The fundamental theorem of asset pricing under transaction costs

Abstract

This paper proves the fundamental theorem of asset pricing with transaction costs, when bid and ask prices follow locally bounded càdlàg (right-continuous, left-limited) processes.

The robust no free lunch with vanishing risk condition (RNFLVR) for simple strategies is equivalent to the existence of a strictly consistent price system (SCPS). This result relies on a new notion of admissibility, which reflects future liquidation opportunities. The RNFLVR condition implies that admissible strategies are predictable processes of finite variation.

The Appendix develops an extension of the familiar Stieltjes integral for càdlàg integrands and finite-variation integrators, which is central to modelling transaction costs with discontinuous prices.

Appendices

Appendix A: Predictable Stieltjes integral

The usual Stieltjes integral ∫S  dθ is well defined for continuous integrands S and integrators θ of finite variation. This section defines an extension of the Stieltjes integral to càdlàg integrands, which makes the integral operator continuous with respect to the pointwise convergence of integrators.²

The discussion is divided into three parts. The first subsection recalls some properties of predictable finite-variation processes, available in most textbooks under the extra assumption of càdlàg processes, relaxed here to làdlàg. The second subsection defines the integral and establishes its Lebesgue- and Fatou-type properties. Since the integral is defined pathwise, the first two subsections are probability-free, requiring only a filtered measurable space with a right-continuous filtration. Thus, the special case of a deterministic integrand and integrator is included in this setting. The third subsection establishes an approximation result for the predictable integral, which requires an underlying probability measure, since it relies on the decomposition of a stopping time into its accessible and totally inaccessible parts. This part requires a filtered probability space satisfying the usual conditions.

1.1 A.1 Predictable finite-variation processes

This subsection recalls some properties of finite-variation processes, dropping the extra assumption of right-continuity which appears in most textbooks.

Consider a measurable space , endowed with a right-continuous filtration such that . On the set Ω×ℝ₊, the predictable σ-algebra is generated by the class of adapted and left-continuous processes. Denote by the class of predictable processes with pathwise finite variation, by the class of adapted càdlàg processes and by the class of adapted làglàd (right-limited and left-limited) processes. Since a function of finite variation has right and left limits at all points, it follows that . A sequence of processes X ⁿ converges pointwise to a process X if X ⁿ(ω,t)→X(ω,t) for all (ω,t)∈Ω×ℝ₊.

Proposition A.1

Let and let ∥θ∥ denote its pathwise total variation. Then:

• (i) ;

• (ii) if converges pointwise to θ, then

  $$ \|\theta\| \le\liminf_{n\rightarrow\infty}\bigl\|\theta^n\bigr\| \quad\mathit{pointwise}. $$

The proof requires a representation of the set of jumps of predictable finite-variation processes. The finite-variation property avoids the use of section theorems. Recall that a random set A⊆Ω×[0,∞) is thin if A⊆⋃ _n≥1〚τ _n 〛 for a sequence of stopping times (τ _n ) _n≥1.

Lemma A.2

Let . Then the sets {X≠X ₋} and {X≠X ₊} are thin, and {X≠X ₋} is predictable.

Proof

For n,p≥0, define the stopping times S _n,p by induction as S _n,0=0 and

$$ S_{n,p+1}= \begin{cases} \inf\{t>S_{n,p}:|X_t-X_{S_{n,p}+}|>2^{-n}\}&\text{for even}\ p,\\[1mm] \inf\{t\ge S_{n,p}:|X_t-X_{S_{n,p}}|>2^{-n}\}&\text{for odd}\ p. \end{cases} $$

By definition, S _n,p <S _n,p+2. Define also the random sets

$$ A_{n,p}= \begin{cases} \{X_{S_{n,p}}\neq X_{S_{n,p}-}, S_{n,p}<\infty\}&\text{for even}\ p,\\[1mm] \{X_{S_{n,p}+}\neq X_{S_{n,p}}, S_{n,p}<\infty\}&\text{for odd}\ p, \end{cases} $$

and observe that . This is obvious for p even, while for p odd the right-continuity of the filtration implies that $X_{S_{n,p}+}$ is -measurable, whence the claim. Then, set $T_{n,p}={S_{n,p}}1_{A_{n,p}}+\infty1_{\varOmega\setminus A_{n,p}}$, which is a stopping time since . As X has finite variation, lim _p→∞ S _n,p =∞ and therefore

so that the sets {X≠X ₊} and {X ₋≠X} are thin. Also, X ₋ is left-continuous, therefore the set {X≠X ₋} is predictable. □

