ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES, AND PASSAGE TIME PROBLEMS

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1 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES, AND PASSAGE TIME PROBLEMS GERARDO HERNÁNDEZ-DEL-VALLE Absrac. We obain explici soluions for he densiy ϕ T of he firs-ime T ha a one-dimensional Brownian process B reaches he wice, coninuously differeniable moving boundary f and such ha f ) for all R +. We do so by finding he expeced value of some funcionals of a 3-dimensional Bessel bridge X and exploiing is relaionship wih firs-passage ime problems as poined ou by Kardaras 7). I urns ou ha his problem is relaed o Schrödinger s equaion wih ime-dependen linear poenial, see Feng ).. Inroducion The main aim of his paper is wo-folded, given ha f C [, ), and such ha f ), for all > : ) we compue ) E [ exp ] f u) X u du, where X is a a 3-dimensional Bessel bridge saring a level a > [where a = f)] and reaching a some fixed ime s, and ) as a corollary, we derive an explici soluion for he densiy ϕ f of he firs ime T ha a one-dimensional Brownian process B reaches he moving boundary f. The derviaion of ϕ f is an old and well sudied problem in probabiliy and sochasic processes. Ye, explici soluions exised only in a limied number of special cases. The appeal of his problem sems no only from is applicaions in finance, physics, biology, ec.) bu also from is mahemaical naure. As i urns ou he problem may be sudied hrough several differen approaches: ) Volerra inegral equaions: Several auhors have derived Volerra inegral equaions in he sudy of he firs-passage ime problems [he reader may consul for insance Peskir ) for a deailed echnical and hisorical accoun on he subjec]. Ye, in he case in which he equaion is no a convoluion, or is kernel is no separable, numerical procedures are ypically employed. Thus he resuls presened in his work provide explici Dae: December 7. Mahemaics Subjec Classificaion. Primary: 6J65,45D5,6J6; Secondary: 45G5, 45G, 45Q5, 45K5. Key words and phrases. Firs-passage ime problems, Schrödinger s equaion, Brownian moion, 3-dimensional Bessel bridge, Volerra inegral equaions of he second kind). Saisics Deparmen, Columbia Universiy, Mail Code 443, New York, N.Y..

2 GERARDO HERNÁNDEZ-DEL-VALLE soluions o previously unknown inegral equaions. ) Parial differenial equaions: Groeneboom 987), Salminen 988), and Marin-Löf 998) ackled he problem of he quadraic boundary by solving a parabolic differenial equaion, derived from Girsanov s and Feynman-Kac heorems. In his paper no only do we generalize his resul an in fac provide an alernaive proof for his special case), bu also show how boundary problems are relaed o Schrödinger s equaion wih ime-dependen linear poenial. 3) Bessel Bridge: Kardaras 7) noes ha he problem of finding ϕ f is equivalen o he sudy of ). Several auhors, for insance Revuz and Yor 5), have analyzed ), ye our resuls are new since we do no rea he squared process X. 4) Monecarlo simulaion. Roughly speaking, he soluion o ), is based upon ransforming a Cauchy problem derived from Feynman-Kac s heorem), equaions 9) and ) ino ha of solving he Schrödinger equaion 3) wih ime-dependen linear poenial [see Feng )]. And hen proceed backwards o consruc he corresponding Green s funcion. In Secion, afer saing he problem we: a) poin ou he exising relaionship beween a hree-dimensional Bessel bridge process and firs-passage ime problems. b) We presen our main resuls, Theorem.3 and Corollary.4, which we will prove in he remainder of he paper. In Secion 3, we sae he problem in erms of he Feynman-Kac equaion. We proceed in solving i by finding paricular soluions wih respec o he backward, a) and forward τ, b) variables, in Secions 4 and 5 respecively. We conclude in Secion 6 by using hese resuls o consruc he necessary Green s funcion G. Boh he backward and forward equaions in Secions 4 and 5 are deal wih in a similar fashion: ) Through algebraic ransformaions we obain Schrödinger s equaion wih ime-dependen linear poenial. ) We solve he corresponding Schrödinger s equaion, by reducing i ino he hea equaion.. The problem Problem.. The main moivaion of his work is finding he densiy ϕ T of he firs ime T, ha a one-dimensional, sandard Brownian moion B reaches he moving boundary f: ) T := inf B = f) where f is wice, coninuously differeniable, and f ) for all >. Alernaively, from Girsanov s heorem, he problem may be resaed as: Proposiion.. Given ha 3) ϕ a ) := a exp a π 3, a >

