On first passage time problems of Brownian motion -- The inverse method of images revisited
Sören Christensen Oskar Hallmann Maike Klein

	
	
	
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	Let W be a standard Brownian motion with W_0 = 0 and let b [0,∞) → be a continuous function with b(0) > 0. In this article, we look at the classical First Passage Time (FPT) problem, i.e., the question of determining the distribution of τinf{ t∈ [0,∞)  W_t ≥ b(t) }. More specifically, we revisit the method of images, which we feel has received less attention than it deserves. The main observation of this approach is that the FPT problem is fully solved if a measure μ exists such that 
		
    ∫_(0,∞)exp(-θ^2/2t+θ b(t)/t)μ(dθ)=1,      t∈(0,∞).

The goal of this article is to lay the foundation for answering the still open question of the existence and characterisation of such a measure μ for a given curve b. We present a new duality approach that allows us to give sufficient conditions for the existence. Moreover, we introduce a very efficient algorithm for approximating the representing measure μ and provide a rigorous theoretical foundation.
	
	
	
	

			2020 MSC     90C05, 60G40, 60J65.
			

			Keywords     first passage time, inverse method of images, linear programming, Brownian motion, boundary hitting.
		
	

	
		
	
	

§ INTRODUCTION

	
	One of the most intuitive questions to ask is when a Brownian motion crosses a given boundary. This problem is widely known as the First Passage Time (FPT) problem. Let W be a one-dimensional Brownian motion with W_0 = 0 and let b [0,∞) → be a continuous function with b(0) > 0. The first passage time of b (from below) is then defined as
	
    τinf{ t ∈[0,∞) W_t ≥ b(t) }.

	One goal of the FPT problem is to determine the cumulative distribution function (c.d.f.) F of τ,  
	i.e., F(t) = (τ≤ t).
At the end of this section we give a survey of the literature on the FPT problem, where it turns out that none of the numerous solution methods answers the question completely satisfactorily.  The method of images takes a special position among these methods because it can be used to obtain a large family of curves for which the first passage time distribution is known quite explicitly. The application in this context goes back at least to ideas of <cit.> and has been widely used for questions of sequential statistics <cit.>. We describe the essential ideas now based on <cit.> as far as necessary for understanding and give more details and an intuition in Section <ref>. 

Let μ be a positive, σ-finite measure on (0,∞) and b [0,∞) → be the unique solution  b(t)=x of the equation

    ∫_(0,∞)exp(-θ^2/2t+θ x/t)μ(dθ)=1,      t∈(0,∞).

 Then the distribution function of τ, the first passage time of the Brownian motion W to b, is given as a function of μ and b, more precisely, 

    (τ≤ t)	   = 1-Φ( b(t)/√(t)) + ∫_(0,∞)Φ( b(t)-θ/√(t)) μ(d θ).

Therefore, the method of images can be seen as a method to generate certain curves with explicitly given first passage time distribution which is why the method has been celebrated. However, explicit solutions remain scarce apart from the linear boundary, cf. <cit.> for more explicit examples.

In practice, it seems more reasonable to ask the inverse question: Given a boundary b, how must μ be chosen to obtain the distribution of τ using the method of images, i.e., such that b is the solution to (<ref>). This is called the inverse method of images.
This question has already been formulated in <cit.>  and was more recently raised again in <cit.>.
	 It should actually be divided into two parts:

	
  * Is it possible to identify classes of curves b that ensure the existence of a representing measure μ in the sense of the method of images, i.e., μ satisfies ∫_(0, ∞)exp(-θ^2/2t+θ b(t)/t)μ(dθ)=1 for all t∈(0,∞)? To the best of our knowledge, there has been no significant progress on this question. 
	
  * Given that a curve is representable, how can we approximate the measure μ numerically in an efficient and in a theoretically sound way? Initial approaches to this, albeit without e.g. a guarantee of convergence, already exist in the literature. We discuss this in more detail in Section <ref>. 

In summary, it can be said that there are only very partial results on the inverse method of images. This may explain why the method has received less attention in recent years than the authors of this article believe it deserves. This paper aims to change this and thus to stimulate new activity in this area. Our main contributions are as follows:

	
  * We formulate and investigate the representability of curves by studying two dual infinite dimensional linear programs. This opens up a new perspective on the problem and provides a basis for formulating sufficient conditions for a curve to be representable by a measure μ. 
	
  * In addition, the adaptation of infinite linear programming algorithms provides an extremely efficient and accurate way to solve the FPT problem numerically. This method is further underpinned by precise convergence results. 


	

 §.§ Structure of the paper


The paper is organised as follows. 
In Section <ref> we introduce the method of images and the inverse method of images in more detail. 
Section <ref> deals with a new infinite  dimensional linear programming approach to the inverse method of images. We give two different linear programs, formulate the corresponding dual programs and prove strong duality results for both set-ups (Sections <ref> and <ref>). 

This forms the basis for our key results. First, we obtain sufficient conditions for representability of a given concave, analytic boundary b, see Section <ref>.
In Section <ref> we provide convergence results for our linear programs, where we only discretise the space axis which allows to present a new algorithm and  give error bounds for the approximation. In Section <ref> we present the numerical results for our algorithm for four boundaries. Finally, we shortly  comment on the inverse method of images for two-sided boundaries in Section <ref>.



 §.§ Literature on the FPT problem

	The FPT problem has been studied extensively and many applications have been developed: In statistics, FPT problems arise for problems in testing, cf. <cit.> or the surveys <cit.>. In finance, the problem emerges in the valuation of barrier options, cf. <cit.>, as well as in default models, cf. <cit.>, and in the evaluation of credit risks, cf. <cit.>. For an overview of the FPT problem and some applications in the field of physics, including a connection to electrostatics, see e.g. <cit.> or the monograph <cit.>.
	
	
	The FPT problem can be traced back to the thesis “Théorie de la spéculation” by Bachelier <cit.>, where the problem is first formulated, but only for constant b. Other early formulations of the problem include <cit.>. 
	
	Even though the problem has been studied for a long time, closed form solutions for the distribution function F of τ given the function b are rare. For linear boundaries b(t) = a+m t with  a>0, m ∈ the well-known Bachelier-Lévy formula gives the density of τ as
	
    a/t^3/2 φ( a+m t/√(t)),      t∈(0,∞),

	where φ is the density of a standard normal distribution, see <cit.> for the case m=0 or <cit.> for the more general case m ∈.
	
	Partial results are available for other kind of boundaries: 
	For the square-root boundary, asymptotic results for the distribution function F of τ are obtained in <cit.>,  a rather lengthy but explicit formula for the c.d.f. F associated to square-root boundaries is derived in <cit.>  and  an infinite power series is presented in <cit.>. 	
	The quadratic boundary was investigated among others in <cit.>, where in both papers a formula for the distribution function of τ is derived which depends on Airy functions.
	
	Another approach to the FPT problem is by integral equations (typically Volterra or Fredholm type integral equations) connecting the boundary b and the distribution function F of τ.
	While there have been many approaches involving Volterra type integral equations for many years, Peskir <cit.> presents a unifying approach via an integral equation – the so-called “master equation” –  to derive these equations.
	These integral equations are in most cases difficult to solve analytically but there are numerical approaches, see e.g. <cit.>.
		There has also been use of Fredholm type integral equations but more rarely, see <cit.>.
			
	In recent years, numerical approaches have rather concentrated on Monte Carlo methods, see e.g. <cit.>.		 Their most important drawbacks are the extensive computation time and the problem of undetected crossings in between discretisation steps. 
	
	

§ THE METHOD OF IMAGES AND THE INVERSE METHOD OF IMAGES


	The method of images 
	 is based on the following idea (cf. <cit.>): consider on the one hand a Brownian motion starting in 0  with unit mass and on the other hand a second Brownian motion with starting points greater than 0 given by some mass distribution μ. 
	 We then observe the “superposition” of the distributions of the two Brownian motions, or more precisely the points (t,x) in space-time at which the Brownian motions are staying with the same intensity. 
	 These are the points (t,x)∈ (0,∞)× such that the density of the standard Brownian motion at time t evaluated in x coincides with the density of the other Brownian motion at time t weighted according to μ also evaluated in x, see Figure <ref>. The main observation in the method of images is that these points form a boundary b for which the first passage time distribution F can be given explicitly in terms of μ (see Proposition <ref> below).


We define

    h(t,x) = 1/√(t)φ( x/√(t)) - ∫_(0,∞)1/√(t)φ( x-θ/√(t)) μ(dθ),

	where φ is the density of the standard normal distribution and

    r_μ(t,x) = r(t,x) = ∫_(0,∞)  r_θ(t,x) μ(dθ),

 with

    r_θ(t,x)=exp(-θ^2/2t+θ x/t).

In most cases we will write r instead of r_μ whenever it is clear which measure μ we are referring to. Then, we have

    h(t,x) = 1/√(t)φ( x/√(t)) (1-r(t,x)).

Now, we consider a given b [0,∞) →. The method of images  refers to the case that μ and b are related in the following way: 

    r(t, b(t))=1,     t∈(0,∞),

which is equivalent to h(t,b(t))=0 for all t∈(0,∞). It can be easily shown that a unique solution b exists to (<ref>) if r_μ is finite, which is guaranteed if μ is a positive measure on (0,∞)
	 such that for all ε > 0
	
    ∫_(0,∞)φ(√(ε)θ ) μ(dθ) < ∞.



All b satisfying (<ref>) for some μ have certain properties:

	Let
	
    θ^∗ = inf{θ∈(0,∞)μ((0, θ]) >0}≥ 0.

	 Furthermore, let b [0,∞)→ satisfy r_μ(t, b(t)) = 1 for all t∈(0,∞). Then,
	
		
  *  b is analytic.
		
  * b is concave.
		
  *  lim_t ↘ 0 b(t) = θ^∗/2.
	

  
	 See the proof of Lemma 1.1 and Lemma 1.2 in  <cit.> and slightly extend the arguments for <ref>.
		


	If there exists t_0∈(0,∞) with r_μ(t, b(t)) = 1 for all t ∈ (0,t_0] only, then (a) and (b) hold on (0,t_0). Moreover, (c) is also satisfied.



Now assume that W is a one-dimensional Brownian motion starting in 0  and let τ be the first passage time of W to b, i.e.,

    τinf{ t ∈[0,∞)  W_t ≥ b(t) }.

 Here b is again a given curve and we will from now on not necessarily assume that b fulfills r(t, b(t))=1 for a given measure μ. 
  If μ is chosen in a suitable way, the function r can be used to approximate the distribution of the first passage time of W to b.

	Let t_0∈(0,∞) and let b [0,∞) → be a continuous function such that there exist ζ_1∈[0,1) and ζ_2≥ 0 with
	
    1-ζ_1≤1/r(t,b(t))≤ 1+ζ_2,      t∈(0,t_0].

	Then, for all s ∈ (0, t_0] and all x < b(s) it holds
	
    (1-ζ_1) r(s,x)≤(τ≤ s|W_s=x)≤ (1+ζ_2) r(s,x).

	In particular, if r(t,b(t))=1 for all t∈(0,t_0], it holds for all s ∈ (0, t_0] and all x < b(s) that
	
    r(s,x)=(τ≤ s|W_s=x).


	
		For the proof slightly extend the arguments in the alternative proof of Theorem 1.1 in <cit.>.
		 For more details we refer to the proof of Proposition 1.3 in <cit.>.


If b satisfies r(t,b(t)) = 1 for all  t ∈ (0,T] with T∈ (0,∞)∪{∞} (with the convention that (0,∞]=(0,∞)), we can recover the distribution function F(t) = (τ≤ t) for t∈ (0,T] from r(t,x) = (τ≤ t  |  W_t =x) by integration. Indeed,

    F(t)    = (W_t ≥ b(t)) + ∫_(-∞,b(t))(τ≤ t  |  W_t = x)  p_t(0,x) dx
       = 1-Φ( b(t)/√(t)) + ∫_(0,∞)Φ( b(t)-θ/√(t)) μ(d θ),

where p_t(0,x)=1/√(2π t)exp(-x^2/(2t)) is the transition kernel of a Brownian motion from 0 at time 0 to x at time t.
 In particular, the density f of τ is given by

    f(t) = 1/2t^3/2∫_(0,∞)θ φ( θ-b(t)/√(t)) μ(d θ).



