General case: continuous local martingales

Let Ω = C_abs := C_abs(R+,[0,∞]) be the space of [0,∞]–valued functions ω that are absorbed in 0 and ∞, and that are continuous on [0, τ_∞(ω)), where τ∞(ω) denotes the

first hitting time of∞byω, to be specified below. The reason for considering this space is that it allows for the application of Parthasarathy’s extension theorem, which we will need below. Let M be the coordinate process, that is, M_t(ω) =ω(t). Define, for the sake of notational simplicity, M∞:=^lim sup_t→∞Mtlim inft→∞Mt(with ∞ ·0 := 1).¹ Denote the canonical filtration by (F_t)_t≥0 withF_t:=σ(Ms:s≤t), and writeF :=^_t≥0F_t. For all a∈[0,∞], defineτ_a as the first hitting time ofa, to wit,

τ_a:= inf{t∈[0,∞] :M_t=a} (2.1) with inf∅ :=T, representing a time “beyond infinity.” The introduction of T allows for a unified approach to treat examples like geometric Brownian motion. We shall extend the natural ordering to [0,∞]∪ {T} byt <T for allt∈[0,∞]. For all stopping times τ, define theσ–algebrasF_τ as

F_τ :={A∈ F :A∩ {τ ≤t} ∈ F_t ∀t∈[0,∞)}

=σ(M_s^τ :s <∞) =σ(M_s^τ∧τ0 :s <∞),

where M^τ ≡M^τ∧τ0 is the process M stopped at the stopping time τ. For the equality between the σ–algebras see Stroock and Varadhan [SV06], Lemma 1.3.3. Let P be a probability measure on (Ω,F), such that M is a nonnegative local martingale with P(M₀= 1) = 1.

2.2.1. Upward conditioning

In this section, we study the law of the local martingaleM conditioned never to hit zero.

This event can be expressed as {τ₀ =T}= ^

a∈[0,∞)

{τ_a≤τ₀} ⊃ ^

a∈(0,∞]

{τ_a∧τ₀=T}. (2.2) The core of this chapter is the following simple observation:

Lemma 2.2.1 (Upward conditioning). If P(τ_a∧τ₀ <T) = 1 for some a∈ (1,∞), we have that

dP(·|τ_a≤τ₀) =M_τadP.

Proof. Note thatM^τa is bounded and thus a uniformly integrable martingale. In partic-ular,

1 =E_P(M_∞^τa) =aP(τ_a≤τ₀) + 0,

1Note that this definition differs from the convention in the remainder of this thesis. The definition of M∞is not further relevant asM converges (or diverges to infinity) almost surely under all measures that we shall consider. We chose this definition ofM∞ since it commutes with taking the reciprocal 1/M∞.

which implies that, for allA∈ F,

P(A|τ_a≤τ₀) = P(A∩ {τ_a≤τ₀})

P(τ_a≤τ₀) = P(A∩ {τ_a≤τ₀})

1 a

=E_P(M_∞^τa1_A), yielding the statement.

Three different probability measures

Consider three possible probability measures:

1. The local martingale M introduces an h–transform Q of P. This is the unique probability measureQ on (Ω,F) that satisfies dQ|_Fτ =M_τdP|_Fτ for all stopping timesτ for whichM^τ is a uniformly integrable martingale. The probability measure Q is called the Föllmer measure of M, see Föllmer [Föl72] and Meyer [Mey72].² Note that the construction of this measure does not require the density processM to be the canonical process on Ω - the extension only relies on the topological structure of Ω =C_abs. This will be important later, when we consider diffusions. We remark that, in the case ofM being aP–martingale, we could also use a standard extension theorem, such as Theorem 1.3.5 in Stroock and Varadhan [SV06].

2. If P(τ₀ = T) = 0, Lemma 2.2.1 in conjunction with (2.2) directly yields the con-sistency of the family of probability measures {P(·|τ_a ≤ τ₀)}_a>1 on the filtration (F_τa)_a>1. By Föllmer’s construction again, there exists a unique probability mea-sureQ on (Ω,F), such that Q| _Fτa =P(·|τ_a≤τ₀)|_Fτa.

3. If P(τ0 = T) >0, we can define the probability measure Q(·) := P(·|τ₀ =T) via the Radon-Nikodym derivative 1_{τ0=T}/P(τ₀ =T).

Since in the case P(τ₀ = T) = 0, we have {τ_a ≤ τ₀} =_P−a.s. {τ_a < τ₀} for all a∈(0,∞], the measureQis also calledupward conditionedmeasure since it is constructed by iteratively conditioning the processM to hit any level abefore hitting 0.