3 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES 3 is he densiy of he firs ime ha B reaches he fixed level a, ), he process X is a 3-dimensional Bessel bridge, which has he following dynamics: d X = dw + X X ) d, s X = a, [, s). Where W is a Wiener process.) Then, he disribuion of T equals PT < ) = [ Ẽ exp exp ] f u) X u du f u)) du f )a ϕ a s)ds Proof. From Girsanov s heorem, and he fac ha he boundary f is wice coninuously differeniable, he following Radon-Nikodym derivaive: dp d P ) := exp := exp f ) B + f s)d B s f s)) ds f s) B s ds f s)) ds, is indeed a maringale and induces he following relaionship: [ ] dp PT ) := Ẽ d P )I T ) [ ] dp = Ẽ d P T )I T ) [ dp ] T = Ẽ d P T ) = s ϕ a s)ds, where he second equaliy follows from he opional sampling heorem, and he hird from condiioning wih respec o T under P, and ϕ a is defined in 3). Nex, Kardaras 7) shows ha calculaing [see for insance Chaper in Revuz and Yor 5), for a deailed overview of 3-dimensional Bessel bridges] : [ dp ] T 4) Ẽ d P T ) = s

4 4 GERARDO HERNÁNDEZ-DEL-VALLE is equivalen o finding he expeced value of a funcional of a 3-dimensional Bessel bridge X, which a ime sars a a and a ime s equals. Indeed, [ dp ] T Ẽ d P T ) = s [ ] T = s 5) = Ẽ exp f T )a + [ = Ẽ exp f s)a + [ = Ẽ exp [ = Ẽ exp T f u) B u du T f u)a X u )du f s)a + f s) f ))a f u)) du ] ] f u) X u du exp f u)) du ] f u)) du f u) X u du f u)) du f )a hus he process a X equals a =, and a s reaches level a for he firs ime, as required. We conclude from equaion 5) ha our goal is o compue he following expeced value: [ ] 6) Ẽ exp f u) X u du. Equaion 6) has he following soluion: Theorem.3. Le he boundary f C [, ) saisfy f ) for all. Then: [ ] Ẽ exp f u) X u du = v, a) where 7) given ha where v, a) = G, a; s, b)db, < s, a R + \ G, a; τ, b) = ϕ b) H, a; τ, b), ϕ a s ) H, a; τ, b) = πτ ) [ exp Where ϕ a is defined in 3). b a f u)du ) τ ) < τ < s f u)) du f τ)b + f )a exp Thus he densiy ϕ T of he sopping T in equaion ) equals: b + a f u)du ) ]. τ )

5 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES 5 Corollary.4. Le B be a one-dimensional sandard Brownian moion B and le he sopping ime T be as in ). Then, he densiy ϕ T equals: ϕ T s) = v, a)ϕ a s) exp f u)) du f )a. where v and ϕ a are defined in 7) and 3) respecively. 3. The Feyman-Kac equaion As he reader may suspec from equaion 6) and he Green s funcion appearing in equaion 7), in Theorem.3, we will follow a P.D.E approach o solve our problem. To do so, we will firs inroduce Feynman-Kac s heorem and some necessary ools, and will conclude his secion by inroducing he corresponding Cauchy problem see Problem 3.3). If we le he operaor A saisfy A v), a) = σ, a) v, a) + b, a) v, a) a a = v, a) + a a a ) v s a, a) < s, a R+ \, Then, we may inroduce a simplified version of he celebraed Feynman-Kac heorem which relaes parabolic parial differenial equaions wih sochasic differenial equaions [for a deailed and general overview of his and relaed opics he reader may consul for insance: Chaper 5, in Karazas and Shreve 998)]: Theorem 3. Feynman-Kac). Given he Cauchy problem v + kv = A v + g; [, T ) R vs, a) = fa); a R + \ where k, x) : [, s] R + [, ), and funcions f and g saisfy he sandard assumpions [see Theorem in Karazas and Shreve 998)], and v saisfies he polynomial growh condiion. Then, he funcion v, a) admis he sochasic represenaion v, a) = Ẽ,a [ f X s ) exp + gv, X v ) exp v on [, s] R + ; in paricular, such a soluion is unique. ku, X u )du ] ku, X u )du dv The following definiion, will guide us on he consrucion of our Green s funcion G: Definiion 3.. [Definiion in Karazas and Shreve 998)] A fundamenal soluion of he second-order parial differenial equaion 8) u + ku = A u

