The general case

We start the treatment of the non-predictable case with two examples that illustrate why it is natural to consider dominating local martingale measures forS^τ−rather than forS.

Example 1.4.14. If S is optional and if Q is a dominating local martingale measure for S rather than for S^τ−, then S does not need to satisfy (NA1): Let τ be exponentially distributed with parameter 1 under Q. Define S_t = e^t1_{t<τ} for t ∈ [0,1]. Since time is finite, S is a uniformly integrable martingale. Therefore, dP = S1dQ is absolutely continuous with respect toQ. But under P we have St=e^t for allt∈[0,1]. Clearly S does not satisfy (NA1) underP, even thoughQ is a dominating martingale measure for S. Note that S^τ− is not a local martingale underQbecause S_t^τ−=e^t for allt∈[0,1].

Recall that a stopping time τ is called foretellable under a probability measure P if there exists an increasing sequence (τn) of stopping times, such that P(τn < τ) = 1 for every n, and such that P(sup_nτ_n = τ) = 1. In this case (τ_n) is called an announcing sequenceforτ. Every predictable time is foretellable under any probability measure, see Theorem I.2.15 and Remark I.2.16 of [JS03].

Example 1.4.15. Let S be a semimartingale under P and let Q ≫ P be a dominat-ing measure with Kunita-Yoeurp decomposition (Z, τ) with respect to P. Assume that

τ is not foretellable under Q. Then there exists an adapted process S which is P– indistinguishable fromS, such thatS^is not aQ–local martingale: Letx∈R^dand define S_t^x =S_t1_{t<τ}+x1_{t≥τ}, which is P–indistinguishable from S since P(τ = ∞) = 1. If S^x is aQ–local martingale, then τ_n^x = inf{t≥0 :|S_t^x| ≥n}, n∈N, defines a localizing sequence. In particular, (τ_n^x) converges Q–almost surely to infinity asntends to ∞, and thusQ(limn→∞τ_n^x≥τ) = 1. Since τ is not foretellable underQ, there must existn∈N for which Q(τ_n^x≥τ)>0. Moreover, we have

E_Q(S₀) =E_Q(S_τ^xx

n) =E_Q(S_τn^x1_{τx

n<τ}) +xQ(τ_n^x ≥τ).

Sinceτ_n^x=τ_n^y for all |x|< n,|y|< n, we obtain a contradiction by lettingxvary through the ball of radius n−1.

These two examples show that givenQ≫P, it is important to choose a good version of S if we want to obtain a Q–local martingale. All the results obtained so far indicate that this good version should beS^τ−. Maybe somewhat surprisingly, this is not true in general, as we demonstrate in the following example.

Example1.4.16. Let (Lt)_t∈[0,1]be a Lévy process underQ, with jump measureν =δ₁+δ−1

and driftb∈R. That is,L_t=N_t¹−N_t²+bt, whereN¹ and N² are independent Poisson processes. Let a > |b| and let ρ be an exponential random variable with parameter a, such that ρ is independent from L. Define τ = ρ if ρ ≤ 1, and τ = ∞ otherwise.

Then (e^at1_{t<τ})_t∈[0,1] is a uniformly integrable martingale, and therefore it defines a probability measure dP =e^a1_{1<τ}dQ. Since τ and L are independent, L has the same distribution underP as under Q. The Kunita-Yoeurp decomposition of Q with respect toP is given by ((e^−at)_t∈[0,1], τ).

We claim thatZ =e^−a·is a supermartingale density forL. Let (π_tW_t−^π) be a strategy for L, where W^π is the wealth process obtained by investing in this strategy. Such a strategy is 1–admissible if and only if |π_t| ≤1 for all t ∈ [0,1]. Moreover, we get from (1.22) that

d(ZW^π)_t∼ −W_t−^π Zt−adt+Zt−πtW_t−^π bdt=W_t−^π Zt−(πtb−a)dt.

Since W^πZ ≥ 0 and since πtb−a < 0 (recall that a > |b|), the drift rate is negative.

Therefore, ZW^π is a local supermartingale, and since it is a nonnegative process, it is a supermartingale.

Nowτ is independent from Lunder Q, andL has no fixed jump times. Hence Q(∆Lτ ̸= 0, τ ≤1) =



[0,1]

Q(∆Lt̸= 0)(Q◦τ⁻¹)(dt) = 0, which implies that L^τ−=L^τ, and this is clearly noQ–local martingale.

Remark 1.4.17. In the preceding example it is possible to show that the modified process Lt=L^τ−_t − b

a1_{t≥τ} (1.24)

is a Q–martingale. More generally, we expect that given a semimartingale S, a super-martingale densityZ for S, and a measureQ ≫ P with Kunita-Yoeurp decomposition (Z, τ) with respect toP, there should always exist a versionSthat isP–indistinguishable from S, such that S^ is a Q–local martingale. But as (1.24) shows, we will need to take different S for different supermartingale densities. Therefore, this seems somewhat un-natural, and we will not pursue it further.

Note that all three examples had one thing in common: τ was not foretellable under Q. It turns out that things get much simpler ifτ is foretellable under Q. But if (τ_n)_n∈N is an announcing sequence forτ, then we obtain from (1.11) that

1 =Q(τ_n< τ) =E_P(Z_τn1_{τn<∞}) for all n∈N, and 0 =Q

 sup

n

τ_n< τ



=E_P(Z_supnτn1_{sup

nτn<∞}).

SinceZ is strictly positive, we conclude that (τ_n)n∈Nis a localizing sequence forZ under P, i.e. Z is aP–local martingale.

