Diffusion transformations, Black-Scholes equation and optimal stopping
aa r X i v : . [ m a t h . P R ] F e b DIFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION ANDOPTIMAL STOPPING
UMUT C¸ ETIN
Abstract.
We develop a new class of path transformations for one-dimensional diffusions that aretailored to alter their long-run behaviour from transient to recurrent or vice versa. This immedi-ately leads to a formula for the distribution of the first exit times of diffusions, which is recentlycharacterised by Karatzas and Ruf [26] as the minimal solution of an appropriate Cauchy problemunder more stringent conditions. A particular limit of these transformations also turn out to be in-strumental in characterising the stochastic solutions of Cauchy problems defined by the generatorsof strict local martingales, which are well-known for not having unique solutions even when onerestricts solutions to have linear growth. Using an appropriate diffusion transformation we showthat the aforementioned stochastic solution can be written in terms of the unique classical solutionof an alternative
Cauchy problem with suitable boundary conditions. This in particular resolvesthe long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing inthe presence of bubbles . Finally, we use these path transformations to propose a unified frameworkfor solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting,which in particular is relevant for the pricing and the computation of optimal exercise boundariesof perpetual American options. Introduction
Conditioning the paths of a given Markov process X to stay in a certain subset of the pathspace is a well-studied subject which has become synonymous with the term h -transform. If onewants to condition the paths of X to stay in a certain set, the classical recipe consists of findingan appropriate excessive function h , defining the transition probabilities of the conditioned processvia h , and constructing on the canonical space a Markov process X h with these new transitionprobabilities. This procedure is called an h -transform. In particular if h is a minimal excessivefunction with a pole at y (see Section 11.4 of [10] for definitions), then X h is the process X conditioned to converge to y and killed at its last exit from y . We refer the reader to Chapter 11of [10] for an in-depth analysis of h -transforms.This paper proposes a new class of path transformations for one-dimensional regular diffusionswith stochastic differential equation (SDE) representation. The new transformations are aimed atswitching the behaviour of the diffusion from transient to recurrent or vice versa. We introducethe concept of recurrent transformation in Section 3 and characterise these transforms via weaksolutions of SDEs. Roughly speaking, a recurrent transformation adds a drift term to the originalSDE of X so that the resulting process is a recurrent regular diffusion with the same state spacewhose law is locally absolutely continuous with respect to the original law. Although the recurrenttransformation is at first sight meaningful only for transient diffusions, we note a special class ofrecurrent transformations in Theorem 3.3 that is applicable not only to transient diffusions but alsoto recurrent ones. This transform, by adding again a certain drift, results in a positively recurrent diffusion. For example, this transformation turns a standard Brownian motion to a Brownian Date : February 2, 2018. motion with alternating drift, which appears in the studies of the bang-bang control problem (seeExample 3.2).As a first application of the recurrent transformation, we compute in Corollary 3.1 the distri-bution of the first exit time from an interval for a given diffusion. Although the formula does notprovide an expression in closed form in general, a simple Monte Carlo algorithm will provide asufficiently close estimate.The distribution of first exit times has attracted the attention of researchers working on problemsarising in the Monte Carlo simulation of stochastic processes (see, e.g., [1], [2], [21], [22], and thereferences therein). Yet precise formulas for the distribution of exit times of diffusions have rarelybeen the subject of a thorough investigation. The recent paper of Karatzas and Ruf [26] seemsto be the only work in the literature that addresses this problem in the general framework ofone-dimensional diffusions. With an additional assumption on the local H¨older continuity of thecoefficients of the SDE satisfied by X they have shown that the distribution function of the firstexit time was the minimal nonnegative solution of a particular Cauchy problem. Although this isa useful characterisation from a theoretical perspective, finding the smallest solution of a Cauchyproblem is in general not a feasible numerical task. Our formula in Corollary 3.1 thus provides away of computing the minimal solutions of the class of Cauchy problems considered by Karatzasand Ruf.As described briefly in Remark 3.4 recurrent transformations can also be used to improve theaccuracy of discrete Euler approximations of a diffusion killed when exiting a bounded interval.As shown by Gobet [21] the discretisation error for such Euler schemes is of order N − , where N is the number of discretisations, as opposed to N − , which is the rate of convergence for discreteEuler schemes for diffusions without killing. As the recurrent transformation removes the killingby passing to a locally absolutely continuous probability measure, it can be used to bring theconvergence rate back to N − using the recipe in Remark 3.4. This important application ofrecurrent transformations will be studied rigorously in a subsequent paper.Section 4 is devoted to the convergence of certain recurrent transforms when X is nonnegativeand on natural scale. Under a mild condition on the diffusion coefficient of X we show that aparticular sequence of recurrent transformations converges monotonically to the h -transform of X ,where h ( x ) = x . We observe that the nature of this convergence depends crucially on whether X is a strict local martingale or not. In particular, we construct on a single probability space asequence of recurrent transforms that increases a.s. to a diffusion that has the same law as theaforementioned h -transform. The limiting diffusion is non-exploding on [0 , ∞ ) if and only if X is atrue martingale.Our interest in local martingales in fact stems from the financial models with bubbles. If afinancial model admits no arbitrage opportunities, the discounted stock price X must follow anonnegative local martingale under a so-called risk-neutral measure by the Fundamental Theoremof Asset Pricing [14]. When X is not a martingale but a strict local martingale, the stock priceexhibits a bubble and many results in the arbitrage pricing theory become invalid (see [12] and[34] for some examples). One particular issue concerns the Black-Scholes pricing equation for aEuropean option that pays the amount of g ( X T ) to its holder at time T for some g ≥
0. Thearbitrage pricing theory suggests that the fair price of this option at time t is v ( T − t, X t ), where v ( t, x ) := E x [ g ( X t )]. Under mild conditions on X and a continuity and linear growth assumptionon g , Ekstr¨om and Tysk [17] have shown that v satisfies the Cauchy problem u t = Au, u (0 , · ) = g, (1.1) IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 3 where A is the infinitesimal generator of X . As a consequence, Ekstr¨om and Tysk have observedin [17] that (1.1) admits multiple nonnegative solutions when X is a strict local martingale and g ( x ) = x . Namely, they have identified u ( t, x ) := x and v ( t, x ) = E x [ X t ] as such two distinctsolutions. Note that X being a strict local martingale implies v ( t, x ) = E x [ X t ] < x = u ( t, x ). Thus, x − E x [ X t ] is a solution of (1.1) when g ≡
0. However, this immediately leads to the conclusionthat there are infinitely many solutions of at most linear growth to (1.1) whenever g is of at mostlinear growth. Indeed, by the above discussion for any α >
0, ˜ u ( t, x ) := E x [ g ( X t )] + α ( x − E x [ X t ])is a nonnegative solution of (1.1) when g is of at most linear growth. Moreover, Ekstr¨om and Tyskhave also shown that E x [ g ( X t )] is of at most linear growth when g is continuous function of atmost linear growth. This in turn renders ˜ u of linear growth. Hence, restricting solutions to haveat most linear growth does not yield uniqueness for the above Cauchy problem.Bayraktar and Xing [5] have followed up this question by showing that the uniqueness of theCauchy problem is determined by the martingale property of X . Later, Bayraktar et al. [4] haveextended the scope of these conclusions to Markovian stochastic volatility models.The absence of uniqueness for solutions of (1.1) is especially problematic if one wants to computethe option prices by solving (1.1) numerically. Also note that one will also fail to compute E x [ g ( X t )]using a Monte-Carlo simulation when g is of linear growth and X is a strict local martingale. Indeed,if, e.g., g ( x ) = x , the Monte-Carlo algorithm will yield x for E x [ X t ] since the discretisation of X viathe Monte-Carlo scheme will result in a true martingale for the approximating process. To resolvethis issue we establish in Section 6 a new characterisation of E x [ g ( X t )] in terms of the uniquesolution of an alternative Cauchy problem. We show that the function ( t, x ) E x [ g ( X t )], after anappropriate scaling, becomes the unique solution of w t = ˜ Aw with certain initial and boundary conditions, when ˜ A is the generator of a suitable h -transform of X . More precisely, this h -transform coincides with the one that is obtained as the limit of recurrenttransforms in Section 4. One interesting corollary of the main result of this section is that for any t > E x [ g ( X t ) is of strictly sublinear growth at ∞ when g is of at most lineargrowth and the stock price is given by a strict local martingale. In particular, lim x →∞ E x [ X t ] x = 0for any t > X is a strict local martingale.While Section 6 is on the valuation of European options, Section 7 considers the pricing ofperpetual American options. In order to price such an option with payoff g , one needs to solve theoptimal stopping problem sup τ ≤ ζ E x h e − λτ g ( X τ ) i , where ζ is the (possibly finite) lifetime of the diffusion X and the discount rate λ > X . Another approach is via the characterisation of λ -excessive functions of X as thevalue function for the optimal stopping problem is the least λ -excessive majorant of g . This isthe path taken by Dayanik and Karatzas in [13]. Beibel and Lerche [6] have also proposed a newmethodology based on simple martingale arguments, which can also be interpreted as change ofmeasure arguments as observed by [31]. While the approach based on the solution of a free boundaryproblem rarely provides explicit solutions, the other two have the potential to offer explicit or semi-explicit solutions. However, these solutions crucially depends on the assumption that one has thesolutions of a family of Sturm-Liouville equations at hand. Moreover, the solution techniques offered UMUT C¸ ETIN in [13] and [6] differ for different boundary behaviour exhibited by X , i.e. whether the boundariesof the state space of X are absorbing or natural, etc. Furthermore, how the function g behaves nearthe boundaries also matters. For instance, Beibel and Lerche [6] have to check five conditions onthe behaviour of g to determine the solution. It is also worth to note the recent work of Lambertonand Zervos [30] who analyse a large class of optimal stopping problems via variational equalitiesdefined by the generator of X and g without the assumption that g is continuous.Section 7 presents a unified solution to the above optimal stopping problem that does not varydepending on the behaviour of g or X near the boundaries. We use the specific recurrent transformof Proposition 3.2, which is applicable to transient as well as recurrent diffusions, to determinewhether the value function is finite. We show that the value function is finite if and only if g satisfies the single condition (7.8), which depends only on the knowledge of u λ ( · , y ), the λ -potentialdensity, for some y . This recurrent transform also changes the optimal stopping problem to onewithout discounting. However, the new problem becomes two-dimensional. In order to reduce thedimension of the problem to one, we apply the transformation that is defined in Section 5, whichis aimed at conditioning the recurrent transformation to have a certain behaviour at the boundarypoints and become transient. After this transformation all that remains to do is to solvesup τ ˜ E x [¯ g ( X τ )] , where ¯ g is a function that depends only on g and u λ ( · , y ), and ˜ E corresponds to the expectationoperator with respect to the law of the diffusion after the final transformation. Solution to theabove is easy and well-known since Dynkin [16]: After a change of scale, the value function of theabove optimal stopping problem is the smallest concave majorant of ¯ g .It has to be noted that Cisse et al. [11] have attacked this problem using h -transforms. However,as we explain in detail in Remark 7.1 the authors make some implicit assumptions regarding theboundary behaviour of X as well as the function g in the proof of their key arguments. Theseassumptions in particular exclude the diffusion processes with infinite lifetime. As we mentionedabove, our approach is general and do not impose any conditions on X other than the regularityand the Engelbert-Schmidt conditions that ensures an SDE representation for X .In essence our framework is fundamentally different in spirit from [11] and [6] in the sense thatit gives a probabilistic interpretation of the value function and the optimal stopping boundariesunder a locally absolutely continuous measure in the classical framework of Dynkin [16] with nodiscounting. The works of [11] and [6], on the other hand, obtain the solution by a clever algorithmof maximisation provided one has the solutions of a family of Sturm-Liouville equations.Differently from our treatment in Section 6 we do not investigate the impact of martingaleproperty of X on the valuation of perpetual American options as the methodology is the same forthe martingales as well as the local martingales. We refer the reader to [3] for a thorough analysisof the influence of the martingale property in a general framework.An outline of this paper is as follows. Section 2 gives a brief overview of several concepts related toone-dimensional diffusions that will be used throughout the paper. Section 3 introduces the conceptof recurrent transformations while Section 4 considers their limit in relation to the local martingaleproperty of X . Section 5 defines a transform designed specifically for recurrent diffusions that isdifferent than the typical h -transform but will still render them transient, which will be useful inSection 7. Section 6 provides a resolution to the non-uniqueness issue of the Black-Scholes pricingequation and Section 7 addresses the optimal stopping problem. Section 8 concludes. Proofs ofcertain results that are not contained in the main body is included in the Appendix. Acknowledgements:
I’d like to thank Johannes Ruf for the useful discussions and the anony-mous referees for their comments that led to several improvements.
IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 5 Preliminaries
Let X be a regular diffusion on ( l, r ), where −∞ ≤ l < r ≤ ∞ . We assume that if any of theboundaries are reached in finite time, the process is absorbed at that boundary. This is the onlyinstance when the process can be ‘killed’, we do not allow killing inside ( l, r ). Such a diffusion isuniquely characterised by its scale function s and speed measure m , defined on the Borel subsetsof the open interval ( l, r ). The set of points that can be reached in finite time starting from theinterior of ( l, r ) and the entrance boundaries will be denoted by I . That is, I is the union of ( l, r )with the regular, exit or entrance boundaries. The law induced on C ( R + , I ), the space of I -valuedcontinuous functions on [0 , ∞ ), by X with X = x will be denoted by P x as usual, while ζ willcorrespond to its lifetime, i.e. ζ := inf { t > X t ∈ { l, r }} . For concreteness we assume that X is the coordinate process on the canonical space Ω := C ( R + , I ), i.e. X t ( ω ) = ω ( t ) for all t ≥ P x ) x ∈ I are properly defined. The filtration ( F t ) t ≥ will correspond to theuniversal completion of the natural filtration of X and, therefore, is right continuous since X isstrong Markov by definition (see Theorem 4 in Section 2.3 in [10]). We will also set F := W t ≥ F t .If µ is a measure on some open interval ( a, b ) and f is a nonnegative or µ -integrable measurablefunction, the integral of f with respect to µ will be denoted by R ( a,b ) f ( x ) µ ( dx ) unless µ is absolutelycontinuous with respect to the Lebesgue measure dx , in which case we shall write R ba f ( x ) µ ( dx ).In what follows we will often replace ζ with ∞ when dealing with the limit values of the processesas long as no confusion arises. Recall that in terms of the first hitting times, T y := inf { t > X t = y } for y ∈ ( l, r ), the regularity amounts to P x ( T y < ∞ ) > x and y belongs to the openinterval ( l, r ). This assumption entails in particular that s is strictly increasing and continuous (seeProposition VII.3.2 in [36]) and 0 < m (( a, z )) < ∞ for all l < a < z < r (see Theorem VII.3.6 andthe preceding discussion in [36]).Recurrence or transience of X depends on the behaviour of s near the boundary points. Moreprecisely, X is transient if and only if at least one of s ( l ) and s ( r ) is finite. Since s is unique onlyup to an affine transformation, we will use the following convention throughout the text: • s ( l ) = 0 whenever finite, • s ( r ) = 1 whenever finite.Note that in view of our foregoing assumptions one can easily deduce that X ζ − ∈ { l, r } when X is transient. We refer the reader to [9] for a summary of results and references on one-dimensionaldiffusions. The definitive treatment of such diffusions is, of course, contained in [25]. The recentmanuscript of Evans and Hening [19] contains a detailed discussion with proofs of some aspects ofthe potential theory of one-dimensional diffusions. Remark 2.1.