6 6 GERARDO HERNÁNDEZ-DEL-VALLE is a nonnegaive funcion G, a; τ, b) defined for τ s, a R +, b R + wih he propery ha for every f C R + ), τ, s], he funcion u, a) := is bounded, of class C,, saisfies 8), and G, a; τ, b) fb)db; < τ, a R + lim u, a) = fa), a R +. τ For fixed τ, b), s] R +, he funcion ϕ, a) := G, a; τ, b) is of class C, and saisfies he backward Kolomogorov equaion in he backward variables, a). And, for fixed, a) [, s) R + he funcion ψτ, b) := G, a; τ, b) is of class C, and saisfies he forward Kolmogorov equaion also known as he Fokker-Planck equaion). Hence, from equaion 5) i follows ha: ] Ẽ [exp,a f u) X u du = v, a). where [ v, a) := E,a f) exp for ku, a) := f u) a, and vt, a) = fa), i.e., ku, X u )du ], Problem 3.3 Bessel s Cauchy problem). Equaion 6) saisfies he following Cauchy problem: 9) ) v, a) + f )av, a) = v, a) a + a a ) v s a, a); in [, s) R+ \ vs, a) = ; a R + \. Definiion 3. hins us owards he pah we should follow. We should firs find a general soluion o he backward-kolmogorov equaion, in Secion 4: ϕ, a) + f )aϕ, a) = ϕ, a) a + a a ) ϕ s a, a); in [, s) R+ \ and o he forward-kolmogorov equaion, in Secion 5: ψτ, b) τ, b) + f k τ)bψτ, b) = ψ τ, b) τ b [ b b b ) ] ψτ, b) ; in, s] R + \.

7 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES 7 Which afer proper change s of variable and algebraic ransformaions will equal respecively: ϕ3, y) = ψ 3 τ, z) = ϕ 3, y), y < s ψ 3 τ, z), τ s, z and y and z are funcion of a and b respecively and will be deermined laer). Since boh ϕ and ψ may be reduced o he Hea equaion, and he variables a, b are only defined on he half line. Then he Green s funcion G which relaes hese wo soluions will be of he form: G, y; τ, z)= exp πτ z y) τ ) exp z + y) τ ) ) < τ s. Thus, he consrucion of he paricular soluion G will jus be a backwards procedure. 4. Schrödinger s backward equaion In his secion we obain a paricular soluion o he backward equaion 9) by firs ransforming i ino Schrödinger s backward-equaion wih ime-dependen linear poenial hrough algebraic ransformaions. Proposiion 4.. Equaion 9) saisfies he following relaionship ) ϕ, a) = ϕ, a) A) exp [ B)a ], where B) = s ) A) = c s ) 3/ ) af )ϕ, a) ϕ aa, a) a ϕ a, a) ϕ, a) = Proof. Le us firs sar by analyzing he ransformaion in equaion ): ϕ a, a) = ϕ a, a)a) exp [ B)a ] +aϕ, a)a)b) exp [ B)a ] ϕ a, a) = ϕ aa, a)a) exp [ B)a ] +aϕ a, a)a)b) exp [ B)a ] +ϕ, a)a)b) exp [ B)a ] +aϕ a, a)a)b) exp [ B)a ] +4a ϕ, a)a)b ) exp [ B)a ] ϕ, a) = ϕ, a)a ) exp [ B)a ] +a ϕ, a)a)b ) exp [ B)a ] +ϕ, a)a) exp [ B)a ].