The method of images is traditionally applied as follows: One starts with a measure μ with associated function r = r_μ and then considers a curve b that is the implicit solution to the equation

    r(t,x)=1,

i.e., b is chosen such that r(t, b(t)) = 1 for all t ∈ (0, t_0] for some t_0 >0. Proposition <ref> then yields that, under certain assumptions, r can be used to derive the passage time distribution of the curve b using (<ref>).

	Using the method of images one can generate curves with explicit hitting probabilities. The easiest examples are linear boundaries b(t)=a+m t with a>0, m ∈, which are generated by μ=exp(-2am)δ_2a, where δ_2a denotes the Dirac measure in 2a,  cf. <cit.>. 



As discussed in the introduction, we are interested in the inverse method of images: given a curve b, does there exist a measure μ such that the method of images applied to μ yields b? 
We approach this question in the following section by methods of linear program. 


In order to do so, we start the following definition.

	A function b [0,∞) → is called representable (in the sense of the method of images) up to T for some T∈(0,∞)∪{∞} if there exists a  measure μ
	 such that b is the solution of
	
    ∫_(0,∞)  r_θ(t,x) μ(dθ) = r(t,x) = 1,      t∈(0,T]

	i.e., r(t, b(t)) = 1 for all t∈(0,T]. In this case, we call μ the representing measure of b up to T and say μ represents b on [0,T]. 


			Here we again use the convention that (0,∞]=(0,∞).


Next we  analyse the existence of representing measures.
  

	Let b[0,∞)→  be analytic with b(0) >0. Assume that μ is a positive σ-finite measure on (0,∞) 
	with μ((0,2b(0)))=0 satisfying 
		
    ∫_(0, ∞)exp(-θ^2/2 t^∗) μ(dθ) < ∞

	for some t^*∈ (0,∞). Furthermore, let b  be analytic with b(0) >0. Assume that 
		there exists a sequence (t_n)_n∈ with accumulation point t̃∈ (0, t^*) such that
		
    ∫_(0,∞)  r_θ(t_n, b(t_n)) μ(dθ) = 1     for all  n ∈.

	Then, b is representable by μ on [0,	t^*).


	We show that the function 
	
    t ↦ r(t,b(t))=∫_(0, ∞) r_θ((t,b(t))μ(dθ)

 is analytic on (0,t̂ ) for every t̂∈ (t̃, t^*). Then the
	claim follows directly from the identity theorem for analytic functions, e.g., cf. <cit.>. 
	
	Let t̂∈ (t̃, t^∗). For all t ∈ (0, t̂ ) observe that
		
    r(t,b(t))    = ∫_[2b(0), ∞)exp(-θ^2/2t+ θ b(t)/t) μ(dθ)
       = ∫_[2b(0), K]exp(-θ^2/2t+ θ b(t)/t) μ(dθ) + ∫_(K, ∞)exp(-θ^2/2t+ θ b(t)/t) μ(dθ),

	where 
	
    K sup_t ∈[0, t̂ ]2t^∗ b(t)/t^∗ -t≥ 2b(0) .

	Then, K is finite and μ is σ-finite by assumption and therefore the first integral is finite. Observe that for  θ > K we have 
	
    exp(-θ^2/2t+ θ b(t)/t) ≤exp(-θ^2/2 t^∗),      t ∈ (0, t̂ ),

	and so the second integral is finite by assumption. Thus, t ↦ r(t, b(t)) is analytic on (0, t̂ ). 





	
	
	

§ A LINEAR PROGRAMMING APPROACH FOR THE INVERSE METHOD OF IMAGES

	
	In order to be able to answer the question of the existence of a representing measure, a candidate must be found. Our approach to this is inspired by <cit.> and consists of identifying μ as the solution to a linear optimisation problem. More precisely, we investigate two linear programs that give upper and lower bounds for the first passage time distribution of a given boundary b. Then, we show duality results for these programs as a basis to give sufficient conditions under which a representing measure μ for b exists.
	In order to have any chance of representability at all, we require the properties of Lemma <ref>. 
	
	We fix an analytic, concave boundary b [0, ∞) → with b(0) > 0 and finite slope at0, i.e., | b'(0)| < ∞. 
	

	For a  locally compact Hausdorff space X  let (X) be the set of all regular signed measures on X with finite total variation and denote by ^+(X) the cone of all non-negative measures in (X). For λ∈(X) denote by λ its total variation. For λ∈^+(X) it holds that λ=∫_X 1  dλ. 
	
	 Moreover, we fix a point (t_0,x_0) with t_0>0 and x_0< b(t_0). In the light of Proposition <ref>, it seems natural to consider the following linear problem
	
    maximise      ∫ r_θ(t_0,x_0)μ(dθ) 
       subject to      μ∈^+([2b(0),∞)),
             ∫ r_θ(t,b(t))μ(dθ)≤ 1     for any  t∈ (0,t_0],
    		D_1

		where we approximate the measure μ representing b “from below”.
	 
	Alternatively, we could also consider the following linear program
	
    minimise      ∫ r_θ(t_0,x_0)μ(dθ) 
       subject to      μ∈^+([2b(0),∞)),
             ∫ r_θ(t,b(t))μ(dθ)≥ 1     for any  t∈ (0,t_0],
    		P_2

	where we approximate the measure μ representing b “from above”. We omitted the bounds of integration as we integrate over the whole space (0,∞) in each case. 
	
		By Lemma <ref> <ref> and Remark <ref>  it is enough to consider measures μ on [2b(0), ∞) for a given boundary b.
		 
	
	
		Observe that both programs are linear but infinite-dimensional.
	
	
	
	The labelling of the first program as (<ref>) and the second as (<ref>) may at first glance be confusing. The notation will make more sense when we consider the associated formal dual problems as well as weak and strong duality.
	
	
	The reader may ask herself which program to prefer over the other, i.e., whether there is a “natural” or “better” choice for one or the other. While we need both programs in order to find sufficient conditions for a representing measure μ to exist, the most immediate use is that admissible measures to these programs lead to lower and upper bounds for the probability (τ≤ t_0 | W_t_0=x_0) as the following lemma shows.

	For each (<ref>)-admissible μ_1 we find that
	
    r_μ_1(t_0, x_0) = ∫ r_θ(t_0,x_0)μ_1(dθ)≤(τ≤ t_0 | W_t_0=x_0);

	and for each (<ref>)-admissible μ_2 it holds  that
	
    r_μ_2(t_0, x_0) = ∫ r_θ(t_0,x_0)μ_2(dθ)≥(τ≤ t_0 | W_t_0=x_0).

	Moreover, if there exists a representing measure μ^* on [0,t_0], i.e., if
	
    ∫ r_θ(t,b(t))μ^*(dθ)= 1     for any  t∈ (0,t_0],

	then μ^∗ is a maximiser in (<ref>) and a minimiser in (<ref>).


	Let μ_1 be (<ref>)-admissible. Then,
	
    r_μ_1(t, b(t)) = ∫ r_θ(t,b(t))μ_1(dθ)≤ 1     for any  t∈ (0,t_0].

	
	Setting ζ_1=0 in Proposition <ref>  yields 
	
    r(t_0, x_0) ≤(τ≤ t|W_t_0=x_0).

	The second inequality follows the same way.
	
	Now let μ^∗ be a representing measure up to t_0, i.e.,
	
    ∫ r_θ(t,b(t))μ^*(dθ)= 1     for any  t∈ (0,t_0].

	We conclude again by Proposition <ref> that
	
    r(t_0, x_0) = ∫ r_θ (t_0, x_0) μ^∗ (d θ) = (τ≤ t|W_t_0=x_0)

	and so μ^∗ is a maximiser in (<ref>) and a minimiser in (<ref>).


Since we do not use that b is concave or analytic in the proof of Lemma <ref>, the linear programs give upper and lower bounds for the first passage time distribution to boundaries b that do not need to be concave or analytic.

By Lemma <ref> optimal solutions to the programs (<ref>) and (<ref>) are candidates for a measure μ representing b on [0,t_0]. Thus, we now investigate the existence of optimal solutions to (<ref>) and (<ref>)  and whether these solutions fulfil r(t,b(t)) =1 for every t ∈ (0,t_0]. 
For the analysis, we follow the standard procedure for linear problems, namely the study and interpretation of (formal) dual problems. However, due to the infinite-dimensional structure of the problems, the question arises as to the choice of the correct spaces in order to obtain duality results. It turns out that our choice of 
	spaces is appropriate for our purposes. We refer to <cit.> or <cit.>  for an overview on general linear programming. 
	

For (<ref>), the dual program is given by

    minimise      λ
       subject to      λ∈^+((0,t_0]),
             ∫ r_θ(t,b(t))λ(dt)≥ r_θ(t_0,x_0)     for any θ∈ [2b(0),∞)
    		P_1

and for (<ref>), the dual program is

    maximise      λ
       subject to      λ∈^+((0,t_0]),
             ∫ r_θ(t,b(t))λ(dt)≤ r_θ(t_0,x_0)     for any θ∈ [2b(0),∞).
    		D_2

	
	The maximising problem is always tagged with a “D” and the minimising problem gets tagged with a “P”. This is in line with the “usual” notation for linear programs where the minimising problem is often regarded as the canonical primal problem and the maximising problem is often called the canonical dual problem. 


Denote by d_1, p_1, d_2 and p_2  the optimal values of (<ref>), (<ref>), (<ref>) and (<ref>), respectively. Then, we have 

    d_1 ≤ p_1    and     d_2 ≤ p_2,

 respectively, i.e., weak duality holds. Indeed, for any (<ref>)-admissible μ and any (<ref>)-admissible λ the constraints of the linear program imply
 
    λ=∫ 1  λ(dt)
    	≥∫∫ r_θ(t,b(t))μ(dθ) λ(dt)
    	= ∫∫ r_θ(t,b(t))λ(dt) μ(dθ)
    	≥∫ r_θ(t_0,x_0)μ(dθ).

Hence, d_1≤ p_1. A similar argument shows d_2≤ p_2.




	

 §.§ Strong duality of (<ref>) and its dual (<ref>)

	
	In this section we establish strong duality for the programs (<ref>) and (<ref>), i.e., we show d_1=p_1. 
	 
Let Ω= (0, t_0] and
	
    T([2b(0), ∞)) → C(Ω),           Tμ(t) = ∫_[2b(0), ∞) r_θ(t, b(t)) μ(d θ), 
       T' (Ω) → C_0([2b(0), ∞)),        T'λ(θ) = ∫_Ω r_θ(t, b(t)) λ (dt),

	where C(I) and C_0(I) denote the space of continuous functions on an interval I and the space of  continuous functions on I vanishing at infinity, respectively.

	
	
	 In addition, let 
	
    ⟨ f, ν⟩∫ f dν

	 on C_0(Ω) ×(Ω) and C_0([2b(0), ∞) )×([2b(0), ∞)), respectively.
	 Moreover, define 
	 
    g(θ) = r_θ(t_0, x_0).

	  We can now reformulate our programs as follows: 
	
    maximise      ⟨ g, μ⟩
       subject to      μ∈^+([2b(0),∞)),
              1 - Tμ∈ C^+(Ω)
    		D_1

	and
	
    minimise      λ
       subject to      λ∈^+(Ω),
              T'λ - g ∈ C_0^+([2b(0),∞)),
    		P_1

	where C^+, C_0^+ denote the cones of non-negative elements in the spaces C and C_0, respectively. 
	
		

	Investigating strong duality between (<ref>) and (<ref>) in our infinite-dimensional setting, is anything but trivial. One main technical problem here is that the underlying set Ω is not compact. Therefore, we first consider auxiliary problems:
	
	
	

  §.§.§ Strong duality of the restricted linear programs

	We now derive a sequence of measures that are the solutions to problems with weaker constraints than in (<ref>) and (<ref>). In Section <ref> we show that these measures converge along a subsequence to a measure μ_1 ∈^+([2b(0), ∞)) and λ_1∈^+(Ω) which serve as candidates for optimisers in (<ref>) and (<ref>), respectively.
	
		To define these restricted linear programs, set for any ε∈Ω
	
    Ω_ε= [ε, t_0]

	and  define
	
    T^'_(Ω_) → C_0([2b(0), ∞)),      T^'_ελ(θ) = ∫_Ω_ε r_θ(t, b(t)) λ(dt).