It has to be noted that notion of recurrence that we consider here excludes somerecurrent solutions of one-dimensional SDEs with time-homogeneous coefficients since we kill ourdiffusion as soon as it reaches a regular boundary point. A notable example is a squared Besselprocess with dimension δ < , which solves the following SDE: X t = x + 2 Z t p X s dB s + δt. The above SDE has a global strong solution, i.e. solution for all t ≥ , which is recurrent (seeSection XI.1 of [36] ). However, the point is reached a.s. and is instantaneously reflecting byProposition XI.1.5 in [36] . As such, it violates our assumption of a diffusion being killed at aregular boundary. According to our assumption, a squared Bessel process of dimension < δ < has to be killed as soon as it reaches and, thus, is a transient diffusion. UMUT C¸ ETIN
As our focus is on diffusions that are also solutions of SDEs, we further impose the so-called
Engelbert-Schmidt conditions . That is, we shall assume the existence of measurable functions σ : ( l, r ) → R and b : ( l, r ) → R such that σ ( x ) > ∃ ε > Z x + εx − ε | b ( y ) | σ ( y ) dy < ∞ for any x ∈ ( l, r ). (2.1)Under this assumption (see [18] or Theorem 5.5.15 in [27]) there exists a unique weak solution (upto the exit time from the interval ( l, r )) to the SDE X t = x + Z t σ ( X s ) dB s + Z t b ( X s ) ds, t < ζ, (2.2)where ζ = inf { t > X t ∈ { l, r }} and l < x < r . Moreover, condition (2.1) further implies one cantake s ( x ) = Z xC exp (cid:18) − Z zc b ( u ) σ ( u ) du (cid:19) dz and m ( dx ) = 2 s ′ ( x ) σ ( x ) dx, for some ( c, C ) ∈ ( l, r ) . (2.3)We collect the assumptions on X in the following: Assumption 2.1. X is a regular one-dimensional diffusion on ( l, r ) such that X t = X + Z t σ ( X s ) dB s + Z t b ( X s ) ds, t < ζ, where σ : ( l, r ) → R and b : ( l, r ) → R satisfy (2.1), ζ = inf { t > X t ∈ { l, r }} . In the sequel the extended generator of X will be denoted by A . Following Definition VII.1.8 ofRevuz and Yor [36] we will write g = A f for a given Borel measurable function f , if there existsBorel function g such that, for each x ∈ I , i) P x -a.s. R t | g ( X s ) | ds < ∞ for every t >
0, and ii) f ( X t ) − f ( X ) − Z t g ( X s ) ds is P x -local martingale. In this case f is said to be in the domain of A . If f is C on I , then A becomes a second order differential operator, i.e. A f ( x ) = 12 σ ( x ) f ′′ ( x ) + b ( x ) f ′ ( x ) . Any regular transient diffusion on ( l, r ) has a finite potential density, u : ( l, r ) → R + , withrespect to its speed measure (see Paragraph 11 in Section II.1 of [9]). That is, for any nonnegativeand measurable f vanishing at accessible boundaries U f ( x ) := Z ∞ E x [ f ( X t )] dt = Z rl f ( y ) u ( x, y ) m ( dy ) . The above implies that the potential density can be written in terms of the transition density ,( p ( t, · , · )) t ≥ , of X with respect to its speed measure: u ( x, y ) = Z ∞ p ( t, x, y ) dt. The above in particular implies that u ( x, y ) = u ( y, x ) since p ( t, · , · ) is symmetric for each t > X is recurrent , either U f ≡ ∞ or U f ≡ For the existence of this transition density and its boundary behaviour see Mc Kean [33].
IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 7
We will denote by ( L xt ) x ∈ ( l,r ) the family of semimartingale local times associated to X . Recallthat the occupation times formula for the semimartingale local time is given by Z t f ( X s ) σ ( X s ) ds = Z rl f ( x ) L xt dx. In the case of one-dimensional transient diffusions the distribution of L y ∞ is known explicitly interms of the potential density (see p.21 of [9]). In particular, P y ( L y ∞ > t ) = exp (cid:18) − s ′ ( y ) t u ( y, y ) (cid:19) . (2.4)Note that if s ( l ) = 0 = 1 − s ( r ), then P x ( X ∞ = r ) = s ( x ) = 1 − P x ( X ∞ = l ) and P x ( T y < ∞ ) = ( s ( x ) s ( y ) , y ≥ x ; − s ( x )1 − s ( y ) , y < x. u ( x, y ) = s ( x )(1 − s ( y )) , x ≤ y. (2.5)On the other hand, if s ( l ) = 0 and s ( r ) = ∞ , then X t → l , P x -a.s. for any x ∈ ( l, r ), which in turnimplies P x ( T y < ∞ )) = ( s ( x ) s ( y ) , y ≥ x ;1 , y < x. u ( x, y ) = s ( x ) , x ≤ y. (2.6)Similarly, if s ( l ) = −∞ and s ( r ) = 1, then X t → r , P x -a.s. for any x ∈ ( l, r ), and P x ( T y < ∞ ) = ( , y ≥ x ; − s ( x )1 − s ( y ) , y < x. u ( x, y ) = 1 − s ( y ) , x ≤ y. (2.7)While the potential density is finite only for transient diffusions, one can define a so-called α -potential density that exists and is finite for all diffusions for all α >
0. For any nonnegative andmeasurable function f vanishing at accessible boundaries, one defines U α f ( x ) := Z ∞ e − αt E x [ f ( X t )] dt. Thus, if we let u α ( x, y ) := Z ∞ e − αt p ( t, x, y ) dt, we obtain U α f ( x ) = Z rl f ( y ) u α ( x, y ) m ( dy ) .u α ( · , · ) is called the α -potential density and is symmetric in ( l, r ) for all α >
0. An alternativeand very useful expression for u α is given in terms of the fundamental solutions of the equation A f = αf . That is, u α ( x, y ) = ψ α ( x ) φ α ( y ) w α , x ≤ y, (2.8)where ψ α and φ α are, respectively, the increasing and decreasing nonnegative solutions of A f = αf subject to certain boundary conditions (see p.19 of [9]), and w α is the Wronskian given by w α = ψ ′ α ( x ) φ α ( x ) − ψ α ( x ) φ ′ α ( x ) s ′ ( x ) , Observe that the diffusion local time, ˜ L , in Paragraph 13 in Section II.2 of [9] is defined via R t f ( X s ) ds = R rl f ( x ) ˜ L xt m ( dx ). Comparing this with the occupation times formula for the semimartingale local time reveals therelationship s ′ ( x ) ˜ L x = L x . UMUT C¸ ETIN which is independent of x . Consequently, using the relationship between the fundamental solutionsof A f = αf and the Laplace transforms of hitting times (see p.18 of [9]), we have E x [exp ( − αT y )] = u α ( x, y ) u α ( y, y ) . (2.9)We refer the reader to Chap. II of Borodin and Salminen [9] for a summary of results concerningone-dimensional diffusions including the ones sketched above.3. Recurrent transformations of diffusions
This section introduces a new kind of path transformation for regular diffusions that producesa recurrent diffusion whose law is locally absolutely continuous with respect to that of the originaldiffusion. To wit, suppose h is a non-negative C -function and M an adapted continuous processof finite variation so that h ( X ) M is a non-negative local martingale. If ( τ n ) is a localising sequencefor this local martingale, using Girsanov’s theorem we arrive at a weak solution on [0 , τ n ] to thefollowing SDE for any given x ∈ ( l, r ): X t = x + Z t σ ( X s ) dB s + Z t (cid:26) b ( X s ) + σ ( X s ) h ′ ( X s ) h ( X s ) (cid:27) ds. (3.1)We can associate to the above SDE the scale function s h ( x ) := Z xc s ′ ( y ) h ( y ) dy, x ∈ ( l, r ) , (3.2)provided that the integral is finite for all x ∈ ( l, r ), which in particular requires h > l, r ).What we would like to achieve is to extend this procedure by taking n → ∞ and obtain a recurrentdiffusion. The latter will require − s h ( l +) = s h ( r − ) = ∞ . We shall see in this section that thisproperty alone is sufficient to obtain a recurrent weak solution of (3.1) on [0 , ∞ ) under some mildconditions on h .Using h and M to get a recurrent process imposes some boundary conditions on h . Indeed, if s ( l ) = 0 (resp. s ( r ) = 1), in order to have s h ( l +) = −∞ (resp. s h ( r − ) = ∞ ), we must havelim x → l h ( x ) = 0 (resp. lim x → r h ( x ) = 0).Moreover, since h ( X ) M is a local martingale, dM t = − M t A h ( X t ) h ( X t ) dt . Thus, M is given by M t = exp (cid:18) − Z t A h ( X s ) h ( X s ) ds (cid:19) . In the light of the above discussion we now introduce the concept of a recurrent transformationof a diffusion . Definition 3.1.
Let X be a regular diffusion satisfying Assumption 2.1 and h : I → [0 , ∞ ) bean absolutely continuous function. Then, ( h, M ) is said to be a recurrent transform (of X ) if thefollowing are satisfied:(1) M is an adapted process of finite variation.(2) h ( X ) M is a nonnegative local martingale.(3) The function s h from (3.2) is finite for all x ∈ ( l, r ) with − s h ( l +) = s h ( r − ) = ∞ .(4) There exists a unique weak solution to (3.1) for t ≥ for any x ∈ ( l, r ) . In the above definition, the defining condition for a recurrent transformation is the function s h and its explosive nature near the boundaries. The function h and the functional M come into playwhen one wants to construct a weak solution of the SDE (3.1) and show that the law of its solutionis locally absolutely continuous with respect to that of the original process X , which satisfies (2.2). IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 9
The next theorem, whose proof is delegated to the Appendix, suggests a general machinery forconstructing recurrent transformations.
Theorem 3.1.
Let X be a regular diffusion satisfying Assumption 2.1. Consider an absolutelycontinuous function h : I → [0 , ∞ ) such that its left derivative h ′ is of finite variation. Supposefurther that the mapping s h given by (3.2) is finite for all x ∈ ( l, r ) and that − s h ( l +) = s h ( r − ) = ∞ .Then, the following statements are valid.(1) h ′ can be chosen to be left-continuous. Moreover, the signed measure defined by h ′ on ( l, r ) admits the Lebesgue decomposition dh ′ ( x ) = h ′′ ( x ) dx + n ( dx ) , where h ′′ denote its Borelmeasurable Radon-Nikodym derivative with respect to the Lebesgue measure on ( l, r ) , and n is a locally finite signed measure on ( l, r ) that is singular with respect to the Lebesguemeasure.(2) The integral [ t<ζ ] (cid:18)Z t (cid:12)(cid:12)(cid:12) ˜ A h ( X s ) (cid:12)(cid:12)(cid:12) ds + Z rl L xt | n ( dx ) | (cid:19) < ∞ , P y -a.s. , (3.3) for every y ∈ ( l, r ) , where ˜ A h ( x ) = σ ( x )2 h ′′ ( x ) + b ( x ) h ′ ( x ) .(3) ( h, M ) is a recurrent transform, where, on [ t < ζ ] , M t := exp − Z t ˜ A h ( X s ) h ( X s ) ds − Z t h ( X s ) d Λ s ( h ) ! and Λ t ( h ) := Z ( l,r ) L xt n ( dx ) . (4) inf { t > h ( X t ) M t = 0 } = ζ, P x -a.s..(5) Let R h,x be the law of the solution of (3.1) and F ∈ F T for some ( F t ) -stopping time T .Then, R h,x ( F, T < ∞ ) = 1 h ( x ) E x [ F h ( X T ) M T ] . (3.4) In particular, h ( X ) M is a P x -martingale.(6) If T is an ( F t ) -stopping time such that R h,x ( T < ∞ ) = 1 , then for any F ∈ F T the followingidentity holds: P x ( ζ > T, F ) = h ( x ) E h,x (cid:20) F h ( X T ) M T (cid:21) , (3.5) where E h,x is the expectation operator with respect to the probability measure R h,x . Example 3.1.
Suppose δ > and consider a δ -dimensional Bessel process on (0 , ∞ ) , i.e. a one-dimensional diffusion with the dynamics dX t = 2 p X t dB t + δdt. The scale function is given by s ( x ) = 1 − x − δ . Thus, X is transient and approaches to ∞ as t → ∞ , while is an inaccessible boundary.Let h ( x ) := x − δ and define M t := exp (cid:18) ( δ − Z t X s ds (cid:19) , t ≥ . Then, it follows from Theorem 3.1 that M is of finite variation. Moreover, s h ( x ) = δ − Z x u du = log x, x > . Thus, − s h (0) = s h ( ∞ ) = ∞ , and we conclude that ( h, M ) is a recurrent transform by invokingTheorem 3.1 again. The transformation yields the following SDE for the resulting process dX t = 2 p X t dB t + 2 dt, which is the SDE for a -dimensional squared Bessel process. Recall (or see p.442 of [36] ) that ispolar for a -dimensional squared Bessel process. The following proposition gives an important example of a recurrent transformation for transientdiffusions, which will be useful in the sequel.
Proposition 3.1.
