Stability under pasting

Let us recall definitions 4.1 and 4.2 on the pasting operation and stability.

Let τ ∈ T be a stopping time and P₁ and P₂ be probability measures equivalent to R. The probability measure defined through

P₃(A) := E_P1[P₂[A | F_τ]], A ∈ F_T is called the pasting of P₁ and P₂ in τ .

A family of probability measures P defined in the probability space (Ω, F , R) is called stable under pasting or simply stable if every P ∈ P is equivalent to R, and if for any P₁ and P₂ in P and any stopping time τ ∈ T , the pasting of P1 and P2 in τ is an element of P.

Trivial examples of stable families of probability measures are {R} and M^e(R) := {P a probability measure | P ∼ R}. In example 4.14 we are going to see a stable family defined in terms of the Girsanov transformation.

In proposition 4.12 we show that the family of equivalent local martingale measures is stable.

The definition of stability under pasting for families of probability measures deserves some comments. First notice that the definition of stability is only formulated for families whose elements are equivalent to the reference prob-ability measure R. Thus, whenever a stable family of probprob-ability measures is given, we implicitely assume that its elements are equivalent to R. In section 6.5 in Föllmer and Schied[27], stability of the family of equivalent martingale measures plays a key role for the analysis of the upper and lower prices π_sup(·) and π_inf(·) of an American option H in discrete time. Another important application of the stability concept appears in the problem of rep-resenting dynamically consistent risk measures, see Föllmer and Penner[26]

for details and references.

Let us now collect some simple properties of the pasting operation.

Lemma 4.8 Let P₁ and P₂ be two equivalent probability measures and let {Zt}0≤t≤T denote a càdlàg version of the density process of P2 with respect to P₁. Let P₃ be the pasting of P₁ and P₂ in σ ∈ T . Then P₃ is equivalent to P₁ and its density is given by

dP₃ dP₁ = Z_T

Z_σ. Moreover, P3 = P₁ in Fσ.

Proof. The proof is similar to lemma 6.42 in [27].

In lemma 4.10 we make use of the following result.

Lemma 4.9 Let P1 and P2 be two equivalent probability measures and let {Z_t}_0≤t≤T denote a càdlàg version of the density process of P₂ with respect to P₁.

Let τ, σ ∈ T be two stopping times with τ ≥ σ. Let Y ≥ 0 be a Fτ -measurable random variable, integrable with respect to P₂. Then

E_P2[Y | F_σ] = 1

Z_σE_P1[Y Z_τ | F_σ].

Proof. See e.g., lemma 3.5.3 in Karatzas and Shreve[36].

Lemma 4.10 Let P1 and P2 be two equivalent probability measures and let P₃ be the pasting of P₁ and P₂ in σ ∈ T . Then the density process of P₃ with respect to R is given by

Z_t³ =







Z_t¹ if t ≤ σ Z_σ^1Z_Z^2t2

σ if t > σ ,

where Z_tⁱ denotes a càdlàg version of the density process of P_i with respect to R, for i = 1, 2.

Proof. Due to lemmas 4.9 and 4.8, the following identity results dP³

dR = Z_T²Z_σ¹ Z_σ².

Now we consider separately the events {t ≤ σ} and {t > σ}. In the event {t ≤ σ} we get

Z_t³ = E_R

"

Z_T²

Z_σ²Z_σ¹ | F_t

#

= E_R[Z_σ¹ | F_t] = Z_t¹. In the event {t > σ}

Z_t³ = Z_σ¹

Z_σ²E_R[Z_T² | F_t] = Z_σ¹ Z_σ²Z_t².

The next lemma is a key result to compute conditional expectations with respect to the pasting of two probability measures.

Lemma 4.11 Let P3 be the pasting of P1 and P2 in σ. Let Y be a positive random variable F_T-measurable and P_i-integrable for i = 1, 2, 3. Then, for any stopping time τ ∈ T we have

E_P3[Y | F_τ] = E_P1[E_P2[Y | F_σ∨τ] | F_τ].

Proof. This is the continuous-time version of lemma 6.43 in [27], and we follow their proof. Let P₃ be the pasting of P₁ and P₂ in σ. The lemma is proved if we show that for any A ∈ F_τ, the following formula holds

E_P3[1_AY ] = E_P3[1_AE_P1[E_P2[Y | F_σ∨τ] | F_τ]].

Lemma 4.8 allows us to write and from lemma 4.9 we deduce that

E_P1[E_P2[Y | F_σ∨τ] | F_τ] = E_P1

 ZT

Z_σ∨τY | F_τ



.

If we put together the right-hand terms, then we see that it is enough to verify In a similar way we get the equality

E_P1 Combining these two equalities for B and B^c we get (4.3).