Proof of Proposition A.1

The proof of (i) is similar to the càdlàg case (Delbaen and Schachermayer [8], Theorem 12.2.1). By Lemma A.2, there is an exhausting sequence of stopping times (σ _k ) _k≥1 for the set of jumps of θ. For a fixed n≥1, denote by $\varPi_{n}=(\tau_{k})_{0\le k\le N_{n}}$ the finite ordered sequence of stopping times obtained from the set $(k 2^{-n})_{0\le k\le2^{n} T}\cup(\sigma_{k})_{k\le n}$, and define the processes

$$ V_t(\theta,\varPi_n)=\sum_{k=1}^{N_n}|\theta_{\tau_{k}\wedge t}-\theta _{\tau_{k-1}\wedge t}|. $$

(A.1)

Then V(θ,Π _n ) is predictable because θ is, and since V(θ,Π _n ) converges to ∥θ∥ pointwise, also ∥θ∥ is predictable and (i) follows.

For (ii), denote by Π=(τ _k )_0≤k≤n a finite ordered sequence of stopping times, and define V(θ,Π) as in (A.1). The pointwise convergence of (θ ⁿ) _n≥1 to θ implies that

$$ \lim_{n\rightarrow\infty}V_t \bigl(\theta^n,\varPi\bigr)=V_t(\theta,\varPi) \quad\text{for all}\ t,\varPi. $$

By (i) it follows that ∥θ∥ _t =sup _Π V _t (θ,Π), and therefore

$$ \liminf_{n\rightarrow\infty}\bigl\|\theta^n\bigr\|_t= \liminf_{n\rightarrow\infty}\sup_{\varPi} V_t\bigl(\theta^n, \varPi\bigr) \ge\sup_{\varPi}\lim_{n\rightarrow\infty}V_t\bigl( \theta^n,\varPi \bigr)=\sup_{\varPi} V_t(\theta, \varPi )=\|\theta\|_t, $$

which proves (ii). □

1.2 A.2 The predictable Stieltjes integral

This subsection defines the integral ∫S  dθ for each and .

Recall (cf. Rudin [30], Chap. 6) that the classical Stieltjes integral ∫f  dg exists if and only if for all ε>0, there exists a δ>0 such that for any pair of partitions Γ,Γ′ with mesh smaller than δ, the corresponding Riemann sums differ by less than ε. Also (Rudin [30], Chap. 6, Exercise 3), if two functions f and g have a finite number of jump discontinuities, then the integrals ∫f  dg and ∫g  df are defined if and only if at each (common) discontinuity point one function is right-continuous and the other is left-continuous.

Thus, because of overlapping discontinuities in θ and S, the integral ∫S  dθ may not exist in the standard Stieltjes sense. A simple example demonstrates the problem.

Example A.3

Consider the deterministic S=θ=1_〚1,2〚. Then S is not Stieltjes-integrable with respect to θ, because there are arbitrarily fine subdivisions such that the corresponding Riemann sums are either 0 or 1. Indeed, some Riemann sums include the jump ΔS ₁Δθ ₁, while others do not.

However, ∫S  dθ is a Stieltjes integral if θ is left-continuous:

Lemma A.4

Let S be a càdlàg function and θ a function of finite variation. Then ∫_[0,T] S  dθ ₋ exists as a Stieltjes integral.

The proof requires a simple lemma.

Lemma A.5

Let S be a càdlàg (deterministic) function such that |ΔS _t |≤s for all t. Then for all η>s, there exists δ>0 such that |S _u −S _t |≤η for all u,t such that |t−u|≤δ.