Suppose X is a regular transient diffusion satisfying Assumption 2.1. Let y ∈ ( l, r ) be fixed and consider the pair ( h, M ) defined by h ( x ) := u ( x, y ) , x ∈ ( l, r ) , and M t = exp (cid:18) s ′ ( y ) L yt u ( y, y ) (cid:19) . Then, the following hold:(1) ( h, M ) is a recurrent transform for X .(2) There exists a unique weak solution to X t = x + Z t σ ( X s ) dB s + Z t (cid:26) b ( X s ) + σ ( X s ) u x ( X s , y ) u ( X s , y ) (cid:27) ds, t ≥ , (3.6) for any x ∈ ( l, r ) , where u x denotes the first partial left derivative of u ( x, y ) with respect to x .(3) Moreover, if R h,x denotes the law of the solution and T is a stopping time such that R h,x ( T < ∞ ) = 1 , then for any F ∈ F T the following identity holds: P x ( ζ > T, F ) = u ( x, y ) E h,x (cid:20) F u ( X T , y ) exp (cid:18) − s ′ ( y )2 u ( y, y ) L yT (cid:19)(cid:21) , (3.7) where E h,x is the expectation operator with respect to the probability measure R h,x . The above is a direct corollary of Theorem 3.1 since n ( dx ) = − s ′ ( y ) ε y ( dx ) in the Lebesguedecomposition of du x ( x, t ) as in Part (1) of Theorem 3.1 and u ( · , y ) is twice differentiable with σ ( x ) u xx ( x, y ) + b ( x ) u x ( x, y ) = 0 for all x = y .Proposition 3.1 is in fact a special case of a more general result that will allow us to construct alarge family of recurrent transformations. In order to motivate this more general result note that u ( · , y ) is the potential of the Dirac measure at point y . Moreover, it is uniformly integrable beingbounded. Conversely, since X in Assumption 2.1 is a symmetric diffusion, it is well-known (see,e.g., Theorem VI.2.11 in [8]) any uniformly integrable potential h is the potential of some measure µ on ( l, r ), i.e. h ( x ) = R ( l,r ) u ( x, y ) µ ( dy ). Also note that if h h = u ( · , y ), then h ( X ) is a supermartingale, which is not a martingale. As a matter of fact, inview of the Riesz representation of excessive functions (see Theorem VI.2.11 in conjunction withProposition IV.5.4 in [8]) the greatest uniformly integrable harmonic function dominated by h is 0. If µ is a measure on ( l, r ), the potential of µ is the function x R ( l,r ) u ( x, y ) µ ( dy ) and is denoted by Uµ . SeeSection VI.2 of [8] for details. IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 11
The next result, whose proof is in the Appendix, shows that the potential of a probability measureon ( l, r ) gives rise to a recurrent transform under an integrability condition.
Theorem 3.2.
Let µ be a Borel probability measure on ( l, r ) such that R ( l,r ) | s ( y ) | µ ( dy ) < ∞ .Suppose X is a regular transient diffusion satisfying Assumption 2.1 and define h ( x ) := Z ( l,r ) u ( x, y ) µ ( dy ) . (1) The left derivative h ′ of h exists and ( h, M ) is a recurrent transform of X , where M t := exp (cid:18)Z t h ( X s ) dA s (cid:19) and A t := Z ( l,r ) s ′ ( x ) L xt µ ( dx ) . (2) If R h,x denotes the law of the solution of (3.1) and T is a stopping time such that R h,x ( T < ∞ ) = 1 , then for any F ∈ F T the following identity holds: P x ( ζ > T, F ) = h ( x ) E h,x (cid:20) F h ( X T ) exp (cid:18) − Z t h ( X s ) dA s (cid:19)(cid:21) , where E h,x is the expectation operator with respect to the probability measure R h,x . Remark 3.1.
Note that u ( · , y ) satisfies the assumptions of the above theorem since µ = ε y and R ( l,r ) u ( x, z ) µ ( dz ) = u ( x, y ) < ∞ for all x ∈ ( l, r ) . Thus, Proposition 3.1 is a direct consequence ofTheorem 3.2 as well. The next example of a recurrent transform that we shall consider in this paper is obtained viathe α -potential density, u α of X . In contrast with the previous transform, which only exists fortransient diffusions, the next transform can be applied to all regular diffusions. Moreover, theresulting diffusion will be positive recurrent . Proposition 3.2.
Suppose X is a regular diffusion satisfying Assumption 2.1. Let y ∈ ( l, r ) and α > be fixed and consider the pair ( h, M ) defined by h ( x ) := u α ( x, y ) , x ∈ ( l, r ) , and M t = exp (cid:18) − αt + s ′ ( y ) L yt u α ( y, y ) (cid:19) . Then, the following hold:(1) ( h, M ) is a recurrent transform for X .(2) There exists a unique weak solution to X t = x + Z t σ ( X s ) dB s + Z t (cid:26) b ( X s ) + σ ( X s ) u αx ( X s , y ) u α ( X s , y ) (cid:27) ds, t ≥ , (3.8) for any x ∈ ( l, r ) , where u αx denotes the first partial left derivative of u α ( x, y ) with respectto x .(3) Moreover, the diffusion defined by the solutions of (3.8) is positive recurrent and its sta-tionary distribution on ( l, r ) is given by π ( dx ) = ( u α ( x, y )) R ∞ se − αs p ( s, y, y ) ds m ( dx ) , (3.9) where ( p ( t, · , · )) t> is the transition density of the original diffusion with respect to its speedmeasure m . As in the case of Proposition 3.1, parts (1) and (2) of the above result is a direct corollary ofTheorem 3.1 but will also be a special case of a more general theorem in terms of α -potentials.Analogously, u α ( · , y ) of X is the α -potential of the Dirac measure at y and ( e − αt u α ( X t , y )) is auniformly integrable supermartingale converging a.s. to 0 as t → ζ . Moreover, any uniformlyintegrable α -potential is of the form R rl u α ( x, y ) µ ( dy ) for some measure on ( l, r ). Theorem 3.3.
Suppose X is a regular diffusion satisfying Assumption 2.1 and α > . Let µ be aBorel probability measure on ( l, r ) such that R ( l,r ) u α ( y, y ) dµ ( y ) < ∞ . Define h ( x ) := Z ( l,r ) u α ( x, y ) µ ( dy ) . (1) The left derivative h ′ of h exists and ( h, M ) is a recurrent transform of X , where M t := exp (cid:18) − αt + Z t h ( X s ) dA s (cid:19) and A t := Z ( l,r ) s ′ ( x ) L xt µ ( dx ) . (2) Moreover, if there exists ε > such that R ( l,r ) u α − ε ( y, y ) µ ( dy ) < ∞ , then the diffusiondefined by the solutions of (3.1) is positive recurrent and its stationary distribution on ( l, r ) is given by π ( dx ) = h ( x ) R rl h ( y ) m ( dy ) m ( dx ) . Remark 3.2.
Note that u α ( · , y ) satisfies the assumptions of the above theorem since µ = ε y and R ( l,r ) u α − ε ( x, z ) µ ( dz ) = u α − ε ( x, y ) < ∞ for all ε ∈ [0 , α ) . Thus Proposition 3.2 follows directlyfrom Theorem 3.3.Moreover, if X is transient, the potential density u exists and is finite. In this case the condition R ( l,r ) u ( y, y ) µ ( dy ) < ∞ is equivalent to R ( l,r ) | s ( y ) | µ ( dy ) < ∞ under the assumption that µ is aprobability measure. Thus, the condition R ( l,r ) u α ( y, y ) µ ( dy ) < ∞ in Theorem 3.3 is the exactanalogue of the condition R ( l,r ) | s ( y ) | µ ( dy ) < ∞ of Theorem 3.2.If f is nonnegative, R rl f ( x ) m ( dx ) = 1 , and R rl f ( x ) u α ( x, x ) m ( dx ) < ∞ , Theorems 3.2 and 3.3show that h ( x ) := U α f ( x ) will define a recurrent transform for α ≥ . In this case the finitevariation process A will be given by A t = Z t f ( X s ) ds. For instance, in Example 3.1 it can be verified using the scale function and the speed measureof squared Bessel processes that h ( x ) = ( δ − R ∞ u ( x, y ) y − δ +24 m ( dy ) leading to dA t = X − δ +24 t dt inthe notation of Theorem 3.2. Example 3.2.
Suppose X is a standard Brownian motion. It is well-known that u α ( x, y ) = 1 √ α exp (cid:16) −√ α | x − y | (cid:17) . Thus, if we use the transform in Proposition 3.2 with y = 0 , the recurrent transform is the solutionto the following SDE: dX t = dB t − √ α sgn( X t ) dt, where sgn( x ) = − [ x< + [ x ≥ . This is a Brownian motion with alternating state-dependent drift,which plays a key role in the so-called bang-bang control problem (see Section 6.6.5 in [27] and thereferences therein). IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 13
We shall consider in subsequent sections the applications of the above recurrent transforms tooptimal stopping as well as some pricing issues arising in Black-Scholes models when the stock pricefollows a strict local martingale. However, one can find an immediate application of the recurrenttransform to the computation of the distribution of the first exit time for a one-dimensional diffusionfrom an interval. Indeed, such a first exit time can always be viewed as the life time of a transientdiffusion by killing the original one as soon as it exits the given interval. Thus, the problem reducesto finding P x ( ζ > t ) for all t >
0, where P x is the law of the transient diffusion starting at x and ζ is its lifetime, i.e. the first time it exits the given interval. The following is a direct consequenceof Proposition 3.1. Corollary 3.1.
Let X be a regular transient diffusion satisfying Assumption 2.1. Then P x ( ζ > t ) = u ( x, y ) E h,x (cid:20) u ( X t , y ) exp (cid:18) − s ′ ( y )2 u ( y, y ) L yt (cid:19)(cid:21) , where E h,x is the expectation operator with respect to the law of the recurrent transform given by(3.6). Although the above formula does not in general give P x ( ζ > t ) in closed-form, it is neverthelesspractical. Indeed, by running a Monte-Carlo simulation of the solution of (3.6), one can get a closeestimate of E h,x (cid:20) u ( X t , y ) exp (cid:18) − s ′ ( y )2 u ( y, y ) L yt (cid:19)(cid:21) by approximating the local time using the occupation times formula.Karatzas and Ruf [26] have shown that the function v ( t, x ) := P x ( ζ > t ) is the smallest nonneg-ative classical supersolution of v t = Av, v (0 , · ) = 1 (3.10)under the assumption that σ and b are locally uniformly H¨older continuous on ( l, r ). Thus, com-bining their Proposition 5.4 and Corollary 3.1 we deduce the following. Corollary 3.2.
Let X be a regular transient diffusion satisfying Assumption 2.1. Assume furtherthat σ and b that appears in (2.2) are locally uniformly H¨older continuous on ( l, r ) . Define v ( t, x ) := u ( x, y ) E h,x (cid:20) u ( X t , y ) exp (cid:18) − s ′ ( y )2 u ( y, y ) L yt (cid:19)(cid:21) , where E h,x is the expectation operator with respect to the law of the recurrent transform given by(3.6). Then, v is the smallest nonnegative classical supersolution of (3.10). Remark 3.3.
In fact there is not a unique way of representing the minimal nonnegative classicalsupersolutions of (3.10). Indeed, if h is the potential of a probability measure µ on ( l, r ) satisfyingthe hypothesis of Theorem 3.2, then P x ( ζ > t ) = h ( x ) E h,x h h ( X t ) exp (cid:16) − Z ( l,r ) s ′ ( y ) L yt h ( y ) µ ( dy ) (cid:17)i . In particular if µ ( dy ) = f ( y ) m ( dy ) for some f , R ( l,r ) s ′ ( y ) L yt h ( y ) µ ( dy ) = R t f ( X s ) h ( X s ) ds . Remark 3.4.
The recurrent transformation of a transient diffusion can be used to improve the ac-curacy of discrete Euler approximations of diffusions that are killed when leaving a bounded interval [ a, b ] . Suppose ζ represents the first exit time from this interval and one is interested in the MonteCarlo simulation of E x [ F ( X T ) [ T <ζ ] ] for some suitable F via a discrete Euler scheme applied tothe SDE (2.2) for X . Gobet [21] has shown that the discretisation error is of order N − , where N is the number of discretisations. This order of convergence is exact and intrinsic to the killing.However, this corresponds to a loss of accuracy compared to the standard Euler scheme appliedto a diffusion without killing, where the error is of order N − . On the other hand, the recurrenttransformation from Theorem 3.2 can be used to improve the convergence rate back to N − since E x [ F ( X T ) [ T <ζ ] ] = h ( x ) E h,x h F ( X T ) 1 h ( X T ) exp (cid:16) − Z T f ( X s ) h ( X s ) ds (cid:17)i , where h ( x ) = R rl u ( x, y ) f ( x ) m ( dx ) for a nonnegative f with R rl f ( x ) m ( dx ) = 1 . This is due to thefact that there is no killing under R h,x , i.e. R h,x ( ζ = ∞ ) = 1 . We will study in more detail theimprovement of the discrete Euler scheme for killed diffusions in a subsequent paper. Connection with Doob’s h -transform. It is trivial to check that ( h, M )-recurrent trans-form of X has h dm as its speed measure. In the specific case considered in Proposition 3.1 therecurrent transform is a one-dimensional diffusion with scale s h ( x ) = Z xc s ′ ( z )( u ( z, y )) dz, and the speed measure ( u ( z, y )) m ( dz ). Note that this is not the only diffusion with this scalefunction and the speed measure. Indeed, if one considers the h -transform of X via h ( x ) = u ( x,y ) u ( y,y ) ,one obtains a diffusion which amounts to conditioning the paths of X to converge to y and killedat its last exit from y . The resulting diffusion is obviously a transient diffusion but has the samescale and the speed (see, e.g. Theorem 6.2 in [19] or Paragraph 31 in Section II.5 of [9]). Thecrucial difference between the two transformations is that the h -transform involves killing while therecurrent transform does not.Killing of the trajectories in the h -transform is also apparent from the following representation.Denoting the law of the h -transform by ˜ P u,x we deduce˜ E u,x [ F [ ζ>t ] ] = E x [ F u ( X t , y )] u ( x, y ) = E x (cid:2) F [ G y >t ] (cid:3) h ( x ) . In the above F is an F t -measurable random variable and G y := sup { t : X t = y } (see Section 3.9– in particular the expression (3.211)– in [32] for the details). The above identity in particularimplies ˜ P u,x ( ζ > t ) = P x ( G y > t ) h ( x ) , ∀ t ≥ , i.e., ˜ P u,x -distribution of ζ coincides with the law of G y under P x after a normalisation. Observethat P x ( G y < ζ ) = 1 since X is transient under P x .Given this close relationship between the recurrent transform and the h -transform one maywonder whether it is possible to obtain the latter from the former via a killing. This is in factpossible. Indeed, for any F t -measurable bounded random variable F , one has E h,x (cid:20) F exp (cid:18) − s ′ ( y )2 u ( y, y ) L yt (cid:19)(cid:21) = E x (cid:20) F u ( X t , y ) u ( x, y ) (cid:21) = ˜ E u,x (cid:2) F [ ζ>t ] (cid:3) . (3.11)Thus, if one kills the trajectories of the recurrent transform at rate s ′ ( y )2 u ( y,y ) L yt , then one obtainsthe h -transform. As such, h -transform is subordinate (see Section III.2 of [8] for a description ofsubordinate semigroups) to the recurrent transform, i.e. ˜ E u,x [ F ] ≤ E h,x [ F ] for all nonnegative F t -measurable F that vanishes on [ ζ, ∞ ). IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 15
We shall next describe how one can implement this killing in practice. To this end define S a := inf { t ≥ L yt > u ( y,y ) as ′ ( y ) } for a > α thatis independent from the recurrent process. Then, for any F t -measurable bounded random variable F E h,x (cid:2) F [ t
Motivation of this section comes from the financial models that we shall treat in more detail inSection 6. Consistent with the setting therein X will assumed to be a non-negative diffusion innatural scale in this section. As our focus is on strict local martingales this necessitates the choiceof r = ∞ . We also translate X so that l = 0. Consequently, u ( x, y ) = x ∧ y and the recurrenttransform in (3.6) reads X t = x + Z t σ ( X s ) dB s + Z t σ ( X s ) X s [ X s ≤ y ] ds, x > . (4.1)We have established in Proposition 3.1 that the above SDE has a non-explosive weak solution thatis unique in law. Moreover, the solution never hits 0. If ( X y ) y> denotes the solutions of (4.1)indexed by y , we notice immediately that the drift term associated to X y is increasing in y . Thus, ifthe solutions are strong, we may hope that the solutions are increasing in y under a mild hypothesison σ . Then, if we let Y t := lim y →∞ X yt , the resulting limit is expected to satisfy Y t = x + Z t σ ( Y s ) dB s + Z t σ ( Y s ) Y s ds, x > . (4.2)Since Y is obtained as an increasing limit of X y , it will never hit 0. However, its behaviour nearthe infinite boundary, and in particular whether it may explode in finite time, requires a furtherlook. We shall in fact see that Y is the SDE satisfied by the h -transform of X , where h ( x ) = x ,and its explosive behaviour depends exclusively on the strict local martingale property of X .The next assumption will be sufficient to ensure that the solutions of (4.1) are strong and increasein y . Note that one could get the existence and uniqueness of strong solutions under weakerhypothesis. However, the following stronger condition is imposed since we are also interested in acomparison result for the strong solutions. Assumption 4.1.