Relationship of probability measures

We are now ready to relate the three probability measures constructed above:

Theorem 2.2.2 (Identity of measures). Set p := P(τ0 =T) = P(M∞ >0). If p = 0, thenQ=Q. If p >0, then Q=Q if and only if M is a uniformly integrable martingale withP(M∞∈ {0,1/p}) = 1.

Proof. First, consider the casep= 0. Both Qand Q satisfy, for all a >1, dQ| _Fτa =M_τadP|_Fτa = dQ|_Fτa.

2See also Delbaen and Schachermayer [DS95a] for a discussion of this measure, Pal and Protter [PP10]

for the extension to infinite time horizons and Carr, Fisher, and Ruf [CFR12] for allowing nonnegative local martingales.

Thus,Qand Q agree on ^_a>1F_τa =^_a>1σ(M_t^τa :t≥0) =F.

Next, consider the casep >0. Then,Q=Qand dQ/dP |_Ft ≤1/pimply thatM_t≤1/p, yielding thatM is a uniformly integrable martingale withM∞= dQ/dP ∈ {0,1/p}. For the reverse direction, observe thatM∞ = 1_{τ0=T}/p. This observation together with its uniform integrability completes the proof.

This theorem implies, in particular, that in finite time the three dimensional Bessel process cannot be obtained by conditioning a Brownian motion not to hit zero. However, over finite time horizons, a Bessel-process can be constructed via theh–transformMTdP, whenM isP–Brownian motion started in 1 and stopped in 0. Over infinite time horizons, one has two choices; the first one is using an extension theorem for theh–transforms, the second one is conditioningM not to hit 0 by approximating this null set by the sequence of events that M hits anya >0 before it hits 0.

Remark 2.2.3 (Conditioning on null sets). We remark that the interpretation of the measureQ^ asP conditioned on a null set requires specifying an approximating sequence of that null set. In Appendix C we illustrate this subtle but important point.

Remark 2.2.4 (The trans-infinite timeT). The introduction ofTin this subsection allows us to introduce the upward-conditioned measure Q and to show its equivalence to the h–transform Q if M converges to zero but not necessarily hits zero in finite time, such asP–geometric Brownian motion. If one is only interested in processes as, say, stopped Brownian motion, then one could formulate all results in this subsection in the standard way when inf∅ := ∞ in (2.1). One would then need to exchange T by ∞ throughout this subsection; in particular, one would have to assume in Lemma 2.2.1 thatP(τa∧τ₀<

∞) = 1 and replace the conditionP(τ₀=T) = 0 byP(τ₀ =∞) = 0 for the construction of the upward-conditioned measureQ.

2.2.2. Downward conditioning

In this subsection, we consider the converse case of conditioningM downward instead of upward. Towards this end, we first provide a well-known result; see for example [CFR12].

For the sake of completeness, we provide a proof.

Lemma 2.2.5 (Local martingale property of 1/M). Under the h–transformed measure Q, the process1/M is a nonnegative local martingale and Q(τ∞=T) =E_P(M∞).

Now we consider the two events {τ_1/n ≤ t} and {τ_1/n > t} separately and used theP– martingale property ofM^τm after conditioning on F_t and F_τ1/n, respectively (note that A∩ {τ_m> t} ∩ {τ_1/n > t} ∈ F_τ1/n), to obtain

Plugging this back into (2.3), we have E_Q

The local martingale property of 1/M then follows from Q^lim

Therefore, 1/M convergesQ–almost surely to some random variable 1/M∞. We observe that

where we use that M converges P–almost surely, since it is a nonnegative supermartin-gale.

The last lemma directly implies the following observation.

Corollary 2.2.6(Mutual singularity). We have P(M∞= 0) = 1if and only ifQ(M∞=

∞) = 1.

This observation is consistent with our understanding that either condition implies that the two measures are supported on two disjoint sets. Corollary 2.2.6 is also consistent with Theorem 2.2.2, which yields that P(M∞ = 0) = 1 implies the identity Q = Q, whereQ denotes the upward conditioned measure.

Lemma 2.2.5 indicates that we can conditionM downward underQ, corresponding to conditioning 1/M upward. The proof of the next result is exactly along the lines of the arguments in Subsection 2.2.1; however, now with the Q–local martingale 1/M taking the place of the P–local martingale M.

Theorem 2.2.7 (Downward conditioning). If p of Theorem 2.2.2 satisfiesp= 0, then dQ(·|τ_1/a≤τ∞) = 1

M_τ1/adQ

for all a > 1. In particular, there exists a unique probability measure P, such that P|_Fτ

1/a =Q(·|τ_1/a<T); in fact, P =P.