Proof

By the càdlàg assumption, for all t∈[0,T] there exists some δ _t >0 such that |S _u −S _t |<(η−s)/2 for u∈[t,t+δ _t ) and |S _u −S _t−|<(η−s)/2 for u∈(t−δ _t ,t). Therefore |S _u −S _t |<η for all u∈U _t =(t−δ _t ,t+δ _t ). By compactness, the open covering of (U _t ) _t∈[0,T] admits a finite subcovering $(U_{t_{i}})_{i=1}^{n}$, and the assertion follows with $\delta =\min_{1\le i\le n}\delta_{t_{i}}$. □

Proof of Lemma A.4

For a given ε>0, set $J^{\varepsilon}_{t}=\sum_{s\le t}\Delta S_{s} 1_{\{ |\Delta S_{s}|>\varepsilon\}}$ and S ^ε=S−J ^ε. Since S is càdlàg, it has a finite number of jumps larger than any given size; therefore J ^ε is piecewise constant and right-continuous. By construction, S ^ε is càdlàg with jumps bounded by ε. Since S is càdlàg and θ ₋ is left-continuous, by the previous observation J ^ε is Stieltjes-integrable with respect to θ ₋. For S ^ε, consider two partitions $\varGamma=(t_{i})_{i=0}^{n},~\varGamma'=(s_{i})_{i=0}^{m}$, and write $\varGamma\cup\varGamma'=(r_{i})_{i=0}^{l}$. Let u _i ∈[t _i−1,t _i ], v _i ∈[s _i−1,s _i ] be arbitrary points. The corresponding Riemann sums satisfy

where t(r _j )=u _i with i such that [r _j−1,r _j ]⊆[t _i−1,t _i ] and s(r _j ) is defined analogously. If the meshes of Γ and Γ′ are smaller than δ, then |t(r)−s(r)|≤δ for all r. Applying Lemma A.5 with s=ε and η=2ε for some small δ, it follows that

$$ \bigl|I_T\bigl(S^\varepsilon,\theta_-;\varGamma\bigr)-I_T \bigl(S^\varepsilon,\theta_-;\varGamma'\bigr)\bigr|\le \sum _{r_i\le T} |S_{t(r_i)}-S_{s(r_{i-1})} | |\theta_{r_i-}- \theta_{r_{i-1}-} |\le2\varepsilon\|\theta\|_T . $$

Since J ^ε is Stieltjes-integrable, |I _T (J ^ε,θ ₋;Γ)−I _T (J ^ε,θ ₋;Γ′)|≤2ε∥θ∥ _T up to a smaller δ′<δ. Adding up, for any ε>0 there is some δ′>0 such that

$$ \bigl|I_T(S,\theta_-;\varGamma)-I_T\bigl(S,\theta_-;\varGamma'\bigr)\bigr|\le 4\varepsilon\|\theta\|_T $$

for any pair of partitions Γ,Γ′ with mesh smaller than δ′, which amounts to pathwise integrability in the sense of Stieltjes. □

The predictable Stieltjes integral is defined as follows.

Definition A.6

For an arbitrary finite-variation θ and càdlàg S, define

$$ I_T(S,\theta):= \int_{[0,T]}S_u \,d\theta_u:= \int_{[0,T]} S\,d\theta _--\sum _{s\le T}(\theta_s-\theta_{s-})\Delta S_s. $$

(A.2)

The first term is a well-defined Stieltjes integral by Lemma A.4, and the second term is absolutely convergent,

$$\sum_{s\le T}\bigl|(\theta_s- \theta_{s-})\Delta S_s\bigr|\le\sum_{s\leq T} |\theta_s-\theta_{s-}\vert2 S^*_T \leq2\| \theta\|_T S^*_T<\infty, $$

since $S_{T}^{*}=\sup_{t\in[0,T]} \vert S_{t}\vert$ is finite by the càdlàg assumption.

Remark A.7

The case for definition (A.2) rests on both mathematical and economic arguments. Proposition A.16 below justifies (A.2) from the viewpoint of stochastic analysis, connecting the predictable Stieltjes integral to the usual stochastic integral via integration by parts.

From the economic viewpoint, consider and , regarding S as the asset price and θ _t (resp. θ _t−, θ _t+) as the number of shares held at time t (resp. immediately before, after t). Intuitively, a jump of S at a predictable time τ corresponds to a shock whose size is unknown but whose timing is known before its occurrence (e.g. the announcement of macroeconomic data). Vice versa, a totally inaccessible τ corresponds to a shock with sudden timing (e.g. a natural catastrophe). The definition of ∫S  dθ should reflect this difference.