There exists a strictly increasing function ρ : [0 , ∞ ) → [0 , ∞ ) with Z ∞ ρ ( a ) da = ∞ such that ( σ ( x ) − σ ( y )) ≤ ρ ( | x − y | ) , x = y. As we will be working with strong solutions in this section let us fix a Brownian motion, β , ona fixed probability space (Ω , F , ( F t ) , P ), where ( F t ) t ≥ is as in Section 2, so that X t = x + Z t σ ( X s ) dβ s , x > . (4.3)It follows from Theorem IX.3.5 in [36] and Corollary 5.3.23 in [27] that X is the unique strongsolution of (4.3) under Assumptions 2.1 and 4.1.What we would like to achieve next is to pass to a locally absolutely continuous measure, whichwill support all the solutions of (4.1). The next result does not need Assumption 4.1. Proposition 4.1.
Suppose that Assumption 2.1 is in force and X satisfies (4.3) on (Ω , F , ( F t ) , P ) supporting the Brownian motion, β . There exists a Q on (Ω , F ) and a sequence of stopping times ( τ n ) n ≥ such that i) lim n →∞ Q ( τ n ≤ t ) = 0 , ii) Q | F τn ≪ P | F τn and iii) X t = x + Z t σ ( X s ) dB s + Z t σ ( X s ) X s [ X s ≤ ds, where B is a (Ω , F , ( F t ) , Q ) -Brownian motion.Proof. Consider the ( h, M ) transform in Proposition 3.1, where y = 1, and set τ n := inf { t ≥ L yt ≥ n } . Then, h ( X t ∧ τ n ) M t ∧ τ n is a bounded martingale that defines a Q n on F τ n . Note that Q n ( τ n ≤ t ) = R h,x ( L yt ≥ n ) using the notation of Proposition 3.1. Thus, lim n →∞ Q n ( τ n ≤ t ) =lim n →∞ R h,x ( L yt ≥ n ) = R h,x ( L yt = ∞ ) = 0, and i) and ii) follow from Theorem 1.3.5 in [40].Moreover, since Q agrees with Q n on F τ n , we have X t = x + Z t σ ( X s ) dB s + Z t σ ( X s ) X s [ X s ≤ ds, t < τ n , where B t = β t − Z t σ ( X s ) X s [ X s ≤ ds, t < τ n . As such, B is a Brownian motion stopped at τ n . Invoking the fact that lim n →∞ Q ( τ n ≤ t ) = 0yields iii). (cid:3) The above proposition constructs a locally absolutely continuous probability measure, Q , anda Q -Brownian motion, B . Thus, once we impose Assumption 4.1, (4.1) will possess the pathwiseuniqueness property by virtue of Proposition IX.3.1 and Lemma IX.3.1 in [36]. Combining thiswith Corollary 5.3.23 in [27] we arrive at the following. Proposition 4.2.
Suppose that Assumptions 2.1 and 4.1 hold. Let B and (Ω , F , ( F t ) , Q ) be as inProposition 4.1. Then, for each y > , there exists a unique strong solution to (4.1). As mentioned earlier Assumption 4.1 will also imply that the solutions of (4.1) are increasing in y . Proposition 4.3.
Suppose that Assumptions 2.1 and 4.1 hold. Let B and (Ω , F , ( F t ) , Q ) be as inProposition 4.1 and denote by X y the unique strong solution of (4.1). Then, Q ( X y t ≤ X y t , ∀ t ≥
0) = 1 whenever y ≤ y . IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 17
Proof.
Let b i ( x ) = σ ( x ) x [ x ≤ y i ] for i = 0 ,
1, and define b ε ( x ) = b ( x ) + ε. Observe that for anysufficiently small δ > g , such that b ( x ) ≤ g ( x ) ≤ b ε ( x ) for x > δ due to the continuity of σ . Thus, it follows from Theorem 1.1 in Chap. VI of [24] that X y t ≤ Z εt for all t < T δ , where T δ = inf { t ≥ X y t ≤ δ } and Z εt = x + Z t σ ( Z εs ) dB s + Z t b ε ( Z εs ) ds. Note that since σ satisfies (2.1) and Assumption 4.1 the above SDE has a unique strong solution.Since δ is arbitrary and lim δ → T δ = ∞ , Q -a.s., we immediately deduce that X y t ≤ Z εt for all t ≥ Z εt → X y t as ε → t < T δ .Indeed, we can again find a Lipschitz continuous function between b ε and b ε whenever ε < ε on ( δ, ∞ ) for any δ >
0. Therefore, the same theorem in [24] yields that Z εt is increasing in ε foreach t >
0. Set Z t = lim ε → Z εt . It follows from the continuity of σ and the dominated convergencetheorem for stochastic integrals thatlim ε → Z t σ ( Z εs ) dB s = Z t σ ( Z s ) dB s . Also observe that Z t < y if and only if Z εt < y for all but finitely many ε (number possiblydepending on ω ) since Z ε is decreasing to Z as ε →
0. Thus, b ε ( Z εt ) → b ( Z t ) as ε → t >
0. Since b ε n is uniformly bounded on ( δ, ∞ ) given any ( ε n ) n ≥ converging to 0, we deduce fromLebesgue’s dominated convergence theorem thatlim ε → Z t ∧ S δ b ε ( Z εs ) ds = Z t ∧ S δ b ( Z s ) ds, where S δ = inf { t > Z t < δ } . Thus, we have shown that Z solves (4.1) with y = y up to S δ .Since X y is the unique solution of this equation, we therefore establish that X y t = lim ε → Z εt for t ≤ S δ = inf { t > X y t < δ } . Therefore, X y t ≤ X y t for t < T δ . As before, we can pass to thelimit as δ → t ≥ X y t ≤ X y t . Moreover, due to the continuity of X y i s, we may choose a null set independent of t to deduce Q ( X y t ≤ X y t , ∀ t ≥
0) = 1. (cid:3)
Thanks to the above result X y is increasing in y and we can define Y t = lim y →∞ X yt . Moreover,the arguments used in the proof of the above proposition yields the following corollary. Corollary 4.1.
Suppose that Assumptions 2.1 and 4.1 hold and let X y be the unique strong solutionof (4.1), where B and (Ω , F , ( F t ) , Q ) are as in Proposition 4.1. Then, Y is the unique strongsolution of (4.2), where Y t = lim y →∞ X yt . It can be checked easily that the scale function of the diffusion in (4.2) is 1 − x . Thus, thesolution never hits 0 and diverges to ∞ as t → ∞ . Whether the explosion happens in finite timedepends on the martingale property of X . Note that if one is content with weak solutions, (4.2)has a unique weak solution when σ satisfies (2.1). Proposition 4.4.
Suppose that σ satisfies (2.1) and consider a weak solution, Y , of (4.2). Let Q x be the law of the solution of (4.2). Then, Q x (lim t →∞ Y t = ∞ ) = Q x ( Y t > , ∀ t >
0) = 1 . Inparticular, ζ = inf { t : Y t = ∞} , Q x -a.s. for each x > . Moreover, Q x ( ζ = ∞ ) = 1 if and only if X is a martingale, where X is given by (4.3).Proof. Note that the scale function of Y after our normalisation is given by s ( x ) = 1 − x . Thus,(2.7) applies and we deduce Q x (lim t →∞ Y t = ∞ ) = Q x ( Y t > , ∀ t >
0) = 1. Since ζ is the lifetimeof the diffusion, this also implies that ζ = inf { t : Y t = ∞} , Q x -a.s.. Next, it follows from Theorem 5.5.29 and Problem 5.5.27 in [27] that Q x ( ζ = ∞ ) = 1 if and onlyif lim x →∞ Z x x − zx zσ ( z ) dz = ∞ . However, Z x x − zx zσ ( z ) dz = 1 x Z xc Z yc zσ ( z ) dzdy. Thus, the above limit is valid if and only if Z ∞ zσ ( z ) dz = ∞ , which is well-known to be equivalent to the martingale property of X (see, e.g., Theorem 1.4 in[15] under a mild assumption on σ or Theorem 1 in [29] for a general result). (cid:3) Remark 4.1.
Using the methods employed in the proof of Theorem 3.1 one can show that the lawof (4.2) is equal to that of the h -transform of X , where h ( x ) = x . The relationship between themartingale property of X and the finiteness of the explosion time of its h -transform, i.e. Proposition4.4, has already been observed in the literature (see, e.g., [20] or, more recently, [28] ). As observed earlier 1 − /x is a scale function of Y . Consequently, 1 /Y is a nonnegative localmartingale. It turns out that the martingale property of 1 /Y is determined by whether X hits 0or not. Proposition 4.5.
Suppose that σ satisfies (2.1) and let Y be a weak solution of (4.2), whose lawis denoted by Q x . Then, Y is a Q x martingale if and only if P ( X t > , ∀ t >
0) = 1 , where X isgiven by (4.3).Proof. Denote Y by ξ . Then, dξ t = σ ( ξ t ) ξ t dB t for some Brownian motion B . It follows fromTheorem 1 in [29] that ξ is a martingale if and only if Z ∞ σ ( z ) z dz = ∞ . However, after a change of variable the above condition is equivalent to Z xσ ( x ) dx = ∞ , which is equivalent to the strict positivity of X by Theorem 5.5.29 in [27]. (cid:3) Yet another transform for recurrent diffusions
We have noted in Section 3 a remarkable transform that turned any regular diffusion into apositively recurrent one. This section will present a particular type of transformation for recurrentdiffusions that will render them transient. This transformation will be especially useful when weconsider the optimal stopping problems in Section 7.When X is a transient diffusion with s ( l ) = 0, it converges to l with positive probability. If onewants to condition this process to converge to r with probability 1, it suffices to use the h -transformwith h = s (see, e.g. Section 6 in [19]). If X is recurrent, on the other hand, the range of s is thewhole real line so one needs to consider taking absolute values to obtain a positive local martingaleusing s . The next proposition introduces a particular conditioning for recurrent diffusions thatconditions X ∞ to exist and take values in the set { l, r } . Similar to the martingale characterisationof a positive diffusion in natural scale in terms of the explosion time of its h -transform that we IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 19 have seen in Proposition 4.4, the resulting diffusion will turn out to have a finite explosion time ifand only if s ( X ) is a strict local martingale. Proposition 5.1.
