A NOTE ON STOCHASTIC INTEGRATION WITH RESPECT TO OPTIONAL SEMIMARTINGALES

In this note we discuss the extension of the elementary stochastic Itô-integral w.r.t. an optional semimartingale. The paths of an optional semimartingale possess limits from the left and from the right, but may have double jumps. This leads to quite interesting phenomena in integration theory. We ﬁnd a mathematically tractable domain of general integrands. The simple integrands are embedded into this domain. Then, we characterize the integral as the unique continuous and linear extension of the elementary integral and show completeness of the space of integrals. Thus our integral possesses desirable properties to model dynamic trading gains in mathematical ﬁnance when security price processes follow optional semimartingales.


Introduction
In this note we discuss the extension of the elementary stochastic Itô-integral in a general framework where the integrator is an optional semimartingale. The paths of an optional semimartingale possess limits from the left and from the right, but may have double jumps. Such processes have been studied extensively by Lenglart [9] and Galtchouk [3,4,5,6]. It turns out that the extension of the elementary integral to all predictable integrands is too small. Namely, the space of integrals for (suitably integrable) predictable integrands is still not complete (even w.r.t. the uniform convergence). This is of course in contrast to the standard framework with a càdlàg integrator, cf. [2]. Galtchouk [4] has introduced a stochastic integral w.r.t. an optional martingale with a larger domain. But the integral of [4] is not the unique (continuous and linear) extension of the elementary integral. There are stochastic integrals that can in no way be approximated by elementary integrals. This is an undesirable feature in some applications, e.g. if one wants to model trading gains from dynamic strategies by the integral. As real-world investment strategies are of course piecewise constant, it would not make sense to optimize over a set of integrals including some elements that cannot be approximated by elementary integrals. In this note we introduce a mathematically tractable domain of integrands which is somehow between the small set of predictable integrands and the large domain in [4]. The latter is a two-dimensional product space of predictable and optional processes. The simple strategies are embedded into our domain. Then, in the usual manner, we characterize the integral defined on this domain as the unique continuous and linear extension of the elementary integral and show its completeness. In mathematical finance completeness of the space of achievable trading gains guarantees that the supremum in a portfolio optimization problem is attained and in "complete markets" derivatives can be replicated and not only be approximated by gains from dynamic trading in the underlying securities. In addition, this note may also provide another abstract view to the extension of the elementary integral and the identification of 1 ]]τ 1 ,τ 2 ]] • X t with X t∧τ 2 − X t∧τ 1 in the usual situation of a càdlàg integrator X .

Notation
Let (Ω, , ( t ) t∈[0,T ] , P) be a complete filtered probability space, where the family ( t ) t∈[0,T ] is not necessarily right-continuous. and denote the predictable resp. the optional σ-algebra on is generated by all left-continuous adapted processes and is generated by all càdlàg adapted processes (considered as mappings on Ω × [0, T ]). If X and Y are two optional processes and we write X = Y , we mean equality up to indistinguishability. The following definitions are from [6]. Adjusted to our finite time horizon setting, we repeat them here for convenience of the reader. We add a localization procedure based on stopping which preserves the martingale property of a process. The results of Galtchouk that we use still hold when localization is done in the way chosen here. Definition 2.1. A stochastic process X = (X t ) t∈[0,T ] is called an optional martingale (resp. square integrable optional martingale), and we write X ∈ (resp. X ∈ 2 ), if X is an optional process and there exists an T Galtchouk has shown in [3] that for any T -measurable integrable random variable Z there exists an optional martingale (X t ) t∈[0,T ] with terminal value X T = Z. Almost all paths of X possess limits from the left and the right (see e.g. Theorem 4 in Appendix I of [2]). Thus if one considers general filtrations, optional martingales emerge quite naturally. For a làglàd process X we denote ∆ − X t := X t − X t− and ∆ + X t := X t+ − X t . Let be a class of stochastic processes. A stochastic process X with right-hand limits is in the localized class of , and we write X ∈ l oc if there exists an increasing sequence (τ n , σ n ) n∈ ⊂ × + such that lim n→∞ P(τ n ∧ σ n = T ) = 1 and the stopped processes X (τ n ,σ n ) defined by X (τ n ,σ n ) t := X t 1 {t≤τ n ∧σ n } + X τ n 1 {t>τ n , τ n ≤σ n } + X σ n + 1 {t>σ n , τ n >σ n } are in for all n.
Galtchouk has shown that it is possible to uniquely decompose a local martingale M into a càdlàg part M r and an orthogonal part M g , i.e. M g M is a local martingale for any càdlàg martingale M . M g possesses càglàd paths (see Theorem 4.10 in [4] for details). Furthermore, any A ∈ can obviously be decomposed uniquely into a càglàd part A g := 0≤s<t ∆ + A s and a càdlàg part A r := A − A g . Note however that for processes which are both local martingales and of finite variation the decompositions usually differ.