On a predictable jump, the investor may rebalance her portfolio “immediately” before τ (and change from position θ _τ− to θ _τ ) and react after the price has changed from S _τ− to S _τ (by moving from θ _τ to θ _τ+). On an unpredictable (i.e., τ totally inaccessible) jump, necessarily θ _τ−=θ _τ , because θ is predictable (there is no possibility to prepare for τ), but there is still a possibility of rebalancing from θ _τ to θ _τ+. Thus, the cost of the portfolio θ should be

$$ S_{\tau-}(\theta_{\tau}-\theta_{\tau-})+S_{\tau}( \theta_{\tau +}-\theta_{\tau})= S_{\tau}(\theta_{\tau+}- \theta_{\tau-})-\Delta S_{\tau}(\theta_{\tau }- \theta_{\tau-}) $$

(A.3)

in the case of predictable τ and

$$ S_{\tau}(\theta_{\tau+}-\theta_{\tau})=S_{\tau}( \theta_{\tau +}-\theta_{\tau-}) $$

(A.4)

in the unpredictable case. These formulas are indeed consistent with (A.2).³

Looking at (A.3) and (A.4) from a different angle, the predictability of θ dictates that ∫S  dθ should include ΔS _τ (θ _τ+−θ _τ ) at all predictable stopping times, and should not include it at totally inaccessible times.

The pathwise definition of the integral (A.2) extends to a linear map from processes into processes.

Proposition A.8

The function (S,θ)↦I _⋅(S,θ),

$$ I_T(S,\theta)=\int_{[0,T]} S\,d \theta_--\sum_{s\le T}(\theta_s- \theta_{s-})\Delta S_s $$

(A.5)

maps into .

Proof

It is necessary to check that ω↦I _T (S,θ)(ω) is -measurable for all T>0, and that the paths t↦I _t (S,θ) have right and left limits. For the latter property, observe that the integral in (A.5) is left-continuous with right limits, while the summation in (A.5) is right-continuous with left limits.

For the measurability part, note that the second term in (A.5) is trivially -measurable; hence it suffices to check the first term, which is a pathwise Stieltjes integral by Lemma A.4. Thus, it is the limit of Riemann sums along a deterministic grid,

$$ \int_{[0,T]}S\,d\theta_-= \lim_{n\rightarrow\infty}\sum _{k=0}^{\lfloor n T\rfloor} S_{\frac{k}{n}} \bigl(\theta_{{\frac{k+1}{n}}-}- \theta_{{\frac{k}{n}}-}\bigr), $$

and therefore it is -measurable. □

The following theorem establishes the main properties of the predictable Stieltjes integral. First, if θ is simple (i.e., constant on a sequence of intervals, some of which may collapse to points), then ∫S  dθ is consistent with the definition of cost process in the main text. Second, the integral satisfies the natural bound (A.7). Finally, the integral satisfies a dominated convergence theorem and a Fatou property with respect to pointwise convergence of the integrator θ.

Theorem A.9

The map I in (A.2) satisfies the following properties:

• (i) If (τ _n ) _n≥0 is a sequence of stopping times and

  

  is predictable, then

  $$ I_T(S,\theta)= \sum_{\tau_i\le T} S_{\tau_i-}(\theta_{\tau_i}-\theta_{\tau_i-}) +\sum _{\tau_i< T} S_{\tau _i}(\theta_{\tau_i+}- \theta_{\tau_i}). $$

  (A.6)

• (ii) I _T is linear both in S and in θ, and with $S^{*}_{T}=\sup _{t\in [0,T]}|S_{t}|$, we have

  $$ \bigl|I_T(S,\theta)\bigr|\le\|\theta\|_T S^*_T . $$

  (A.7)

• (iii) If θ ⁿ→θ pointwise and sup _n≥1∥θ ⁿ∥ _T <∞, then I(S,θ ⁿ)→I(S,θ) pointwise.

• (iv) If (θ ⁿ) _n≥1 are as in (iii) and S≥0, then lim inf _n→∞ I(S,∥θ ⁿ∥)≥I(S,∥θ∥) pointwise.

