Different Types of Structure Conditions of Semimartingale with Jacod Decomposition

The objective of this article is to use Jacod decomposition to develop different types of semimartingale structure conditions. We make the following contributions to that end: When a continuous semimartingale meets the structure condition (SC), we prove that there is a minimal martingale density and a predictable variation part. When a special semimartingale meets the minimal structure condition (MSC) and the natural structure condition (NSC), we derive a Radon-Nikodym decomposition and a Natural Kunita-Watanabe decomposition from a given sigma martingale density, which is written under the Jacod decomposition.


Introduction
According to [1], the fundamental theorem of asset pricing (FTAP) gives an economic meaning to the no-arbitrage condition: the no-free lunch with vanishing risk (NFLVR). This theorem guarantees that (NFLVR) is necessary and sufficient for the presence of a particular pricing operator, an equivalent σ-martingale measure. The weaker requirements that make up (NFLVR) are the no-arbitrage condition (NA) and the no unbounded profit with bounded risk condition (NUPBR).
Several researchers have proposed reformulations of the (NUPBR) condition that are similar. For example, [2] recently demonstrated the equivalence of (NUPBR) and the existence of a rigorous sigma-martingale density for one-dimensional but in general semimartingale. The (NUPBR) condition is also equivalent to the existence of a strictly positive σ-martingale density for the underlying semimartingale, as [3] demonstrated. The fundamental benefit of these equivalent reformulations is that they all guarantee the existence of a fair pricing operator to price the terminal wealth of all 1-admissible trading strategies.
However, how can we identify a natural candidate for a strictly positive σ-martingale density for an arbitrary, locally square-integrable semimartingale S M A = + , where M denotes the local martingale part and A denotes a predictable variation part? A weak structure condition (SC') that leads to a structure condition (SC) for a continuous semimartingale was proposed by [4] [5]. Structure condition (SC) is a useful tool that leads to the minimal martingale density, which is a natural candidate for a strictly positive σ-martingale density.
Although (SC) and its related structure theorem are stable for a large class of semimartingales, they have some flaws. This is because they're only useful for finding σ-martingale densities for a continuous semimartingale that are strictly positive. Furthermore, under the (proper) measure, the structure condition (SC) is not invariant. To address these flaws, [6] chose to take a fresh look at the special semimartingale and its unique decomposition, resulting in the creation of more than one new type of structure condition: minimal structure condition (MSC) and natural structure condition (NSC).
The goal of this paper is to use Jacod decomposition to establish the concept of several types of semimartingale structure conditions. We'll start by deriving the minimal martingale density from a positive exponential sigma martingale density, which is stated using Jacod decomposition when a continuous semimartingale S meets the structure condition. For the circumstances when P-minimal martingale density and Q-minimal martingale density exist, we will additionally show a predictable variation part of a continuous semimartingale. Finally, when a special semimartingale meets the minimal and natural structure conditions, we will derive Radon-Nikodym decomposition and natural Kunita-Watanabe decomposition from sigma martingale density.
The following is how the rest of the article is structured: Important definitions, theorems, assertions, and Lemmas are discussed in Section 2. In Section 3, we show that a continuous semimartingale has minimal martingale density and drift part. We derive Radon-Nikodym decomposition from a given sigma martingale density and prove a drift variation part of minimal structure condition in Section 4. We derive natural Kunita-Watanabe decomposition (NKWD) and prove the existence of sigma martingale from (NKWD) in Section 5. Finally, conclusions and suggestions are in Section 6.

Important Definitions, Notations, Theorems and Propositions
i) According to [7] ( )  is the set of all locally square-integrable martingale.
iii) According to ([7], Proposition 1.14) the random measure µ associated with its jumps is defined by where a δ denotes the Dirac measure at point a. iv) According to ([7], Definition 1.27a) Therefore the set vii) According to ([9], Lemma 3.72) if Definition 2.1 (σ-martingale density) According to [5] a σ-martingale density (or local martingale density) for S is a local P-martingale with the following properties Also in addition Z is called in addition strictly positive-σ martingale density if 0 Z > .
A strictly positive σ-martingale density is in general only a local martingale but not a true P-martingale. But According to [5] if semimartingale S satisfies the weak structure condition is called the mean-variance tradeoff process of semimartingale S. Because M is positive semidefinite, the process K is increasing and null at 0, but note that it may take the value ∞ in general.
Therefore we say that a semimartingale S satisfies the structure condition (SC) if S satisfies ( SC′ ) and ˆT K < ∞ P-a.s. . We call Ẑ the minimal martingale density for S. Since it is in a sense the simplest martingale density. Therefore, the perceptive Ẑ is minimal because it is obtained from the simplest choice, 0 R = .
Theorem 2.5. ( [11], Theorem 1). If S admits a strict martingale density Z * and that either S is continuous or S is a special semimartingale satisfying . Then S satisfies the structure condition (SC). Further Z * can be written as is strongly orthogonal to M. If S is continuous we can simplify where Ẑ is minimal martingale density which is obtained for the simplest choice 0 L = .
According to [7] [12] any local martingale M admits a unique decomposition According to [7] [12], a cadlag, adapted stochastic process ( M is a local martingale and t A is an adapted cadlag process with a finite variation.
A special semimartingale is a semimartingale t X that admits a decomposi- The presence of arbitrage is not allowed in modelling asset prices. Therefore it is important that the underlying asset price process be arbitrage-free.
be a strictly positive local Q-martingale. Then Z is a strictly positive Q-σ-martingale (local Q-martingale) density for S if and only if Lemma 2.2. According to [6] if we let Q P  ,