A semimartingale X is called special if there exists a representation (2.1) with a strongly predictable process A ∈ l oc .
Note that any optional semimartingale has limits from the left and the right, i.e. almost all paths are làglàd (again by [2] this assertion holds for the local martingale component; for the finite variation component the assertion is trivial).

Results
Suppose X is the (for simplicity deterministic) evolution of a stock price given by is the time of a double jump. ]t 0 , T ] denotes an interval on whereas for τ 1 , τ 2 stopping times ]]τ 1 , interval. Now consider the strategies A n where we buy one unit of the stock at time t 0 − 1/n and sell it at time t 0 . The (negative) trading gain would be 1/n − 1, and as n → ∞ the trading loss would go to 1 and occur exactly at time t 0 . Other possible strategies B n would be to buy one unit of the stock at time t 0 and sell it at time t 0 + 1/n. The trading gain would be 2 + 1/n, which would converge to a trading gain of 2 also occuring at time t 0 . If we wanted the space of trading strategies to be complete, for the two sequences of trading strategies there should be limit trading strategies A and B reproducing the limit trading gain such that it occured exactly at time t 0 . If we wanted to use one-dimensional processes to specify our trading strategy, we would run into a dilemma because something like 1 [t 0 ] would have to represent both A and B, but this is clearly impossible since the trading gains from A and B are completely different. Put differently, since the process has double jumps, there might be a left jump ∆ − X t and a right jump ∆ + X t at the same time. Using a one-dimensional integrand, an investor cannot differentiate between what should be invested in the left jump and what should be invested in the right jump, because at each point in time he only has a single value of the integrand at his disposal. For example, in the considerations above, the limit strategy A would have to invest 1 in ∆ − X t 0 but 0 in ∆ + X t 0 . This explains why Galtchouk [4] introduced two-dimensional integrands (H, G) where H is ameasurable process and G is an -measurable process. Unfortunately, this expansion of the space of integrands to two dimensions leads to a new problem. The integrals of these two-dimensional integrands can in general no longer be approximated by integrals of simple predictable integrands as the following example shows.
i.e. the symmetric difference A∆B has to be a thin set. Note that τ is Our general integrands will be / ( )-measurable functions.
Proof. Obvious as and are σ-fields and countable unions of thin sets are thin sets.
where τ i ∈ , τ 1 ≤ τ 2 . . . ≤ τ n+1 , Z 0 is 0 -measurable, and each Z i is a τ i -measurable random variable. Let denote the class of simple integrands. Note that the simple integrands are indeed -measurable, and that there is a one-to-one correspondence between the simple integrands defined in (3.2) and the usual one-dimensional simple predictable integrands. By Proposition 3. 3 generates the σ-field on {1, 2} × Ω. We call simple integrands simple -measurable. We now define for H ∈ the elementary stochastic integral in the usual way by