Suppose X is a recurrent diffusion satisfying Assumption 2.1. Let c > be fixedand y ∗ be the unique point in ( l, r ) such that s ( y ∗ ) = 0 . Then, the following statements are valid:(1) N is a local martingale, where N t := (1 + c | s ( X t ) | ) exp (cid:16) − cs ′ ( y ∗ ) L y ∗ t (cid:17) . (2) For any x ∈ ( l, r ) there exists a unique weak solution to X t = x + Z t σ ( X s ) dB s + Z t (cid:26) b ( X s ) − c s ′ ( X s )1 − cs ( X s ) [ X s ≤ y ∗ ] + c s ′ ( X s )1 + cs ( X s ) [ X s >y ∗ ] (cid:27) ds, t < ζ, (5.1) where ζ := inf { t : X t − ∈ { l, r }} .(3) The regular diffusion defined by (5.1) has scale function ˜ s ( x ) := 1 + c ( s ( x ) + | s ( x ) | )2(1 + c | s ( x ) | ) , (5.2) and speed measure ˜ m ( dx ) = 4(1 + c | s ( x ) | ) cσ ( x ) s ′ ( x ) dx = 2(1 + c | s ( x ) | ) c m ( dx ) . ˜ P x ( X ζ = r ) = ˜ s ( x ) = 1 − ˜ P x ( X ζ = l ) , where ˜ P x denotes the law of (5.1). Moreover, ˜ P x ( ζ = ∞ ) = 1 if and only if s ( X ) is a P x -martingale.(4) For any F ∈ F t the following absolute continuity relationship holds. ˜ P x ( F, ζ > t ) = E x [ F N t ]1 + c | s ( x ) | . (5.3) Consequently, N is a martingale if and only if s ( X ) is.Proof. Note that 1 + c | s ( x ) | is absolutely continuous with a jump in its left derivative at x = y ∗ with size 2 cs ′ ( y ∗ ). Thus, N is a local martingale due Itˆo-Tanaka formula as in Proposition 3.1.Moreover, the arguments used in the proof of Theorem 3.1 also yields the existence of a weaksolution to (5.1), which is unique in law.By direct manipulation one can also verify that ˜ s and ˜ m are a scale function and a speed measurefor the solutions of (5.1). That ˜ P x ( X ζ = r ) = ˜ s ( x ) follows directly from the statement preceding(2.5).According to Theorem 5.5.29 in [27] ˜ P x ( ζ = ∞ ) = 1 if and only iflim x → r Z xy ∗ ˜ s ( x ) − ˜ s ( z ) ˜ m ( dz ) = lim x → l Z y ∗ x ˜ s ( z ) − ˜ s ( x ) ˜ m ( dz ) = ∞ . However, the above hold if and only if Z ry ∗ s ( z ) σ ( z ) s ′ ( z ) dz = − Z y ∗ l s ( z ) σ ( z ) s ′ ( z ) dz = ∞ , which is equivalent to the martingale property of s ( X ) by Theorem 1 of [29]. In order to prove the remaining assertions let l < a < b < r and T a,b := inf { t : X t / ∈ ( a, b ) } .Then, N T a,b is a bounded positive martingale. Therefore,˜ P x ( t < T a,b , F ) = E x [ [ t
Suppose that X is a Brownian motion so that y ∗ = 0 . Then, taking c = 1 inProposition 5.1 implies that the transformed process is a weak solution of X t = x + B t + Z t sgn( X s )1 + | X s | ds, t > , where sgn( x ) = − [ x< + [ x ≥ . Roughly speaking | X | behaves like a -dimensional Besselprocess when X is away from . Observe that the above SDE has a non-exploding solution sinceBrownian motion is a martingale. Moreover, X ∞ exists and equals ∞ or −∞ with probabilities ˜ s ( x ) and − ˜ s ( x ) , respectively. Non-uniqueness of the Black-Scholes equation
As promised earlier we will now apply the results of Sections 3 and 4 to financial models, wherethe stock price movements are governed by a regular one-dimensional diffusion. To simplify theexposition we shall assume that the interest rate is 0. Our interest is in the pricing equationfor a derivative contract written on this stock. The Fundamental Theorem of Asset Pricing (see[14]) stipulates that the stock price must follow a local martingale under an equivalent probabilitymeasure, i.e. risk-neutral measure, and the price of the derivative contract equals the expectationof its terminal payoff under this measure if it is replicable.Throughout this section we will assume that the stock price under the unique risk-neutral measureis given by X t = X + Z t σ ( X s ) dB s , X > , (6.1)on (Ω , F , ( F t ) t ≥ , P ), where X is deterministic and σ satisfies (2.1) on (0 , ∞ ) as well as Assumption4.1. In particular X is the unique strong solution of the above equation. We also impose thecondition that X is a strict local martingale, i.e. Z ∞ zσ ( z ) dz < ∞ . (6.2) Remark 6.1.
Note that we do not assume X is always strictly positive, i.e. X can hit in finitetime with positive probability. IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 21
The strict local martingale assumption places a bubble on the stock price in the sense that itis valued higher in the market than its expected future cash flows. Appearance of bubbles causesmany standard results in derivative pricing theory become invalid (see [12] and [34]). In particular,the Cauchy problem associated to the prices of European options do not admit a unique solution.
Definition 6.1.
Let a > and b be measurable functions on (0 , ∞ ) and D be an interval in [0 , ∞ ) .Consider a continuous function g : D R . A continuous function u : [0 , ∞ ) × D → R is said tobe a classical solution on [0 , ∞ ) × D of u t ( t, x ) = 12 a ( x ) u xx ( t, x ) + b ( x ) u x ( t, x ) (6.3) u (0 , x ) = g ( x ) , (6.4) if u ∈ C , ((0 , ∞ ) × int( D )) , (6.3) is satisfied for all ( t, x ) ∈ (0 , ∞ ) × int( D ) while (6.4) is valid forall x ∈ D . Given the above definition the following is an easy consequence of Theorem 3.2 in Ekstr¨om andTysk [17]. For the rest of this section D ∗ will denote [0 , ∞ ) if P x (inf { t : X t = 0 } < ∞ ) > x >
0. On the other hand, if 0 is not accessible in finite time, D ∗ := (0 , ∞ ). Theorem 6.1.
Suppose that σ satisfies (2.1) on (0 , ∞ ) , (6.2) and Assumption 4.1. Consider acontinuous function g : [0 , ∞ ) → [0 , ∞ ) of at most linear growth and define on [0 , ∞ ) × D ∗ thefunction v ( t, x ) := E x [ g ( X t )] , where X is the unique solution of (6.1). Assume further that g (0) = 0 if ∈ D ∗ . Then, v is a classical solution on [0 , ∞ ) × D ∗ of the Cauchy problem v t = 12 σ v xx , v (0 , · ) = g. (6.5)Non-uniqueness of the Cauchy problem is implicit in the above theorem. Indeed, if we let g ( x ) = x and w ( x ) = x , both w and v are solutions of (6.5). Yet, E x [ X t ] = x since X is a strictlocal martingale.The equation (6.5) is called the Black-Scholes pricing equation in the literature. If g is the time- T payoff of a European derivative written on the stock, v ( T − t, X t ) gives the time- t price of thisderivative, where v is the solution of (6.5). On the other hand the arbitrage pricing theory statesthat the price of the derivative at time t equals E X t [ g ( X T − t )] for a sufficiently well-behaved payoffsince the risk-neutral measure is unique. Although Theorem 6.1 shows that the function definedby this alternative pricing formula still satisfies the Black-Scholes equation, non-uniqueness of theCauchy problem is problematic especially when one has to rely on numerical methods to find theprice of the derivative.The goal of this section is to identify the stochastic solution , E x [ g ( X t )] in terms of the uniquesolution of some Cauchy problem. The discussion following Theorem 6.1 shows that there is nohope if we work with the differential operator associated to the generator of X . However, thesolutions of (4.2), which can be interpreted as the limit of recurrent transforms of X , or in view ofRemark 4.1 as an h -transform of X , come to our rescue. Theorem 6.2.
Suppose that σ satisfies (2.1) on (0 , ∞ ) , (6.2) and Assumption 4.1. Consider acontinuous function g : [0 , ∞ ) → [0 , ∞ ) of at most linear growth at infinity and g (0) = 0 whenever ∈ D ∗ . Let v ( t, x ) := E x [ g ( X t )] , where X is the unique solution of (6.1), for ( t, x ) ∈ [0 , ∞ ) × D ∗ .Then, the following statements are valid:(1) If ∈ D ∗ , v ( t,
0) = 0 for all t ≥ . (2) For x > , v ( t, x ) = xw ( t, x ) , where w is the unique classical nonnegative solution on [0 , ∞ ) × (0 , ∞ ) of w t ( t, x ) = 12 σ ( x ) w xx ( t, x ) + σ ( x ) x w x ( t, x ) (6.6) w (0 , x ) = g ( x ) x (6.7) among the class of functions satisfying the following conditions:(a) w is of O ( x − ) as x → : lim x → sup s ≤ t xw ( s, x ) < ∞ , ∀ t > . (6.8) Moreover, if X reaches in finite time lim x → sup s ≤ t xw ( s, x ) = 0 , ∀ t > . (6.9) (b) w approaches to near infinity: ∀ t > , lim n →∞ w ( t n , x n ) = 0 if x n ↑ ∞ and t n → t. (6.10) (3) If Y is a weak solution of (4.2) and Q x is its law, w ( t, x ) = Q x (cid:20) g ( Y t ) Y t [ ζ>t ] (cid:21) , (6.11) where ζ corresponds to the lifetime of Y . Note that we do not require g ( x ) x to be bounded near 0 in the above theorem. In particular, if D ∗ = (0 , ∞ ) and g ≡ w will be the solution of a Cauchy problem with the unbounded initialcondition x . In this case the unique solution is given by x = Q x [ Y t ] since Y is a martingale when X is strictly positive as observed in Proposition 4.5. Remark 6.2.
In Theorem 6.2 the conditions (6.8) and (6.9) are natural growth conditions near for the problem at hand given that we want w satisfy xw ( t, x ) = v ( t, x ) = E x [ g ( X t )] . Indeed, g ( x ) ≤ K (1 + x ) implies v ( t, x ) ≤ K (1 + x ) since X is a nonnegative local martingale, which inturn implies (6.8). Moreover, when D ∗ = [0 , ∞ ) , v will be uniformly continuous on [0 , t ] × [0 , x ] forall x > in view of the definition of a classical solution, which will lead to (6.9).On the other hand, (6.10) must be imposed to achieve the intended uniqueness. Indeed, supposethat X is a strictly positive strict local martingale and g ( x ) = x . Then both and v ( t,x ) x are classicalsolutions of (6.6) with the initial condition (6.7) and satisfy the growth condition (6.8). However,only v ( t,x ) x satisfies (6.10) as v ( t,x ) x = Q x ( ζ > t ) . . We end this section with the following immediate corollary to Theorem 6.2, which implies thatthe function x E x [ X t ] is of strictly sublinear growth at infinity for t > Corollary 6.1.
Suppose that σ satisfies (2.1) on (0 , ∞ ) , (6.2) and Assumption 4.1. Let g be as inTheorem 6.2. Then for every t > x →∞ E x [ g ( X t )] x = 0 , i.e. the function x E x [ g ( X t )] is of strictly sublinear growth at infinity. IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 23 Optimal stopping
In this section we will consider the following optimal stopping problem for a regular diffusion on( l, r ) satisfying Assumption 2.1: V ( x ) := sup τ ≤ ζ E x [ e − λτ g ( X τ )] , (7.1)where λ > τ is any stopping time with the usual convention that e − λτ g ( X τ ( ω )) = lim sup t →∞ e − λt g ( X t ( ω ))if τ ( ω ) = ζ ( ω ) = ∞ . Here g is taken to be a nonnegative function that is continuous on I . From afinancial perspective V can be interpreted as the price of a perpetual American option with payoff g on a stock whose dynamics are governed by X and is currently priced at x while λ equals theconstant interest rate. Remark 7.1.
This problem has been considered by Cisse et al. in [11] , where the authors also usechange of measure techniques with certain implicit assumptions on their way towards a solution.For instance, the proof of the key Lemma 3.5 is based on a result of Shiryaev, which requires thecontinuity of the function f (of their Lemma 3.5) in the one-point compactifaction of the statespace by adding the cemetery state. This in particular requires the boundedness of g in (7.1) witha certain behaviour at the boundary points. Moreover, the Sturm-Liuoville equation on p.1251 thatdefines the family of excessive functions φ B ( x ) = E x [ e − qT B ] , where T B = inf { t > X t / ∈ ( a, b ) } ,stipulates that φ B ( a +) = φ B ( b − ) = 1 , for all a and b satisfying l ≤ a < b ≤ r . However, thisimmediately rules out the case when X has infinite lifetime or an entrance boundary. Indeed, if a = l and b = r , T B = ζ . Thus, if the diffusion has infinite lifetime, φ B ( x ) = 0 for all x ∈ ( l, r ) ,which leads to φ B ( l +) = φ B ( r − ) = 0 by continuity. Similarly, if l is an entrance boundary, a = l ,and b < r , then φ B ( l +) = 1 implies E l [ e − qT B ] = 1 , i.e. P l ( T B = 0) = 1 . This is a contradiction tothe assumption that l is an entrance boundary which entails that the diffusion immediately entersthe open interval ( l, r ) right after time and never returns to l . Consequently, P l ( T B = T b ) = 1 ,where T b := inf { t > X t = b } . Clearly, P l ( T b >
0) = 1 .The method that is described below is applicable to all regular one-dimensional diffusions satis-fying Assumption 2.1. Aside from the above restrictions the method of Cisse et al. requires theknowledge of all φ B for all open sets B . As we shall see later, our solution only requires theknowledge of u λ ( · , y ) for some y ∈ ( l, r ) . To ease the exposition and simplify the proofs we shall assume from now on that X is on naturalscale. We will solve the above problem using mainly the λ -potential kernel, u λ , and the recurrenttransform introduced in Proposition 3.2. We start with the following lemma, which is a directconsequence of Proposition 3.1. Lemma 7.1.
Let X be a regular diffusion satisfying Assumption 2.1 on ( l, r ) , y ∈ ( l, r ) be fixed,and g be a nonnegative measurable function on I . Then, for any stopping time τ and λ > , wehave E x h e − λτ g ( X τ ) [ τ<ζ ] i = u λ ( x, y ) E h,x (cid:20) g ( X τ ) u λ ( X τ , y ) exp (cid:18) − L yτ u λ ( y, y ) (cid:19) [ τ< ∞ ] (cid:21) , (7.2) where E h,x is the expectation with respect to R h,x , which is the law of the recurrent transform inProposition 3.2. Thus, the recurrent transform associated to u λ removes the discounting in the optimal stoppingproblem making it more tractable. We shall apply one more transformation to get rid of the localtime factor in order to make the problem one-dimensional again. However, this recurrent transformwill already give us the necessary condition for the finiteness of the optimal stopping problem in (7.1) once we have the result from the next lemma. Throughout this section E h,x and R h,x willcorrespond to the expectation operator and the law associated to the solutions of (3.8), whose scalefunction can be chosen as follows for a given y ∈ ( l, r ): s h ( x ) = Z xy u λ ( z, y )) dz. (7.3) Lemma 7.2.