In the next proposition we illustrate how to use lemma 4.11 to show the well-known fact that the family of local martingale measures M is stable under pasting.

Proposition 4.12 The family of equivalent local martingale measures M for a price process X, specified as in section 1.1, is stable under pasting.

Proof. Delbaen[7] proved that M is m-stable, a concept equivalent to stabil-ity under pasting. His setup is in continuous time and infinite horizon for a locally bounded price process; see his proposition 9.1. Föllmer and Schied[27]

proved the proposition in discrete time and finite horizon; see their proposi-tion 6.45. Notice that in [27] they denoted the family of martingale measures by P.

We now adapt the argument in the proof of proposition 6.45 of [27] to continuous time and finite horizon. Let P₁, P₂ ∈ M and σ ∈ T . Let P₃ be the pasting of P₁ and P₂ in σ. Let {θ^k_i}^∞_i=1be a localizing sequence of X with respect to P_k, for k = 1, 2. This means that (in our finite horizon setting)

1. θ^k_i is a stopping time in T .

2. The sequence converges R-a.s. to the horizon T : limi→∞θ_i^k = T . 3. The stopped process {X_t∧θ^k

i}_0≤t≤T is a P_k-martingale, for k = 1, 2.

We define θ³_i := θ¹_i ∧ θ²_i and show that {θ_i³}^∞_i=1 is a localizing sequence of X with respect to P₃. The first two properties of a localizing sequence are obvious for {θ³_i}_0≤t≤T. To prove the last property, take s, t ∈ [0, T ] with where in the first equality we have applied lemma 4.11. In the second equality we have applied Doob’s optional sampling theorem for martingales. We are allowed to do so, since the stopping time s ∧ θ³_i is bounded and θ_i³ ≤ θ²_i. We apply once more Doob’s optional sampling theorem to conclude that

EP1[X_(s∧θ³

In example 4.14 below, we construct a stable family of probability mea-sures. It is a special case of theorem 1.3 in Delbaen[7]. We will need the stochastic exponential of a continuous martingale.

Definition 4.13 Let M := {M_t}_0≤t≤T be a continuous local martingale with M₀ = 0. The stochastic exponential of M denoted {Et(M )}_0≤t≤T is defined by

where {hM i_t}_0≤t≤T is the quadratic variation process of the local martingale M .

Example 4.14 Let {Wt}_0≤t≤T be a standard Brownian motion defined in the probability space (Ω, F , F = {Ft}_{t∈[0,T ]}, R) where F is the augmented Brown-ian filtration. Let furthermore {ξ_t⁰}_0≤t≤T be a predictable process with

E_R

Notice that this inequality implies that the stochastic integral ξ⁰· dW_t:=

Z t 0

ξ_s⁰dW_s,

is well defined and is a square integrable martingale. We furthermore require that

2. Let P be the family of probability measures obtained from the family of densities with respect to R given by:

dens(P) := {E_T(ξ · W ) | {ξ_t}_0≤t≤T is a predictable process satisfying (4.6)} . Then the probability measures in P are equivalent to R and P is a convex stable family.

Proof. Note that (4.5) is the Novikov criterion for Girsanov transformation theorem; see for example corollary 3.5.13 and theorem 3.5.1 in Karatzas and Shreve[36]. The first assertion in the proposition follows from (4.4) and (4.6).

Now we verify the second assertion.

1. Let ξ be a predictable process satisfying (4.6). We show that

ER[ET(ξ · W )] = 1 (4.7) R(E_T(ξ · W ) > 0) = 1. (4.8) From the first step we know that ξ · W is a uniformly integrable mar-tingale. From (4.6) and (4.5) we deduce that

E_R

Hence, by Novikov’s criterion for Girsanov transformation theorem, the process {E_t(ξ ·W )}_0≤t≤T is a uniformly integrable R-martingale. In par-ticular this implies (4.7).

In order to prove (4.8), we first apply Itô isometry:

2. In order to prove that P is convex, it is enough to show that dens(P) is convex. Take two elements in dens(P), Z_T¹ = E_T(ξ¹· W ) and Z_T² = W ), and thus Z_T³ ∈ dens(P). This proves the required convexity.

3. We now show that P is stable under pasting. Let P_i ∈ P for i = 1, 2, and let Z_tⁱ = E_t(ξⁱ· W ) be the density process of P_i with respect to R.

Let σ ∈ T be a stopping time and P₃ be the pasting of P₁ and P₂ in σ.

We must show that P₃ ∈ P. The process

ξ_t³ := ξ¹_t1{t≤σ}+ ξ_t²1{t>σ},

is predictable (since F is the augmented Brownian filtration) and satis-fies (4.6). Let {Z_t³}_0≤t≤T be a càdlàg version of the density of P₃ with respect to R, lemma 4.10 implies that Z_T³ = E_T(ξ³· W ).