Construction of Martingale Measure in the Hazard Process Model of Credit Risk
CConstruction of Martingale Measure in theHazard Process Model of Credit Risk
Marek Capi´nski ∗ and Tomasz Zastawniak †
28 August 2019
Abstract
In credit risk literature, the existence of an equivalent martingale mea-sure is stipulated as one of the main assumptions in the hazard processmodel. Here we show by construction the existence of a measure thatturns the discounted stock and defaultable bond prices into martingalesby identifying a no-arbitrage condition, in as weak a sense as possible,which facilitates such a construction.
No arbitrage is the principal condition in mathematical finance, a basis forpricing derivative securities. In the literature on the hazard process model ofcredit risk (for example, [BieRut02], [BieJeaRut09] and references therein) theexistence of an equivalent martingale measure is assumed, the lack of arbitragefollowing as an immediate consequence. Here we work in the opposite direction.A construction of a martingale measure in the relatively straightforward caseof the hazard function model of credit risk in the absence of simple arbitragewas accomplished in [CapZas14]. In the much more general setting of the haz-ard process model, the construction of a martingale measure from a suitablyweak no-arbitrage condition turns out to be far from trivial, and constitutesthe main result of the present paper. This no-arbitrage condition, referred toas the no-quasi-simple-arbitrage principle later in the paper, implies that thepre-default value of the defaultable bond is a strict submartingale with valuesbetween 0 and 1 under the Black–Scholes measure. It makes it possible to applythe Doob–Meyer type multiplicative decomposition for positive submartingales,leading to a new definition of the survival process, hence of the hazard pro-cess, as the unique up to indistinguishability strictly positive previsible (with ∗ [email protected] ; Faculty of Mathematics, AGH–University of Science and Technol-ogy, Al. Mickiewicza 30, 30–059 Krak´ow, Poland. † [email protected] ; Department of Mathematics, University of York, Hes-lington, York YO10 5DD, United Kingdom. a r X i v : . [ q -f i n . M F ] A ug espect to the Black–Scholes filtration) process that features in the multiplica-tive decomposition. The martingale measure in the hazard process model isthen constructed with the aid of the survival (or hazard) process by a methodresembling the classical construction of Wiener measure on path space. We consider three assets, a non-defaultable bond B ( t, T ) = e − r ( T − t ) for t ∈ [0 , T ] growing at a constant rate r ≥
0, a stock with prices S ( t ) for t ≥ D ( t, T ) for t ∈ [0 , T ], where T > S ( t ) and D ( t, T ) are defined on a probability space (Ω , Σ , P ), where P is the physicalprobability. Throughout this paper, equalities and inequalities between randomvariables on (Ω , Σ , P ) as well as pathwise properties of stochastic processes suchas, for example, continuity of paths will be understood to hold P -a.s.The stock price process S ( t ) is assumed to follow the Black–Scholes modelwith driving Brownian motion W ( t ). We write ( F t ) t ≥ for the augmented fil-tration generated by the Brownian motion.We also take a random variable τ > , Σ , P ) to play the role of thetime of default. Let ( I t ) t ≥ be the filtration generated by the default indicatorprocess I ( t ) = { τ ≤ t } and let ( G t ) t ≥ be the enlarged filtration, G t := σ ( F t ∪ I t )for each t ≥ D ( t, T ) = c ( t ) { t<τ } , (1)where c ( t ), t ∈ [0 , T ], called the pre-default value of D ( t, T ), is an ( F t ) t ∈ [0 ,T ] -adapted process with continuous paths such that c ( t ) ∈ (0 ,
1) for t ∈ [0 , T ) and c ( T ) = 1. In particular, the payoff of this bond is D ( T, T ) = { T <τ } at time T ,that is, the defaultable bond has zero recovery. Remark 1
In Appendix 7 we show that expression (1) follows from certainweaker assumptions by means of a no-arbitrage argument within a family ofsimple strategies in the BD section of the market only. This is similar tothe hazard function model consisting of two bonds B and D only, consideredin [CapZas14]. However, while c ( t ) is a deterministic strictly increasing functionin the toy model in [CapZas14], here it is an ( F t ) t ∈ [0 ,T ] -adapted process, whichturns out to be a strict supermartingale under a suitable no-arbitrage conditionas shown in Section 3.Additionally, we assume that P ( s < τ ≤ t |F T ) > s, t ≥ s < t . In other words, for every A ∈ F T of positivemeasure P , the event A ∩ { s < τ ≤ t } is also of positive measure P . Thiscondition means that there are no gaps in the set of values of τ , i.e. default canhappen at any time, no matter what the stock price process is doing. Since the stock S follows the Black–Scholes model, there is a unique proba-bility measure Q BS equivalent to P such that the discounted stock price process e − rt S ( t ) is an ( F t ) t ≥ -martingale under Q BS . Since all the processes will needto be considered up to time T