For any l < a < b < r and x, y ∈ ( a, b ) we have E h,x " [ T a Suppose y ≤ x . Let us kill the recurrent transform as soon as it hits a or b and then applyan h -transform via R h,x ( T a < T b ) = s h ( b ) − s h ( x ) s h ( b ) − s h ( a ) . This h -transform conditions the diffusion to converge to a . Thus, if we denote the law of this h -transform by R h,a,x and its potential kernel by u a (by dropping the dependence on b to ease thenotation), then E h,x (cid:2) [ T a Let x ∈ ( l, r ) be fixed and consider the value function, V , defined in (7.1). If V ( x ) is finite, then lim inf a → l g ( a ) u λ ( a, x ) s h ( a ) > −∞ and lim sup b → r g ( b ) u λ ( b, x ) s h ( b ) < ∞ . (7.6) IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 25 Proof. Suppose that (7.6) is violated. Then, either lim inf a → l g ( a ) u λ ( a,x ) s h ( a ) = −∞ or lim sup b → r g ( b ) u λ ( b,x ) s h ( b ) = ∞ or both. Suppose it is the former statement and, thus, there exists a sequnce ( a n ) with a n → l and lim n →∞ g ( a n ) u λ ( a n , x ) s h ( a n ) = −∞ . (7.7)Then, we claim that lim n →∞ E x h e − λT n g ( X T n ) i = ∞ , where T n := T a n ∧ ζ , which is in contradiction with the hypothesis that V ( x ) < ∞ .Indeed, by Lemma 7.1 and taking y = x , we have E x h e − λT n g ( X T n ) [ T n <ζ ] i = u λ ( x, x ) E h,x (cid:20) g ( X T n ) u λ ( X T n , x ) exp (cid:18) − L xT n u λ ( x, x ) (cid:19) [ T n < ∞ ] (cid:21) = u λ ( x, x ) g ( a n ) u λ ( a n , x ) E h,x (cid:20) exp (cid:18) − L xT n u λ ( x, x ) (cid:19)(cid:21) , where the last line is due to the recurrence of X under R h,x . However, Lemma 7.2 together withthe nonnegativity of g now yield E x h e − λT n g ( X T n ) i ≥ u λ ( x, x ) g ( a n ) u λ ( a n , x ) lim b → r 11 + s a n ( b ; x ) s h ( b ) u λ ( x, x ) . On the other hand,lim b → r s a n ( b ; x ) s h ( b ) = lim b → r Z xa n s ′ h ( z ) s h ( b )( s h ( b ) − s h ( z )) dz = Z xa n lim b → r s ′ h ( z ) s h ( b )( s h ( b ) − s h ( z )) dz = − s h ( a n )by the dominated convergence theorem. Recall that, since x = y , s h ( x ) = 0 by (7.3). Thus, theclaim follows from (7.7).If, instead, lim sup b → r g ( b ) u λ ( b,x ) s h ( b ) = ∞ , a similar construction shows that V ( x ) = ∞ in that case,too. (cid:3) The above result shows that the boundedness of z g ( z ) u λ ( z, x )(1 + | s h ( z ) | ) (7.8)is necessary in order for V ( x ) to be finite. In fact the condition (7.8) is independent of x andensures V ( x ) < ∞ for all x , as one can also guess from the strong Markov property of X . Lemma 7.3. The mapping in (7.8) is bounded if and only if for some y ∈ ( l, r ) z g ( z ) u λ ( z, y )(1 + | s h ( z ) | ) (7.9) is bounded, where s h is defined by (7.3).Proof. It suffices to show that sup a u λ ( a, x ) u λ ( a, y ) < ∞ . Indeed, by the symmetry property of the potential kernels and (2.9)lim a → l u λ ( a, x ) u λ ( a, y ) = lim a → l u λ ( x, a ) u λ ( y, a ) = lim a → l E x [ e − λT a ] E y [ e − λT a ] . Moreover, if x > y , E x [ e − λT a ] = E y [ e − λT a ] E x [ e − λT y ] by the strong Markov property. Thus, for a < y < x , lim a → l u λ ( a,x ) u λ ( a,y ) = E x [ e − λT y ]. Similarly, for a < x < y , lim a → l u λ ( a,x ) u λ ( a,y ) = E y [ e − λTx ] . Thestrong Markov property can be used also to show lim a → r u λ ( a,x ) u λ ( a,y ) < ∞ , concluding the proof. (cid:3) Remark 7.2. A similar condition for the finiteness of the value function can be found in Part (I)of Theorem 6.3 in [30] . Namely, the value function is finite if and only if lim sup x → l g ( x ) φ α ( x ) < ∞ and lim sup x → r g ( x ) ψ α ( x ) < ∞ , where φ α and ψ α are the fundamental solutions appearing in (2.8). On the other hand, (7.9) isequivalent to lim sup x → l g ( x ) ψ α ( x ) | s h ( x ) | < ∞ and lim sup x → r g ( x ) φ α ( x ) s h ( x ) < ∞ . Combining the two conditions allows us to conclude that φ α ( x ) ψ α ( x ) | s h ( x ) | (resp. ψ α ( x ) φ α ( x ) s h ( x ) ) remainbounded as x → l (resp. x → r ) when the above limits are nonzero. The above discussion justifies the following Assumption 7.1. For some (thus, for all) y ∈ ( l, r ) the mapping in (7.9) is bounded. The denominator in (7.9) should remind us of the transformation discussed in Section 5. Indeed,let us fix a y ∈ ( l, r ) and remind ourselves that ( R h,x ) x ∈ ( l,r ) corresponds to the recurrent transformin Proposition 3.2 for α = λ . Note that we can choose its scale function to be s h that is defined in(7.3) and satisfies s h ( y ) = 0. The following follows immediately from Proposition 5.1 and Lemma7.1. Proposition 7.2. Suppose X is a regular diffusion on natural scale satisfying Assumption 2.1 andlet c = u λ ( y,y )2 . Then(1) For any x ∈ ( l, r ) there exists a unique weak solution to X t = x + Z t σ ( X s ) dB s + Z t (cid:26) σ ( X s ) u λx ( X s , y ) u λ ( X s , y ) − c s ′ h ( X s )1 − cs ( X s ) [ X s ≤ y ] + c s ′ h ( X s )1 + cs ( X s ) [ X s >y ] (cid:27) , t < ζ, (7.10) where ζ := inf { t : X t − ∈ { l, r }} .(2) The regular diffusion defined by (7.10) has scale function ˜ s ( x ) := 1 + c ( s h ( x ) + | s h ( x ) | )2(1 + c | s h ( x ) | ) , (7.11) and speed measure ˜ m ( dx ) = 4(1 + c | s h ( x ) | ) cσ ( x ) s ′ r ( x ) dx. ˜ P x ( X ζ = r ) = ˜ s ( x ) = 1 − ˜ P x ( X ζ = l ) , where ˜ P x denotes the law of (5.1).(3) For any F ∈ F t the following absolute continuity relationship holds. ˜ P x ( F, ζ > t ) = E h,x h F (cid:16) u λ ( y,y )2 | s h ( X t ) | (cid:17) exp (cid:16) − L yt u λ ( y,y ) (cid:17)i c | s h ( x ) | . (7.12) In particular, for any nonnegative continuous function g on I and stopping time τ , E x h e − λτ g ( X τ ) [ τ<ζ ] i = u λ ( x, y ) (cid:0) c | s h ( x ) | (cid:1) ˜ E x (cid:20) g ( X τ ) u λ ( X τ , y ) (1 + c | s h ( X τ ) | ) [ τ<ζ ] (cid:21) . (7.13) IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 27 The identity (7.13) together with Assumption 7.1 allows us to solve (7.1), which is the contentof the next theorem whose proof is delegated to the Appendix. Theorem 7.1. Let X be a regular diffusion on natural scale satisfying Assumption 2.1. Considera nonnegative continuous function g on I satisfying Assumption 7.1. Let ˜ s be as in (7.11) and G be the smallest concave majorant on (˜ s ( l ) , ˜ s ( r )) of the function ˆ g ( x ) := g (˜ s − ( x )) u λ (˜ s − ( x ) , y ) (cid:16) u λ ( y,y )2 | s h (˜ s − ( x )) | (cid:17) , and define Γ := { x ∈ (˜ s ( l ) , ˜ s ( r )) : ˆ g ( x ) ≥ G ( x ) } . Then, V ( x ) = u λ ( x, y ) (cid:16) u λ ( y, y )2 | s h ( x ) | (cid:17) G (˜ s ( x )) < ∞ . Moreover, the optimal stopping time for (7.1) is τ ∗ := inf { t ≥ s ( X t ) ∈ Γ } . An immediate corollary to the above theorem is the following converse to the statement inProposition 7.1. Corollary 7.1. Let x ∈ ( l, r ) be fixed and consider the value function, V , defined in (7.1). V ( x ) is finite if and only if the mapping in (7.9) is bounded.Proof. The necessity has already been proved in Proposition 7.1 in view of Lemma 7.3. Sufficiencyfollows from Theorem 7.1. (cid:3) Remark 7.3. Note that the sole purpose of the assumption that X is on natural scale in the abovetheorem is to simplify the exposition. If X is not on natural scale, then one can define Y = s ( X ) ,which will be on natural scale, and consider instead the problem sup τ E x [ e − λτ g ( s − ( Y τ ))] . Conclusion We have introduced a new class of path transformations for one-dimensional regular diffusionsaimed at modifying their behaviour towards recurrence. As a first application these transformationsare used to compute the distribution of the first exit time from an interval for any diffusion. Thesetransforms turned out to be instrumental in understanding strict local martingales better as well.In Theorem 6.2 we give a novel characterisation of the Black-Scholes valuation formula in terms ofthe unique solution of an alternative Cauchy problem when the stock price is a local martingale andthus resolve the longstanding issue with the numerical computation of the option price when theoption payoff is unbounded with linear growth. Finally, using the path transformations developedin this paper, we propose a unified framework for solving explicitly the optimal stopping problemfor one-dimensional diffusions with discounting in Section 7. Following Remark 3.4 application ofrecurrent transformations to study the discrete Euler schemes for killed diffusion is left for futureresearch. References [1] P. Baldi , Exact asymptotics for the probability of exit from a domain and applications to simulation , The Annalsof Probability, (1995), pp. 1644–1670.[2] P. Baldi, L. Caramellino, and M. G. Iovino , Pricing general barrier options: a numerical approach usingsharp large deviations , Mathematical Finance, 9 (1999), pp. 293–321.[3] E. Bayraktar, C. Kardaras, and H. Xing , Strict local martingale deflators and valuing american call-typeoptions , Finance and Stochastics, 16 (2012), pp. 275–291.[4] , Valuation equations for stochastic volatility models , SIAM Journal on Financial Mathematics, 3 (2012),pp. 351–373.[5] E. Bayraktar and H. Xing , On the uniqueness of classical solutions of cauchy problems , Proceedings of theAmerican Mathematical Society, 138 (2010), pp. 2061–2064.[6] M. Beibel and H. R. Lerche , Optimal stopping of regular diffusions under random discounting , Theory ofProbability & Its Applications, 45 (2001), pp. 547–557.[7] P. Billingsley , Probability and measure , Wiley Series in Probability and Mathematical Statistics, John Wiley& Sons, Inc., New York, third ed., 1995. A Wiley-Interscience Publication.[8] R. M. Blumenthal and R. K. Getoor , Markov processes and potential theory , Pure and Applied Mathematics,Vol. 29, Academic Press, New York-London, 1968.[9] A. N. Borodin and P. Salminen , Handbook of Brownian motion—facts and formulae , Probability and itsApplications, Birkh¨auser Verlag, Basel, second ed., 2002.[10] K. L. Chung and J. B. Walsh , Markov processes, Brownian motion, and time symmetry , vol. 249 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer,New York, second ed., 2005.[11] M. Ciss´e, P. Patie, and E. Tanr´e , Optimal stopping problems for some markov processes , The Annals ofapplied probability, (2012), pp. 1243–1265.[12] A. M. Cox and D. G. Hobson , Local martingales, bubbles and option prices , Finance and Stochastics, 9 (2005),pp. 477–492.[13] S. Dayanik and I. Karatzas , On the optimal stopping problem for one-dimensional diffusions , StochasticProcesses and their Applications, 107 (2003), pp. 173–212.[14] F. Delbaen and W. Schachermayer , A general version of the fundamental theorem of asset pricing , Mathe-matische annalen, 300 (1994), pp. 463–520.[15] F. Delbaen and H. Shirakawa , No arbitrage condition for positive diffusion price processes , Asia-PacificFinancial Markets, 9 (2002), pp. 159–168.[16] E. Dynkin , Optimal choice of the stopping moment of a markov process , in Dokl. Akad. Nauk SSSR, vol. 150,1963.[17] E. Ekstr¨om and J. Tysk , Bubbles, convexity and the black-scholes equation , The Annals of Applied Probability,(2009), pp. 1369–1384.[18] H. J. Engelbert and W. Schmidt , Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations. III , Math. Nachr., 151 (1991), pp. 149–197.[19] S. N. Evans and A. Hening , Markov processes conditioned on their location at large exponential times , Workingpaper.[20] H. F¨ollmer , The exit measure of a supermartingale , Zeitschrift f¨ur Wahrscheinlichkeitstheorie und verwandteGebiete, 21 (1972), pp. 154–166.[21] E. Gobet , Weak approximation of killed diffusion using euler schemes , Stochastic processes and their applica-tions, 87 (2000), pp. 167–197.[22] E. Gobet and S. Menozzi , Exact approximation rate of killed hypoelliptic diffusions using the discrete eulerscheme , Stochastic Processes and their Applications, 112 (2004), pp. 201–223.[23] P. Halmos , Measure Theory , Graduate Texts in Mathematics, Springer New York, 2014.