Diffusion transformations, Black-Scholes equation and optimal stopping

# Diffusion transformations, Black-Scholes equation and optimal stopping

Umut Çetin Department of Statistics, London School of Economics and Political Science, 10 Houghton st, London, WC2A 2AE, UK
October 3, 2019
###### Abstract.

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf [26] as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an alternative Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing in the presence of bubbles. Finally, we use these path transformations to propose a unified framework for 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 boundaries of perpetual American options.

## 1. Introduction

Conditioning the paths of a given Markov process to stay in a certain subset of the path space is a well-studied subject which has become synonymous with the term -transform. If one wants to condition the paths of to stay in a certain set, the classical recipe consists of finding an appropriate excessive function , defining the transition probabilities of the conditioned process via , and constructing on the canonical space a Markov process with these new transition probabilities. This procedure is called an -transform. In particular if is a minimal excessive function with a pole at (see Section 11.4 of [10] for definitions), then is the process conditioned to converge to and killed at its last exit from . We refer the reader to Chapter 11 of [10] for an in-depth analysis of -transforms.

This paper proposes a new class of path transformations for one-dimensional regular diffusions with stochastic differential equation (SDE) representation. The new transformations are aimed at switching the behaviour of the diffusion from transient to recurrent or vice versa. We introduce the concept of recurrent transformation in Section 3 and characterise these transforms via weak solutions of SDEs. Roughly speaking, a recurrent transformation adds a drift term to the original SDE of so that the resulting process is a recurrent regular diffusion with the same state space whose law is locally absolutely continuous with respect to the original law. Although the recurrent transformation is at first sight meaningful only for transient diffusions, we note a special class of recurrent transformations in Theorem 3.3 that is applicable not only to transient diffusions but also to 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 motion with alternating drift, which appears in the studies of the bang-bang control problem (see Example 3.2).

As a first application of the recurrent transformation, we compute in Corollary 3.1 the distribution of the first exit time from an interval for a given diffusion. Although the formula does not provide an expression in closed form in general, a simple Monte Carlo algorithm will provide a sufficiently close estimate.

The distribution of first exit times has attracted the attention of researchers working on problems arising in the Monte Carlo simulation of stochastic processes (see, e.g., [1], [2], [21], [22], and the references therein). Yet precise formulas for the distribution of exit times of diffusions have rarely been the subject of a thorough investigation. The recent paper of Karatzas and Ruf [26] seems to be the only work in the literature that addresses this problem in the general framework of one-dimensional diffusions. With an additional assumption on the local Hölder continuity of the coefficients of the SDE satisfied by they have shown that the distribution function of the first exit time was the minimal nonnegative solution of a particular Cauchy problem. Although this is a useful characterisation from a theoretical perspective, finding the smallest solution of a Cauchy problem is in general not a feasible numerical task. Our formula in Corollary 3.1 thus provides a way of computing the minimal solutions of the class of Cauchy problems considered by Karatzas and Ruf.

As described briefly in Remark 3.4 recurrent transformations can also be used to improve the accuracy 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 , where is the number of discretisations, as opposed to , which is the rate of convergence for discrete Euler schemes for diffusions without killing. As the recurrent transformation removes the killing by passing to a locally absolutely continuous probability measure, it can be used to bring the convergence rate back to using the recipe in Remark 3.4. This important application of recurrent transformations will be studied rigorously in a subsequent paper.

Section 4 is devoted to the convergence of certain recurrent transforms when is nonnegative and on natural scale. Under a mild condition on the diffusion coefficient of we show that a particular sequence of recurrent transformations converges monotonically to the -transform of , where . We observe that the nature of this convergence depends crucially on whether is a strict local martingale or not. In particular, we construct on a single probability space a sequence of recurrent transforms that increases a.s. to a diffusion that has the same law as the aforementioned -transform. The limiting diffusion is non-exploding on if and only if is a true martingale.