Proof

In (i), (A.6) follows immediately from (A.2). For (ii), the linearity in θ and S is immediate from (A.2), which also implies the estimate

(A.8)

Now, define the sequence of stopping times $(\sigma^{\varepsilon}_{n})_{n\ge0}$ as

$$ \sigma^\varepsilon_0=0 \quad\text{and}\quad \sigma^\varepsilon_{n+1}=\inf\bigl\{t>\sigma^\varepsilon_n: |S_t-S_{\sigma^\varepsilon_n}|>\varepsilon\bigr\} $$

and set , which is piecewise constant and right-continuous, and satisfies $|S_{t}-S^{\varepsilon}_{t}|\le\varepsilon$ for all t∈[0,T]. By linearity in S and (A.8), for any ε>0, we have

$$\bigl|I_T(S,\theta)\bigr|\le \bigl|I_T\bigl(S^\varepsilon,\theta\bigr)\bigr|+3 \varepsilon\| \theta\|_T . $$

Fix T>0, and from now on assume θ ₀=θ _s =0 for s≥T without loss of generality. To calculate I _T (S ^ε,θ), first observe that (with the convention σ ₋₁=0)

and therefore

(A.9)

Consequently,

$$\bigl|I_T \bigl(S^\varepsilon,\theta\bigr)\bigr|=\biggl|\sum_{\sigma_n\le T}S_{\sigma _{n-1}}(\theta _{\sigma_n}-\theta_{\sigma_{n-1}})\biggr|\le S^*_T \|\theta\|_T $$

and (ii) follows since ε>0 is arbitrary.

(iii) By the linearity in S, it suffices to consider the case of θ ⁿ converging pointwise to zero. Property (ii) implies that

$$ \bigl|I_T\bigl(S,\theta^n\bigr)\bigr|\le \bigl|I_T\bigl(S^\varepsilon,\theta^n\bigr)\bigr|+\varepsilon\bigl\| \theta^n\bigr\|_T. $$

The calculations in the proof of (ii) show that the term |I _T (S ^ε,θ ⁿ)| depends only on finitely many values of θ; thus it becomes arbitrarily small for large n. Since M:=sup _n≥1∥θ ⁿ∥ _T is finite,

$$ \limsup_{n\to\infty} \bigl|I_T\bigl(S,\theta^{n}\bigr)\bigr|\le\varepsilon M $$

and convergence follows since ε is arbitrary.

(iv) By the same argument as in the proof of Proposition A.1, we have for s(ω)<t(ω) that

$$\liminf_{n\to\infty} \bigl(\bigl\|\theta^n\bigr\|_t-\bigl\|\theta^n\bigr\|_s\bigr)\geq \|\theta\|_t-\|\theta\|_s, $$

in the pointwise sense. Then, recalling (A.9), S≥0 implies that

As |I _T (S,∥θ ⁿ∥)−I _T (S ^ε,∥θ ⁿ∥)|≤ε∥θ ⁿ∥ _T and |I _T (S,∥θ∥)−I _T (S ^ε,∥θ∥)|≤ε∥θ∥ _T , it follows that

$$\varepsilon \Bigl(\sup_n\|\theta_n\|_T+\| \theta\|_T \Bigr)+ \liminf_{n\to \infty}\int_{[0,T]}S\, d\|\theta_n\|\geq \int_{[0,T]}S \,d\|\theta\|, $$

and the assertion follows as ε↓0. □

1.3 A.3 Approximations of predictable integrals

The previous subsection has shown that the integral θ↦I _T (S,θ) is continuous with respect to pointwise convergence. This result is probability-free.

A much more precise result holds in a filtered probability space satisfying the usual conditions, if S is locally bounded. Then, both θ and I _T (S,θ) admit uniform approximations, almost surely, but the approximating sequence may depend on the probability measure. In this regard, approximating sequence are analogous to announcing sequences for predictable times.

Consider a filtered probability space .

Theorem A.10

Let and be locally bounded. Then, for all ε>0 there exist a strictly increasing sequence of stopping times (σ _n ) _n≥0 with sup _n≥1 σ _n >T and a predictable process θ″ of the form

(A.10)

satisfying , |θ″−θ|≤ε, |∫S  dθ″−∫S  dθ|≤ε and ∥θ″∥≤∥θ∥ pointwise on [0,T] (outside a P-zero set).

The proof of this theorem requires a property of predictable finite-variation processes.

Proposition A.11

Any is locally bounded.