only, it will suffice if Q BS is understood as ameasure defined on the σ -algebra F T . In the BS segment of the market a self-financing strategy with rebalancing incontinuous time can be defined in the usual manner in terms of the stochasticintegral with respect to the Black–Scholes stock price process S ( t ). Such astrategy is said to be admissible whenever its discounted value process is an( F t ) t ≥ -martingale under Q BS .On the other hand, the above properties of the defaultable bond are a priori not enough to consider a stochastic integral with respect to the process D ( t, T ).Hence, for the time being at least, we consider a class of self-financing strategiessuch that continuous rebalancing is allowed within the BS segment of the mar-ket, while the position in D can only be rebalanced at a finite set of times. Wewill show that lack of arbitrage opportunities within the class of such strategiesis equivalent to the existence of a martingale measure.A strategy of this kind can be constructed as follows. Take 0 = s < s < · · · < s N = T to be the defaultable bond rebalancing times for the strategy, andlet y n be F s n − -measurable random variables representing the positions in D within the time intervals from s n − to s n for n = 1 , . . . , N . At time 0 we startan admissible self-financing Black–Scholes strategy x = ( x B , x S ) in the BS segment of the market, and follow this strategy up to time s . Then, if τ ≤ s ,that is, if default has already occurred and the defaultable bond D has becomeworthless, we follow the same strategy x up to time T . But if s < τ , that is, ifno default has occurred yet, we rebalance the position in D from y to y , whichmeans a (positive or negative) cash injection into the BS segment of the market.We add this cash injection to the value of the strategy x at time s , and starta new self-financing Black–Scholes strategy x = ( x B , x S ) from this new valueat time s . Then, at time s we either continue following the same strategy x up to time T if τ ≤ s , or else we rebalance the position in D from y to y ,adjust the value of the BS segment accordingly, and start a new self-financingBlack–Scholes strategy x = ( x B , x S ) from the adjusted value at time s . Inthis manner, we proceed step by step up to and including time s N − . This isformalised in the next definition. The following slightly weaker condition is in fact sufficien t: P ( T < τ |F T ) > P ( s < τ ≤ t |F T ) > s, t ∈ [0 , T ] such that s < t . efinition 2 By a quasi-simple self-financing strategy we understand an R -valued ( G t ) t ∈ [0 ,T ] -adapted process ϕ = (cid:0) ϕ B , ϕ S , ϕ D (cid:1) representing positions in B, S, D such that there are sequences of times 0 = s < s < · · · < s N = T , R -valued ( F t ) t ∈ [0 ,T ] -adapted processes x , . . . , x N and R -valued random variables y , . . . , y N satisfying the following conditions:1. x n = (cid:0) x Bn , x Sn (cid:1) is an admissible self-financing Black–Scholes strategy inthe time interval [ s n − , T ] and y n is an F s n − -measurable random variablesuch that ϕ B ( t ) = x Bn ∧ µ ( t ) , ϕ S ( t ) = x Sn ∧ µ ( t ) , ϕ D ( t ) = y n ∧ µ (3)for each n = 1 , . . . , N and t ∈ ( s n − , s n ], where µ := max { m = 1 , . . . , N : s m − < τ } ;2. The value process V ϕ ( t ) := ϕ B ( t ) B ( t, T ) + ϕ S ( t ) S ( t ) + ϕ D ( t ) D ( t, T )satisfies the following self-financing condition for each n = 0 , . . . , N − V ϕ ( s n ) = lim t (cid:38) s n V ϕ ( t ) . Remark 3
The minimum n ∧ µ in (3) captures the fact that we switch to a newBlack–Scholes strategy x n and a new defaultable bond position y n at time s n − only if no default has yet occurred at that time. Definition 4
We say that the no-quasi-simple-arbitrage (NQSA) principle holdsif there is no quasi-simple self-financing strategy ϕ = (cid:0) ϕ B , ϕ S , ϕ D (cid:1) such that V ϕ (0) = 0, V ϕ ( T ) ≥
0, and V ϕ ( T ) > P .The following result provides a characterisation of the NQSA principle interms of the process c ( t ). Theorem 5
Under the assumptions in Section 2, the following conditions areequivalent:1. The NQSA principle holds;2. The process e − rt c ( t ) , t ∈ [0 , T ] is a strict ( F t ) t ∈ [0 ,T ] -submartingale un-der Q BS . Proof.
Since we can work with discounted values, it is enough to consider thecase when r = 0, that is, B ( t, T ) = 1 for all t ∈ [0 , T ].We begin by showing that 1 ⇒
2. Suppose that the NQSA principle holds.To verify that c ( t ) is a strict ( F t ) t ∈ [0 ,T ] -submartingale under Q BS , we take any4 , t ∈ [0 , T ] such that t < t and need to show that c ( t ) < E Q BS ( c ( t ) |F t ).Let A := { c ( t ) ≥ E Q BS ( c ( t ) |F t ) } . Because c ( t ) is a random variable with values in (0 , x = ( x B , x S ) in theBlack–Scholes model that replicates the contingent claim c ( t ) at time t , thatis, x B ( t ) + x S ( t ) S ( t ) = c ( t ) . The value of the strategy x B ( t ) + x S ( t ) S ( t ) = E Q BS ( c ( t ) |F t )for any t ∈ [0 , t ] is an ( F t ) t ∈ [0 ,t ] -martingale under Q BS and has continuouspaths. We extend this strategy by putting x ( t ) := ( c ( t ) ,