	Note that one difference between T^'_ and T' is the area of integration. On  C(Ω_ε) ×(Ω_ε) and C_0([2b(0), ∞)) ×([2b(0), ∞)) we also consider the algebraic pairing, i.e., the bilinear mapping,
	
    ⟨ f, ν⟩ = ∫ f dν,

which is real-valued and point separating. Moreover, if C_0(Ω_ε)=C(Ω_ε) and C_0([2b(0), ∞)) are endowed with the weak topologies σ(C_0, ) and the spaces (Ω_ε) and ([2b(0), ∞)) with the vague topologies σ(, C_0) induced by the algebraic pairing, then all four spaces are locally convex Hausdorff spaces. Note that the space C_0(Ω_ε) is the continuous dual of (Ω_ε) and vice versa. The same holds for C_0([2b(0), ∞)) and ([2b(0), ∞)). 

	Fubini's theorem implies that 
	
    ⟨ T μ, λ⟩ = ⟨μ, T^' _λ⟩

	with the slight abuse of notation ⟨μ, T^' _λ⟩:=⟨  T^' _λ, μ⟩
	for every μ∈([2b(0), ∞)) and λ∈(Ω_ε). 
	Hence, the operators T and T^'_ are adjoint operators. Moreover,  T and T^'_ are σ(, C_0)-σ(C_0, )-continuous (with respect to the corresponding spaces), see  Lemma 5.17 in <cit.>. 
	

	
 
 Now consider the restricted linear program
	
    maximise      ⟨ g, μ⟩
       subject to      μ∈^+([2b(0),∞)),
              1 - Tμ∈ C^+(Ω_ε).
    		D_1, ε


To show the existence of a maximiser μ_ in (<ref>), we start by considering the formal Lagrange dual problem associated to (<ref>) which is given by 
	
    minimise      λ
       subject to      λ∈^+(Ω_ε),
              T^'_λ - g ∈ C_0^+([2b(0),∞)).
    		P_1, ε

	Denote the optimal values of (<ref>) and (<ref>) by d_1, ε and p_1, ε, respectively. Then similar to (<ref>) we see that 
	
    0 ≤ d_1, ε≤ p_1, ε,

	 i.e., weak duality holds. 
	Now we prove the existence of an optimiser in (<ref>) and (<ref>).
	
		There exist a (<ref>)-admissible μ_1, ε such that d_1, ε = ⟨ g, μ_1, ε⟩ and a (<ref>)-admissible λ_1, ε such that p_1, ε = λ_1, ε, i.e., μ_1,ε and λ_1,ε are optimal in (<ref>) and (<ref>), respectively.
		
	
	Let ψ[2b(0),∞)→, θ↦exp(-θb(t_0)-x_0/t_0) and observe that ψ∈ C_0([2b(0),∞)) since  x_0< b(t_0). 
	Consider the following modified version of (<ref>): 
		
    maximise      ⟨ψ, μ⟩
       subject to      μ∈^+([2b(0),∞)),
              1- T_mod μ∈ C^+(Ω_ε),
    			D_1, ε, mod

		where  
		
    T_mod([2b(0),∞))→ C(Ω_ε),      T_mod μ(t) = ∫_[2b(0), ∞)r_θ(t, b(t))/r_θ(t_0, b(t_0)) μ(dθ).

		Note that T_mod is  continuous 		and that (<ref>) is equivalent to (<ref>). Moreover, for any (<ref>)-admissible μ we have
		
    1 ≥ T_mod μ (t_0)   =∫_[2b(0), ∞)r_θ(t_0, b(t_0))/r_θ(t_0, b(t_0)) μ(dθ)
    		= μ.

		Hence, every (<ref>)-admissible μ is contained in B_([2b(0), ∞))	(1), the σ(, C_0)-closed ball of radius 1 around 0 in ([2b(0),∞)).
		
		Instead of solving the maximisation problem (<ref>) we can equivalently maximise 
		the mapping  μ↦⟨ψ, μ⟩ over the set
		
    C_d^ε = T_mod^-1( 1 - C^+(Ω_ε) ) ∩^+([2b(0),∞)) ∩ B_([2b(0),∞))(1).

					Here we intersect with the set  B_([2b(0),∞))(1) which can be shown to be compact (see below) and allows us to conclude that C_d^ε itself is compact as a closed subset of a compact set. 
						 Indeed,  first observe that 1 - C^+(Ω_ε) is closed as it is homeomorphic to the σ(C_0, )-closed cone
		
    C^+(Ω_ε) = 	C_0^+(Ω_ε)=⋂_λ∈^+(Ω_ε){ f ∈ C(Ω_ε) ⟨ f, λ⟩≥ 0 }.

		The continuity of T_mod implies that T_mod^-1(1-C^+(Ω_ε)) is σ(, C_0)-closed as well. Rewriting
		
    ^+([2b(0),∞)) = ⋂_f ∈ C_0^+([2b(0),∞)){μ∈([2b(0),∞)) ⟨ f, μ⟩≥ 0 }

		yields that ^+([2b(0),∞)) is σ(, C_0)-closed, too. Finally, we find that B_([2b(0),∞))(1) is σ(, C_0)-compact due to the Alaoglu-Bourbaki theorem, cf. <cit.>. Hence, C_d^ is compact as a closed subset of a compact set. 
		
		Since ψ∈ C_0([2b(0),∞)) the mapping μ↦⟨ψ,μ⟩ is continuous with respect to the topology σ(, C_0)  and thus attains its maximal value d_1, ε  on C_d^ε at some measure μ_ε, mod∈ C_d^ε. Finally, the optimal value d_1, ε of (<ref>) is attained at μ_1, ε, where
		
    d μ_1, ε/d μ_ε, mod(·)   =1/r_·(t_0,b(t_0)).

		
		Now, turning our attention to (<ref>), we define λ̃ = δ_t_0. λ̃ is admissible in (<ref>) for all ε∈[0,t_0] since it holds that  
		
    T^'_λ̃(θ) = exp( -θ^2/2t_0 + θ b(t_0)/t_0) > exp( -θ^2/2t_0 + θ x_0/t_0) = g(θ),    for all θ∈ [2b(0), ∞)

		as b(t_0) > x_0 by assumption. Thus, λ̃ is admissible and λ̃ = 1. Therefore, it suffices to minimise the σ(, C)-continuous mapping λ↦λ over the set
		
    C_p^ε = (T^'_)^-1( g+ C^+_0([2b(0), ∞)) ) ∩^+(Ω_ε) ∩ B_(Ω_ε)(1).

		The σ(, C)-compactness of C_p^ε now follows along the same lines as the compactness of C_d^ε. Thus, we  conclude 
		 that the continuous function λ↦λ attains its minimum p_1, ε at some measure λ_1, ε∈ C_p^ε.
	
	
	To prove strong duality we first introduce the Lagrange function 
	
    L_1(Ω_ε) ×([2b(0), ∞)) →[-∞, ∞]

	 associated with the (<ref>)-(<ref>)-duality  defined as
	
    L_1(λ, μ)  = λ + ⟨ g, μ⟩ - ⟨ T^'_λ, μ⟩ + _^+(Ω_ε)(λ) - _^+([2b(0), ∞))(μ),

	where we set L_1(λ, μ)=-∞ for (λ,μ)∈^-(Ω_)×^-([2b(0),∞)) with ^-(I)=(I)\^+(I) for an interval I and where
	
    _M(x)  =
    		
    			0,        if  x ∈ M,
    ∞,    if  x ∉ M,

	for any set M. 
	We will use the following simplifications later on:
		
    inf_λ∈(Ω_ε)sup_μ∈([2b(0), ∞)) L_1(λ, μ)   = inf_λ∈^+(Ω_ε)sup_μ∈^+([2b(0), ∞))( λ + ⟨ g-T^'_λ, μ⟩) 
       = inf_λ∈^+(Ω_ε)( λ + sup_μ∈^+([2b(0), ∞))⟨ g - T^'_λ, μ⟩)
       = inf_λ∈^+(Ω_ε)
     T^'_λ≥ gλ = p_1, ε,
    		
    sup_μ∈([2b(0), ∞))inf_λ∈(Ω_ε) L_1(λ, μ)    = sup_μ∈^+([2b(0), ∞))inf_λ∈^+(Ω_ε)( λ + ⟨ g-T^'_λ, μ⟩) 
       = sup_μ∈^+([2b(0), ∞))( ⟨ g, μ⟩ + inf_λ∈^+(Ω_ε)⟨λ, 1- Tμ⟩)
       = sup_μ∈^+([2b(0), ∞))
     Tμ≤ 1⟨ g, μ⟩ = d_1, ε.


	Moreover, define the dual value function v_1 by 
	
    v_1(f)  = inf_μ∈([2b(0), ∞)) L^∗_1,μ (f),      f∈  C(Ω_ε),

	where L^∗_1,μ with
	
    L^∗_1,μ(f)=sup_λ∈(Ω_ε){⟨ f,λ⟩ - L_1,μ(λ)}
 
		is the convex conjugate of the mapping L_1,μ=L_1(·,μ).
	

	
	
	
		The dual value function v_1 is convex and we have v_1(0)=-d_1, ε and v_1^∗∗(0) = -p_1, ε, where v_1^**=(v_1^*)^* is the convex  biconjugate of v_1. In particular, it holds -d_1, ε≥ -p_1, ε.
	
	
		We find with Lemma 5.25 in <cit.> and our calculations in (<ref>) that
		
    v_1^∗∗(0) ≤ v_1(0)    = inf_μ∈([2b(0), ∞)) L^∗_1, μ(0)
       = inf_μ∈([2b(0), ∞))sup_λ∈(Ω_ε)( ⟨ 0, λ⟩ - L_1, μ(λ) )
       = - sup_μ∈([2b(0), ∞))inf_λ∈(Ω_ε) L_1(λ, μ) = -d_1, ε.

		The conjugate v_1^∗(Ω_ε) → [-∞, ∞] of v_1 is given by
		
    v_1^∗(λ)    = sup_f ∈ C(Ω_ε)( ⟨ f, λ⟩ - v_1(f) )
       = sup_μ∈([2b(0), ∞))sup_f ∈ C(Ω_ε)( ⟨ f, λ⟩ - L_1,μ^∗(f) )
       = sup_μ∈([2b(0), ∞)) L_1,μ^∗∗(λ)
       = sup_μ∈([2b(0), ∞)) L_1,μ (λ),

		where for the last equality we used the Fenchel-Moreau theorem (see e.g. Theorem 5 in <cit.>) which is applicable since for all μ∈([2b(0), ∞)) the mapping (Ω_ε) ∋λ↦ L_1, μ(λ) is closed, i.e., either lower semi-continuous and  L_1,μ(λ)>-∞ for all λ or L_1,μ≡- ∞, and convex. 
		
		Moreover, we obtain for the biconjugate	v_1^∗∗ of v_1 
		
    v_1^∗∗(f)    = sup_λ∈(Ω_ε)( ⟨ f, λ⟩ - v_1^∗ (λ) ) 
       = sup_λ∈(Ω_ε)inf_μ∈([2b(0), ∞))( ⟨ f, λ⟩ - L_1(λ, μ) ).

		The calculation in (<ref>)  yields
		
    v_1^∗∗(0)    = sup_λ∈(Ω_ε)inf_μ∈([2b(0), ∞))( ⟨ 0, λ⟩ - L_1(λ, μ) )
       = - inf_λ∈(Ω_ε)sup_μ∈([2b(0), ∞)) L_1(λ, μ) = -p_1, ε.

		
		
		Note that  v_1(f)>-∞ for all f∈ C(Ω_ε). Indeed, assume by contradiction that there exists f ∈ C(Ω_ε) such that v_1(f) = - ∞. Then also v_1^∗∗(f) = -∞ since v_1^∗∗≤ v_1. Then (<ref>) implies that sup_μ L_1(λ, μ) = ∞ for any λ∈(Ω_ε) and thus p_1, ε = ∞. But by Lemma <ref>  there exists a (<ref>)-admissible solution and therefore,  v_1(f)>-∞ for all f∈ C(Ω_ε).
		