Proof

Let τ _n =inf{t>0:|θ _t |>n}, so that lim _n→∞ τ _n =∞ a.s. because |θ|≤∥θ∥ is finite-valued. Then is bounded, i.e., |θ| is prelocally bounded. A prelocally bounded predictable process is also locally bounded by VIII.11 in Dellacherie and Meyer [10]. □

Proof of Theorem A.10

Fix T>0. Note that implies that ∥θ∥ is locally bounded by Proposition A.11, while S is locally bounded by assumption. Thus, there exists a sequence of stopping times (υ _M ) _M≥1,υ _M ≤T such that |S _t |,∥θ∥ _t ≤M for t∈〚0,υ _M 〛, and it suffices to prove the claim on 〚0,υ _M 〛. Indeed, if θ ^M satisfies the claim for ε ^M=ε2^−(M+1), then satisfies the claim on [0,T]. Thus, from now on assume that . Define the sequence of stopping times $(\rho_{m})_{m=0}^{\infty}$ as

$$ \rho_0=0, \qquad\rho_{m+1}=\inf\bigl\{t>\rho_m: |S_t-S_{\rho_m}|\ge \delta\bigr\} \wedge T. $$

Since S is càdlàg, sup _m ρ _m >υ _M a.s. on {υ _M <T}.

Step 1: Construction of the approximation on 〛ρ _m−1,ρ _m 〛.

For m≥1, set $\varepsilon_{m}=\frac{\varepsilon}{2M} 2^{-(m+1)}$, and define the stopping times (σ _n ) _n≥0 by⁴

Since θ has finite variation, the random set ⋃ _n≥0〚σ _n 〛 has finite sections. Consider a set such that $\tau _{n}:=(\sigma_{n})_{A_{n}}$ is the totally inaccessible part of σ _n . The sequence (τ _n ) _n≥1 is strictly increasing because $(\sigma _{n})_{G_{n}}$ is predictable for the choice $G_{n}:=\{\|\theta\|_{\sigma_{n}}\ge v_{n-1}+\varepsilon_{m}\}$, hence $A_{n}\subseteq\{\|\theta\|_{\sigma_{n}}< v_{n-1}+\varepsilon _{m}, \| \theta\|_{\sigma_{n}+}\ge v_{n-1}+\varepsilon_{m}\}$.

The predictable set B=⋃ _n≥0(〚σ _n 〛∖〚τ _n 〛) admits an exhausting sequence of predictable times (π _n ) _n≥0. Since B has finite sections, this sequence is strictly increasing up to ordering.⁵ Consider first the process

which is predictable because each π _n is predictable (cf. Jacod and Shiryaev [18], I.2.12). Then, set

which is again predictable, since 〚σ _n 〛∖〚τ _n 〛⊆⋃ _k≥0〚π _k 〛. Thus, θ′ constructs an approximation on the accessible part of 〚σ _n 〛, and θ″ adjusts it on the totally inaccessible part of 〚σ _n 〛. This two-step procedure is necessary, because the accessible part of 〚σ _n 〛 is not identified by a single predictable time. θ″ satisfies, for all n,

(A.11)

(A.12)

(A.13)

(A.14)

Indeed, (A.11)–(A.13) hold by construction, and (A.14) holds because τ _n is totally inaccessible and θ,θ″ are predictable. Finally, note that (A.12) implies that $\theta_{\rho_{m}}=\theta ''_{\rho_{m}}$ on the accessible part of ρ _m .

Step 2: Prove the equality

(A.15)

To see this equality, observe that

where the second equality exploits (A.11). Hence, we obtain the equality

as follows. First, recall that $\theta''_{\sigma_{n}-}=\theta_{\sigma_{n-1}+}$. On 〚τ _n 〛=〚σ _n 〛∩A _n the first term on the right-hand side vanishes by (A.14), and the claim follows. On 〚σ _n 〛∖〚τ _n 〛, the second term on the right-hand side vanishes by (A.12), while the first one reduces to $S_{\sigma_{n}-}(\theta''_{\sigma_{n}-}-\theta_{\sigma_{n}-})$.