Our interest in local martingales in fact stems from the financial models with bubbles. If a financial model admits no arbitrage opportunities, the discounted stock price must follow a nonnegative local martingale under a so-called risk-neutral measure by the Fundamental Theorem of Asset Pricing [14]. When is not a martingale but a strict local martingale, the stock price exhibits 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 a European option that pays the amount of to its holder at time for some . The arbitrage pricing theory suggests that the fair price of this option at time is , where . Under mild conditions on and a continuity and linear growth assumption on , Ekström and Tysk [17] have shown that satisfies the Cauchy problem

 ut=Au,u(0,⋅)=g, (1.1)

where is the infinitesimal generator of . As a consequence, Ekström and Tysk have observed in [17] that (1.1) admits multiple nonnegative solutions when is a strict local martingale and . Namely, they have identified and as such two distinct solutions. Note that being a strict local martingale implies . Thus, is a solution of (1.1) when . However, this immediately leads to the conclusion that there are infinitely many solutions of at most linear growth to (1.1) whenever is of at most linear growth. Indeed, by the above discussion for any , is a nonnegative solution of (1.1) when is of at most linear growth. Moreover, Ekström and Tysk have also shown that is of at most linear growth when is continuous function of at most linear growth. This in turn renders of linear growth. Hence, restricting solutions to have at 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 the Cauchy problem is determined by the martingale property of . Later, Bayraktar et al. [4] have extended 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 compute the option prices by solving (1.1) numerically. Also note that one will also fail to compute using a Monte-Carlo simulation when is of linear growth and is a strict local martingale. Indeed, if, e.g., , the Monte-Carlo algorithm will yield for since the discretisation of via the Monte-Carlo scheme will result in a true martingale for the approximating process. To resolve this issue we establish in Section 6 a new characterisation of in terms of the unique solution of an alternative Cauchy problem. We show that the function , after an appropriate scaling, becomes the unique solution of

 wt=~Aw

with certain initial and boundary conditions, when is the generator of a suitable -transform of . More precisely, this -transform coincides with the one that is obtained as the limit of recurrent transforms in Section 4. One interesting corollary of the main result of this section is that for any the valuation function is of strictly sublinear growth at when is of at most linear growth and the stock price is given by a strict local martingale. In particular, for any if is a strict local martingale.

While Section 6 is on the valuation of European options, Section 7 considers the pricing of perpetual American options. In order to price such an option with payoff , one needs to solve the optimal stopping problem

 supτ≤ζEx[e−λτg(Xτ)],

where is the (possibly finite) lifetime of the diffusion and the discount rate corresponds to the constant interest rate.

Peskir and Shiryaev [35] give an excellent survey of available methods to tackle this problem. One approach to the above consists of solving a free boundary problem associated to the infinitesimal generator of . Another approach is via the characterisation of -excessive functions of as the value function for the optimal stopping problem is the least -excessive majorant of . This is the path taken by Dayanik and Karatzas in [13]. Beibel and Lerche [6] have also proposed a new methodology based on simple martingale arguments, which can also be interpreted as change of measure arguments as observed by [31]. While the approach based on the solution of a free boundary problem 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 the solutions of a family of Sturm-Liouville equations at hand. Moreover, the solution techniques offered in [13] and [6] differ for different boundary behaviour exhibited by , i.e. whether the boundaries of the state space of are absorbing or natural, etc. Furthermore, how the function behaves near the boundaries also matters. For instance, Beibel and Lerche [6] have to check five conditions on the behaviour of to determine the solution. It is also worth to note the recent work of Lamberton and Zervos [30] who analyse a large class of optimal stopping problems via variational equalities defined by the generator of and without the assumption that is continuous.

Section 7 presents a unified solution to the above optimal stopping problem that does not vary depending on the behaviour of or near the boundaries. We use the specific recurrent transform of Proposition 3.2, which is applicable to transient as well as recurrent diffusions, to determine whether the value function is finite. We show that the value function is finite if and only if satisfies the single condition (7.8), which depends only on the knowledge of , the -potential density, for some . This recurrent transform also changes the optimal stopping problem to one without discounting. However, the new problem becomes two-dimensional. In order to reduce the dimension of the problem to one, we apply the transformation that is defined in Section 5, which is aimed at conditioning the recurrent transformation to have a certain behaviour at the boundary points and become transient. After this transformation all that remains to do is to solve

 supτ~Ex[¯g(Xτ)],

where is a function that depends only on and , and corresponds to the expectation operator with respect to the law of the diffusion after the final transformation. Solution to the above is easy and well-known since Dynkin [16]: After a change of scale, the value function of the above optimal stopping problem is the smallest concave majorant of .