		Finally, we show that v_1 is convex. To this end, let α∈ (0,1) and f_1, f_2 ∈ C(Ω_ε). Note that L_1 is concave in its second component. For any measures μ̃, μ̂∈([2b(0), ∞)) it holds that 
		
    v_1(α f_1 + (1-α) f_2)    = inf_μ∈([2b(0), ∞)) L^∗_1,μ(α f_1 + (1-α) f_2)
       = inf_μ∈([2b(0), ∞))sup_λ∈(Ω_ε)( ⟨α f_1 + (1-α) f_2, λ⟩ - L_1(λ, μ) )
       ≤sup_λ∈(Ω_ε)( ⟨α f_1 + (1-α) f_2, λ⟩ - L_1(λ, αμ̃ + (1-α)μ̂))
       ≤αsup_λ∈(Ω_ε)( ⟨ f_1, λ⟩ - L_1(λ, μ̃) ) + (1-α) sup_λ∈(Ω_ε)( ⟨ f_2, λ⟩ - L_1(λ, μ̂) )
       =α L_1,μ̃(f_1)+(1-α) L_1,μ̂(f_2).

		Taking the infimum over μ̃, μ̂∈([2b(0), ∞)) we obtain the convexity of v_1.
			
	
		From the proof of Lemma <ref> we conclude that v_1 C(Ω_ε)→ (-∞,∞]. 
	
	
			Lemma <ref> implies that -d_1, ε≥ -p_1, ε which is equivalent to weak duality. Of course, we already showed weak duality above without using the additional structure of the dual value function v_1.
			
		
	
	Now, we can prove  strong duality.
	

	
		There exist a (<ref>)-admissible λ_1, ε and a (<ref>)-admissible μ_1, ε  such that the optimal values p_1, ε and d_1, ε of (<ref>) and (<ref>) are attained by λ_1, ε and μ_1, ε, respectively. Strong duality holds  between the programs (<ref>) and (<ref>), i.e., d_1,ε=p_1,ε, and we have the following complementary slackness conditions
		
    Tμ_1, ε   = 1,       λ_1, ε-almost surely on Ω_ε and
    	
    		T^'_λ_1, ε   = g,       μ_1, ε-almost surely on  [2b(0), ∞).

	
	




	
		Lemma <ref> gives the existence of a (<ref>)-admissible λ_1, ε with p_1=λ_1, and of  a (<ref>)-admissible μ_1, ε with d_1=⟨ g, μ_1,⟩.
		
		By	Lemma <ref> strong duality holds if v_1^∗∗(0) = v_1(0). From the Fenchel-Moreau theorem (see e.g., Theorem 5 in <cit.>) and the convexity of v_1 (see Lemma <ref>), we deduce that
		
    v_1^∗∗(0) = ((v_1))(0) = (v_1)(0),

		where ,  and  denote the closure, the convex hull and the lower semi-continuous hull of a function, for more details see, e.g., <cit.>. 
		Now, use Lemma 5.25 in <cit.> to obtain
		
    v_1^∗∗(0) = sup_O ∈(0)inf_f ∈ O  v_1(f),

		where (0) is the set of all σ(C_0, )-open neighbourhoods of 0. Hence, we want to show that
		
    v_1(0) = sup_O ∈(0)inf_f ∈ O  v_1(f),

		which is in particular satisfies if v_1 is continuous in 0 with respect to the topology σ(C_0, ).
		Consider O  = { f∈ C_0(Ω_ε) f _∞ < 1}, which is a σ(C_0, )-open neighbourhood of 0.  Then, for any f ∈ O it holds that
		
    v_1(f)    = inf_μ∈([2b(0), ∞)) L^∗_1, μ(f)
        = inf_μ∈([2b(0), ∞)) sup_λ∈(Ω_ε)( ⟨ f, λ⟩ - L_1,μ(λ) )
       = inf_μ∈^+([2b(0), ∞)) sup_λ∈^+(Ω_ε)( ⟨ f, λ⟩ - λ - ⟨ g, μ⟩ + ⟨ T^'_λ, μ⟩)
       ≤sup_λ∈^+(Ω_ε)(  f _∞ λ - λ) = 0.

	 		Now, 
		 <cit.> yields that v_1 is continuous in 0 and therefore 
		
    p_1, ε = -v_1^∗∗(0) = -v_1(0) = d_1, ε.

		
		Moreover, the optimisers λ_1, ε and μ_1, ε in (<ref>) and (<ref>), respectively,  fulfil the complementary slackness conditions. Indeed, using (<ref>) and that T and T^'_ are adjoint operators   we find that
		
    0    ≤⟨ T^'_λ_1, ε - g, μ_1, ε⟩ = ⟨ T^'_λ_1, ε, μ_1, ε⟩ - d_1, ε= ⟨λ_1, ε, T μ_1, ε⟩ - p_1, ε = ⟨λ_1, ε, Tμ_1, ε - 1 ⟩≤ 0.

		In particular, T^'_λ_1, ε = g holds μ_1, ε-almost surely on [2b(0), ∞) and Tμ_1, ε = 1 holds λ_1, ε-almost surely on Ω_ε.
	

	

  §.§.§ Strong duality of the unrestricted programs 

	
	We now return to the original programs (<ref>) and (<ref>). We consider suitable sequences of solutions
	of the restricted programs (<ref>) and (<ref>) for ε↘ 0 which then allow to prove strong duality and the existence of optimisers (under some assumptions). 
	
	We now consider measures λ_ε∈(Ω_) as measures on Ω=(0, t_0] by continuing them as the null measure outside of Ω_ε.
	
	
	Recall that 
	
		
  * in (<ref>) we can restrict to admissible λ with λ≤ 1 and
		
  * for the modified program (<ref>) any admissible  μ̃ satisfies μ̃≤ 1, see (<ref>). 

		Any admissible μ of the unmodified program (<ref>) can be recovered from an admissible μ̃ of the modified program by dμ/ dμ̃(·) =1/r_·(t_0,b(t_0)), cf. the proof of Lemma <ref>. 
		
		In particular, the bounds on the total variation are independent of ε. Observe that the metrisation of the vague topology is possible on the total variation unit balls in both spaces (for example, cf. <cit.>). From the Alaoglu-Bourbaki theorem, cf. <cit.>, we deduce that these unit balls are vaguely compact. Thus, there exists a sequence  ε_n ↓ 0 and measures μ̃_1 and λ_1 with μ̃_1 ∨λ_1 ≤ 1 such that μ̃_1, ε_n→μ̃_1 and λ_1, ε_n→λ_1 vaguely as n→∞.
	
	Moreover, since for every function f[2b(0), ∞) →∞ with compact support, the function f/r_·(t_0,b(t_0)) also has compact support, we conclude that 
	
    μ_1, ε_nμ_1,      vaguely,

	where
	
    dμ_1, ε_n/dμ̃_1, ε_n(·) :=1/r_·(t_0,b(t_0)),     dμ_1/dμ̃_1(·)= 1/r_·(t_0,b(t_0)).
	
	Thus, we have vague convergence of the optimal measures for the restricted measures to some measures μ_1 and λ_1. These measures are good candidates for being the optimal measures in (<ref>) and (<ref>). However, we do not yet know whether μ_1 and λ_1 are even admissible in the respective programs, much less whether these solutions are optimal or whether strong duality holds.
	
	Let us start by looking at admissibility. We know that μ_1, ε_n converges vaguely to μ_1 and that the mapping θ↦ r_θ(t, b(t)) is continuous on [2b(0), ∞) for any t ∈Ω and vanishes at infinity. Thus, by <cit.> we have
	
    T μ_1 (t) = ∫_[2b(0), ∞) r_θ(t, b(t)) μ_1(dθ) = lim_n →∞∫_[2b(0), ∞) r_θ(t, b(t)) μ_1, ε_n(dθ) ≤ 1

	and so μ_1 is indeed (<ref>)-admissible.
	
	For λ_1 to be (<ref>)-admissible, we additionally assume that λ_1, ε_n→λ_1 as n→∞. Since then the Portemanteau theorem (see e.g., <cit.>) implies that λ_1, ε_n→λ_1 weakly and not only vaguely. The only thing that could go wrong is that λ_1, ε_n pushes mass into 0 for n →∞ since the total mass of λ_1 would then be smaller than the mass of the λ_1, ε_n. In an application the assumption  λ_1, ε_n→λ_1 as n→∞ can be numerically investigated by computing λ_1, ε for smaller and smaller ε.
		
	Moreover, in order to guarantee that t ↦ r_θ(t,b(t)) is bounded on (0, t_0] for all θ∈[2b(0),∞), we use the assumption that b'(0) < ∞. Indeed, for  θ > 2b(0) we find that
	
    r_θ(t, b(t)) = exp( -θ/2t(θ-2b(t)) )  0

 since θ - 2b(t) > 0 for t sufficiently small by the continuity of b. For  θ = 2b(0) it holds that
	
    r_2b(0)(t, b(t)) = exp( 2b(0) ( b(t)-b(0)/t) )  exp( 2b(0)b'(0) ) < ∞

	 since we assumed 0 < b(0) < ∞ and b'(0) < ∞. In both cases we find that t ↦ r_θ(t, b(t)) is bounded. Then, the Portmanteau theorem yields that
	
    ∫_(0, t_0] r_θ(t, b(t)) λ_1(dt) = lim_n →∞∫_(0, t_0] r_θ(t, b(t)) λ_1, ε_n(dt) ≥ g(θ)

	and so λ_1 is (<ref>)-admissible.
	
	Finally,  <cit.> implies lim_n →∞⟨ g, μ_1, ε_n⟩ = ⟨ g, μ_1 ⟩. Moreover, by our assumption we have lim_n →∞λ_1, ε_n = λ_1. Thus,
	
    p_1    ≤λ_1 ≤lim_n →∞λ_1, ε_n = lim_n →∞ p_1, ε_n= lim_n →∞ d_1, ε_n = lim_n →∞⟨ g, μ_1, ε_n⟩ = ⟨ g, μ_1 ⟩≤ d_1 ≤ p_1,

	where we use that weak duality (cf. (<ref>)) holds in the last inequality.
		In particular,
	
		
  * λ_1 and μ_1 are (<ref>)- and (<ref>)-optimal, respectively,
		
  * d_1 = p_1, i.e., strong duality holds. 
	
		Finally, we again use the adjointness of T and T' as well as  d_1=p_1 to obtain
	
    0 ≤⟨ T' λ_1 - g, μ_1 ⟩ = ⟨ T' λ_1, μ_1 ⟩ - d_1 = ⟨λ_1, T μ_1 ⟩ - p_1 = ⟨λ_1, Tμ_1 - 1 ⟩≤ 0

	and therefore we have T^'λ_1 = g holds μ_1-almost surely on [2b(0), ∞) and Tμ_1 = 1 holds λ_1-almost surely on Ω. We summarise our findings.
			
		There exists a sequence (ε_n)_n ∈ and measures λ_1 ∈^+(Ω) and μ_1 ∈^+([2b(0), ∞)) such that λ_1, ε_n→λ_1 and μ_1, ε_n→μ_1 vaguely. Moreover, μ_1 is (<ref>)-admissible.
			
		In addition, if λ_1, ε_n→λ_1, 
		then λ_1 is (<ref>)-admissible, the optimal values d_1 and p_1 in (<ref>) and (<ref>) are attained by μ_1 and λ_1, respectively, the optimal values coincide (i.e., strong duality holds) and the following complementary slackness conditions are satisfied: 
			
    ∫ r_θ(t,b(t))μ_1(dθ)   = 1        for λ_1-almost all  t∈ (0,t_0],
    ∫ r_θ(t,b(t))λ_1(dt)   =r_θ(t_0,x_0)        for μ_1-almost all θ∈ [2b(0),∞).

		
	
	
	

 §.§ Strong duality of (<ref>) and its dual (<ref>)

	As in Section <ref> we consider the restricted linear program
	
	
    minimise      ⟨ g, μ⟩
       subject to      μ∈^+([2b(0),∞)),
              Tμ - 1 ∈ C^+(Ω_ε)
    		P_2, ε

	and its formal dual
	
    maximise      λ
       subject to      λ∈^+(Ω_ε),
              g - T^'_λ∈ C^+_0([2b(0),∞)).
    		D_2, ε

	 Denote the optimal values of (<ref>) and (<ref>) by p_2, ε and d_2, ε, respectively.
	 Again we have weak duality, i.e.,
	 
    d_2,≤ p_2,.
 



  §.§.§ Strong duality of the restricted linear programs

	Now we show that there exist optimal solutions to the restricted primal and dual problem, compare Lemma <ref>. Moreover, strong duality holds. 