0) for any t ∈ ( t , T ].Using this, we can construct a quasi-simple self-financing strategy as follows: • Do nothing until time t . • At time t , if the event A has occurred but no default has happenedyet, that is, t < τ , then sell a single defaultable bond D for c ( t ), in-vest the amount x B ( t ) + x S ( t ) S ( t ) = E Q BS ( c ( t ) |F t ) in the strategy x = ( x B , x S ), and put the balance of these transactions into the non-defaultable bonds B . Then follow the self-financing strategy x = ( x B , x S )in the BS segment of the market up to time t . Otherwise do nothing. • At time t close all positions and invest the balance in the non-defaultablebons B until time T .The precise formulas defining this strategy ϕ = ( ϕ B , ϕ S , ϕ D ) are ϕ B ( t ) := ϕ S ( t ) := ϕ D ( t ) := 0 for t ∈ [0 , t ] ,ϕ B ( t ) := (cid:0) x B ( t ) + c ( t ) − E Q BS ( c ( t ) |F t ) (cid:1) A ∩{ t <τ } ,ϕ S ( t ) := x S ( t ) A ∩{ t <τ } , ϕ D ( t ) := − A ∩{ t <τ } for t ∈ ( t , t ] ,ϕ B ( t ) := c ( t ) A ∩{ t <τ ≤ t } + ( c ( t ) − E Q BS ( c ( t ) |F t )) A ∩{ t <τ } ,ϕ S ( t ) := 0 , ϕ D ( t ) := − A ∩{ t <τ ≤ t } for t ∈ ( t , T ] . Consider the case when 0 < t < t < T (the other cases when t = 0 or t = T are similar and will be omitted for brevity). In Definition 2 we take N := 3 and s := 0, s := t , s := t , s := T . We also put x ( t ) := (0 , , y := 0 ,x ( t ) := 1 A ( x B ( t ) + c ( t ) − E Q BS ( c ( t ) |F t ) , x S ( t )) , y := − A ,x ( t ) := 1 A ( c ( t ) − E Q BS ( c ( t ) |F t ) , , y := 0 . ϕ = ( ϕ B , ϕ S , ϕ D ). Its initial value is V ϕ (0) = 0and final value is V ϕ ( T ) = c ( t ) A ∩{ t <τ ≤ t } + ( c ( t ) − E Q BS ( c ( t ) |F t )) A ∩{ t <τ } . Since the NQSA principle holds and c ( t ) ≥ E Q BS ( c ( t ) |F t ) on A , we musthave P ( A ∩ { t < τ ≤ t } ) = 0 given that c ( t ) >
0. Because A ∈ F s ⊂ F T ,it follows by assumption (2) that P ( A ) = 0, proving that c ( t ) is indeed astrict ( F t ) t ∈ [0 ,T ] -submartingale under Q BS . This completes the proof of theimplication 1 ⇒ ⇒
1, we assume that c ( t ) is a strict ( F t ) t ∈ [0 ,T ] -submartingale under Q BS , and take any quasi-simple self-financing strategy ϕ = (cid:0) ϕ B , ϕ S , ϕ D (cid:1) with V ϕ (0) = 0 and V ϕ ( T ) ≥
0. To verify that the NQSA principleholds, we need to show that V ϕ ( T ) = 0. Let s n , x n , y n be the correspondingsequences as in Definition 2.For each n = 1 , . . . , N and t ∈ [ s n − , T ], we put U n ( t ) := x Bn ( t ) + x Sn ( t ) S ( t ) . Then, for each n = 1 , . . . , N and t ∈ ( s n − , s n ], we have V ϕ ( t ) = ϕ B ( t ) + ϕ S ( t ) S ( t ) + ϕ D ( t ) D ( t, T ) = U n ∧ µ ( t ) + y n ∧ µ c ( t ) { t<τ } = n − (cid:88) k =1 U k ( t ) { s k − <τ ≤ s k } + U n ( t ) { s n − <τ } + y n c ( t ) { t<τ } . (4)First, we consider the case when V ϕ ( s n ) ≥ n = 0 , . . . , N , andproceed by induction on n to show that V ϕ ( s n ) = 0 for each n = 0 , . . . , N .For n = 0 we have V ϕ ( s ) = V ϕ (0) = 0. Suppose that V ϕ ( s n − ) = 0 for some n = 1 , . . . , N . Self-financing at s n − means that0 = V ϕ ( s n − ) = lim t (cid:38) s n − V ϕ ( t )= n − (cid:88) k =1 U k ( s n − ) { s k − <τ ≤ s k } + ( U n ( s n − ) + y n c ( s n − )) { s n − <τ } . It follows that U k ( s n − ) { s k − <τ ≤ s k } = 0 for each k = 1 , . . . , n − , ( U n ( s n − ) + y n c ( s n − )) { s n − <τ } = 0 . Because U k ( s n − ) and U n ( s n − ) + y n c ( s n − ) are F T -measurable, it follows byassumption (2) that U k ( s n − ) = 0 for each k = 1 , . . . , n − ,U n ( s n − ) + y n c ( s n − ) = 0 . s n is0 ≤ V ϕ ( s n )= n − (cid:88) k =1 U k ( s n ) { s k − <τ ≤ s k } + U n ( s n ) { s n − <τ } + y n c ( s n ) { s n <τ } = n (cid:88) k =1 U k ( s n ) { s k − <τ ≤ s k } + ( U n ( s n ) + y n c ( s n )) { s n <τ } . Hence U k ( s n ) { s k − <τ ≤ s k } ≥ k = 1 , . . . , n, ( U n ( s n ) + y n c ( s n )) { s n <τ } ≥ . Because U k ( s n ) and U n ( s n ) + y n c ( s n ) are F T -measurable, it follows by assump-tion (2) that U k ( s n ) ≥ k = 1 , . . . , n,U n ( s n ) + y n c ( s n ) ≥ . Since the value U n ( t ) of the admissible self-financing strategy x n ( t ) in the Black–Scholes model is an ( F t ) t ≥ -martingale under Q BS , it follows that0 ≤ E Q BS ( U n ( s n ) + y n c ( s n ) |F s n − ) = U n ( s n − ) + y n E Q BS (cid:0) c ( s n ) |F s n − (cid:1) = y n (cid:0) E Q BS (cid:0) c ( s n ) |F s n − (cid:1) − c ( s n − ) (cid:1) . Because E Q BS (cid:0) c ( s n ) |F s n − (cid:1) > c ( s n − ), we can see that y n ≥
0. On the otherhand, 0 ≤ E Q BS ( U n ( s n ) |F s n − ) = U n ( s n − ) = − y n c ( s n − ) . Since c ( s n − ) >
0, it follows that y n ≤
0. Hence, we have shown that y n = 0.As a result, E Q BS ( U n ( s n ) |F s n − ) = U n ( s n − ) = − y n c ( s n − ) = 0. Because U n ( s n ) ≥
0, it follows that U n ( s n ) = 0. Moreover, for each k = 1 , . . . , n −
1, wehave E Q BS ( U k ( s n ) |F s n − ) = U k ( s n − ) = 0and U k ( s n ) ≥