It has to be noted that Cisse et al. [11] have attacked this problem using -transforms. However, as we explain in detail in Remark 7.1 the authors make some implicit assumptions regarding the boundary behaviour of as well as the function in the proof of their key arguments. These assumptions in particular exclude the diffusion processes with infinite lifetime. As we mentioned above, our approach is general and do not impose any conditions on other than the regularity and the Engelbert-Schmidt conditions that ensures an SDE representation for .

In essence our framework is fundamentally different in spirit from [11] and [6] in the sense that it gives a probabilistic interpretation of the value function and the optimal stopping boundaries under a locally absolutely continuous measure in the classical framework of Dynkin [16] with no discounting. The works of [11] and [6], on the other hand, obtain the solution by a clever algorithm of 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 martingale property of on the valuation of perpetual American options as the methodology is the same for the martingales as well as the local martingales. We refer the reader to [3] for a thorough analysis of 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 to one-dimensional diffusions that will be used throughout the paper. Section 3 introduces the concept of recurrent transformations while Section 4 considers their limit in relation to the local martingale property of . Section 5 defines a transform designed specifically for recurrent diffusions that is different than the typical -transform but will still render them transient, which will be useful in Section 7. Section 6 provides a resolution to the non-uniqueness issue of the Black-Scholes pricing equation and Section 7 addresses the optimal stopping problem. Section 8 concludes. Proofs of certain 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 anonymous referees for their comments that led to several improvements.

## 2. Preliminaries

Let be a regular diffusion on , where . We assume that if any of the boundaries are reached in finite time, the process is absorbed at that boundary. This is the only instance when the process can be ‘killed’, we do not allow killing inside . Such a diffusion is uniquely characterised by its scale function and speed measure , defined on the Borel subsets of the open interval . The set of points that can be reached in finite time starting from the interior of and the entrance boundaries will be denoted by . That is, is the union of with the regular, exit or entrance boundaries. The law induced on , the space of -valued continuous functions on , by with will be denoted by as usual, while will correspond to its lifetime, i.e. . For concreteness we assume that is the coordinate process on the canonical space , i.e. for all . However, this assumption is only for convenience and one can work with other measurable spaces as long as the measures are properly defined. The filtration will correspond to the universal completion of the natural filtration of and, therefore, is right continuous since is strong Markov by definition (see Theorem 4 in Section 2.3 in [10]). We will also set . If is a measure on some open interval and is a nonnegative or -integrable measurable function, the integral of with respect to will be denoted by unless is absolutely continuous with respect to the Lebesgue measure , in which case we shall write .

In what follows we will often replace with when dealing with the limit values of the processes as long as no confusion arises. Recall that in terms of the first hitting times, for , the regularity amounts to whenever and belongs to the open interval . This assumption entails in particular that is strictly increasing and continuous (see Proposition VII.3.2 in [36]) and for all (see Theorem VII.3.6 and the preceding discussion in [36]).

Recurrence or transience of depends on the behaviour of near the boundary points. More precisely, is transient if and only if at least one of and is finite. Since is unique only up to an affine transformation, we will use the following convention throughout the text:

• whenever finite,

• whenever finite.

Note that in view of our foregoing assumptions one can easily deduce that when is transient. We refer the reader to [9] for a summary of results and references on one-dimensional diffusions. The definitive treatment of such diffusions is, of course, contained in [25]. The recent manuscript of Evans and Hening [19] contains a detailed discussion with proofs of some aspects of the potential theory of one-dimensional diffusions.

###### Remark 2.1.

It has to be noted that notion of recurrence that we consider here excludes some recurrent solutions of one-dimensional SDEs with time-homogeneous coefficients since we kill our diffusion as soon as it reaches a regular boundary point. A notable example is a squared Bessel process with dimension , which solves the following SDE:

 Xt=x+2∫t0√XsdBs+δt.

The above SDE has a global strong solution, i.e. solution for all , which is recurrent (see Section XI.1 of [36]). However, the point is reached a.s. and is instantaneously reflecting by Proposition XI.1.5 in [36]. As such, it violates our assumption of a diffusion being killed at a regular boundary. According to our assumption, a squared Bessel process of dimension has to be killed as soon as it reaches and, thus, is a transient diffusion.