	There exist a (<ref>)-admissible μ_2, ε and a (<ref>)-admissible λ_2, ε  such that the optimal values p_2, ε and d_2, ε of (<ref>) and (<ref>) are attained by μ_2, ε and λ_2, ε, respectively. Strong duality holds, i.e., d_2,ε=p_2,ε, and we have the following complementary slackness conditions
	
    Tμ_2, ε = 1,     λ_2, ε-almost surely on Ω_ε,
       T^'_λ_2, ε = g,     μ_2, ε-almost surely on  [2b(0), ∞).



 
	To show the existence of an optimal solution to (<ref>) and (<ref>), respectively, we proceed similar to the proof of Lemma <ref>. Note that it is enough to show that there exists a (<ref>)-admissible μ and that all (<ref>)-admissible λ are bounded by some constant. 
		
	Consider μ̅ = c ·δ_2b(0) with
	
    c > exp( -2b(0) b(t_0)-b(0)/t_0).

	Then, for any t ∈Ω_ε
	
    Tμ̅(t)    = c ·exp( 2b(0) b(t)-b(0)/t_0)
    		≥ c ·exp( 2b(0) b(t_0)-b(0)/t_0) > 1,

	by the concavity of b. Thus,  Tμ̅ - 1 ∈ C^+(Ω_ε) and hence, μ̅ is (<ref>)-admissible with μ̅ = c < ∞.
	
	Observe that for any dual feasible λ and for θ = 2b(0) the constraint yields
	
    g(2b(0))
    		   = ∫_Ω_εexp( 2b(0)  b(t)-b(0)/t)  λ(dt)
       ≥∫_Ω_εexp( 2b(0) b(t_0)-b(0)/t_0) λ(dt)
    		=exp( 2b(0) ( b(t_0)-b(0)/t_0) ) λ,

	by the concavity of b. In particular, for any feasible λ we have that
	
    λ≤exp( 2b(0) x_0-b(t_0)/t_0) =  C.


	
	In order to show strong duality in the same way as in the first set-up, we consider modified programs. Strong duality for the modified programs follows with the same arguments as in the first set-up and the results carry over to the unmodified programs (<ref>) and (<ref>). For more details we refer to <cit.>.
	

		The modification of the programs are necessary for our lines of proof (cf. the proof of Proposition <ref>): our dual value function for the unmodified programs (<ref>) and (<ref>) would have been
		
    v_2,unmod(f)    = inf_λ∈^+(Ω_ε)sup_μ∈^+([2b(0), ∞))( ⟨ f, μ⟩ - ⟨ g, μ⟩ - λ + ⟨ T μ, λ⟩)
       ≤sup_μ∈^+([2b(0), ∞))(  f _∞μ - ⟨ g, μ⟩).

			We could not have defined a σ(C_0, )-open neighbourhood O of 0 with  f≤ g for all f∈ O which then guarantees that v_2,unmod(f) is bounded on O. This necessitated our detour.
	
	
	

  §.§.§ Strong duality of the unrestricted programs

	
	As before, we lift the results about the restricted programs (<ref>) and (<ref>) to the unrestricted programs (<ref>) and (<ref>). Recall that p_2 and d_2 denote the optimal values of the unrestricted programs and that weak duality holds, i.e.,
	
    0 ≤ d_2 ≤ p_2.

   
   
    For any ε > 0 the optimisers μ_2, and λ_2, in (<ref>) and (<ref>) satisfy
   
    μ_2, ε   ≤ c
    λ_2, ε   ≤exp( 2b(0) x_0 - b(t_0)/t_0) =  C

	for some c> exp( -2b(0) b(t_0)-b(0)/t_0). In particular, the bounds c and C are independent of , thus, with the same metrisation argument as in the first set-up there exists a sequence (ε_n)_n∈ℕ and measures μ_2∈^+([2b(0),∞)) with μ_2 ≤ c and λ_2∈^+(Ω) with λ_2 ≤ C such that 
	
    μ_2, ε_n μ_2,  vaguely,
    λ_2, ε_n λ_2,   vaguely.

	
	Since for any t ∈Ω the mapping θ↦ r_θ(t, b(t)) is continuous and vanishes at infinity, it follows from <cit.>  that
	
    T μ_2(t) = ∫_[2b(0), ∞) r_θ(t, b(t)) μ_2 (dθ) = lim_n →∞∫_[2b(0), ∞) r_θ(t, b(t)) μ_2, ε_n (d θ) ≥ 1.

	Hence, μ_2 is (<ref>)-admissible. 
	
	To show that λ_2 is (<ref>)-admissible, observe that for any κ∈(0,t_0/4) we have ∅≠Ω_2κ⊂Ω_κ⊂Ω and so by Urysohn's lemma (e.g., cf. <cit.>) there exists  a continuous function φ^κΩ→ [0,1] such that φ^κ(t) = 1 for all t ∈Ω_2 κ and φ^κ(t) = 0 for all t ∈cl(Ω∖Ω_κ). Since t ↦ r_θ(t, b(t))φ^κ(t) is continuous for any θ∈ [2b(0), ∞), we conclude that for any θ∈ [2b(0), ∞)
	
    ∫_(0, t_0] r_θ(t, b(t)) λ_2(dt)    = lim_κ↓ 0∫_(0, t_0] r_θ(t, b(t)) _Ω_2κ(t) λ_2(dt)
       ≤lim_κ↓ 0∫_(0, t_0] r_θ(t, b(t))φ^κ(t) λ_2(dt)
       = lim_κ↓ 0lim_n →∞∫_(0, t_0] r_θ(t, b(t))φ^κ(t) λ_2, ε_n(dt)
       ≤lim sup_κ↓ 0lim sup_n →∞∫_(0, t_0] r_θ(t, b(t)) λ_2, ε_n(dt) ≤ g(θ),

	where we use <cit.> in the second to last line. Therefore,  λ_2 is (<ref>)-admissible.

	Since g is continuous and vanishes at infinity,  <cit.> yields ⟨ g, μ_2 ⟩ = lim_n →∞⟨ g, μ_2, ε_n⟩ and therefore
	
    d_2 ≤ p_2 ≤⟨ g, μ_2 ⟩ = lim_n →∞⟨ g, μ_2, ε_n⟩ = lim_n →∞ p_2, ε_n = lim_n →∞ d_2, ε_n≤ d_2.

	The last inequality is true as any (<ref>)-feasible λ is also (<ref>)-feasible. Thus, we conclude that μ_2 is (<ref>)-optimal,  d_2 = p_2, i.e., strong duality holds and lim_n →∞ d_2, ε_n = d_2.
	
	To show that λ_2 is (<ref>)-optimal (and not only admissible) we assume – similar to the first set-up – that λ_2, ε_n→λ_2 as n→∞.  Then, 
	
    λ_2  = lim_n →∞λ_2, ε_n = lim_n →∞ d_2, ε_n = d_2.

 Finally, we have
	
    0 ≤⟨ T' λ_2 - g, μ_2 ⟩ = ⟨ T'λ_2, μ_2 ⟩ - d_2 = ⟨λ_2, T μ_2 ⟩ - p_2 = ⟨λ_2, Tμ_2 - 1 ⟩≤ 0

	which implies that T' λ_2 = g holds μ_2-almost surely on [2b(0), ∞) and Tμ_2 = 1 holds λ_2-almost surely on Ω. We summarise our findings.
	
	
		 There exists a sequence (ε_n)_n ∈ and measures λ_2 ∈^+(Ω) and μ_2 ∈^+([2b(0), ∞)) such that λ_2, ε_n→λ_2 and μ_2, ε_n→μ_2 vaguely. The measures μ_2 and λ_2 are (<ref>)- and (<ref>)-admissible, respectively, strong duality holds, i.e., d_2 = p_2 and μ_2 is (<ref>)-optimal.
		
		In addition, if λ_2, ε_n→λ_2,
		then λ_2 is (<ref>)-optimal and the following complementary slackness conditions are satisfied: 
			
    ∫ r_θ(t,b(t))μ_2(dθ)   = 1        for λ_2-almost all  t∈ (0,t_0],
    ∫ r_θ(t,b(t))λ_2(dt)   =r_θ(t_0,x_0)       for μ_2-almost all θ∈ [2b(0),∞).


	
	
	
		Note that in this second set-up, we have strong duality and optimality of μ_2 without any assumption, while in Theorem <ref> we needed the additional assumption λ_1, ε_n→λ_1 in order to obtain strong duality and optimality of μ_1. In both cases, the additional assumption is necessary for the optimality of λ_1 and λ_2, respectively, as well as for the complementary slackness conditions to hold. This is due to the fact that any (<ref>)-feasible λ is also (<ref>)-admissible as we consider λ to be the null measure outside of Ω_ε. However, a (<ref>)-admissible μ does not have to be (<ref>)-admissible, as μ may not fulfil Tμ = 1 outside of Ω_ε.
		
	

	
	

§ ON THE EXISTENCE OF REPRESENTING MEASURES

	In this section, we investigate the existence of a representing measure. More precisely, given an analytic, concave boundary b with b(0)>0 we want to prove the existence of a measure μ such that for all t ∈(0,t_0]
	
    1 = r(t, b(t))=∫_[2b(0),∞)r_θ(t,b(t))μ(dθ).



	 Now we see why the approach via linear programs is so fruitful for the question of representability: The complementary slackness conditions in Theorem <ref> already state that
	
    r(t, b(t))=∫_[2b(0),∞)r_θ(t,b(t))μ_1(dθ) = 1      for λ_1-almost all  t∈ (0,t_0],

	where μ_1 and λ_1 are optimal measures for (<ref>) and (<ref>), respectively. The analogous result holds for μ_2 and λ_2 by Theorem <ref>. Thus, if λ_1 or λ_2 put mass everywhere on Ω = (0, t_0], then b is representable on [0,t_0]. 
	
	As we want to make use of the complementary slackness conditions as well as of strong duality throughout this section, we assume 
	
	Let b[0,∞)→ satisfy Assumption (A) and let the assumptions of Theorems <ref> and <ref> hold true, in particular,  λ_1, ε^(1)_n→λ_1, λ_2, ε^(2)_n→λ_2 as n→∞, where (^(i)_n)_n∈ is a sequence   such that  λ_i,^(i)_n  is an optimisers in (P_i,_n) and λ_i the vague limit of (λ_i,^	(i)_n)_n∈, i=1,2.



		First we show that there exists a measure λ̅ which is admissible in (<ref>) as well as in (<ref>) and already fulfils all constraints in both programs with equality.
	
		Let σ_b be the last passage time of the Brownian motion W to b before time t_0, i.e.,
		
    σ_b=sup{t∈[0,t_0] W_t=b(t)},

			where we set σ_b=t_0 on {W_t<b(t)  for all  t∈[0,t_0]}, 
			 and define
		
    λ̅(dt)=(σ_b∈ dt|W_0=0, W_t_0=x_0).

		Then, λ̅ is admissible for (<ref>) as well as for (<ref>) and 
		
    ∫ r_θ(t,b(t))λ̅(dt)= r_θ(t_0,x_0)      for any θ∈ [2b(0),∞).

	
	
		First, note that λ̅∈^+((0, t_0]). 
	 Let θ≥ 2b(0) and consider the additional linear boundary 
	 
    ℓ_θ(t)=θ/2+m_θ t,

		where m_θ≥ b^'(0). Note that ℓ_θ lies above the boundary b. Let τ_θ be the first passage time of W to ℓ_θ. Then by Remark <ref> the boundary ℓ_θ is representable by μ_ℓ_θ=exp(-θ m_θ)δ_θ and by Proposition <ref> it holds for   t∈(0,t_0] and x< ℓ_θ(t) that 
		
    (τ_θ≤ t |  W_0=0,W_t=x)=(τ_θ<t  | W_t=x)=exp(-θ  m_θ) r_θ(t,x).

		To show (<ref>), denote by _(0,0)^(s,x) the law of a Brownian bridge from (s, x) to (0, 0). Here, we consider the Brownian bridge from (s, x) to (0, 0)   as a process in reversed time. In particular, for a Brownian bridge from (t_0,x_0) to (0,0) note that the first passage time to b coincides with the last passage time σ_b  of b on (0,t_0] for a Brownian bridge from (0,0) to (t_0,x_0). Using the strong Markov property of the Brownian bridge gives
		
    r_θ(t_0,x_0)   =exp(θ  m_θ) (τ_θ≤ t_0 | W_0=0,W_t_0=x_0)
       =exp(θ m_θ) ∫_(0,t_0]_(0,0)^(t,b(t))(τ_θ≤ t) _(0,0)^(t_0,x_0)(σ_b∈ dt)
       =exp(θ m_θ) ∫_(0,t_0]exp(-θ m_θ)  r_θ(t,b(t)) _(0,0)^(t_0,x_0)(σ_b∈ dt)
       =∫_(0,t_0] r_θ(t,b(t)) λ̅(dt).