0, which means that U k ( s n ) = 0. Hence, V ϕ ( s n ) = n (cid:88) k =1 U k ( s n ) { s k − <τ ≤ s k } + ( U n ( s n ) + y n c ( s n )) { s n <τ } = 0 , completing the induction step.It remains to consider the case when V ϕ ( s n ) < P for some n = 0 , . . . , N . Let m be the largest integer n among 0 , . . . , N such that V ϕ ( s n ) < P . Clearly, 0 < m < N since V ϕ (0) = 07nd V ϕ ( s N ) = V ϕ ( T ) ≥
0. Because, by (4),0 ≤ V ϕ ( s m +1 )= m (cid:88) k =1 U k ( s m +1 ) { s k − <τ ≤ s k } + U m +1 ( s m +1 ) { s m <τ } + y m +1 c ( s m +1 ) { s m +1 <τ } = m +1 (cid:88) k =1 U k ( s m +1 ) { s k − <τ ≤ s k } + ( U m +1 ( s m +1 ) + y m +1 c ( s m +1 )) { s m +1 <τ } , we can see that for each k = 1 , . . . , m + 1 we have U k ( s m +1 ) { s k − <τ ≤ s k } ≥ U k ( s m +1 ) ≥ ≤ E Q BS ( U k ( s m +1 ) |F s m ) = U k ( s m ) . Moreover, by (4), we also have V ϕ ( s m ) = m − (cid:88) k =1 U k ( s m ) { s k − <τ ≤ s k } + U m ( s m ) { s m − <τ } + y m c ( s m ) { s m <τ } = m (cid:88) k =1 U k ( s m ) { s k − <τ ≤ s k } + ( U m ( s m ) + y m c ( s m )) { s m <τ } . As a result, V ϕ ( s m ) { τ ≤ s m } = m (cid:88) k =1 U k ( s m ) { s k − <τ ≤ s k } ≥ . This implies that P ( V ϕ ( s m ) <
0) = P ( { V ϕ ( s m ) < } ∩ { s m < τ } ) . Since { V ϕ ( s m ) < } ∈ G s m , it follows by the properties of the enlarged filtration(for example, Lemma 5.27 in [CapZas16]) that there is an A ∈ F s m such that { V ϕ ( s m ) < } ∩ { s m < τ } = A ∩ { s m < τ } . We define a new quasi-simple self-financing strategy ψ = ( ψ B , ψ S , ψ D ) by ψ B ( t ) := ψ S ( t ) := ψ D ( t ) := 0 for t ∈ [0 , s m ] ,ψ B ( t ) := (cid:0) ϕ B ( t ) − V ϕ ( s m ) (cid:1) A ∩{ s m <τ } ,ψ S ( t ) := ϕ S ( t ) A ∩{ s m <τ } , ψ D ( t ) := ϕ D ( t ) A ∩{ s m <τ } for t ∈ ( s m , T ] . For this strategy, we put˜ x Bn ( t ) := m 8n place of x Bn ( t ) , x Sn ( t ) , y n in Definition 2. Then, for n = 1 , . . . , m and t ∈ ( s n − , s n ], we have ψ B ( t ) = ˜ x Bn ∧ µ ( t ) = 0 ,ψ S ( t ) = ˜ x Sn ∧ µ ( t ) = 0 ,ψ D ( t ) = ˜ y n ∧ µ = 0 . Moreover, for n = m + 1 , . . . , N and t ∈ ( s n , s n +1 ], we have ψ B ( t ) = ˜ x Bn ∧ µ ( t ) = { m Without loss of generality, we can take r = 0 to simplify the proof.Suppose that c ( t ) is an ( F t ) t ∈ [0 ,T ] -submartingale under Q BS . Because c ( t ) isbounded, strictly positive, with continuous paths and satisfies c ( T ) = 1, all itspaths are bounded away from 0 on the closed interval [0 , T ]. Moreover, the aug-mented filtration ( F t ) t ∈ [0 ,T ] generated by Brownian motion satisfies the usualconditions. Therefore, Theorem 33 in [Aze78] (or Theorem 2 of [YoeMey76])gives a unique multiplicative decomposition of c ( t ) on [0 , T ]. Namely, there isa unique (up to indistinguishability) strictly positive non-increasing ( F t ) t ∈ [0 ,T ] -previsible process G ( t ) on [0 , T ] such that G (0) = 1 and c ( t ) G ( t ) is an ( F t ) t ∈ [0 ,T ] -martingale under Q BS . Because c ( T ) = 1, it follows that for each t ∈ [0 , T ] c ( t ) G ( t ) = E Q BS ( G ( T ) |F t ) . Since G (0) = 1 and G ( t ) is positive and non-increasing, it follows that G ( T )is bounded, so it is square integrable. Hence, c ( t ) G ( t ) = E Q BS ( G ( T ) |F t ) hascontinuous paths by the martingale representation theorem. It follows that G ( t )also has continuous paths given that c ( t ) does and is positive.It remains to verify that, if G ( t ) is a non-increasing process and c ( t ) G ( t ) isa martingale, then c ( t ) is a strict submartingale if and only if G ( t ) is a strictsupermartingale. Let 0 ≤ s < t ≤ T . Because E Q BS ( c ( t ) G ( t ) |F s ) = c ( s ) G ( s ),the inequality c ( s ) < E Q BS ( c ( t ) |F s )holds if and only if E Q BS ( c ( t ) G ( t ) |F s ) < E Q BS ( c ( t ) G ( s ) |F s ) , that is, if and only if E Q BS ( c ( t ) G ( t ) A ) < E Q BS ( c ( t ) G ( s ) A )for every A ∈ F s of positive measure. Because G ( t ) ≤ G ( s ) and c ( t ) is strictlypositive, this is so if and only if { c ( t ) G ( t ) A < c ( t ) G ( s ) A } = { G ( t ) A < G ( s ) A } is a set of positive measure for every A ∈ F s of positive measure. Using, theinequality G ( t ) ≤ G ( s ) once again, we can see that this is, in turn, equivalentto E Q BS ( G ( t ) |F s ) < G ( s ) , G ( t ) whose existence isasserted in 2. Indeed, since every left-continuous adapted process is previsible,it follows that G ( t ) is previsible. The uniqueness (up to indistinguishability) ofthe multiplicative decomposition of the submartingale c ( t ) therefore gives thatof the process G ( t ). (cid:4) Definition 7 We call the unique process G ( t ) in Theorem 6 the survival process and Γ( t ) := − log G ( t ) the hazard process .In the next section we use G ( t ) (or, equivalently, Γ( t )) to construct a mea-sure Q extending Q BS from the σ -algebra F T to G T such that e − rt S ( t ) and e − rt D ( t, T ) become {G ( t ) } t ∈ [0 ,T ] -martingales under Q . To achieve this we con-struct Q in such a way that G ( t ) and Γ( t ) will be expressed in terms of Q asin (5). This justifies calling them the survival process and hazard process inDefinition 7. A probability measure Q extending Q BS