8 8 GERARDO HERNÁNDEZ-DEL-VALLE Subsiuing hese equaions ino 9) we ge equaing o zero we have A ) A) a B ) ϕ ϕ + af ) = +B) + a ϕ a ϕ B) + a B ) + + a a ) ab), s ϕ aa ϕ a + aϕ a ϕ B) a ) ϕ a s ϕ A ) A) a B ) ϕ ϕ + af ) ϕ aa ϕ aϕ a ϕ B) B) a B ) ϕ a a ϕ + a ϕ a s ϕ B) + a s B) = finally, we facorize ) A ) A) + 3B) a a ϕ a ϕ B) s This equaliy holds for every a > if and only if as claimed. B ) + B ) s B) ) + af ) ϕ aa ϕ ϕ a a ϕ ϕ ϕ =. B) = s ) A) = c s ) 3/ af )ϕ, a) ϕ aa, a) a ϕ a, a) ϕ, a) = ) 4.. The Schrödinger s backward equaion. If we le ϕ = a ϕ, hen ϕ = a ϕ ϕ a = a ϕ a a ϕ afer subsiuing in equaion 7): or, equivalenly ϕ aa = a ϕ aa a ϕ a + a 3 ϕ a ϕ a, af )ϕ, a) ϕ aa, a) ϕ, a) =. 3) ϕ, a) = ϕ a, a) af )ϕ, a) which is Schrödinger s backward-equaion wih ime-dependen linear poenial [see for insance, Feng )].

9 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES 9 Theorem 4.. Given he backward Kolmogorov equaion in 3): 4) ϕ, a) = ϕ a, a) af )ϕ, a) we have he following fundamenal soluion, ϕ I a f)) 5), a) = π where I =. f u)) du + f )a, Proof of 5). Firs se ϕ, a) = λ, a)e β)a, where β will be deermined laer and compue ϕ, a) = aβ )λ, a)e β)a + λ ϕ a, a) = β)λ, a)eβ)a + λ, a)eβ)a a ϕ a, a) = β )λ, a)e β)a + β) λ a + λ a, a)eβ)a + β) λ a eβ)a nex, subsiue ino equaion 4), a)eβ)a, a)eβ)a = β )λs, a)e β)a + β) λ, a)eβ)a a + λ, a)eβ)a a λ, a) = λ a, a) + β )λ, a) + β) λ, a) a +aβ ) f ))λ, a) Nex, if we le y = a v) and se λ, a) = u, y), hen u, y) + v ) u y, y) = u y, y) + β )u, y) + β) u, y) y +y + v))β ) f ))λ, y) o delee he erm se v ) = β), hence u, y) y u, y) = u y, y) + β )u, y) +y + v))β ) f ))λ, y) furhermore, seing β ) = f ) we ge u, y) = u y, y) + β )u, y)

10 GERARDO HERNÁNDEZ-DEL-VALLE which is a parabolic differenial equaion which now may be ransformed ino he hea equaion by leing Gu) = /β u) and i follows ha ϕ 3, y) = u, y)e R Gu)du ϕ3, y) = ϕ 3, y) y which has general soluion equal o: ϕ 3, y) = I y exp π i.e., and Hence u, y) = = e R Gu)du ϕ 3, y) = e R f u)) du ϕ 3, y) I y = π f u)) du λ, a) = ϕ, a) = = = I a v)) π I a f)) π I a f)) π I a f)) π 5. Schrödinger s Forward equaion f u)) du f u)) du. f u)) du + β)a f u)) du + f )a From equaion 9) we will now sudy is dual, he Fokker-Planck equaion: ψτ, b) τ, b) + f k τ)bψτ, b) = τ ψ τ, b) b [ b b b ) ] ψτ, b). The echniques used in his secion are equivalen o hose used when reaing he backward-equaion, wih he obvious modificaions.) Tha is 6) ψτ, b) τ, b) = τ ψ τ, b) b b + b ) ψ b b + f k τ)b τ, b) ) ψτ, b)