		Hence, (<ref>) holds and in particular, λ̅ is admissible in both (<ref>) and (<ref>).
	
	 
		Note that λ̅ is a strong contender to be optimal in both (<ref>) and (<ref>) as it already fulfils the constraint not only with inequality but even with equality everywhere. If λ̅ really is optimal in both programs, then this yields a stochastic interpretation of (<ref>) and (<ref>) as the programs having the (conditional) last passage time distribution of a standard Brownian motion to b as optimisers. This would be a nice symmetry with the programs (<ref>) and (<ref>), where in both programs an optimiser is given by the (conditional) first passage time distribution. 
		
	
	Recall that p_1,d_1, p_2 and d_2 denote the optimal values of (<ref>), (<ref>), (<ref>) and (<ref>), respectively. If we assume strong duality in both set-ups and since λ̅ is admissible in both (<ref>) and (<ref>) by Lemma <ref> we obtain
	
    d_1 = p_1 ≤λ̅≤ d_2 = p_2.

	With the help of (<ref>) we can provide sufficient conditions for a general concave, analytic b to be representable.
	
		Assume that Assumption (B) is satisfied.  If either
		
			
  * d_1 = p_2 or 
			
  * p_1 = d_2, 
		
		then, b is representable on [0,t_0].
	
	
		The proof follows directly from (<ref>):  If one of the conditions is fulfilled, we immediately have that λ̅ is the optimal measure in both (<ref>) and (<ref>). As λ̅ puts mass everywhere in (0, t_0], we conclude from either Theorem <ref> or Theorem <ref> that r(t, b(t)) = 1 for all t ∈ (0, t_0] and so b is representable on [0,t_0].
	
	
	 Of course, the conditions d_1=p_2 or p_1=d_2  in Theorem <ref> are equivalent by (<ref>). We state the theorem in this way to stress that if the optimal values of the “μ-problems” (<ref>) and (<ref>) or the optimal values of the “λ-problems” (<ref>) and (<ref>) agree, then we have representability.
		
	
	
		The conditions d_1=p_2 or p_1=d_2  from Theorem <ref> are a substantive improvement over the conditions that we usually impose to guarantee that b is representable. Usually we would have to prove r(t,b(t)) =1 for all t ∈ (0, t_0], now it suffices to show d_1 = p_2 or p_1 = d_2. In particular, these conditions can easily be checked in implementations (see Section <ref> below).
			
	

	Moving away from the measure λ̅, we can also give more sufficient conditions such that b is representable on [0,t_0]. To this end, we first derive some properties of admissible measures in (<ref>) and (<ref>), respectively.
	
		Let λ be admissible for (<ref>). Then, for every ε∈(0,t_0) we have  λ([t_0-ε, t_0]) > 0.
	
	
		Let λ be admissible in (<ref>). In particular,
		
    ∫_Ω r_θ(t,b(t))λ(dt)≥ r_θ(t_0,x_0)      for any θ∈ [2b(0),∞),

		which implies
		
    ∫exp(-1/2θ^2(1/t-1/t_0)+θ(b(t)/t-x_0/t_0) )λ(dt)≥ 1       for any θ∈ [2b(0),∞).

		Observe that we may write 
		
    ∫exp(-1/2θ^2(1/t-1/t_0)+θ(b(t)/t-x_0/t_0) )λ(dt)= ∫α(t)exp(-β(t)(θ-γ(t))^2)λ(dt)

		for certain positive functions α, β and γ being bounded on [0,t_0-ε] for each ε∈(0,t_0). Now, if ε∈(0,t_0) is such that λ([t_0-ε,t_0])=0, then dominated convergence yields the contradiction 
		
    ∫exp(-1/2θ^2(1/t-1/t_0)+θ(b(t)/t-x_0/t_0) )λ(dt)→0θ→∞.

	
	
		Let λ be admissible for (<ref>). Then, λ({t_0}) = 0.
	
	
		Let λ be admissible for (<ref>). In particular, we have that for all θ∈ [2b(0), ∞)
		
    ∫_(0, t_0]exp( -θ^2/2( 1/t -1/t_0) + θ( b(t)/t - x_0/t_0) ) λ (dt) ≤ 1.

		Assume that λ({t_0}) > 0, then we find
		
    1    ≥∫_(0,t_0]exp( -θ^2/2( 1/t -1/t_0) + θ( b(t)/t - x_0/t_0) ) λ (dt)
       ≥exp( θ( b(t_0)-x_0/t_0) ) ·λ({t_0})  ∞

		as b(t_0) > x_0 by assumption. This is a contradiction. So λ({t_0}) = 0.
	
	With this results we derive sufficient conditions guaranteeing that b is representable.
	
		Let Assumption (B) be satisfied. Moreover, 
		assume that the integrability condition from Theorem <ref> is satisfied, i.e., assume that there exist some t^∗>t_0 such that
		
    ∫_(0, ∞)exp(-θ^2/2 t^∗) μ_i(dθ) < ∞

		for i =1, 2, where  μ_1 and μ_2 are optimisers in (<ref>) and (<ref>) respectively.  Moreover, assume that one of the following conditions is met:
		
			
  * λ_1({t_0}) = 0,
			
  * for every ε∈(0,t_0) it holds λ_2([t_0-ε, t_0]) > 0, 
		
		where λ_1 and λ_2 are optimisers in (<ref>)  and (<ref>), respectively. Then, b is representable on [0,t_0].
	
	 
		First assume condition (i) holds. Lemma <ref> yields that λ_1 puts mass in every interval [t_0-ε, t_0] for every ε∈ (0, t_0) but by assumption λ_1({t_0}) = 0. Due to the complementary slackness conditions from Theorem <ref>, we conclude that there exists a strictly increasing sequence t_1, t_2, …↗ t_0 such that
		
    ∫ r_θ(t_n, b(t_n)) μ_1(dθ) = 1      for all  n ∈.

		Theorem <ref> then implies that b is representable on [0,t_0]. 
		
		If condition (ii) holds the proofs follows along the same lines with the help of Lemma <ref> and Theorem <ref>.
	
	
		Note that Theorem <ref> also offers sufficient conditions for b to be representable that are easier to check than the usual condition r(t, b(t))=1 for all t∈ (0,t_0]. In Chapter <ref>, we investigate a new method to obtain numerical candidates for μ_1 and λ_1 or μ_2 and λ_2, respectively. Then, it is rather straightforward to check whether these measures fulfil the conditions from Theorem  <ref> at least numerically.
		
	
	

§ COMPUTATIONAL METHOD FOR THE LINEAR PROGRAMMING APPROACH

	After a short introduction to existing numerical approaches for the inverse method of images, we give a convergence result for discretised versions of our programs and based on that new algorithm. We also provide error bounds for the numerical distribution function of the first passage time to a boundary b. Moreover, we numerically investigate representability, i.e., we present a numerical study of the assumptions in Theorem <ref>. 
	
	

 §.§ Existing computational approaches for the inverse method of images

	Existing computational approaches
	We will now discuss some approximation methods for the inverse method of images that have already been investigated. 
	 
	In <cit.>, 
 the authors consider 	for a given a boundary b, the equation r_μ(t,b(t))=1
	for t ∈ (0,∞), where μ has to be found in the set of signed measure. To approximate the actual but unknown 
	μ with a signed measure μ̃ such that the boundary b̃ generated by μ̃ is close to b, it is assumed that
	μ=∑_r=1^N w_r  δ_θ_r
		with unknown weights w_r∈, r = 1, …, N for some N ∈. Moreover, they choose a set of increasing time points t_s, s = 1, …, 2N, to obtain the simplified equations
	
    1 = r_μ(t_s,b(t_s)) =∑_r=1^N w_r exp( θ_r b(t_s)/t_s - θ_r^2/2t_s),    s=1, …, 2N.

	If the values of θ_r are pre-assigned, only N time points are required to solve (<ref>) which is linear in w_r. The idea behind the discretisation in (<ref>) is that the given boundary b and the boundary b̃ generated by the signed measure μ̃ through the method of images are equal at the time points t_s, s=1,…, 2N.  
	The authors emphasize that a higher number of time points may be desirable to increase the accuracy of the approximation of b with b̃ but this may of course turn the system (<ref>) singular.
	
	The authors investigate the approximation error by showing that for the first passage times τ_b and τ_b̃ to the boundaries b and b̃, respectively, it holds for all t >0 that
	
    | (τ_b < t) - (τ_b̃ < t) | → 0   as  ε̅_t sup_0<s<t|b̃(s) -b(s) |→ 0.

		In other words: the distribution functions of τ_b and τ_b̃ are close if b and b̃ are close. However, no convergence result for the algorithm itself is provided. The investigated examples of square-root and parabolic boundaries show fairly good approximations with deviations of b̃ from b especially for small t. 
	
	In <cit.>, Zipkin refines the numerical methods developed by among others <cit.>.
		Given a boundary b, <cit.> uses the same discretisations as <cit.> but requires non-negative weights w_r. Here the number of points with non-negative mass and the number of time discretisations are denoted by J and I, respectively, and do not satisfy I=2J in general. Then, for w = (w_j)_j=1, …, J and  =(1,…,1)∈^I Equation (<ref>) can be restated as Mw = for a suitable matrix M. <cit.> relaxes this setting slightly by choosing slack variables s∈^I and by fixing a vector p∈ (0,∞)^I. Then, the following linear program has to be solved
	
		   minimise         p^Ts

		   subject to        Mw +s = ,

		            w ≥0,

		            s ≥0.
	
	The linear program's objective function minimises the p-weighted deviations s. Solving the linear program then give the weights w_j, j=1, …, J, such that b̃ generated by μ̃ = ∑_j w_j δ_θ_j most closely resembles b. <cit.> does not give any convergence results for his algorithm. However, he finds in examples that the approximation b̃ of b generally works well on the discretised time interval [t_1, t_I] but deviates from b outside that interval. The algorithm can nevertheless be seen as a substantial improvement on the algorithm from <cit.>. <cit.> also presents a generalisation of how to include not only point measures but also more general positive, σ-finite measures with densities.
	
	Although the algorithms set out by <cit.> and <cit.> provide good approximations, they suffer from some drawbacks: in both cases, the measure has to be discretised as a weighted sum of point measures as well as the time axis has to be discretised. Moreover, for neither algorithm a convergence result is provided. Both problems are addressed in the following section.
	
	

 §.§ Convergence results and a new algorithm

	
	In this section, we present an algorithm for the inverse method of images and provide convergence results. This section is inspired by methods for American options set out in <cit.>. 
	
	Let us start by discretising the linear problem. To this end, let μ_i ∈^+([2b(0), ∞)), i ∈,  and for n∈ let 
	
    U_n  = {∑_i=1^n a_iμ_i   a ∈ [0,∞)^n}

	 denote the positive cone generated by the measures μ_i, i=1, …, n, and denote by U_∞ the closure of ⋃_n ∈ U_n with respect to the vague topology.  Restricting the linear program (<ref>) to measures in  U_n results in
	
    minimise      ∫ r_θ(t_0,x_0)μ(dθ) 
       subject to      μ∈ U_n,
             ∫ r_θ(t,b(t))μ(dθ)≥ 1     for any  t ∈ (0,t_0].
    		P_2,n

	
	
	Now we prove the following existence and consistency result for our simplified program (<ref>).
	
			Assume that there exists C ∈ (0,∞) such that C ·μ_1 is admissible in (P_2,1) and therefore in any (<ref>) for n ∈∪{∞}. 
		
			
  * For n ∈∪{∞} the optimal value p_2, n of the linear program (<ref>) is attained at some admissible measure μ_2, n^∗ and satisfies p_2, n≤ C ‖μ_1 ‖. Moreover, for m ≤ n the measure μ_2,m^∗ is (<ref>)-admissible and it holds that p_2, m≥ p_2, n.
			