from F T to G T such that G ( t ) is givenby (5) would need to satisfy Q ( A ∩ { s < τ ≤ t } ) = E Q BS ( A ( G ( s ) − G ( t ))) , (6) Q ( A ∩ { T < τ } ) = E Q BS ( A G ( T )) (7)for each s, t ∈ [0 , T ] such that s < t and each A ∈ F T .With the aim of constructing such a measure Q , we first construct a mea-sure ˜ Q on the space ˜Ω := ˙ R [0 ,T ] × (0 , ∞ ) , where ˙ R := R ∪{∞} is the one-point compactification of R , and then pull it backto Ω by the map ( W, τ ) : Ω → ˜Ω to obtain Q . Here ˙ R [0 ,T ] can be regarded asthe space of paths of the Brownian motion W driving the Black–Scholes model,and (0 , ∞ ) as the space of values of τ .If the pulled-back measure Q is to satisfy (6) and (7), then we need to put˜ Q ( A z,Z × ( s, t ]) := E Q BS ( { W ∈ A z,Z } ( G ( t ) − G ( s ))) , (8)˜ Q ( A z,Z × ( T, ∞ )) := E Q BS ( { W ∈ A z,Z } G ( T )) (9)for each s, t ∈ [0 , T ] such that s < t and each cylindrical set A z,Z in ˙ R [0 ,T ] ofthe form A z,Z := { x ∈ ˙ R [0 ,T ] : ( x ( z ) , . . . , x ( z n )) ∈ Z } for some non-negative integer n , some z = ( z , . . . , z n ) ∈ [0 , T ] n and some Borelset Z ∈ B ( ˙ R n ).Next, ˜ Q can be extended in the standard manner to an additive set functionon the algebra ˜ A consisting of finite unions of disjoint sets of the form A z,Z × ( s, t ]11r A z,Z × ( T, ∞ ), where s, t ∈ [0 , T ] with s < t , and where A z,Z is a cylindricalset in ˙ R [0 ,T ] . The next step is to show that ˜ Q is a measure (that is, it is countablyadditive) on the algebra ˜ A . Lemma 8 ˜ Q is a countably additive set function on ˜ A . Proof. The following argument resembles in some respects the standard proofof countable additivity of Wiener measure on path space.Let K be the class of subsets in ˜Ω of the form K u,U ∪ ( L v,V × ( T, ∞ )), where L u,U := { ( x, t ) ∈ ˙ R [0 ,T ] × (0 , T ] : ( x ( u ) , . . . , x ( u k ) , t ) ∈ U } ,M v,V := { x ∈ ˙ R [0 ,T ] : ( x ( v ) , . . . , x ( v l )) ∈ V } , for some positive integers l, m , some u = ( u , . . . , u l ) ∈ [0 , T ] l , v = ( v , . . . , v m ) ∈ [0 , T ] m and some compact sets U ⊂ R l × (0 , T ], V ⊂ R m .We claim that K is a compact class in ˜Ω (see Definition 1.4.1 in [Bog07]).To prove this, take any sequence of sets K n ∈ K such that (cid:84) ∞ n =1 K n = ∅ .Then K n = L u n ,U n ∪ ( M v n ,V n × ( T, ∞ )) for some positive integers l n , m n , some u n ∈ [0 , T ] l n , v n ∈ [0 , T ] m n and some compact sets U n ⊂ R l n × (0 , T ], V n ⊂ R m n .We can see (c.f. Theorem A5.17 in [Ash72]) that U n , V n are closed subsets in˙ R l n × [0 , T ] and, respectively, ˙ R m n . Hence L u n ,U n , M v n ,V n are closed in theproduct topology in ˙ R [0 ,T ] × [0 , T ] and, respectively, ˙ R [0 ,T ] . It also follows that (cid:84) ∞ n =1 L u n ,U n = ∅ and (cid:84) ∞ n =1 M v n ,V n = ∅ . By the Tikhonov theorem (the productof any family of compact sets is compact in the product topology), ˙ R [0 ,T ] × [0 , T ]and ˙ R [0 ,T ] are compact. Therefore, there is an N such that (cid:84) Nn =1 L u n ,U n = ∅ and (cid:84) Nn =1 M v n ,V n = ∅ , hence (cid:84) Nn =1 K n = ∅ . This proves that K is a compactclass.We also claim that K is a class approximating the additive set function ˜ Q on the algebra ˜ A , that is, for any A ∈ ˜ A and ε > K ε ∈ K and A ε ∈ ˜ A such that A ε ⊂ K ε ⊂ A and ˜ Q ( A \ A ε ) < ε . (This definition of anapproximating class comes from Theorem 1.4.3 in [Bog07].) Take any A ∈ ˜ A and ε > 0. We can write A as A = m (cid:91) i =1 ( A z,Z i × ( s i , t i ]) ∪ ( A z,Z m +1 × ( T, ∞ )) (10)for some non-negative integer m , some s , t , . . . , s m , t m ∈ [0 , T ] such that0 ≤ s < t ≤ · · · ≤ s m < t m ≤ T and some cylindrical sets A z,Z , . . . , A z,Z m and A z,Z m +1 in ˙ R [0 ,T ] . In particular,note that the cylindrical sets can be chosen to share the same tuple z ∈ R n forsome non-negative integer n , with Z , . . . , Z m +1 ∈ B ( ˙ R n ).Let η := ε m +1 . Regularity of the Borel sets Z , . . . , Z m +1 (see Theorem 1.4.8in [Bog07]) implies that there are compact sets F , . . . , F m +1 ⊂ R n such thatfor each i = 1 , . . . , m + 1 we have F i ⊂ Z i and Q BS { W ∈ A z,A i \ F i } < η. G has non-increasing continuous paths, it follows that for each i = 1 , . . . , m we have G ( s i ) − G ( t ) (cid:38) t (cid:38) s i . By monotone convergence,it follows that E Q BS ( G ( s i ) − G ( t )) (cid:38) t (cid:38) s i . Hence there is a v i ∈ ( s i , t i )such that E Q BS ( G ( s i ) − G ( v i )) < η. Next, we take some w i ∈ ( s i , v i ) for each i = 1 , . . . , m and put K ε := m (cid:91) i =1 ( A z,F i × [ w i , t i ]) ∪ ( A z,F m +1 × ( T, ∞ )) ,A ε := m (cid:91) i =1 ( A z,F i × ( v i , t i ]) ∪ ( A z,F m +1 × ( T, ∞ )) . Clearly, A ε ∈ ˜ A , K ε ∈ K and A ε ⊂ K ε ⊂ A . Since A \ A ε = m (cid:91) i =1 ( A z,Z i × ( s i , v i ]) ∪ m (cid:91) i =1 (cid:0) A z,Z i \ F i × ( v i , t i ] (cid:1) ∪ ( A z,Z m +1 \ F m +1 × ( T, ∞ )) , which is a union of disjoint sets, it follows that ˜ Q ( A \ A ε ) is the sum of thefollowing three terms: m (cid:88) i =1 ˜ Q ( A z,Z i × ( s i , v i ]) = m (cid:88) i =1 E Q BS (1 { W ∈ A z,Zi } ( G ( s i ) − G ( v i ))) ≤ m (cid:88) i =1 E Q BS [ G ( s i ) − G ( v i )] < mη, m (cid:88) i =1 ˜ Q (cid:0) A z,Z i \ F i × ( v i , t i ] (cid:1) = m (cid:88) i =1 E Q BS (1 { W ∈ A z,Zi \ Fi } ( G ( v i ) − G ( t i ))) ≤ m (cid:88) i =1 Q BS ( W ∈ A z,Z i \ F i ) ≤ mη, ˜ Q ( A z,Z m +1 \ F m +1 × ( T, ∞ )) = E Q BS (1 { W ∈ A z,Zm +1 \ Fm +1 } G ( T )) ≤ Q BS ( W ∈ A z,Z m +1 \ F m +1 ) < η. Hence ˜ Q ( A \ A ε ) < mη + mη + η = (2 m + 1) η < ε , which shows that K is a classapproximating ˜ Q on the algebra ˜ A . By Theorem 1.4.3 in [Bog07], it followsthat ˜ Q is a countably additive set function on ˜ A , completing the proof. (cid:4) Next, we introduce the algebra A of subsets of Ω of the form { ( W, τ ) ∈ A } such that A ∈ ˜ A , and define a set function Q on A by Q (( W, τ ) ∈ A ) := ˜ Q ( A ) (11)for each A ∈ ˜ A . The following lemma is needed to show that Q is well definedand countably additive on A . 13 emma 9 Let A ∈ ˜ A be such that { ( W, τ ) ∈ A } = ∅ . Then ˜ Q ( A ) = 0 . Proof. We can write A as in (10). Then { ( W, τ ) ∈ A } = m (cid:91) i =1 ( { W ∈ A z,Z i } ∩ { s i < τ ≤ t i } ) ∪ ( { W ∈ A z,Z m +1 } ∩ { T < τ } ) . Since { ( W, τ ) ∈ A } = ∅ , it follows that { W ∈ A z,Z i } ∩ { s i < τ ≤ t i } = ∅ foreach i = 1 , . . . , m and { W ∈ A z,Z m +1 } ∩ { T < τ } = ∅ . Assumption (2) impliesthat P ( W ∈ A z,Z i ) = 0, hence Q BS ( W ∈ A z,Z i ) = 0 for each i = 1 , . . . , m + 1.As a result,˜ Q ( A ) = m (cid:88) i =1 ˜ Q ( A z,Z i × ( s i , t i ]) + ˜ Q ( A u,Zm +1 × ( T, ∞ ))= m (cid:88) i =1 E Q BS (1 { W ∈ A z,Zi } ( G ( s i ) − G ( t i ))) + E Q BS (1 { W ∈ A z,Zm +1 } G ( T )) = 0 , completing the proof. (cid:4) Proposition 10 The set function Q is well defined on A by (11) , that is, if { ( W, τ ) ∈ A } = { ( W, τ ) ∈ B } for some A, B ∈ ˜ A , then ˜ Q ( A ) = ˜ Q ( B ) . Proof. If { ( W, τ ) ∈ A } = { ( W, τ ) ∈ B } , then { ( W, τ ) ∈ A (cid:77) B } = ∅ , where A (cid:77) B = ( A \ B ) ∪ ( B \ A ) denotes the symmetric difference. By Lemma 9, itfollows that ˜ Q ( A (cid:77) B ) = 0, hence ˜ Q ( A ) = ˜ Q ( B ). (cid:4) Proposition 11 Q is a countably additive function on the algebra A . Proof. Let B n ∈ A be a sequence of disjoint sets such that (cid:83) ∞ n =1 B n ∈ A .For each n we can write B n = { ( W, τ ) ∈ A n } for some A n ∈ ˜ A . For any n (cid:54) = m , since { ( W, τ ) ∈ A n ∩ A m } = B n ∩ B m = ∅ , it follows by Lemma 9 that˜ Q ( A n ∩ A m ) = 0. We put D n := A n \ (( A ∩ A n ) ∪ · · · ∪ ( A n − ∩ A n )) , so that ˜ Q ( A n ) = ˜ Q ( D n ) for each n . The sets D n are pairwise disjoint and (cid:83) ∞ n =1 A n = (cid:83) ∞ n =1 D n . By the countable additivity of ˜ Q in ˜ A (see Lemma 8), itfollows that Q (cid:32) ∞ (cid:91) n =1 B n (cid:33) = ˜ Q (cid:32) ∞ (cid:91) n =1 A n (cid:33) = ˜ Q (cid:32) ∞ (cid:91) n =1 D n (cid:33) = ∞ (cid:88) n =0 ˜ Q ( D n ) = ∞ (cid:88) n =0 ˜ Q ( A n ) = ∞ (cid:88) n =0 Q ( B n ) , proving countable additivity of Q on A . (cid:4) Q on the algebra A gives a non-negative measure, denoted by the same symbol Q , on the σ -algebra σ ( A ) generated by A . The final step is to extend Q to G T = σ ( F T ∪ I T ). Weput H T := σ ( W t , t ∈ [0 , T ]) , N T := { A ∈ Σ : A ⊂ B for some B ∈ H T such that P ( B ) = 0 } . Then F T = σ ( H T ∪ N T ) , G T = σ ( H T ∪ N T ∪ I T ) , σ ( A ) = σ ( H T ∪ I T ) . Observe that G T = { A ∈ Σ : A (cid:77) B ∈ N T for some B ∈ σ ( A ) } . Now, for any A ∈ G T , we put Q ( A ) := Q ( B )for any B ∈ σ ( A ) such that A (cid:77) B ∈ N T . This does not depend on the choiceof such B and defines a non-negative measure Q on the σ -algebra G T . It is aprobability measure since Q (Ω) = Q (0 < τ ) = Q BS ( G (0)) = 1 . In the next three propositions we show that the probability measure Q con-structed above has the desired properties, namely: • Q coincides with the Black–Scholes risk neutral measure Q BS on the σ -algebra F T ; • Q satisfies (5); • the discounted stock price and defaultable bond price processes e − rt S ( t )and e − rt D ( t, T ) are ( G t ) t ∈ [0 ,T ] -martingales under Q . Proposition 12 Q = Q BS on F T . Proof. Because the family of sets of the form { W ∈ A z,Z } , where A z,Z is acylindrical set in R [0 ,T ] , is closed under finite intersections and generates the σ -algebra H T , it suffices to show (see Lemma 1.9.4 in [Bog07]) that Q ( W ∈ A z,Z ) = Q BS ( W ∈ A z,Z )for any such set to prove that Q = Q BS on H T . Indeed, this equality holdssince Q ( W ∈ A z,Z ) = Q (( W, τ ) ∈ A z,Z × (0 , ∞ ))= Q (( W, τ ) ∈ A z,Z × (0 , T ]) + Q (( W, τ ) ∈ A z,Z × ( T, ∞ ))= ˜ Q ( A z,Z × (0 , T ]) + ˜ Q ( A z,Z × ( T, ∞ ))= E Q BS ( { W ∈ A z,Z } ( G (0) − G ( T ))) + E Q BS ( { W ∈ A z,Z } G ( T ))= E Q BS ( { W ∈ A z,Z } ) = Q BS ( W ∈ A z,Z ) , G (0) = 1. Augmenting by the null setsfrom N T preserves the equality Q = Q BS , which therefore also holds on F T = σ ( H T ∪ N T ). (cid:4) Proposition 13 For each t ∈ [0 , T ] , G ( t ) = Q ( t < τ |F T ) . Proof. We need to show that, for each A ∈ F T and t ∈ [0 , T ], E Q (1 A G ( t )) = Q ( A ∩ { t < τ } ) . In fact, it suffices to show this equality for any A ∈ H T . Because the family ofsets of the form { W ∈ A z,Z } , where A z,Z is a cylindrical set in R [0 ,T ] , is closedunder finite intersections and generates the σ -algebra H T , it suffices to show(see Lemma 1.9.4 in [Bog07]) that E Q ( { W ∈ A z,Z } G ( t )) = Q ( { W ∈ A z,Z } ∩ { t < τ } )for any such set. Indeed, since { W ∈ A z,Z } G ( t ) is an F T -measurable randomvariable and, by Proposition 12, Q = Q BS on F T , it follows that E Q ( { W ∈ A z,Z } G ( t )) = E Q BS ( { W ∈ A z,Z } G ( t ))= E Q BS ( { W ∈ A z,Z } ( G ( t ) − G ( T ))) + E Q BS ( { W ∈ A z,Z } G ( T ))= ˜ Q ( A z,Z × ( t, T ]) + ˜ Q ( A z,Z × ( T, ∞ ))= ˜ Q ( A z,Z × ( t, ∞ )) = Q ( { ( W, τ ) ∈ A z,Z × ( t, ∞ ) } )= Q ( { W ∈ A z,Z } ∩ { t < τ } ) , as required. (cid:4) Proposition 14 Both the discounted stock price process e − rt S ( t ) and discounteddefaultable bond price process e − rt D ( t, T ) are ( G t ) t ∈ [0 ,T ] -martingales under Q . Proof. Without loss of generality, we can assume that r = 0. Because theprocesses S ( t ) and c ( t ) G ( t ) are ( F t ) t ∈ [0 ,T ] -martingales under Q BS and Q BS = Q on F T , they are also ( F t ) t ∈ [0 ,T ] -martingales under Q . Thus, by Proposition 13,for any s, t ∈ [0 , T ] such that s ≤ t and any A ∈ F t , we have E Q ( A ∩{ s<τ } S ( t )) = E Q ( A { s<τ } S ( t )) = E Q ( A E Q ( { s<τ } |F T ) S ( t ))= E Q ( A G ( s ) S ( t )) = E Q ( A G ( s ) E Q ( S ( T ) |F t ))= E Q ( A G ( s ) S ( T )) = E Q ( A E Q ( { s<τ } |F T ) S ( T ))= E Q ( A ∩{ s<τ } S ( T ))and, since D ( t, T ) = c ( t ) { t<τ } and c ( T ) = 1, we also have E Q ( A ∩{ s<τ } D ( t, T )) = E Q ( A c ( t ) { t<τ } ) = E Q ( A c ( t ) E Q ( { t<τ } |F T ))= E Q ( A c ( t ) G ( t )) = E Q ( A E Q ( c ( T ) G ( T ) |F t ))= E Q ( A c ( T ) G ( T )) = E Q ( A G ( T )) = Q ( A ∩ { T < τ } )= E Q ( A ∩{ s<τ } { T <τ } ) = E Q ( A ∩{ s<τ } D ( T, T )) . A ∩{ s < τ } , where A ∈ F t and s ∈ [0 , t ], is closed underfinite intersections and generates the σ -algebra G t , it follows (see Lemma 1.9.4in [Bog07]) that E Q ( C S ( t )) = E Q ( C S ( T )) and E Q ( C D ( t, T )) = E Q ( C D ( T, T ))for every C ∈ G t . We can conclude that S ( t ) = E Q ( S ( T ) |G t ) and D ( t, T ) = E Q ( D ( T, T ) |G t ) , as required. (cid:4) One simple example is the hazard process model with constant hazard rate λ > 0. In this case the default time τ is exponentially distributed under Q withparameter λ , the survival process is given by G ( t ) = e − λt , and the defaultablebond price is D ( t, T ) = { t<τ } e − ( r + λ )( T − t ) .We modify this simple example by stipulating two constants λ + , λ − > 0, andtaking the hazard rate to be the process λ ( t ) equal to λ + whenever W ( t ) ≥ λ − otherwise. This gives G ( t ) = e − (cid:82) t λ ( s ) ds = e − λ + γ + ( t ) − λ − γ − ( t ) = e − ( λ + − λ − ) γ + ( t ) − λ − t , where γ + ( t ) := (cid:90) t [0 , ∞ ) ( W ( s )) ds, γ − ( t ) := (cid:90) t ( −∞ , ( W ( s )) ds are the sojourn times of W ( t ) above and below 0, which satisfy γ + ( t )+ γ − ( t ) = t .Since W ( t ) is the Brownian motion driving the stock price process S ( t ), thismeans that the hazard rate depends on whether S ( t ) is above or below a certainlevel.The survival process G ( t ) gives rise to a martingale measure Q as in theconstruction in Section 5. The probability distribution of the default time τ under Q can be found by computing the expectation Q ( t < τ ) = Q BS ( G ( t )) = e − λ − t E Q BS ( e − ( λ + − λ − ) γ + ( t ) ) . A formula for the Laplace transform of the probability distribution of the sojourntime γ + ( t ) can be found, for example, in [BorSal02], part II, formula 1.1.4.3. Itgives Q ( t < τ ) = e − ( λ ++ λ − ) t I (cid:18) ( λ + − λ − ) t (cid:19) , where I ( x ) = 1 π (cid:90) π e x cos θ dθ c ( t ) of D ( t, T )is the modified Bessel function of the first kind. Hence the price of the default-able bond at time 0 is D (0 , T ) = e − rT Q ( T < τ ) = e − rT e − ( λ ++ λ − ) T I (cid:18) ( λ + − λ − ) T (cid:19) . It is also interesting to compute the pre-default value c ( t ), hence the default-able bond price D ( t, T ) = { t<τ } c ( t ) for any t ∈ [0 , T ]. We have c ( t ) = e − r ( T − t ) G ( t ) − E Q BS ( G ( T ) |F t )= e − ( r + λ − )( T − t ) E Q BS ( e − ( λ + − λ − )( γ + ( T ) − γ + ( t )) |F t ) . Observe that γ + ( T ) − γ + ( t ) = (cid:90) Tt { W ( s ) ≥ } ds = (cid:90) Tt { W ( s ) − W ( t ) ≥− W ( t ) } ds can be regarded as the sojourn time above − W ( t ) of the Brownian motion W ( s ) − W ( t ) starting at time t . Hence formula 1.1.4.3 in part II of [BorSal02] forthe Laplace transform of the probability distribution of the sojourn time abovea given level makes it possible to compute the above conditional expectation,and gives c ( t ) = e − ( r + λ + )( T − t ) erf (cid:32) | W ( t ) | (cid:112) T − t ) (cid:33) + 1 π (cid:90) Tt e ( λ + − λ − )( T − s ) e − W ( t )22( s − t ) (cid:112) ( T − s ) ( s − t ) ds W ( t ) ≥ 0, and c ( t ) = e − ( r + λ − )( T − t ) erf (cid:32) | W ( t ) | (cid:112) T − t ) (cid:33) + 1 π (cid:90) Tt e ( λ − − λ + )( T − s ) e − W ( t )22( s − t ) (cid:112) ( T − s ) ( s − t ) ds if W ( t ) ≤ 0, where erf( x ) = 1 √ π (cid:90) x − x e − t dt is the error function.Several sample paths of the pre-default value c ( t ) of the defaultable bond D ( t, T ) are shown in Figure 1 for T = 2, r = 0 . 