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 and such that

 σ(x)>0 and ∃ε>0 s.t. ∫x+εx−ε1+|b(y)|σ2(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 (up to the exit time from the interval ) to the SDE

 Xt=x+∫t0σ(Xs)dBs+∫t0b(Xs)ds,t<ζ, (2.2)

where and . Moreover, condition (2.1) further implies one can take

 s(x)=∫xCexp(−2∫zcb(u)σ2(u)du)dz and m(dx)=2s′(x)σ2(x)dx, for some (c,C)∈(l,r)2. (2.3)

We collect the assumptions on in the following:

###### Assumption 2.1.

is a regular one-dimensional diffusion on such that

 Xt=X0+∫t0σ(Xs)dBs+∫t0b(Xs)ds,t<ζ,

where and satisfy (2.1), .

In the sequel the extended generator of will be denoted by . Following Definition VII.1.8 of Revuz and Yor [36] we will write for a given Borel measurable function , if there exists Borel function such that, for each , i) -a.s. for every , and ii)

 f(Xt)−f(X0)−∫t0g(Xs)ds

is -local martingale. In this case is said to be in the domain of . If is on , then becomes a second order differential operator, i.e.

 Af(x)=12σ2(x)f′′(x)+b(x)f′(x).

Any regular transient diffusion on has a finite potential density, , with respect to its speed measure (see Paragraph 11 in Section II.1 of [9]). That is, for any nonnegative and measurable vanishing at accessible boundaries

 Uf(x):=∫∞0Ex[f(Xt)]dt=∫rlf(y)u(x,y)m(dy).

The above implies that the potential density can be written in terms of the transition density111For the existence of this transition density and its boundary behaviour see Mc Kean [33]. , , of with respect to its speed measure:

 u(x,y)=∫∞0p(t,x,y)dt.

The above in particular implies that since is symmetric for each (see p. 520 of [33]). If is recurrent , either or (see Theorem 1 in Section 3.7 of [10]). Therefore, potential density only makes sense for transient diffusions.

We will denote by the family of semimartingale local times222Observe that the diffusion local time, , in Paragraph 13 in Section II.2 of [9] is defined via . Comparing this with the occupation times formula for the semimartingale local time reveals the relationship . associated to . Recall that the occupation times formula for the semimartingale local time is given by

 ∫t0f(Xs)σ2(Xs)ds=∫rlf(x)Lxtdx.

In the case of one-dimensional transient diffusions the distribution of is known explicitly in terms of the potential density (see p.21 of [9]). In particular,

 Py(Ly∞>t)=exp(−s′(y)t2u(y,y)). (2.4)

Note that if , then and

 Px(Ty<∞)=⎧⎪ ⎪⎨⎪ ⎪⎩s(x)s(y),y≥x;1−s(x)1−s(y),y

On the other hand, if and , then , -a.s. for any , which in turn implies

 Px(Ty<∞))=⎧⎨⎩s(x)s(y),y≥x;1,y

Similarly, if and , then , -a.s. for any , and

 Px(Ty<∞)=⎧⎨⎩1,y≥x;1−s(x)1−s(y),y

While the potential density is finite only for transient diffusions, one can define a so-called -potential density that exists and is finite for all diffusions for all . For any nonnegative and measurable function vanishing at accessible boundaries, one defines

 Uαf(x):=∫∞0e−αtEx[f(Xt)]dt.

Thus, if we let

 uα(x,y):=∫∞0e−αtp(t,x,y)dt,

we obtain

 Uαf(x)=∫rlf(y)uα(x,y)m(dy).

is called the -potential density and is symmetric in for all . An alternative and very useful expression for is given in terms of the fundamental solutions of the equation . That is,

 uα(x,y)=ψα(x)ϕα(y)wα,x≤y, (2.8)

where and are, respectively, the increasing and decreasing nonnegative solutions of subject to certain boundary conditions (see p.19 of [9]), and is the Wronskian given by

 wα=ψ′α(x)ϕα(x)−ψα(x)ϕ′α(x)s′(x),

which is independent of . Consequently, using the relationship between the fundamental solutions of and the Laplace transforms of hitting times (see p.18 of [9]), we have

 Ex[exp(−αTy)]=uα(x,y)uα(y,y). (2.9)