11 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES Proposiion 5.. Equaion 6) saisfies he following relaionship 7) ψτ, b) = ψ τ, b) Aτ) exp [ Bτ)b ], where 8) Bτ) = ) Aτ) = c ) / bf τ) b ) ψ τ, b) ψ bbτ, b) + b ψ b τ, b) +ψ τ τ, b) = Proof. Le us firs sar by analyzing he ransformaion in equaion 8): ψ b τ, b) = ψ b τ, b)aτ) exp [ Bτ)b ] +bψ τ, b)aτ)bτ) exp [ Bτ)b ] ψ b τ, b) = ψ bbτ, b)aτ) exp [ Bτ)b ] +bψ b τ, b)aτ)bτ) exp Bτ)b ) +ψ τ, b)aτ)bτ) exp [ Bτ)b ] +bψ b Aτ)Bτ) exp [ Bτ)b ] +4b ψ τ, b)aτ)b τ) exp [ Bτ)b ] ψ τ τ, b) = ψ τ, b)a τ τ) exp [ Bτ)b ] +b ψ τ, b)aτ)b τ τ) exp [ Bτ)b ] +ψ τ τ, b)aτ) exp [ Bτ)b ]. Subsiuing hese equaions ino 6) we ge A τ τ) Aτ) + b B τ τ) + ψ τ ψ + bf τ) b ) equaing o zero we have = ψbb ψ + bψ b ψ Bτ) + Bτ) + bψ b ψ Bτ) + b B τ) b b ) ψ b ψ b b ) bbτ), A τ τ) Aτ) + b B τ τ) + ψ τ ψ + ψ bb ψ + b bf τ) b ) bψ b ψ Bτ) Bτ) b B τ) b ) ψ b ψ + b b ) bbτ) =,

12 GERARDO HERNÁNDEZ-DEL-VALLE finally, we facorize Aτ τ) Aτ) + Bτ) b ψ b ψ ψ bb ψ Bτ) + + b ) + b B τ τ) B τ) ψ b ψ + ψ τ ψ =. This equaliy holds for every b > if and only if as claimed. ) + bf τ) b ) Bτ) ) Bτ) = ) Aτ) = c ) / bf τ) b ) ψ τ, b) ψ bbτ, b) + b ψ b τ, b) +ψ τ τ, a) = 5.. The Schrödinger s forward equaion. We firs sar wih he following ransformaion ψ = b ψ, i.e. ψb = ψ + b ψb ψbb = ψb + bψbb ψτ = bψτ which afer subiuion in 8) equals 9) b f τ)ψ b ψ b ψ ψb bψ bb + b ψ + ψb + bψτ = b f τ)ψ b ψ bψ bb + bψτ = bf τ) ) ψ ψ bb + ψτ =. Nex, seing ψ equal o: we have ha ψτ = ψb = ψbb = ψ = ψ, ) ψ + ψ τ ψ b ψ bb.

13 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES 3 Thus subiuing ino 8) ) bf τ) ψ ) ψ ψ bb + ) ψ + ψ τ = bf τ) ψ ψ bb + ψ τ = bf τ) ψ ψ bb + ψ τ = which is Schrödinger s forward equaion. Theorem 5.. Given he forward equaion as in ) ) ψ τ τ, b) = ψ b τ, b) bf τ) ψ τ, b) we have he following fundamenal soluion ψ b fτ)) ) τ, b) = + πτ τ f u)) du f τ)b Proof of ). Firs se ψ τ, b) = λτ, b)e βτ)b, where β will be deermined laer and compue ψ τ τ, b) = bβ τ τ)λτ, b)e βτ)b + λ τ ψ b τ, b) = βτ)λτ, b)eβτ)b + λ b ψ b τ, b) = β τ)λτ, b)e βτ)b βτ) λ b + λ a τ, b)eβτ)b βτ) λ b eβτ)b nex, subsiue ino equaion ) τ, b)eβτ)b τ, b)eβτ)b τ, b)eβτ)b = β τ)λτ, b)e βτ)b βτ) λ τ, b)eβτ)b b + λ τ, b)eβτ)b b λ τ τ, b) = λ b τ, b) + β τ)λτ, b) βτ) λ τ, b) b +bβ τ τ) f τ))λτ, b) Nex, if we le z = b vτ) and se λτ, b) = uτ, z), hen u τ τ, z) v τ τ) u y τ, z) = u z τ, z) + β τ)uτ, z) βτ) u τ, z) z +z + vτ))β τ τ) f τ))uτ, z) o delee he erm u τ, z) z