  *  There exists a subsequence of optimisers (μ_2, n_k^∗)_k∈ and a (P_2,∞)-admissible measure ν_∞ such that μ_2, n_k^∗→ν_∞ vaguely as k→∞. Moreover,
			
    p_2, ∞≤∫ r_θ(t_0, x_0) ν_∞ (dθ) ≤inf_n ∈ p_2, n = lim_n →∞ p_2, n.

			
  * If there exists a sequence (ξ_n)_n∈ with ξ_n ∈ U_n and   sup_n∈ξ_n<∞ such that (ξ_n)_n∈ converges vaguely to some (P_2,∞)-optimal measure μ_∞^∗ as n →∞ and if
			
    lim_n →∞ sup_t ∈ (0, t_0]|∫ r_θ(t, b(t)) μ_∞^∗ (dθ) - ∫ r_θ(t, b(t)) ξ_n (dθ) |/∫ r_θ(t, b(t)) μ_1(dθ) = 0,

			then ν_∞ from (b) is (P_2,∞)-optimal. 
			Moreover, we have
			
    p_2, ∞ = ∫ r_θ(t_0, x_0) ν_∞ (dθ) = inf_n ∈ p_2, n = lim_n →∞ p_2, n.

		
	
	
		
			
  * For any n ∈∪{∞} we reformulate the program (<ref>) by absorbing r_θ(t_0, x_0) into μ. The linear program now reads
			
    minimise      ‖μ‖
       subject to      μ∈ U_n,
             ∫r_θ(t,b(t))/r_θ(t_0, x_0)μ(dθ) ≥ 1     for any  t ∈ (0,t_0].
    				P_2,n'

			Note that μ is admissible in (<ref>) if and only if μ' given by 
			
    dμ'/dμ(·)=r_·(t_0,x_0)

			is admissible in (<ref>). 
			
			Let _P'_2,n be the set of all admissible measures in (<ref>). 
			Since μ_a  = C ·μ_1 ∈ U_n is (<ref>)-admissible for every n ∈∪{∞} by assumption, the measure μ'_a defined by
			
    d	μ'_a/d μ_a (·) = r_· (t_0, x_0)

			 is (<ref>)-admissible. Moreover, every potential minimiser μ^∗∈_P'_2,n satisfies
			
    ‖μ^∗‖≤‖μ'_a ‖ρ,

			where ρ < ∞. In particular, any potential solution μ^∗∈_P'_2,n is contained in the vaguely compact ball B_(ρ) = {μ∈([2b(0), ∞)) ‖μ‖≤ρ}. Therefore, in the linear program (<ref>) it is sufficient to consider measure from the set
			
    _P'_2,n∩ B_(ρ) = ⋂_t ∈ (0, t_0] H(t) ∩ U_n ∩ B_(ρ),

			where 
			
    H(t)  = {μ∈() ∫r_θ(t,b(t))/r_θ(t_0, x_0)μ(dθ) ≥ 1 }.

				Since H(t) and U_n are closed with respect to the vague topology,  and B_  is vaguely compact due to the Alaoglu-Bourbaki theorem, cf. <cit.>, also _P'_2,n∩ B_(ρ) is vaguely compact. Then, the optimal value p'_2,n of the linear program (<ref>) is attained by some measure μ'_2,n∈_P'_2,n∩ B_(ρ) as μ↦‖μ‖ is lower semi-continuous with respect to the vague topology. Then, the optimal value p_2,n in (<ref>) is attained by μ^∗_2,n where 
				 
    dμ^∗_2,n/  dμ'_2,n(·)= 1/r_·(t_0, x_0)

				 	 and it holds that 
				 
    p_2,n=∫ r_θ(t_0,x_0) μ^∗_2,n(dθ)=μ_2,n^'≤ρ=Cμ_1.

			
  * Since ‖μ'_2,n‖≤ρ for any n ∈ and as the total variation unit ball is vaguely compact, there exists a subsequence (μ_2,n_k')_k∈ and a measure ν'_∞∈ U_∞∩ B_(ρ)  such that μ_2,n_k' →ν_∞' vaguely as k→∞. Then, we obtain that the subsequence (μ^∗_2,n_k)_k∈ 
			converges vaguely to ν_∞, where 
			
    d ν_∞/dν_∞'(·)=1/r_·(t_0,x_0).

			Let  t ∈ (0, t_0). Then, the mapping [2b(0), ∞) →, θ↦r_θ(t,b(t))/r_θ(t_0, x_0) vanishes at infinity (which does not hold true for t=t_0), so <cit.> yields
			
    ∫r_θ(t,b(t))/r_θ(t_0, x_0) ν_∞'(dθ) = lim_k→∞∫r_θ(t,b(t))/r_θ(t_0, x_0) μ_2,n_k'(dθ) ≥ 1

			as μ'_2,n_k is (P'_2,n_k)-admissible and hence, 
			
    Tν_∞(t)=∫ r_θ(t,b(t))ν_∞(dθ)=∫r_θ(t,b(t))/r_θ(t_0, x_0) ν_∞'(dθ)≥ 1     for all t∈(0,t_0).

			Since Tν_∞ is continuous, we conclude that
			
    Tν_∞(t_0)=∫ r_θ(t_0,b(t_0))ν_∞(dθ)≥ 1.

			
		Thus, ν_∞ is (P_2,∞)-admissible. In particular,  p_2,∞≤∫ r_θ(t_0, x_0) ν_∞ (dθ). Moreover, using vague convergence and <cit.> we find that ‖ν_∞' ‖≤lim inf_k →∞‖μ_2, n_k' ‖. As (p_2,n)_n∈ is monotone, we conclude that
			
    p_2, ∞≤∫ r_θ(t_0, x_0)  ν_∞ (dθ) = ‖ν_∞' ‖≤lim inf_k →∞‖μ_2, n_k' ‖ = lim inf_k→∞ p_2,n_k = lim_n →∞ p_2, n.

			
  * Let n ∈ and define η_n  = ξ_n + ε_n μ_1, where
			
    ε_n  = sup_t ∈ (0, t_0]|∫ r_θ(t, b(t)) μ_∞^∗ (dθ) - ∫ r_θ(t, b(t)) ξ_n (dθ) |/∫ r_θ(t, b(t)) μ_1(dθ).

			Since μ_∞^∗ is (P_2,∞)-optimal by assumption and therefore (P_2,∞)-admissible, we find for all t ∈ (0, t_0] that
			
    ∫ r_θ(t, b(t))  η_n(dθ) -1
    				≥    ∫ r_θ(t, b(t))  η_n(dθ) - ∫ r_θ(t, b(t)) μ_∞^∗(dθ)
    
    				=     ∫ r_θ(t, b(t)) μ_1(dθ) (ε_n - ∫ r_θ(t, b(t)) μ_∞^∗(dθ) - ∫ r_θ(t, b(t)) ξ_n(dθ)/∫ r_θ(t, b(t))  μ_1(dθ))
    ≥    0,

			which means that η_n is (<ref>)-admissible. Using <ref> we obtain
			
    p_2,∞≤∫ r_θ (t_0, x_0) ν_∞(dθ) ≤ p_2,n   ≤∫ r_θ (t_0, x_0) η_n(dθ)
       = ∫ r_θ (t_0, x_0) ξ_n(dθ) + ε_n ∫ r_θ (t_0, x_0) μ_1(dθ).

			Since  (ξ_n)_n∈ converges  vaguely to μ_∞^∗ and  sup_n∈ξ_n<∞ by assumption, we conclude from <cit.> that 
			
    lim_n →∞∫ r_θ (t_0, x_0) ξ_n(dθ) = ∫ r_θ (t_0, x_0) μ_∞^∗ (dθ) = p_2,∞.

			Combining (<ref>), (<ref>) and lim_n →∞ε_n = 0  yields
			
    p_2,∞≤∫ r_θ (t_0, x_0) ν_∞(dθ) ≤lim_n →∞ p_2,n≤lim_n →∞∫ r_θ (t_0, x_0) η_n(dθ) = p_2,∞.

			In particular, ν_∞ is (P_2,∞)-optimal. 
 
	
		An analogous statement holds for (<ref>).

		 For n ∈∪{∞} consider the linear program
		
    maximise      ∫ r_θ(t_0,x_0)μ(dθ) 
       subject to      μ∈ U_n,
             ∫ r_θ(t,b(t))μ(dθ)≤ 1     for any  t ∈ (0,t_0]
    			D_1,n

		and assume that there exists C ∈ (0,∞) such that C ·μ_1 is admissible in (D_1,1).
		
			
  * For n ∈∪{∞}, the optimal value d_1, n of the linear program (<ref>)  is attained at some admissible measure μ_1, n^∗. Moreover, it holds that d_1, m≤ d_1, n≤ 1 for m ≤ n.
			
  * There exists a subsequence of optimisers (μ_1, n_k^∗)_k∈ and a (D_1,∞)-admissible measure ν_∞ such that μ_1, n_k^∗→ν_∞ vaguely as k→∞. Moreover,
			
    d_1, ∞≥∫ r_θ(t_0, x_0) ν_∞ (dθ) ≥sup_n ∈ d_1, n = lim_n →∞ d_1, n.

			
  * If there exists a sequence (ξ_n)_n∈  with ξ_n ∈ U_n and   sup_n∈ξ_n<∞ such that (ξ_n)_n∈ converges vaguely to some (D_1,∞)-optimal measure μ_∞^∗ as n →∞ and if
			
    lim_n →∞ inf_t ∈ (0, t_0]|∫ r_θ(t, b(t)) μ_∞^∗ (dθ) - ∫ r_θ(t, b(t)) ξ_n (dθ) |/∫ r_θ(t, b(t)) μ_1(dθ) = 0,

			then ν_∞ from (b) is (D_1,∞)-optimal.  
			Moreover, it holds that
			
    d_1, ∞ = ∫ r_θ(t_0, x_0) ν_∞ (dθ) = sup_n ∈ d_1, n = lim_n →∞ d_1, n.

		
	
	
		As in the proof of Lemma <ref> consider the modified program
		
    maximise      ⟨ψ ,μ⟩
       subject to      μ∈ U_n,
             ∫r_θ(t,b(t))/r_θ(t_0, b(t_0))μ(dθ) ≤ 1     for any  t ∈ (0,t_0], 
    			D_1,n'

		where ψ[2b(0),∞)→, θ↦exp(-θb(t_0)-x_0/t_0). By (<ref>) any (<ref>)-admissible μ satisfies ‖μ‖≤ 1. 
		Thus, all admissible measures in (<ref>) are contained in B_(1) ={μ∈([2b(0), ∞)) ‖μ‖≤ 1 }. The rest of the proof follows quite similar to the proof of Proposition <ref>.
	
	Similar results can be formulated and proven for the programs (<ref>) and (<ref>), for more details see Appendix A.2 in <cit.>. 
	

	Observe that for μ_i=δ_θ_i with {θ_i}_i∈ a dense subset of [2b(0), ∞) the set U_∞   is vaguely dense in ^+([2b(0), ∞)) by <cit.>. Thus, we can restrict ourselves to the point measures  μ_i=δ_θ_i in Propositions <ref> and <ref>  and still are able to approximate every possible representing measure μ arbitrarily close.  The inclusion of measures with densities, as done in <cit.>, is therefore not necessary for the convergence of our algorithm but may be useful for numerical reasons.
	
	For our algorithm we further simplify the discretised linear problem (<ref>) and only consider the interval I=[2b(0), 2b(0) +l] for some  l>0. 
	 Let μ_i=δ_θ_i be the point measures in θ_i=2b(0)+i-1/nl, i=1, …, n, and let again U_n  = {∑_i=1^n a_iμ_i  a ∈ [0,∞)^n}  be the positive cone generated by the measures μ_i, i=1, …, n. The restriction (<ref>) of the linear program (<ref>) to measures in U_n	 can then be simplified to
	
    minimise      ∑_i=1^n r_θ_i(t_0,x_0) a_i 
       subject to       a ∈ [0,∞)^n,
             ∑_i=1^n r_θ_i(t,b(t)) a_i ≥ 1     for any  t ∈ (0,t_0].

	For the implementation we make use of the cutting plane algorithm described in <cit.> and <cit.>, where  also good convergence results are provided. We  now present our algorithm:
	
	Step 1: Let the set of initial constraints Γ_1 ⊂ (0,t_0] be the set {t_0}. Choose a maximum number of iterations k_max and set k = 1.
	