We refer the reader to Chap. II of Borodin and Salminen [9] for a summary of results concerning one-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 produces a recurrent diffusion whose law is locally absolutely continuous with respect to that of the original diffusion. To wit, suppose is a non-negative -function and an adapted continuous process of finite variation so that is a non-negative local martingale. If is a localising sequence for this local martingale, using Girsanov’s theorem we arrive at a weak solution on to the following SDE for any given :

 Xt=x+∫t0σ(Xs)dBs+∫t0{b(Xs)+σ2(Xs)h′(Xs)h(Xs)}ds. (3.1)

We can associate to the above SDE the scale function

 sh(x):=∫xcs′(y)h2(y)dy,x∈(l,r), (3.2)

provided that the integral is finite for all , which in particular requires on . What we would like to achieve is to extend this procedure by taking and obtain a recurrent diffusion. The latter will require . We shall see in this section that this property alone is sufficient to obtain a recurrent weak solution of (3.1) on under some mild conditions on .

Using and to get a recurrent process imposes some boundary conditions on . Indeed, if (resp. ), in order to have (resp. ), we must have (resp. ).

Moreover, since is a local martingale, . Thus, is given by

 Mt=exp(−∫t0Ah(Xs)h(Xs)ds).

In the light of the above discussion we now introduce the concept of a recurrent transformation of a diffusion.

###### Definition 3.1.

Let be a regular diffusion satisfying Assumption 2.1 and be an absolutely continuous function. Then, is said to be a recurrent transform (of ) if the following are satisfied:

1. is an adapted process of finite variation.

2. is a nonnegative local martingale.

3. The function from (3.2) is finite for all with .

4. There exists a unique weak solution to (3.1) for for any .

In the above definition, the defining condition for a recurrent transformation is the function and its explosive nature near the boundaries. The function and the functional come into play when one wants to construct a weak solution of the SDE (3.1) and show that the law of its solution is locally absolutely continuous with respect to that of the original process , which satisfies (2.2). The next theorem, whose proof is delegated to the Appendix, suggests a general machinery for constructing recurrent transformations.

###### Theorem 3.1.

Let be a regular diffusion satisfying Assumption 2.1. Consider an absolutely continuous function such that its left derivative is of finite variation. Suppose further that the mapping given by (3.2) is finite for all and that . Then, the following statements are valid.

1. can be chosen to be left-continuous. Moreover, the signed measure defined by on admits the Lebesgue decomposition , where denote its Borel measurable Radon-Nikodym derivative with respect to the Lebesgue measure on , and is a locally finite signed measure on that is singular with respect to the Lebesgue measure.

2. The integral

 1[t<ζ](∫t0∣∣~Ah(Xs)∣∣ds+∫rlLxt2|n(dx)|)<∞,Py%−a.s., (3.3)

for every , where .

3. is a recurrent transform, where, on ,

 Mt := exp(−∫t0~Ah(Xs)h(Xs)ds−∫t01h(Xs)dΛs(h)) and Λt(h) := ∫(l,r)Lxt2n(dx).
4. -a.s..

5. Let be the law of the solution of (3.1) and for some -stopping time . Then,

 Rh,x(F,T<∞)=1h(x)Ex[1Fh(XT)MT]. (3.4)

In particular, is a -martingale.

6. If is an -stopping time such that , then for any the following identity holds:

 Px(ζ>T,F)=h(x)Eh,x[1F1h(XT)MT], (3.5)

where is the expectation operator with respect to the probability measure .

###### Example 3.1.

Suppose and consider a -dimensional Bessel process on , i.e. a one-dimensional diffusion with the dynamics

 dXt=2√XtdBt+δdt.

The scale function is given by . Thus, is transient and approaches to as , while is an inaccessible boundary.

Let and define

 Mt:=exp((δ−2)28∫t01Xsds),t≥0.

Then, it follows from Theorem 3.1 that is of finite variation. Moreover,

 sh(x)=δ−22∫x11udu=logx,x>0.

Thus, , and we conclude that is a recurrent transform by invoking Theorem 3.1 again. The transformation yields the following SDE for the resulting process

 dXt=2√XtdBt+2dt,

which is the SDE for a -dimensional squared Bessel process. Recall (or see p.442 of [36]) that is polar for a -dimensional squared Bessel process.

The following proposition gives an important example of a recurrent transformation for transient diffusions, which will be useful in the sequel.