14 4 GERARDO HERNÁNDEZ-DEL-VALLE se v τ τ) = βτ), hence u τ τ, z) = u z τ, z) + β τ)uτ, z) +z + vτ))β τ τ) f τ))uτ, z) furhermore, seing β τ τ) = f τ) we ge u τ τ, z) = u z τ, z) + β τ)uτ, z) which is a parabolic differenial equaion which now may be ransformed ino he hea equaion by leing Gu) = /β u) and ψ 3 τ, z) = uτ, z)e R τ Gu)du i follows ha whose soluion is i.e., ψ 3 τ τ, z) = ψ 3 τ, z) z ψ 3 τ, z) = exp z πτ τ and Hence uτ, z) = = e R τ Gu)du ψ 3 τ, z) = e R τ f u)) du ψ 3 τ, z) = y πτ τ + λτ, b) = ψ τ, b) = = = b vτ)) + πτ τ b fτ)) + πτ τ b fτ)) + πτ τ b fτ)) + πτ τ f u)) du f u)) du f u)) du. f u)) du βτ)b f u)) du f τ)b 6. General soluion From he backward 5) and forward ) soluions of Schrödinger s equaion, and he fac ha a, b R + \ we have Proposiion 6.. Schrödinger s equaion, see equaions 4) and ), wih imedependen linear poenial and such ha a, b R + \ has he following Green s

15 ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES 5 funcion 3) H, a; τ, b) = πτ ) [ exp where < τ s. b a f u)du ) τ ) f u)) du f τ)b + f )a exp b + a f u)du ) ], τ ) Proof. Recall ha he backward and forward equaions may be reduced correspondingly o: ϕ3, y) = ϕ 3, y) y ψ 3 τ τ, z) = ψ 3 τ, z), z where y = a f), and z = b fτ), for < τ < s, and a, b R + \. Thus, he Green s funcion G ha links hese wo soluions should be of he form: ) z y) z + y) G, y; τ, z) = exp exp. πτ ) τ ) τ ) Finally, equaion 3) follows from equaions 4) and ). Which alernaively leads o Proposiion 6.. Our iniial equaion 9) has he following Green s funcion: G, a; τ, b) = ϕ b) H, a; τ, b) ϕ a s ) Proof. I follows from equaions ), 7), and 3) and verificaion of Definiion 3.. Proof of Theorem.3. I follows from Proposiion 6.. References [] Feng, M. ). Complee soluion of he Schrödinger equaion for he ime-dependen linear poenial, Phys. Rev. A 64, 34. [] Groeneboom, P. 987). Brownian moion wih a parabolic drif and Airy funcions. Prob. Theory Relaed Fields 79. [3] Karazas, I. and S. Shreve. 99). Brownian Moion and Sochasic Calculus, Springer- Verlag, New York. [4] Kardaras, K 7). On he densiy of firs passage imes for difussions. on preparaion. [5] Marin-Löf, A. 998). The final size of a nearly criical epidemic, and he firs passage ime of a Wiener process o a parabolic barrier. J. Appl. Prob. 35. [6] Peskir, G. ). On inegral equaions arising in he firs-passage problem for Brownian moion, J. Inegral Equaions Appl., 4. [7] Revuz, D., and M. Yor. 5). Coninuous maringales and Brownian moion, Springer- Verlag, New York. [8] Salminen, P. 988). On he hiing ime and las exi ime for a Brownian moion o/from a moving boundary, Advances in Applied Probabiliy. Saisics Deparmen, Columbia Universiy, Mail Code 443, New York, N.Y. address: gerardo@sa.columbia.edu