Jacod Decomposition
Jacod decomposition was introduced by [ such that the following integral exists is an optional process and.

4)
2 loc N ∈  is a local P-martingale null at 0 with [ ] The difference between Jacod decomposition (1)

Structure Condition
In order to explain the importance of (SC) as a good tool for finding strictly positive σ-martingale densities in continuous paths cases, we recall its definition. In this section first, we are going to prove the existence of minimal martingale density from a given sigma martingale density written in Jacod decomposition.
Also since the existence of structure condition rely on the existence of the predictable process of quadratic variation of the local martingale part, we are going to prove in the second part the existence of a continuous predictable process

Minimal Martingale Density
Since any local martingale admits a unique decomposition of two parts: a continuous part and a discontinuous part, we can decompose M as ( )  (2), then we can call Let c S be a semimartingale with representation where c M is a continuous local martingale part and A is a finite variation process. Then the stochastic integral: The representation (3) is not unique therefore the stochastic integral (4) is also not unique. If is another presentation, then we can have also another stochastic integral 2, 2 Then: So that ( ) means there is no difference between the sharp and square brackets according to [15]. We are going to have This means our strictly positive σ martingale density is equal to minimal martingale density when a semimartingale S is continuous.

A Predictable Variation Part When P-Minimal Martingale Density Exists
Structure condition which is defined with continuous semimartingale S of the form 0 S S M A ′ = + + has two aspects: i) It requires a predictable variation component A to be absolutely continuous with M ′ of martingale component M ′ with density function Z.
ii) It imposes the square integrability condition on M ′ and a specific integrability condition on the density function Z.
All two conditions are reinterpretations under square integrability conditions of the equation That is between the a predictable variation part A, the local martingale part M ′ and a strict σ martingale density Z.
Therefore first we are going to prove the equation Substitute Equations (9) and (10) From Equation (12) we have a local martingale part From Equation (12) we are going to have Due to Yoeurps Lemma [12], the first bracket term on the R.H.S is a P-local martingale, and L.H.S is a (differential of) a P-σ-martingale (a local-P-martingale) if and only if Z is a strictly positive P-σ-martingale (a local P-martingale) density for a semimartingale S. Therefore by application of the product rule, this shows that ZS has a predictable variation part d d , From this finite variation part

A Predictable Variation Part When Q-Minimal Martingale Density Exists
For the case of Q-minimal martingale density ˆQ Z it depends on the ˆP Z by Bayes Rule.
Since we have seen that structure condition is satisfied in ˆP Z it is also be satisfied on ˆQ Z because they depend on each other. According to [6] the predictable quadratic variation of a locally square-integrable martingale is not invariant under equivalent measure changing. Therefore, in this case under ˆQ Z we are going to have

Minimal Structure Condition
Although the SC theorem holds for every locally square-integrable semimartin- now let's add two Equations (16) and (17) ( ) ( ) From Equation (19), let multiply both sides by negative sign then we are going

A Predictable Variation Part of (MSC)
According to the 1 st structure theorem and the minimal structure conditions Z is Journal of Mathematical Finance a strictly positive σ-martingale (

Conclusions and Suggestions
There have been no meaningful uses of Jacod decomposition since the introduction of structure condition (SC), which allows for the existence of minimal martingale measure until the introduction of new types of structure conditions (MSC and NSC) of special semimartingale. According to [20] [21] [22], the necessary results can be obtained by writing a sigma martingale density in terms of a Dolean-Dade exponential of Jacod decomposition.
We were able to prove the existence of minimal martingale density when a continuous semimartingale satisfies SC and derive a Radon-Nikodym decomposition and a Natural Kunita-Watanabe decomposition in our case, by writing a sigma martingale density in terms of Dolean-Dade exponential of Jacod decomposition.
We can suggest that, if the sigma martingale density is stated in a different way from the one we studied, the minimal martingale density can be derived.
However, this will only be achievable if continuous semimartingale satisfies the structure conditions' features.