###### Proposition 3.1.

Suppose is a regular transient diffusion satisfying Assumption 2.1. Let be fixed and consider the pair defined by

 h(x):=u(x,y),x∈(l,r), and Mt=exp(s′(y)Lyt2u(y,y)).

Then, the following hold:

1. is a recurrent transform for .

2. There exists a unique weak solution to

 Xt=x+∫t0σ(Xs)dBs+∫t0{b(Xs)+σ2(Xs)ux(Xs,y)u(Xs,y)}ds,t≥0, (3.6)

for any , where denotes the first partial left derivative of with respect to .

3. Moreover, if denotes the law of the solution and is a stopping time such that , then for any the following identity holds:

 Px(ζ>T,F)=u(x,y)Eh,x[1F1u(XT,y)exp(−s′(y)2u(y,y)LyT)], (3.7)

where is the expectation operator with respect to the probability measure .

The above is a direct corollary of Theorem 3.1 since in the Lebesgue decomposition of as in Part (1) of Theorem 3.1 and is twice differentiable with for all .

Proposition 3.1 is in fact a special case of a more general result that will allow us to construct a large family of recurrent transformations. In order to motivate this more general result note that is the potential333If is a measure on , the potential of is the function and is denoted by . See Section VI.2 of [8] for details. of the Dirac measure at point . Moreover, it is uniformly integrable being bounded. Conversely, since 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 is the potential of some measure on , i.e. . Also note that if is a uniformly integrable potential, e.g. , then is a supermartingale, which is not a martingale. As a matter of fact, in view of the Riesz representation of excessive functions (see Theorem VI.2.11 in conjunction with Proposition IV.5.4 in [8]) the greatest uniformly integrable harmonic function dominated by is . The next result, whose proof is in the Appendix, shows that the potential of a probability measure on gives rise to a recurrent transform under an integrability condition.

###### Theorem 3.2.

Let be a Borel probability measure on such that . Suppose is a regular transient diffusion satisfying Assumption 2.1 and define

 h(x):=∫(l,r)u(x,y)μ(dy).
1. The left derivative of exists and is a recurrent transform of , where

 Mt:=exp(∫t01h(Xs)dAs) and At:=∫(l,r)s′(x)Lxt2μ(dx).
2. If denotes the law of the solution of (3.1) and is a stopping time such that , then for any the following identity holds:

 Px(ζ>T,F)=h(x)Eh,x[1F1h(XT)exp(−∫t01h(Xs)dAs)],

where is the expectation operator with respect to the probability measure .

###### Remark 3.1.

Note that satisfies the assumptions of the above theorem since and for all . Thus, Proposition 3.1 is a direct consequence of Theorem 3.2 as well.

The next example of a recurrent transform that we shall consider in this paper is obtained via the -potential density, of . In contrast with the previous transform, which only exists for transient diffusions, the next transform can be applied to all regular diffusions. Moreover, the resulting diffusion will be positive recurrent.

###### Proposition 3.2.

Suppose is a regular diffusion satisfying Assumption 2.1. Let and be fixed and consider the pair defined by

 h(x):=uα(x,y),x∈(l,r), and Mt=exp(−αt+s′(y)Lyt2uα(y,y)).

Then, the following hold:

1. is a recurrent transform for .

2. There exists a unique weak solution to

 Xt=x+∫t0σ(Xs)dBs+∫t0{b(Xs)+σ2(Xs)uαx(Xs,y)uα(Xs,y)}ds,t≥0, (3.8)

for any , where denotes the first partial left derivative of with respect to .

3. Moreover, the diffusion defined by the solutions of (3.8) is positive recurrent and its stationary distribution on is given by

 π(dx)=(uα(x,y))2∫∞0se−αsp(s,y,y)dsm(dx), (3.9)

where is the transition density of the original diffusion with respect to its speed measure .

As in the case of Proposition 3.1, parts (1) and (2) of the above result is a direct corollary of Theorem 3.1 but will also be a special case of a more general theorem in terms of -potentials. Analogously, of is the -potential of the Dirac measure at and is a uniformly integrable supermartingale converging a.s. to as . Moreover, any uniformly integrable -potential is of the form for some measure on .

