x + exp 2xz ) exp 2xz ) 2z = z x ) 2xz + f x, where fy) = e y y is bounded by on [, ). As e y ds < and sup V s) s) 2 s [,] E[Zs 2 ] V ) +, he resul will follow once we obain. sup s [,] E[Xs 2 [τ>s] ] <, and ) 2. E [τ>s] ds <, Xs 2 demonsraed below.. By Io formula, [τ>] X 2 = [τ>] + 2 X s db s + 2 { Zs X s X 2 s V s) s + f Observe ha he elemenary inequaliy 2ab a 2 + b 2 implies 2 [τ>] 2 X s db s + [τ>] Z s X s X 2 s V s) s ds Z 2 s X 2 s V s) s ds 25 ) 2 τ X s db s + Z 2 s V s) s ds. ) 2Zs X s + } ) ds. V s) s 2 A.32) X s db s ) 2, and