Therefore, we should look for supermartingale densities that are local martingales.

We call such supermartingale densities local martingale densities. If (S_t)_t∈[0,T] is one dimensional with finite terminal timeT <∞, it is shown by Kardaras [Kar12], Theorem 1.1, that local martingale densities exist if and only if (NA1) is satisfied. The proof is in the spirit of the article [KK07]. Takaoka [Tak13] solves the multidimensional case with finite terminal time. More precisely, it is easily deduced from Remark 7 of [Tak13] that for a locally bounded d–dimensional semimartingale (S_t)_t∈[0,T], (NA1) is satisfied if and only if there exists a local martingale density. Takoaka’s proof is based on the insight of Delbaen and Schachermayer [DS95c], that a change of numéraire can induce the (NA) property, even if previously there were arbitrage opportunities in the market. [Tak13]

continues to show that a clever choice of numéraire preserves the (NA1) property, so that then the condition (NA) + (NA1) = (NFLVR) is satisfied, which permits to apply the Fundamental Theorem of Asset Pricing [DS94]. See also the recent preprint Song [Son13], where an alternative proof of Takaoka’s result is given that does not use the Fundamental Theorem of Asset Pricing. Roughly speaking, this is achieved by combining the philosophies behind [KK07] and [Tak13].

Of course [Kar12], [Tak13], and [Son13] all work with complete filtrations, but given a local martingale densityZthat is (F_t^P)–adapted, there exists an indistinguishable process Z that is (F_t)–adapted, see Lemma A.4.

Lemma 1.4.18. Let (St)_t∈[0,T] be a locally bounded semimartingale on a finite time horizonT <∞, and letZ be a local martingale density forS. Letτ be a stopping time and Qbe a probability measure, such that(Z/E_P(Z₀), τ) is the Kunita-Yoeurp decomposition of Q with respect toP. Then S^τ− is a Q–local martingale.

Conversely, if Q≫P has Kunita-Yoeurp decomposition (Z, τ) with respect to P, and if S^τ− is a Q–local martingale, then Z is a supermartingale density for S.

Proof. The proof is very similar to the one of Corollary 1.4.10. Recall from Remark 1.4.11 that we only used the predictability ofS once in the proof of Corollary 1.4.10, to

obtain

E_Q((H·S^τ−)_σn)≤0 (1.25)

for all strategiesH that are 1–admissible for (S^τ−)^σn underQ. Here (σ_n) was a localizing sequence of finite stopping times forM underP, whereZ =Z₀+M−D. Therefore, it suffices to show that (1.25) always holds if Z has the decompositionZ =Z0+M, i.e. if D= 0, even if S is not predictable.

So let (σn) be a localizing sequence of finite stopping times for the local martingale Z under P, and let H be a strategy that is 1–admissible for (S^τ−)^σn under Q (and then also for S^σn underP). We apply Corollary 1.4.4 withD= 0, and obtain

E_Q(1 + (H·S^τ−)_σn) =E_P((1 + (H·S)_σn)Z_σn)≤1,

where the last step follows becauseZ is a supermartingale density. From here on we can just copy the proof of Corollary 1.4.10.

We obtain our main result, a weak fundamental theorem of asset pricing:

Corollary 1.4.19(“Correct formulation of Theorem 1.1.6”). Let(F_t)_t∈[0,T]be the right-continuous modification of a standard system. Let S = (St)_t∈[0,T] be a locally bounded, right-continuous stochastic process. Then S satisfies (NA1_s) if and and only there exists an enlarged probability space (Ω,F,(F_t)_t∈[0,T], P), and a dominating measure Q ≫ P with Kunita-Yoeurp decomposition (Z, τ) with respect to P, such that S^τ− is a Q–local martingale.

Remark 1.4.20. There is another subset of supermartingale densities of which one might expect that they correspond to local martingale measures forS^τ−: the maximal elements among the supermartingale densities. A supermartingale densityZis calledmaximalif it is indistinguishable from any supermartingale densityY that satisfiesYt≥Ztfor allt≥0.

IfSis not continuous, then some maximal supermartingale densities are supermartingales and not local martingales, see Example 5.1’ of Kramkov and Schachermayer [KS99].

But such Z will usually not correspond to local martingale measures for S. Assume for example that we are in the situation described in Theorem 2.2 of [KS99], i.e. we have a dual optimizerZ and a primal optimizerH for a certain utility maximization problem.

Then point iii) of this Theorem 2.2 states that (1 + (H·S))Z is a uniformly integrable martingale. If we assume now thatZ is not a local martingale, as is the case in Example 5.1’ of [KS99], and if (τ_n) is a localizing sequence of finite stopping times for the local martingale partM of Z =Z₀+M −D, then we obtain from Corollary 1.4.4 that

E_Q(1 + (H·S^τ−)_τn) =E_P((1 + (H·S)_τn)Z_τn) +E_P

 _τn

0

(1 + (H·S)s−)dD_s



= 1 +E_P

 τn

0

(1 + (H·S)s−)dD_s



, (1.26)

where we used that (1+(H·S))Zis a uniformly integrable martingale. SinceHis optimal, the wealth process (1 + (H·S)s−)_s≥0 will be strictly positive with positive probability.

Since also dD̸= 0 with positive probability, the expectation in (1.26) is strictly positive for sufficiently large n, and therefore (H ·S^τ−)^τn cannot be a Q–supermartingale, i.e.

S^τ− cannot be aQ–local martingale.