1, and λ + = 0 . λ − = 0 . 1. Thebroken lines marking the envelope of the set of sample paths are the graphs of e − ( r + λ + )( T − t ) and e − ( r + λ − )( T − t ) . In particular, we arrive at the price D (0 , T ) = c (0) = 0 . D ( t, T ) Here we show that expression (1) for the defaultable bond can be obtained fromsome weaker assumptions about D ( t, T ) and the lack of arbitrage in a class ofsimple strategies in the BD section of the market only. This is similar to thehazard function model considered in [CapZas14]. Definition 15 By a BD -simple self-financing strategy we understand an R -valued ( G t ) t ∈ [0 ,T ] -adapted process ψ = (cid:0) ψ B , ψ D (cid:1) representing positions in B and D such that there are sequences of times 0 = s < s < · · · < s N = T andrandom variables x , . . . , x N and y , . . . , y N with the following properties:1. x n and y n are G s n − -measurable and ψ B ( t ) = x n , ψ D ( t ) = y n for each n = 1 , . . . , N and t ∈ ( s n − , s n ];2. The value process V ψ ( t ) := ψ B ( t ) B ( t, T ) + ψ D ( t ) D ( t, T )satisfies the following self-financing condition for each n = 0 , . . . , N − V ψ ( s n ) = lim t (cid:38) s n V ψ ( t ) . Definition 16 We say that the no- BD - simple-arbitrage (NBDSA) principle holds if there is no BD -simple self-financing strategy ψ = (cid:0) ψ B , ψ D (cid:1) such that V ψ (0) = 0, V ψ ( T ) ≥ 0, and V ψ ( T ) > P .19 roposition 17 Suppose that D ( t, T ) is a ( G t ) t ∈ [0 ,T ] -adapted process satisfying D ( T, T ) = { T <τ } , with paths which are continuous on [0 , τ ) ∩ [0 , T ] and right-continuous elsewhere. If the NBDSA principle holds, then there is an ( F t ) t ∈ [0 ,T ] -adapted process c ( t ) defined for all t ∈ [0 , T ] , with continuous paths, such that c ( t ) ∈ (0 , for all t ∈ [0 , T ) , c ( T ) = 1 and D ( t, T ) = c ( t ) { t<τ } (12) for all t ∈ [0 , T ] . Proof. Because we can switch to working with discounted values, it is enoughto consider the case when r = 0, so that B ( t, T ) = 1 for all t ∈ [0 , T ].For t = T , we have D ( T, T ) = c ( T )1 { T <τ } with c ( T ) = 1. For any t ∈ [0 , T ),since D ( t, T ) is a G t -measurable random variable, it follows by well-known prop-erties of the enlarged filtration (for example, Proposition 5.28 in [CapZas16])that there exists an F t -measurable random variable c ( t ) such that D ( t, T ) { t<τ } = c ( t ) { t<τ } . (13)Let A t := (cid:8) D ( t, T ) { τ ≤ t } > (cid:9) , A (cid:48) t := (cid:8) D ( t, T ) { τ ≤ t } < (cid:9) . We consider the BD -simple self-financing strategy ψ B ( u ) := ψ D ( u ) := 0 for u ∈ [0 , t ] ,ψ B ( u ) := D ( t, T ) (cid:0) A t − A (cid:48) t (cid:1) ,ψ D ( u ) := − A t + A (cid:48) t for u ∈ ( t, T ] . To verify that this is indeed a BD -simple self-financing strategy, when t ∈ (0 , T ),we take N := 2, s := 0, s := t , s := T and x := 0, y := 0, x := D ( t, T ) (cid:0) A t − A (cid:48) t (cid:1) , y := − A t + A (cid:48) t in Definition 15 to obtain this strategy.When t = 0, we take N := 1, s := 0, s := T and x := D (0 , T ) (cid:0) A − A (cid:48) (cid:1) , y := − A + A (cid:48) .The initial value of this strategy is V ψ (0) = 0, and its final value V ψ ( T ) = D ( t, T ) (cid:0) A t − A (cid:48) t (cid:1) is strictly positive on A t ∪ A (cid:48) t and 0 otherwise. This wouldviolate the NBDSA principle unless P ( A t ∪ A (cid:48) t ) = 0. Hence we can concludethat D ( t, T ) { τ ≤ t } = 0. Together with (13), this gives (12).We claim that c ( t ) ∈ (0 , 1) for each t ∈ [0 , T ). We put B t := { c ( t ) ≤ } , B (cid:48) t := { c ( t ) ≥ } and observe that ϕ B ( u ) := ϕ D ( u ) := 0 for u ∈ [0 , t ] ,ϕ B ( u ) := c ( t ) (cid:0) B (cid:48) t − B t (cid:1) { t<τ } ,ϕ D ( u ) := − (cid:0) B (cid:48) t − B t (cid:1) { t<τ } for u ∈ ( t, T ]20s a BD -simple self-financing strategy with initial value V ϕ (0) = 0 and finalvalue V ϕ ( T ) = c ( t ) (cid:0) B (cid:48) t − B t (cid:1) { t<τ } − (cid:0) B (cid:48) t − B t (cid:1) { T <τ } = c ( t ) (cid:0) B (cid:48) t − B t (cid:1) { t<τ ≤ T } + ( c ( t ) − (cid:0) B (cid:48) t − B t (cid:1) { T <τ } ≥ . Given that the NBDSA principle holds, we must have P ( B t ∩ { T < τ } ) = 0because ( c ( t ) − (cid:0) B (cid:48) t − B t (cid:1) > B t . Moreover, we must have P ( B (cid:48) t ∩{ t < τ ≤ T } ) = 0 since c ( t ) (cid:0) B (cid:48) t − B t (cid:1) > B (cid:48) t . By assumption (2), since B t , B (cid:48) t ∈ F t ⊂ F T , we obtain P ( B t ) = P ( B (cid:48) t ) = 0, which proves the claim.Finally, we construct a continuous modification ˆ c of c , with ˆ c ( t ) ∈ (0 , 1) forall t ∈ [0 , T ) and ˆ c ( T ) = 1, adapted to the filtration ( F t ) t ∈ [0 ,T ] and such that D ( t, T ) = ˆ c ( t ) { τ