[24] N. Ikeda and S. Watanabe , Stochastic differential equations and diffusion processes , vol. 24, Elsevier, 2014.[25] K. Itˆo and H. P. McKean, Jr. , Diffusion processes and their sample paths , Springer-Verlag, Berlin-New York,1974. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125.[26] I. Karatzas and J. Ruf , Distribution of the time to explosion for one-dimensional diffusions , ProbabilityTheory and Related Fields, 164 (2016), pp. 1027–1069.[27] I. Karatzas and S. E. Shreve , Brownian motion and stochastic calculus , vol. 113 of Graduate Texts inMathematics, Springer-Verlag, New York, second ed., 1991. IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 29 [28] C. Kardaras, D. Kreher, and A. Nikeghbali , Strict local martingales and bubbles , The Annals of AppliedProbability, 25 (2015), pp. 1827–1867.[29] S. Kotani , On a condition that one-dimensional diffusion processes are martingales , in In Memoriam Paul-Andr´e Meyer: S´eminaire de Probabilit´es XXXIX, M. ´Emery and M. Yor, eds., Springer Berlin Heidelberg, 2006,pp. 149–156.[30] D. Lamberton and M. Zervos , On the optimal stopping of a one-dimensional diffusion , Electron. J. Probab,18 (2013), pp. 1–49.[31] H. R. Lerche and M. Urusov , Optimal stopping via measure transformation: the beibel–lerche approach ,Stochastics An International Journal of Probability and Stochastic Processes, 79 (2007), pp. 275–291.[32] M. B. Marcus and J. Rosen , Markov processes, Gaussian processes, and local times , vol. 100 of CambridgeStudies in Advanced Mathematics, Cambridge University Press, Cambridge, 2006.[33] H. P. McKean , Elementary solutions for certain parabolic partial differential equations , Transactions of theAmerican Mathematical Society, 82 (1956), pp. 519–548.[34] S. Pal and P. Protter , Analysis of continuous strict local martingales via h-transforms , Stochastic Processesand their Applications, 120 (2010), pp. 1424–1443.[35] G. Peskir and A. Shiryaev , Optimal stopping and free-boundary problems , Springer, 2006.[36] D. Revuz and M. Yor , Continuous martingales and Brownian motion , vol. 293 of Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, third ed.,1999.[37] W. Rudin , Real and complex analysis , McGraw-Hill Book Co., New York, third ed., 1987.[38] M. Sharpe , General theory of Markov processes , vol. 133 of Pure and Applied Mathematics, Academic Press,Inc., Boston, MA, 1988.[39] A. N. Shiryaev , Optimal stopping rules , vol. 8, Springer Science & Business Media, 2007.[40] D. W. Stroock and S. S. Varadhan , Multidimensional diffusion processes , Springer, 2007. Appendix A. Proof of Theorem 3.1 (1) To show the first assertion it suffices to show that h ′ equals a left-continuous function Lebesguea.e. since the left derivative is defined uniquely only outside a Lebesgue null set. However,since h ′ is assumed to be of finite variation, there exist non-decreasing functions g + and g − such that h ′ = g + − g − . It follows from Exercise 12 in Chap. 7 of [37] that g + and g − areleft-continuous a.e.. Thus, h ′ is equal to a left-continuous function a.e..Since h ′ is of finite variation and can be taken to be left continuous, Exercise 13 in Chap. 7 of[37] shows that h ′ can be viewed as a signed Borel measure on ( l, r ). Then, it follows from theLebesgue decomposition theorem (Theorem C in Section 32 of [23]) and the Radon-Nikodymtheorem (Theorem B in [23]) that the measure dh ′ ( x ) admits the stated decomposition. That h ′′ can be taken Borel measure follows from the fact that every Lebesgue measurable functionis equal to a Borel measurable function a.e..(2) Observe that, in view of occupation times formula, the integral in (3.3) equals on [ t < ζ ] Z rl (cid:12)(cid:12)(cid:12)(cid:12) σ ( x )2 h ′′ ( x ) + b ( x ) h ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) L xt σ ( x ) dx + Z rl L xt | n ( dx ) | = Z rl (cid:12)(cid:12)(cid:12)(cid:12) h ′′ ( x ) + b ( x ) σ ( x ) h ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) L xt dx + Z rl L xt | n ( dx ) | Due to the continuity of X , on [ t < ζ ] and on almost every path L xt would be equal to 0 for all x outside a compact interval in ( l, r ), which is determined by the maximum and the minimumof X on [0 , t ]. Thus, due to the continuity of x L xt , it suffices to check Z K (cid:12)(cid:12)(cid:12)(cid:12) h ′′ ( x ) + b ( x ) σ ( x ) h ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx + Z K | n ( dx ) | < ∞ (A.1) for an arbitrary compact K contained in ( l, r ). First note that Z K (cid:12)(cid:12) h ′′ ( x ) (cid:12)(cid:12) dx + Z K | n ( dx ) | < ∞ since h ′ is of finite variation.Moreover, Z K (cid:12)(cid:12)(cid:12)(cid:12) b ( x ) σ ( x ) h ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx = C + Z K (cid:18)Z yc (cid:12)(cid:12)(cid:12)(cid:12) b ( x ) σ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx (cid:19) | dh ′ ( y ) | , for some C < ∞ and c ∈ K due to the finiteness of h ′ and R yc (cid:12)(cid:12)(cid:12) b ( x ) σ ( x ) (cid:12)(cid:12)(cid:12) dx at the boundary of K . However, the integral in the above representation is finite since dh ′ is of finite variationand R yc (cid:12)(cid:12)(cid:12) b ( x ) σ ( x ) (cid:12)(cid:12)(cid:12) dx is bounded in K . This completes the proof that (A.1) holds for an arbitrarycompact set K , which in turn yields the claim.(3) It follows from the previous part that Λ( h ) is of finite variation. Since ( h ( X s ) s ≤ t is away from0, path by path for t < ζ , it immediately follows that M is of finite variation, too.Since h can be considered as a difference of convex functions, it follows from Itˆo-Tanakaformula that on [ t < ζ ] h ( X t ) = h ( x ) + Z t h ′ ( X s ) dX s + 12 Z ( l,r ) L xt (cid:8) h ′′ ( x ) dx + n ( dx ) (cid:9) = Z t h ′ ( X s ) dX s + 12 Z t σ ( X s ) h ′′ ( X s ) ds + 12 Z rl L xt n ( dx ) . Thus, a simple application of integration by parts formula yields h ( X t ) M t = h ( x ) + Z t h ′ ( X s ) M s σ ( X s ) dB s , t < ζ, proving the local martingale property for h ( X ) M . In particular, h ( X ) M is a continuous non-negative supermartingale with an integrable limit as t → ζ .Finally, due to the hypotheses on s h it follows from Theorem 5.5.15 in [27] that there existsa unique weak solution to (3.1). Moreover, the solution is recurrent by Part a) of Proposition5.5.22 in [27].(4) Since − s h ( l +) = s h ( r − ) = ∞ , it follows that h ( l ) = 0 (resp. h ( r ) = 0) if s ( l ) = 0 (resp. s ( r ) = 0). That is, h vanishes at the accessible boundaries and, thus, h ( X ζ ) = 0 on [ ζ < ∞ ].Consequently, h ( X t ) M t = h ( X t ) M t [ t<ζ ] since M t > t < ζ ] except on a P x -null set by(3.3). Moreover, h must be strictly positive on ( l, r ) in order for s h to be finite on ( l, r ). Thesein turn yield the desired identity that inf { t > h ( X t ) M t = 0 } = ζ, P x -a.s.(5) In view of the previous part h ( X t ) h ( x ) M t is a supermartingale multiplicative functional satisfyingHypothesis 62.9 in [38], that is, a supermartingale vanishing on [ ζ, ∞ ). Then (3.4) followsdirectly from Theorem 62.19 in [38] since R h,x ( ζ = ∞ ) = 1. Note that the space Ω is projectivein the terminology of Section 62 of [38] since it is the path space.To show the martingale property observe that in view of (3.4) and R h,x ( ζ = ∞ ) = 1,1 = R h,x ( t < ζ ) = R h,x ( t < ∞ ) = 1 h ( x ) E x [ h ( X t ) M t ] , yielding the martingale property of h ( X ) M under P x . IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 31 (6) By the virtue of the monotone convergence theorem E h,x (cid:20) F h ( X T ) M T (cid:21) = lim n →∞ E h,x (cid:20) F (cid:16) h ( X T ) M T ∧ n (cid:17)(cid:21) . Thus, employing (3.4) we arrive at E h,x (cid:20) F h ( X T ) M T (cid:21) = lim n →∞ E x h F (cid:0) [ h ( X T ) M T > n ] + nh ( X T ) M T [ h ( X T ) M T ≤ n ] (cid:1)i = P x ( F, h ( X T ) M T > 0) + lim n →∞ E x (cid:2) F nh ( X T ) M T [ T <ζ ] [ h ( X T ) M T ≤ n ] (cid:3) , where the second line follows from the dominated convergence theorem and that h ( X T ) M T = h ( X T ) M T [ T <ζ ] , P x -a.s.. Moreover, E x (cid:2) F nh ( X T ) M T [ T <ζ ] [ h ( X T ) M T ≤ n ] (cid:3) ≤ P x (cid:16) T < ζ, h ( X T ) M T ≤ n (cid:17) , which converges to 0 as n → ∞ since h ( X T ) M T > T < ζ ] except on a P x -null set by theprevious part. Thus, E h,x (cid:20) F h ( X T ) M T (cid:21) = P x ( F, h ( X T ) M T > 0) = P x ( F, T < ζ ) . This completes the proof. Appendix B. Proof of Theorem 3.2 (1) It follows from a simple differentiation of the potential functions in (2.5)-(2.7) that the left-derivative of u ( · , y ), i.e. u x ( · , y ), at x ∈ ( l, r ) is bounded by s ′ ( x ) uniformly in y . Thus, since µ is a probability measure on ( l, r ) and s ′ is continuous under Assumption 2.1, the dominatedconvergence theorem implies the left derivative of h is given by h ′ ( x ) = Z ( l,r ) u x ( x, y ) µ ( dy ) , x ∈ ( l, r ) . (B.1)Next, consider a finite subinterval [ a, b ] of ( l, r ) and recall that u x ( · , y ) is non-decreasing on ( l, y )and non-increasing on ( y, r ). Straightforward computation reveals that the total variation of u x ( · , y ) on [ a, b ], denoted by k u x ( · , y ) k T V ( a,b ) , admits k u x ( · , y ) k T V ( a,b ) ≤ sup x,z ∈ [ a,b ] | s ′ ( x ) − s ′ ( z ) | ≤ K ( a, b ) < ∞ , for some constant K ( a, b ) by the continuity of s ′ and the compactness of [ a, b ]. Consequently, k h ′ k T V ( a,b ) ≤ Z ( l,r ) k u x ( · , y ) k T V ( a,b ) µ ( dy ) ≤ K ( a, b ) , since µ (( l, r )) = 1. Thus, h ′ is of finite variation.Next, let v ( x, y ) := u ( s − ( x ) , s − ( y )) and observe that v ( x, · ) is concave for each x ∈ ( l, r ).Thus, h ( s − ( x )) = Z ( l,r ) v ( x, s ( y )) µ ( dy ) ≤ v x, Z ( l,r ) s ( y ) µ ( dy ) ! , by Jensen’s inequality. Next observe that s ( l ) < R ( l,r ) s ( y ) µ ( dy ) < s ( r ). Indeed, if s ( l ) = −∞ , s ( l ) < R ( l,r ) s ( y ) µ ( dy ) directly follows from the hypothesis that R ( l,r ) | s ( y ) | µ ( dy ) < ∞ . If s ( l ) = 0, since s ( x ) ≥ x ≥ l , we have R ( l,r ) s ( y ) µ ( dy ) ≥ 0. In fact, R ( l,r ) s ( y ) µ ( dy ) > since, otherwise, s = 0, µ -a.s.. However, { x : s ( x ) = 0 } = { l } as s is strictly increasing underAssumption 2.1. Thus, R ( l,r ) s ( y ) µ ( dy ) > µ does not charge { l } .Similarly, we can show R ( l,r ) s ( y ) µ ( dy ) < s ( r ). Moreover, by the continuity of s , there existssome y ∗ ∈ I such that R ( l,r ) s ( y ) µ ( dy ) = s ( y ∗ ).Therefore, h ( x ) ≤ v ( s ( x ) , s ( y ∗ )) = u ( x, y ∗ ). This in turn implies Z y ∗ l s ′ ( y ) h ( y ) dy ≥ | s u ( l +) | , (B.2)where s u ( x ) = Z xy ∗ s ′ ( z )( u ( z, y ∗ )) dz, x ∈ ( l, r ) . Suppose, first, that s ( l ) = 1 − s ( r ) = 0. Then, for x < y ∗ , s u ( x ) = 1(1 − s ( y ∗ )) Z xy ∗ s ′ ( z ) s ( z ) dz = 1(1 − s ( y ∗ )) (cid:18) s ( y ∗ ) − s ( x ) (cid:19) , which in particular shows that lim x → l s u ( x ) = −∞ . Similarly, for x > y ∗ , s u ( x ) = 1 s ( y ∗ )) Z xy ∗ s ′ ( z )(1 − s ( z )) dz = 1 s ( y ∗ ) (cid:18) − s ( x ) − − s ( y ∗ ) (cid:19) , and, thus, s u ( r − ) = ∞ . The other cases are handled the same way to show − s u ( l +) = s u ( r − ) = ∞ . This in turn yields in view of (B.2) that s h ( l +) = − R y ∗ l s ′ ( y ) h ( y ) dy = −∞ . Similarly, s h ( r − ) ≥ s u ( r − ) = ∞ .Thus, h satisfies the conditions of Theorem 3.1. In particular, h ′ can be taken left-continuouswith the Lebesgue decomposition dh ′ ( x ) = h ′′ ( x ) dx + n ( dx ), where n is a locally finite signedmeasure that is singular with respect to the Lebesgue measure. Moreover, ( h, M ) is a recurrenttransform where M t = exp − Z t ˜ A h ( X s ) h ( X s ) ds − Z t h ( X s ) d Λ s ( h ) ! , where Λ t ( h ) = R ( l,r ) L xt n ( dx ).On the other hand, the occupation times formula applied to R t A h ( X s ) h ( X s ) ds yields M t = exp (cid:18) − Z ( l,r ) L xt h ( x ) (cid:16) dh ′ ( x ) + b ( x ) h ′ ( x ) σ ( x ) dx (cid:17)(cid:19) . Thus, we will be done if dh ′ ( x ) + b ( x ) h ′ ( x ) σ ( x ) dx = − s ′ ( x ) µ ( dx ) on ( l, r ). Note that this