	

	 Step 2: Calculate a solution a^(k)∈ [0,∞)^n of the finite dimensional linear program
		
    minimise      ∑_i=1^n r_θ_i(t_0,x_0) a_i^(k)
       subject to       a^(k)∈ [0,∞)^n,
             ∑_i=1^n r_θ_i(t,b(t)) a_i^(k)≥ 1     for all  t ∈Γ_k.

		
			Step 3: Determine  a point t^(k)∈ (0, t_0], where the constraint is most severely violated, i.e.,
		
    t^(k)∈_t ∈ (0, t_0]{∑_i=1^n r_θ_i(t,b(t)) a_i^(k)}.

		
			Step 4: Add the point t^(k)		to the set of constraints, i.e., set Γ_k+1 = Γ_k ∪{t^(k)}.
		
		Step 5: If the maximum number of iterations is reached, i.e., if k = k_max, terminate the algorithm and output the approximate solution
		
    ã = a^(k_max).

		Otherwise, increase the iteration counter k  k+1 and return to the second step.
		
			
		
  * In Proposition <ref> we do not assume the measures μ_i, i ∈, to be point measures as we do in our algorithm. This indicates that the algorithm also works for a much larger class of “auxiliary measures” μ_i. 
	
  * Our algorithm requires fewer assumptions than the algorithm proposed in <cit.> as we only choose the points (θ_i)_i=1,…,n in [2b(0), ∞) where the algorithm can put point masses and the initial constraint at t_0. Then, our algorithm “chooses” the next points where the constraint is evaluated. In contrast, in <cit.>, both the division of [2b(0), ∞) as well as the points t_1, …, t_m where the constraint is evaluated have to be chosen in advance. 


	
	For the linear program (<ref>), we use similar simplifications and an analogous algorithm.
	
	 For the linear programs (<ref>) and (<ref>), i.e., for the “λ”-problems, we divide the interval (0, t_0] into n_λ equidistant points 0<t_1 < … < t_n_λ = t_0 and let λ_i=δ_t_i for i = 1, …, n_λ. We then use analogous algorithms which are initialised with Γ_1 = {2b(0)}.
	
	If we have obtained a candidate representing measure from one of the above outlined algorithms, it is interesting to know the difference of the cumulative distribution function generated by this measure and the actual cumulative distribution function. If a measure μ represents b 
	on (0, t_0], then 
	the cumulative distribution function F of τ is given by
	
    F(t)
    		   = 1-Φ( b(t)/√(t)) + ∫_(-∞,b(t)) r_μ(t,x)  p_t(0,x) dx,

	recall (<ref>).
	If we now have obtained a measure μ̃ from our algorithms, we define r̃(t,x) = r_μ̃(t,x)= ∫_[2b(0), ∞) r_θ(t,x) μ̃(d θ) and approximate the true cumulative distribution function F by
	
    F(t) = 1-Φ( b(t)/√(t)) + ∫_(-∞,b(t))r̃(t,x)  p_t(0,x) dx.

	
		Let F be the true first passage time distribution to a boundary b and μ̃ a measure such that for all t ∈ (0, t_0]
		
    1-ζ_1 ≤1/r̃_μ̃(t, b(t))≤ 1+ζ_2

		for some ζ_1 ∈ [0,1) and ζ_2 >0. Then,
		
    sup_t ∈ (0, t_0]| F(t) - F(t) |≤max( ζ_1, ζ_2 ).

		In particular, we have
		
    lim_ζ_1 ↘ 0
    ζ_2 ↘ 0  sup_t ∈ (0, t_0]| F(t) - F(t)| = 0.

	
	
		Since by assumption, we have for all t ∈ (0, t_0] that
		
    1-ζ_1 ≤1/r̃_μ̃(t, b(t))≤ 1+ζ_2

		for some ζ_1 ∈ [0,1) and ζ_2 >0, Proposition <ref> yields for t ∈ (0, t_0] and x < b(t)
		
    (1-ζ_1) r̃_μ̃(t,x)≤(τ≤ t | W_t=x)≤ (1+ζ_2) r̃_μ̃(t,x).

		 Hence, using (<ref>) we obtain
		
    F(t)    = 1-Φ( b(t)/√(t)) + ∫_(-∞,b(t))(τ≤ t |  W_t = x)   p_t(0,x) dx
       ≤ 1-Φ( b(t)/√(t)) + ∫_(-∞,b(t)) (1+ζ_2)  r̃(t,x)  p_t(0,x) dx
    					≤ (1+ζ_2)  F(t).

		 In particular, we  conclude
		
    F(t) - F(t) ≤ζ_2.

		Analogously, we find
		
    F(t) - F(t) ≥ -ζ_1.

		To summarise, it holds that
		
    sup_t ∈ (0, t_0]| F(t) - F(t) |≤max( ζ_1, ζ_2 ).

	
	
	
	

 §.§ Numerical study of representability

	
	In this section, we present numerical results for our algorithm for three concave, analytic boundaries b  and one convex, analytic boundary b and investigate their representability. 
	For all boundaries, we choose 
	
		
  * t_0 = 1,
		
  * x_0 = b(t_0)-1 and 
		
  * the maximum number of iterations k_max = 20.
		
		 Moreover, for the “μ-problems” (<ref>) and (<ref>) we select a length of l=5 and discretise the interval [2b(0), ∞) by setting n=100 equidistant points in [2b(0), 2b(0) + l] where the algorithm can put point mass.
		 
		  For the “λ-problems” (<ref>) and (<ref>) we discretise the interval (0, t_0] by choosing n_λ=100 equidistant points where the algorithm can put mass. Table <ref> gives the optimal values of all these four programs.
		  
	Moreover,  for the functions 
	
    r_μ_1,n(t, b(t)) ∫_[2b(0), ∞) r_θ(t, b(t)) μ_1,n(dθ)

	 and r_μ_2,n(t, b(t)) with t between 0 and t_0=1 we consider the minima and maxima of r_μ_1,n^-1 and r_μ_2,n^-1. According to Proposition <ref>, this will give us an idea of the quality of the approximation of the true distribution function F with the numerical c.d.f. F. The results can be found in Table <ref>. Also note that the computational times for all algorithms were very fast (less than 1 sec). 
	
		
	
	
	We immediately see in Table <ref> that in the case of the linear boundary b(t) = 1+t the values of d_1,n, p_1,n_λ, d_2,n_λ and p_2,n agree in the first 8 digits. So, we can heuristically confirm strong duality in both set-ups as well as that the conditions for representability from Theorem <ref> are met, i.e., d_1 = p_2 and p_2 = d_1. This is, of course, not surprising as we already know that linear boundaries b are representable, recall Remark <ref>. Therefore, this case can serve as a check of our algorithm.
	
	Moreover, we consider the boundaries b_μ_1,n and b_μ_2,n generated by the numerical solutions μ_1,n and μ_2,n and compare these boundaries to the boundary b that was the input for the optimisation problems. For our four input boundaries Figure <ref> and <ref> depict the graphs of these three boundaries in the top figure, show the value of r_μ_1,n(t, b(t)) and r_μ_2,n(t, b(t)) between 0 and 1 in the middle figure and in the bottom the distribution function obtained by substituting the representing measure μ with the numerical solutions μ_1,n and μ_2,n, respectively, i.e.,
	
    F_μ_i(t)  1-Φ( b(t)/√(t)) + ∫_(0,∞)Φ( b(t)-θ/√(t)) μ_i,n(d θ)

	for i = 1,2. Note that F_μ_i yields an approximation of the true distribution function F in analytical form.
	
	We can see on the left of Figure <ref> that for the linear boundary b(t)=1+t  both numerical boundaries perfectly replicate the original boundary. In particular, the graphs of r_μ_i,n(t, b(t)), i=1, 2, which are equal to 1 support that b is represented by μ_i,n. This is not very surprising as it is well-known that linear boundaries are representable by measures μ which only put mass into 2b(0) which both μ_1,n and μ_2,n do. Moreover, we can see in Table <ref> that r_μ_1,n^-1 and r_μ_2,n^-1 deviate from 1 by less than 10^-10. The deviation should be 0 and can probably be attributed to small numerical rounding errors. In particular, we know by Proposition <ref> that we get a very good approximation of the distribution function F.
	
	Let us now consider the boundaries b(t) = √(1+t) and b(t) = log(2+t), i.e., boundaries which are concave and monotone increasing. For both boundaries, we  observe (compare Table <ref>) that the values of d_1,n, p_1,n_λ, d_2,n_λ and p_2,n agree in the first 8 digits, i.e., we  again numerically confirm both strong duality in both set-ups as well as the conditions for representability from Theorem <ref>.
	
	Even though we choose t_0=1 we observe in the top of Figure <ref> that the boundaries are very well replicated by the boundaries generated by μ_1,n and μ_2,n up to t=2 with just slight deviations between t=2 and t=3. The graphs for r_μ_1,n(t, b(t)) and r_μ_2,n(t, b(t)) are virtually indistinguishable from 1 and we can see in Table <ref> that r_μ_1,n^-1 and r_μ_2,n^-1 deviate from 1 by less than 10^-2. Thus, the graphs of the numerical distribution functions F_μ_i, i=1,2, are very exact approximations for F due to Proposition <ref>. Moreover, note that Proposition <ref> gives a rather conservative estimate of the approximation error, so the true approximation error is probably much less.
	

	Now we turn our attention to b(t) = 1+t^2. We immediately see from Table <ref> that d_1,n and p_1,n_λ agree in the first 8 digits and d_2,n_λ and p_2,n agree in the first 2 digits, so we can safely assume that strong duality holds in both set-ups. However, the gaps between the two different set-ups are very large.

	 We immediately see on the right of Figure <ref> why the algorithms does not produce sensible results: both programs only allow for concave boundaries and the “most convex” the program can do is a linear boundary. Unsurprisingly, r_μ_1,n(t, b(t)) and r_μ_2,n(t, b(t)) are for the most part very far away from 1. 
  Expectedly, the numerical versions of the distribution functions are vastly different from one another.
	
	Finally, we investigate whether the conditions of Theorem <ref> are numerically met. Whether λ_1,n_λ does not put mass into t_0 is hard to verify numerically, since the mass could be so small that it numerically vanishes. But we can investigate the mass which λ_2,n_λ puts close to t_0 for increasing n_λ. Due to the discretisation of the t axis, we look for mass in the interval ((n-1)t_0/n, t_0]. In order to make sure that the mass near t_0 does not vanish, we need to ensure that x_0 is not too far below b(t_0) as the hitting probability might become 0 numerically. So, we choose x_0 = b(t_0) -0.1. All other parameters are chosen in the same way as before. Linear b are omitted in this test, as we already know that they are representable and  λ̅, the conditional last passage time distribution of the Brownian motion W to b, can be shown to be an optimal measure in this case, cf. <cit.>.  Table <ref> shows the values we obtained. 
	
	
	We immediately see in Table <ref> that for the concave boundaries the measure λ_2,n always puts mass near t_0 while we increase n_λ. This is yet another very strong indicator that concave boundaries are representable according to Theorem <ref>. In contrast, for the convex boundary there is no mass near t_0 for any choice of n_λ, which is an indicator that the condition from Theorem <ref> is not only sufficient but might also be necessary.
	
	To sum up, the algorithm shows perfect replication of linear boundaries and very good replication of concave boundaries even beyond the controlled interval (0, t_0].
	

	

§ EXTENSION TO TWO-SIDED BOUNDARIES


	Finally, we shortly comment on the (inverse) method of images for two-sided boundaries. For more details we refer to <cit.>. 
	
	Denote by b_1 the lower bound and by b_2 the upper bound for which we are interested in the first passage time distribution of a one-dimensional Brownian motion W. In particular, assume 
	
		
  * b_1≤ b_2,
		
  * b_1(0) < W_0 < b_2(0)
	
	and define 
	
    τ =inf{t∈[0,∞) W_t ∉(b_1(t), b_2(t))}.

	
	
 An analogue to Proposition <ref> for the problem with two-sided boundaries can be proven which allows to formulate two linear programs which approximate the measure μ representing b_1 and b_2 from below and from above, respectively, and these programs are analogous to the problems for the classic method of images with one boundary.   
In particular, the linear programs and their formal duals can implement in the same way as in Section <ref>, cf. <cit.>.


	

	
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