Proof. Step 1 (uniqueness). Let H • X and H • X be two extensions satisfying (i) and (ii). Then (i) and (ii) imply that := {F ∈
: 1 F • X = 1 F • X } is a Dynkin system. Since ⊂ and is a ∩stable generator of , by a Dynkin argument we have = . A locally bounded -measurable process H can be approximated pointwise by the sequence (H n ) n∈ , where Because of the linearity requirement (i) we know that H n • X = H n • X for all n. In addition it is true that |H n | ≤ |H| + 1. Thus from (ii) follows H • X = H • X and the uniqueness of the extension is established.
Step 2 (existence). Let X = X 0 + M + A with M ∈ l oc and A ∈ be any decomposition of X . Consider the integral (once again denoted by H → H • X ) which is by Galtchouk defined for any locally bounded H 1 ∈ and H 2 ∈ , thus in particular when H is locally bounded and -measurable. Note that (3.3) generally depends on the decomposition of the optional semimartingale into a local martingale and a process of finite variation (Thus in Galtchouk H 1 • M r + H 2 • M g and H 1 • A r + H 2 • A g are seen as separate integrals. But, later on by the uniqueness of the extension it will turn out that for / ( )-measurable integrands the choice of the decomposition is not relevant).
If H is a simple integrand this integral is equal to our definition of the simple integral, i.e. it is an extension. From the standard theory (see e.g. [2], chapter VIII) we know that the first half of the right-hand side of (3.3) fulfils properties (i) and (ii). For the left-continuous parts H 2 • M g and H 2 • A g the same line of argument holds true: M g can be decomposed into a locally square integrable martingale and a local martingale of finite variation (by considering the process 0≤s≤· ∆ + M s 1 {|∆ + M s |>1} ∈ l oc and using the existence of strongly predictable càglàd compensators, see Lemma 1.10 in [6]). Because a version of Doob's inequality still holds for optional square-integrable martingales (see Appendix I in [2] on how to prove such inequalities using the optional section-theorem, which still holds under non-usual conditions), the usual arguments for the càdlàg case can be reproduced for the locally square integrable part. The martingale part of finite variation is treated like ( which is a Lebesgue-Stieltjes integral. Thus it is known that it is linear and has the continuity property.

Remark 3.6. We have shown that it is possible to extend the integral in a unique way from all simple -measurable integrands (which are in a one-to-one correspondence with the (one-dimensional) simple predictable integrands) to all locally bounded -measurable integrands. Note that the elementary integral does not depend on the decomposition in (3.3). In Galtchouk's framework [6] the integral is extended uniquely from all two-dimensional simple ⊗ -measurable integrands to all locally bounded ⊗ -measurable integrands. What cannot be done is to extend the integral uniquely from one-dimensional simple predictable integrands to all locally bounded
⊗ -measurable integrands. To see this note that besides H • X :  all four integrals converge uniformly in probability against the corresponding integrals without truncation. Thus H • X is also well-defined.
Similarly, for A ∈ l oc let By the decomposition of M (resp. A) into a right-and a left-continuous part we ensure that m (resp. n) is a measure. Note that m and n are in general not σ-finite. Let H • M ∈ 2 ; then we have that (3.4) The crucial third equality follows because H 1 • M r and H 2 • M g are orthogonal optional martingales, which is due to fact that (see [4], Theorem 7.11). The fourth equality is valid since there are Itô isometries for both the standard stochastic integral and the optional stochastic integral w.r.t. to a càglàd optional martingale (see [4], Section 7). Let us verify an isometry property for the integrable variation part. Note that for the finite variation part A, the process A g is just the sum of the jumps ∆ + A.   Proof. The proof of the second half of Theorem I.3.18 in [8] can be reproduced without any major changes (using Proposition 4.4 and Lemma 4.2). Note that the associativity of the integral used in the proof holds because where the crucial third equality is true because for any A ∈ we obviously have (A r ) g = (A g ) r = 0. The fourth equality follows from the associativity of the one-dimensional Lebesgue-Stieltjes integral.