willfollow if for any continuous f with a compact support in ( l, r ), we establish12 Z ( l,r ) f ( x ) dh ′ ( x ) + Z rl f ( x ) b ( x ) h ′ ( x ) σ ( x ) dx = − Z ( l,r ) f ( x ) s ′ ( x ) µ ( dx ) . (B.3)First note that u x ( · , y ) is differentiable everywhere except at y under Assumption 2.1. Usingthis observation and the fact that the jump in the (left-continuous) left-derivative u x ( x, y ) at x = y equals u x ( y + , y ) − u x ( y, y ) = − s ′ ( y ), we deduce du x ( x, y ) = u xx ( x, y ) dx − s ′ ( y ) ε y ( dx ) , IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 33 for some function u xx ( · , y ) that is a.e. uniquely determined by the second derivative of u ( · , y ),which exists at each x = y . Moreover,12 σ ( x ) u xx ( x, y ) + b ( x ) u x ( x, y ) = 0 , ∀ x = y. (B.4)Next, the second integral on the left hand side of (B.3) equals Z rl f ( x ) b ( x ) σ ( x ) Z ( l,r ) u x ( x, y ) µ ( dy ) ! dx = Z ( l,r ) µ ( dy ) Z rl f ( x ) b ( x ) σ ( x ) u x ( x, y ) dx = − Z ( l,r ) µ ( dy ) (cid:18)Z yl f ( x ) u xx ( x, y ) dx + Z ry f ( x ) u xx ( x, y ) (cid:19) = − Z ( l,r ) µ ( dy ) (cid:18) f ( y )( u x ( y, y ) − u x ( y + , y )) − Z rl f ′ ( x ) u x ( x, y ) dx (cid:19) = − Z ( l,r ) s ′ ( y ) f ( y ) µ ( dy ) − Z rl f ′ ( x ) Z ( l,r ) u x ( x, y ) µ ( dy ) dx ! , where the first equality follows from Fubini’s theorem since | u x ( x, y ) | ≤ s ′ ( x ) as observed beforeand f has compact support. The second line above is a consequence of (B.4) and the third lineis a straightforward integration by parts. The last line is a consequence of u x ( y + , y ) − u x ( y, y ) = − s ′ ( y ) and another application of Fubini’s theorem due to the aforementioned bound on u x .Since R rl f ′ ( x ) R ( l,r ) u x ( x, y ) µ ( dy ) dx = R rl f ′ ( x ) h ′ ( x ) dx , (B.3) follows from a simple integrationby parts.(2) This is a restatement of the final part of Theorem 3.1 in this special case. Appendix C. Proof of Theorem 3.3 (1) As in the proof of Theorem 3.2, one can differentiate from the left under the integral sign since R ( l,r ) u α ( y, y ) µ ( dy ) < ∞ and u αx ( x, y ) ≤ (cid:18) ψ ′ α ( x ) ψ α ( x ) + φ ′ α ( x ) φ α ( x ) (cid:19) u α ( y, y ) , where ψ α and φ α are the fundamental solutions as in (2.8). The fact that h ′ is of finite variationcan be shown similarly using the representation of (2.8) and the continuity properties of thefundamental solutions.If X is transient, the potential kernel u exists and we have h ( x ) ≤ Z ( l,r ) u ( x, y ) µ ( dy ) . Thus, − s h ( l +) = s h ( r − ) = ∞ by Theorem 3.2.Also note that u αx ( x, y ) is differentiable from left at all x = y with the left-derivative u αxx ( x, y )satisfying σ u αxx ( x, y ) + b ( x ) u αx ( x, y ) = αu α ( x, y ) for x = y (see Paragraphs 10 and 11 inSection II.1 of [9]). Moreover, u αx ( y + , y ) − u αx ( y, y ) = φ ′ α ( y ) ψ α ( y ) − ψ ′ α ( y ) φ α ( y ) w α = − s ′ ( y ). Thus, du αx ( x, y ) = u αxx ( x, y ) dx − s ′ ( y ) ε y ( dx ), and the same arguments of the proof of Theorem 3.2 canbe used to show that ( h, M ) is a recurrent transform.Now, suppose X is recurrent. By applying a scale transformation we may assume withoutloss of generality that X is on natural scale. This in turn implies − l = r = ∞ . Using the fact that u α ( x, y ) ≤ u α ( y, y ) h ( x ) ≤ Z ( −∞ , ∞ ) u α ( y, y ) µ ( dy ) < ∞ , we deduce R ∞ c h ( x ) dx ≥ R ∞ c R ∞−∞ u α ( y,y ) µ ( dy )) dx = ∞ . That is, s h ( r − ) = ∞ . Similarly, s h ( l +) = −∞ and that ( h, M ) is a recurrent transform follows again from the same lines of theproof of Theorem 3.2 in view of the aforementioned properties of u α .(2) Note that the speed measure of the recurrent transform is given by h ( x ) m ( dx ). Thus, we needto show that the speed measure is finite since the stationary distribution of a one-dimensionaldiffusion is given by its speed measure when it is finite (see p.37 of [9]).Using Jensen’s inequality and Fubini’s theorem we get Z rl h ( x ) m ( dx ) ≤ Z rl Z ( l,r ) ( u α ( x, y )) µ ( dy ) m ( dx ) = Z ( l,r ) Z rl ( u α ( x, y )) m ( dx ) µ ( dy ) . Moreover, Z rl ( u α ( x, y )) m ( dx ) = Z rl Z ∞ Z ∞ e − α ( t + s ) p ( t, x, y ) p ( s, x, y ) dsdtm ( dx )= Z ∞ Z ∞ e − α ( t + s ) Z rl p ( t, y, x ) p ( s, x, y ) m ( dx ) dsdt = Z ∞ Z ∞ e − α ( t + s ) p ( t + s, y, y ) dsdt = Z ∞ ue − αu p ( u, y, y ) du ≤ ε Z ∞ e − ( α − ε ) u p ( u, y, y ) du = u α − ε ( y, y ) ε . In above, the first equality follows from (2.8), the second is due to the symmetry of the transitiondensity and the Fubini’s theorem, while the third is a consequence of Chapman-Kolmogorovidentity.Therefore, R rl h ( x ) m ( dx ) ≤ R ( l,r ) u α − ε ( y,y ) ε µ ( dy ) < ∞ . Appendix D. Proof of Theorem 6.2 If 0 ∈ D ∗ , and X = 0, X t = 0 for all t > P -a.s. since 0 is an absorbing boundary. Thus, v ( t, 0) = E [ g ( X t )] = E [ g (0)] = 0 since g (0) = 0 when 0 ∈ D ∗ .As mentioned in Remark 4.1, the process Y can be considered as an h -transform of X with h ( x ) = x . Indeed, if τ n := inf { t : X t / ∈ ( n , n ) } , X τ n is a bounded martingale and a straightforwardapplication of Girsanov’s theorem yields that X is a weak solution of (4.2) up to τ n . Therefore, E x (cid:2) g ( X t ) [ t<τ n ] (cid:3) = E x (cid:20) [ t<τ n ] g ( X t ) X t X t (cid:21) = xQ x (cid:20) [ t<τ n ] g ( X t ) X t (cid:21) , where Q x is the unique law of solutions of (4.2). Observe that τ n converges to the lifetime, ζ , of X under P x and Q x . Moreover, since X is a positive supermartingale, P x ( ζ = inf { t : X t = 0 } ) = 1while Q x ( ζ = inf { t : X t = ∞} ) = 1 by Proposition 4.4 since the scale function of (4.2) is 1 − /x .Thus, the monotone convergence theorem together with the assumption that g (0) = 0 when 0 is IFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING 35 an accessible boundary under P x yields xQ x (cid:20) [ t<ζ ] g ( X t ) X t (cid:21) = E x (cid:2) g ( X t ) [ t<ζ ] (cid:3) = E x [ g ( X t )] = v ( t, x ) . (D.1)Thus, v ( t, x ) = xw ( t, x ), where w is as defined in (6.11), since the law of Y is the same as that of X under Q x .Recall from Theorem 6.1 that v satisfies (6.5). This automatically implies w satisfy (6.6) and(6.7). To prove the other properties for w fix an m > w ( t, x ) = Q x (cid:20) [ t<ζ ] [ Y t >m ] g ( Y t ) Y t (cid:21) + Q x (cid:20) [ t<ζ ] [ Y t 0. Indeed, since Q x ( ζ > t ) is decreasing in t , we have, for any monotone sequence ( t n ) n ≥ with limit s ,lim n →∞ Q x n ( ζ > t n ) ≤ lim n →∞ Q x n ( ζ > s ∧ t ) = 0 . Motivated by the above define w ( t, x ) := Q x ( ζ > t ) and pick y > x . Then Q x ( ζ > t ) = Q x ( ζ > t, T y < t ) + Q x ( ζ > t, T y > t )= E x (cid:2) [ T y 0. Fix a T > f ( t, x ) := w ( T − t, x ). Employing Itˆo’s formula and using the continuity of σ and w x we get w ( T, x ) = Q x (cid:2) f ( T ∧ τ n,m , Y T ∧ τ n,m ) (cid:3) = Q x (cid:20) g ( Y T ) Y T [ T <τ n,m ] (cid:21) + Q x (cid:2) w ( T − τ n,m , Y τ n,m ) [ T >τ n,m ] (cid:3) since Q x ( T = τ n,m ) = 0.Note that the first term of the summation converges to Q x (cid:20) g ( Y T ) Y T [ T <ζ ] (cid:21) by monotone convergence.Let us first suppose that 0 is an accessible boundary for X . Then Q x (cid:2) w ( T − τ n,m , Y τ n,m ) [ T ≥ τ n,m ] (cid:3) = Q x (cid:20) w ( T − T m , m ) [ T ≥ τ n,m ] [ T n >T m ] (cid:21) + Q x h w ( T − T n , n ) [ T ≥ τ n,m ] [ T n Indeed, Q x (cid:2) w ( T − τ n,m , Y τ n,m ) [ T ≥ τ n,m ] (cid:3) = Q x (cid:20) w ( T − T m , m ) [ T ≥ τ n,m ] [ T n >T m ] (cid:21) + Q x h w ( T − T n , n ) [ T ≥ τ n,m ] [ T n Step 1: Let’s first see that if a is an accessible boundary under P x for some x ∈ ( l, r ), thenlim z → a g ( z ) u λ ( z,y )(1+ | s h ( z ) | ) < ∞ . Indeed, since g is continuous on I and a ∈ I , g ( a ) < ∞ .On the other hand, lim z → a u λ ( z, y ) = 0 (see Table 1 in [33]). Thus, a straightforward ap-plication of L’Hospital’s rule yields lim z → a u λ ( z, y )(1 + | s h ( z ) | ) = lim z → a u λx ( z,y ) > a is a regular absorbing boundary (see, again, Table 1 in [33]). This in particular impliesˆ g (˜ s ( a )) = G (˜ s ( a )).Step 2: Note that G is well-defined and bounded due to Assumption 7.1. Let Y = ˜ s ( X ) andobserve that Y is a local martingale under ˜ P x and G is the least excessive majorant of ˆ g on D := (˜ s ( l ) , ˜ s ( r )) for Y . Note that by the continuity of G we can extend it continuouslyto ˜ s ( ˜ I ), where ˜ I is the state space of X under ˜ P x for any x ∈ ( l, r ). Moreover G will be thesmallest concave majorant on ˜ s ( ˜ I ) of ˆ g ( x ) x ∈ D ( x ), which is lower semicontinuous on ˜ s ( ˜ I ).Therefore, for x ∈ ( l, r ) sup τ ˜ E x [ˆ g (˜ s ( X τ )) [ τ<ζ ] ] = G (˜ s ( x ))by Theorem 1 on p. 124 of [39] or Proposition 3.3 in [13]. Moreover, Lemma 8 and Theorem2 in Chapter 3 of [39] establish for any ε > P x ( τ ∗ ε < ζ ) = 1 and ˜ E x [ G (˜ s ( X τ ∗ ε ))] = G (˜ s ( x )), where τ ∗ ε := inf { t ≥ g (˜ s ( X t )) + ε ≥ G (˜ s ( X t )) } . Step 3: Let v ( x ) := u λ ( x, y ) (cid:16) u λ ( y,y )2 | s h ( x ) | (cid:17) G (˜ s ( x )) for x ∈ I . Observe that v ( a ) is well-defined by the continuity whenever a is an accessible boundary in view of Step 1 and that v ( x ) ≥ g ( x ) for all x ∈ I by construction. Moreover, (cid:0) e − λt v ( X t ) (cid:1) is a P x -supermartingale.Also observe that v ( a ) = g ( a ) whenever a is an accessible boundary due to the constructionof G and Step 1. Thus, if P x ( ζ < ∞ ) = 1, ˆ g (˜ s ( X ζ )) = G (˜ s ( X ζ ) implying P x ( τ ∗ ε < ζ ) = 1by the continuity of ˆ g and G . On the other hand, if P x ( ζ < ∞ ) = 0, P x ( τ ∗ ε > t ) = v ( x ) G (˜ s ( x )) ˜ E x h u λ ( X τ ∗ ε , y ) (cid:0) c | s h ( X τ ∗ ε ) | (cid:1) [ τ ∗ ε >t ] i , which converges to 0 as t → ∞ by dominated convergence since ˜ P x ( τ ∗ ε < ζ ) = 1. Thus, inview of Step 2, we obtain E x [ e − λτ ∗ ε v ( X τ ∗ ε )] = v ( x ) G (˜ s ( x )) ˜ E x [ G (˜ s ( X τ ∗ ε )] = v ( x ) . (E.1)Step 4: The above in fact implies E x [ e − λτ ∗ v ( X τ ∗ )] = v ( x ). Indeed, letting ρ n := inf { t ≥ e − λt v ( X t ) ≥ n } , we have in view of (E.1) that v ( x ) = E x [ e − λτ ∗ ε v ( X τ ∗ ε )] ≤ E x [ e − λρ n ∧ τ ∗ ε v ( X ρ n ∧ τ ∗ ε )]since (cid:0) e − λt v ( X t ) (cid:1) is a P x -supermartingale. Thus, by virtue of the dominated convergencetheorem as ε → 0, we deduce v ( x ) ≤ E x [ e − λρ n ∧ τ ∗ v ( X ρ n ∧ τ ∗ )]. However, E x [ e − λρ n ∧ τ ∗ v ( X ρ n ∧ τ ∗ )]converges to E x [ e − λτ ∗ v ( X τ ∗ )] by the monotone convergence theorem as n → ∞ . This shows v ( x ) ≤ E x [ e − λτ ∗ v ( X τ ∗ )], which in turn yields the claim as v ( x ) ≥ E x [ e − λτ ∗ v ( X τ ∗ )] by thesupermartingale property of (cid:0) e − λt v ( X t ) (cid:1) and Fatou’s lemma.Step 5: We will now see that V = v . To this end let τ be an arbitrary stopping time and τ n :=inf { t ≥ X t / ∈ ( a n , b n ) } , where l < a n < b n < r are such that a n → l and b n → r as n → ∞ . Then, by Fatou’s lemma we obtain E x [ e − λτ g ( X τ )] ≤ E x [ e − λτ ∧ τ n g ( X τ ∧ τ n )] = v ( x ) G (˜ s ( x )) ˜ E x [˜ g (˜ s ( X τ ∧ τ n ))] ≤ v ( x ) in view of Step 2.On the other hand, E x [ e − λτ ∗ ε g ( X τ ∗ ε )] = v ( x ) ˜ E x [˜ g (˜ s ( X τ ∗ ε ))] G (˜ s ( x )) ≥ v ( x ) ˜ E x [ G (˜ s ( X τ ∗ ε )) − ε ] G (˜ s ( x )) = v ( x )( G (˜ s ( x ) − ε ) G (˜ s ( x )) . (E.2)By letting ε → 0, we arrive at V ( x ) ≥ v ( x ), which in turn implies V = v . The fact that V is finite is a consequence of Assumption 7.1.Step 6: It remains to show that τ ∗ is optimal. Indeed, since e − λτ ∗ ε v ( X τ ∗ ε ) converges to e − λτ ∗ v ( X τ ∗ )in L ( P x ) as observed in Step 4, and v ≥ g , ( e − λτ ∗ ε g ( X τ ∗ ε )) ε> is a uniformly integrablefamily. Therefore, with the help of (E.2), we arrive at E x h e − λτ ∗ g ( X τ ∗ ) i = lim ε → E x h e − λτ ∗ ε g ( X τ ∗ ε ) i ≥ lim ε → v ( x )( G (˜ s ( x ) − ε ) G (˜ s ( x )) = v ( x ) . Department of Statistics, London School of Economics and Political Science, 10 Houghton st,London, WC2A 2AE, UK E-mail address ::