###### Theorem 3.3.

Suppose is a regular diffusion satisfying Assumption 2.1 and . Let be a Borel probability measure on such that . Define

 h(x):=∫(l,r)uα(x,y)μ(dy).
1. The left derivative of exists and is a recurrent transform of , where

 Mt:=exp(−αt+∫t01h(Xs)dAs) % and At:=∫(l,r)s′(x)Lxt2μ(dx).
2. Moreover, if there exists such that , then the diffusion defined by the solutions of (3.1) is positive recurrent and its stationary distribution on is given by

 π(dx)=h2(x)∫rlh2(y)m(dy)m(dx).
###### Remark 3.2.

Note that satisfies the assumptions of the above theorem since and for all . Thus Proposition 3.2 follows directly from Theorem 3.3.

Moreover, if is transient, the potential density exists and is finite. In this case the condition is equivalent to under the assumption that is a probability measure. Thus, the condition in Theorem 3.3 is the exact analogue of the condition of Theorem 3.2.

If is nonnegative, , and , Theorems 3.2 and 3.3 show that will define a recurrent transform for . In this case the finite variation process will be given by

 At=∫t0f(Xs)ds.

For instance, in Example 3.1 it can be verified using the scale function and the speed measure of squared Bessel processes that leading to in the notation of Theorem 3.2.

###### Example 3.2.

Suppose is a standard Brownian motion. It is well-known that

 uα(x,y)=1√2αexp(−√2α|x−y|).

Thus, if we use the transform in Proposition 3.2 with , the recurrent transform is the solution to the following SDE:

 dXt=dBt−√2αsgn(Xt)dt,

where . 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 the references therein).

We shall consider in subsequent sections the applications of the above recurrent transforms to optimal stopping as well as some pricing issues arising in Black-Scholes models when the stock price follows a strict local martingale. However, one can find an immediate application of the recurrent transform to the computation of the distribution of the first exit time for a one-dimensional diffusion from an interval. Indeed, such a first exit time can always be viewed as the life time of a transient diffusion by killing the original one as soon as it exits the given interval. Thus, the problem reduces to finding for all , where is the law of the transient diffusion starting at and is its lifetime, i.e. the first time it exits the given interval. The following is a direct consequence of Proposition 3.1.

###### Corollary 3.1.

Let be a regular transient diffusion satisfying Assumption 2.1. Then

 Px(ζ>t)=u(x,y)Eh,x[1u(Xt,y)exp(−s′(y)2u(y,y)Lyt)],

where 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 in closed-form, it is nevertheless practical. Indeed, by running a Monte-Carlo simulation of the solution of (3.6), one can get a close estimate of

 Eh,x[1u(Xt,y)exp(−s′(y)2u(y,y)Lyt)]

by approximating the local time using the occupation times formula.

Karatzas and Ruf [26] have shown that the function is the smallest nonnegative classical supersolution of

 vt=Av,v(0,⋅)=1 (3.10)

under the assumption that and are locally uniformly Hölder continuous on . Thus, combining their Proposition 5.4 and Corollary 3.1 we deduce the following.

###### Corollary 3.2.

Let be a regular transient diffusion satisfying Assumption 2.1. Assume further that and that appears in (2.2) are locally uniformly Hölder continuous on . Define

 v(t,x):=u(x,y)Eh,x[1u(Xt,y)exp(−s′(y)2u(y,y)Lyt)],

where is the expectation operator with respect to the law of the recurrent transform given by (3.6). Then, 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 classical supersolutions of (3.10). Indeed, if is the potential of a probability measure on satisfying the hypothesis of Theorem 3.2, then

 Px(ζ>t)=h(x)Eh,x[1h(Xt)exp(−∫(l,r)s′(y)Lyt2h(y)μ(dy))].

In particular if for some , .

###### Remark 3.4.

The recurrent transformation of a transient diffusion can be used to improve the accuracy of discrete Euler approximations of diffusions that are killed when leaving a bounded interval . Suppose represents the first exit time from this interval and one is interested in the Monte Carlo simulation of for some suitable via a discrete Euler scheme applied to the SDE (2.2) for . Gobet [21] has shown that the discretisation error is of order , where 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 applied to a diffusion without killing, where the error is of order . On the other hand, the recurrent transformation from Theorem 3.2 can be used to improve the convergence rate back to since

 Ex[F(XT)1[T<ζ]]=h(x)Eh,x[F(XT)1h(XT)exp(−∫T0f(Xs)h(Xs)ds)],

where for a nonnegative with