Step 3: Prove the estimate

(A.16)

First, rewrite (A.15) as

where . Define N _m =max{n:σ _n <ρ _m+1}, which implies that $\sigma_{N_{m}+1}=\rho _{m+1}$. By (A.15) it follows that

Now note that $|S-S_{\rho_{m+1}-}|\le\delta$ and $|S-S'_{\sigma _{n}}|\le \delta$ for n≤N _m , since $|S-S_{\rho_{m}}|\le\delta$ on 〚ρ _m ,ρ _m+1〚. Moreover, we also have $\|\Delta S_{\rho _{m}}\|\le2 M$ because |S|≤M, and finally $|\theta_{\rho _{m+1}-}-\theta_{\sigma_{N_{m}}+}|\le\varepsilon_{m}=\varepsilon 2^{-(m+1)}$, and so the claim follows.

Step 4: Setting δ=ε/M, and recalling that ∥θ∥ is bounded by M, it follows that

 □

The next corollary shows that the interpolation of θ not only approximates ∫S  dθ, but any finer Riemann sum as well.

Corollary A.12

Let θ,S,(σ _n ) _n≥0, and θ″ be as in Theorem A.10. Let $(\tilde{\sigma}_{n})_{n\ge0}$ be finer than (σ _n ) _n≥0 (i.e., for all n≥0, there exists $\tilde{n}(n)$ such that $\sigma_{n}=\sigma_{\tilde{n}(n)}$ a.s.), and define $\tilde{\theta}''$ as in (A.10), with $\tilde{\sigma}_{n}$ replacing σ _n . Then we have $|\theta''-\tilde{\theta}''|\le\varepsilon$, $|\int S\, d\theta ''-\int S\,d\tilde{\theta}''|\le\varepsilon$ and $\|\theta''\|\leq\|\tilde{\theta}''\|$ pointwise on [0,T].

Proof

$\|\theta''\|\leq\|\tilde{\theta}''\|$ is clear. To see that $|\theta ''-\tilde{\theta}''|\le\varepsilon$, note that $\tilde{\theta}''_{\tilde{\sigma}_{\tilde{n}(n)}}=\theta''_{\sigma_{n}}=\theta_{\sigma_{n}}$, and recall that ∥θ∥ increases by less than ε _m on , whence $|\theta''-\tilde{\theta}''|\le|\theta''-\theta|+|\theta -\tilde{\theta}''|\le2\varepsilon_{m}<\varepsilon$.

To see that $|\int S \,d\theta''-\int S\,d\tilde{\theta}''|\le\varepsilon$, observe that the equality (A.15) and the inequality (A.16) continue to hold with $\tilde{\theta}''$ instead of θ, again because $\theta,\theta'',\tilde{\theta}''$ coincide on 〚σ _n 〛. It follows that also the estimate in Step 4 above continues to hold, whence the claim. □

The next corollary is a straightforward consequence of Theorem A.10.

Corollary A.13

Let and assume that are locally bounded. Then for all ε>0, there exists θ″ as in Proposition A.10, which satisfies |θ″−θ|≤ε, |∫S  dθ″−∫S  dθ|+|∫κ  d∥θ″∥−∫κ  d∥θ∥|≤ε, and ∥θ″∥≤∥θ∥ pointwise on [0,T].

Repeating the above proof in a deterministic setting yields the following deterministic statement.⁶

Corollary A.14

Let θ be a finite-variation function, and S a càdlàg function. Then for all ε>0, there exists a sequence of time instants (τ _n ) _n≥0 such that the integrand θ″ defined in (A.10) satisfies |θ″−θ|≤ε, |∫S  dθ″−∫S  dθ|≤ε and ∥θ″∥≤∥θ∥ pointwise.

A simple consequence is that the integral I in (A.2) is the unique extension of the simple integrals (A.6) which satisfies a Lebesgue-type theorem with respect to pointwise convergence of the integrator.

Proposition A.15

The integral (A.2) is the unique extension of (A.6) to the class of all finite-variation functions θ which satisfies (iii) in Theorem A.9.

Proof

Take θ _n given by Corollary A.14 with the choice ε=1/n. Let $\tilde{I}$ be an arbitrary extension of the restriction of I to simple θ which satisfies (iii) of Theorem A.9. Then necessarily $\tilde{I}(S,\theta)=\lim_{n\to\infty} I(S,\theta_{n})$, and this latter limit is I(S,θ) by Corollary A.14. □

When S is a semimartingale, integration by parts links the predictable Stieltjes integral to the usual stochastic integral.

Proposition A.16

Let and S a càdlàg semimartingale. Then

$$ \int_{0}^T S_u\,d\theta_u=\theta_T S_T-\theta_0 S_0-\int_{0}^T \theta_u\, dS_u, $$

where the first integral is in the predictable Stieltjes sense and the second one is a usual stochastic integral.

Proof

Without loss of generality, assume θ ₀=θ _T =0. Then the stochastic integral exists, as θ is locally bounded by Proposition A.11. Consider first simple θ. By linearity of the integrals, it suffices to treat the left- and right-continuous cases separately, and localization allows to assume ∥θ∥,θ bounded.

First, consider θ left-continuous with finitely many jumps (τ _j )_0≤j≤n . Then

by the definition of (elementary) stochastic integrals, with the convention τ ₋₁=0. This argument shows the statement for θ, and extends easily to arbitrary simple left-continuous θ.

Take now θ right-continuous. Without loss of generality, suppose (see the proof of Theorem A.10) that the jump times τ _j are predictable with announcing sequences $\tau_{j-1}\leq\tau_{j}^{k}< \tau_{j}$, k≥1. Set

Note that $\theta_{\tau_{j+1}}$ is -measurable (Jacod and Shiryaev [18], I.2.4); hence the conditional expectation converges a.s. to $\theta_{\tau_{j+1}}$ as k→∞ by the martingale convergence theorem. Hence γ(k)→θ pointwise (possibly outside a P-zero set).

It follows from the previous step, by the dominated convergence theorem for stochastic integrals, and by Theorem A.9(iii) that

$$ -\int_0^T \theta_u \,dS_u=\lim_{k\to\infty}-\int_0^T \gamma _u(k)\,dS_u=\lim _{k\to\infty} I(S,\gamma(k))= I(S,\theta). $$

For general , take an approximation θ ⁿ constructed in Theorem A.10 with ε:=1/n. The above arguments imply that $I_{T}(S,\theta^{n})=-\int_{0}^{T} \theta^{n}_{u} \,dS_{u}$ for all n. Moreover, I _T (S,θ ⁿ)→I _T (S,θ) by Theorem A.10 and the stochastic integrals of θ ⁿ converge to that of θ by dominated convergence (note that |θ _n |≤|θ|+1/n). □

Appendix B

For completeness, this appendix recalls the statements of some now classical tools in mathematical finance. First, a compactness lemma for bounded sets in $L^{0}_{+}$:

Lemma B.1

(Delbaen and Schachermayer [6], Lemma A1.1)

Let (f _n ) _n≥1 be a sequence of [0,∞)-valued measurable functions on a probability space . There exists a sequence g _n ∈conv((f _k ) _k≥n ) such that (g _n ) _n≥1 converges almost surely to a [0,∞]-valued function g. If conv((f _n ) _n≥1) is bounded in L ⁰, then g is finite almost surely.

Second, an important consequence of the Krein–Smulian theorem (cf. Delbaen and Schachermayer [6], Theorem 2.1; Kabanov and Last [21], Lemma 4.4):

Lemma B.2

Let C⊆L ^∞ be a convex set. Then C is σ(L ^∞,L ¹)-closed if and only if C∩{Z:∥Z∥_∞≤x} is closed in probability for all x>0.

Third, a version of the Kreps [26]–Yan [33] separation theorem (cf. Schachermayer [31] or Kabanov and Stricker [23]):

Theorem B.3

Let $-L^{\infty}_{+}\subseteq C\subseteq L^{\infty}$ be a convex cone, closed in the σ(L ^∞,L ¹)-topology, such that $C\cap L^{\infty}_{+}=\{ 0\}$. Then there exists a probability Q equivalent to P such that E _Q [C]≤0.

Finally, a compactness lemma for finite-variation processes (cf. Guasoni [14], Lemma 3.4; Campi and Schachermayer [2], Proposition 14). Note that Lemma A.2 allows to avoid section theorems.

Lemma B.4

Consider a sequence such that (∥θ ⁿ∥ _T ) _n≥1 is bounded in L ⁰. Then there exist and a sequence of convex combinations η ⁿ∈conv(θ ⁿ,θ ⁿ⁺¹,…) converging to θ pointwise such that also ∥η _n ∥→∥η∥ pointwise.