Semimartingales and Shrinkage of Filtration

TL;DR

This paper investigates how the semimartingale characteristics of a process change when considering a smaller filtration, focusing on both adapted and non-adapted cases, with implications for filtration shrinkage in stochastic processes.

Contribution

It provides formulas for the semimartingale characteristics of processes under filtration shrinkage, extending understanding of optional projections and their characteristics.

Findings

Derived formulas for $$-semimartingale characteristics when $X$ is $$-adapted.

Established methods to compute $$-semimartingale characteristics of optional projections.

Analyzed the relationship between $$- and $g$-semimartingale characteristics in filtration shrinkage.

Abstract

We consider a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, which is endowed with two filtrations, $\mathbb{G}$ and $\mathbb{F}$, assumed to satisfy the usual conditions and such that $\mathbb{F} \subset \mathbb{G}$. On this probability space we consider a real valued special $\mathbb{G}$-semimartingale $X$. The purpose of this work is to study the following two problems: A. If $X$ is $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of $X$ in terms of the $\mathbb{G}$-semimartingale characteristics of $X$. B. If $X$ is not $\mathbb{F}$-adapted, given that the $\mathbb{F}$-optional projection of $X$ is a special semimartingale, compute the $\mathbb{F}$-semimartingale characteristics of $\mathbb{F}$-optional projection of $X$ in terms of the $\mathbb{G}$-canonical decomposition and $\mathbb{G}$-semimartingale characteristics of $X$.

Equations

$$X_t(\chi )=X_t-X_0-\sum _{0<s\le t}(\Delta X_s-\chi (\Delta X_s)),t\ge 0.$$

$$X(\chi )=B^{\mathbb{G}}+M^{\mathbb{G}},$$

$$B^{\mathbb{G}}=b^{\mathbb{G}}\cdot A^{\mathbb{G}},C^{\mathbb{G}}=c^{\mathbb{G}}\cdot A^{\mathbb{G}},{\nu }^{\mathbb{G}}(dt,dx)=K_t^{\mathbb{G}}(dx)d{A}_t^{\mathbb{G}},$$

$$A_t^{\mathbb{G}}={\int }_0^t{a}_u^{\mathbb{G}}du,$$

$$\mathbb{E}(Z_{\tau }\text{mathbbm}{1}_{\tau <\infty }|{\mathcal{F}}_{\tau })={}^{o,\mathbb{F}}{Z}_{\tau }\text{mathbbm}{1}_{\tau <\infty }.$$

$$\mathbb{E}(Z_{\tau }\text{mathbbm}{1}_{\tau <\infty }|{\mathcal{F}}_{\tau -})={{}^{p,\mathbb{F}}Z}_{\tau }\text{mathbbm}{1}_{\tau <\infty }.$$

$$\tilde{\Omega }:=\Omega \times {\mathbb{R}}_{+}\times \mathbb{R},\tilde{\mathcal{F}}:=\mathcal{F}\otimes \mathcal{B}({\mathbb{R}}_{+})\otimes \mathcal{B}(\mathbb{R}).$$

$${\tilde{\mathcal{O}}}_{\mathbb{F}}:={\mathcal{O}}_{\mathbb{F}}\otimes \mathcal{B}(\mathbb{R}),{\tilde{\mathcal{P}}}_{\mathbb{F}}:={\mathcal{P}}_{\mathbb{F}}\otimes \mathcal{B}(\mathbb{R}).$$

$$V(\omega ,t):={\int }_{[0,t]\times \mathbb{R}}W(\omega ,s,x)\pi (\omega ;ds,dx)$$

$$\mathbb{E}\bigg{(}\int_{\mathbb{R}_{+}\times\mathbb{R}}W(s,x)\pi(ds,dx)\bigg{)}=\mathbb{E}\bigg{(}\int_{\mathbb{R}_{+}\times\mathbb{R}}{{}^{q,\mathbb{F}}}W(s,x)\pi(ds,dx)\bigg{)},$$

$$M_{\pi}(\widetilde{B}):=\mathbb{E}\bigg{(}\int_{\mathbb{R}_{+}\times\mathbb{R}}\mathbbm{1}_{\widetilde{B}}(\omega,t,x)\pi(\omega,dt,dx)\bigg{)},\qquad\widetilde{B}\in\widetilde{\mathcal{F}},$$

$$\mathbb{E}\Big{(}\int_{\mathbb{R}_{+}\times\mathbb{R}}W(t,x)\pi(dt,dx)\Big{)}=\mathbb{E}\Big{(}\int_{\mathbb{R}_{+}\times\mathbb{R}}W(t,x)\pi^{o,\mathbb{F}}(dt,dx)\Big{)}.$$

$$\mathbb{E}\Big{(}\int_{\mathbb{R}_{+}\times\mathbb{R}}W(t,x)\pi(dt,dx)\Big{)}=\mathbb{E}\Big{(}\int_{\mathbb{R}_{+}\times\mathbb{R}}W(t,x)\pi^{p,\mathbb{F}}(dt,dx)\Big{)}.$$

$$(g\ast \gamma {)}_t:={\int }_0^t{\int }_{\mathbb{R}}g(x)\gamma (ds,dx),t\ge 0.$$

$$M^{\mathbb{G}}=X(\chi )-B^{\mathbb{G}};Y=(M^{\mathbb{G}}{)}^2-C^{\mathbb{G}}-{\chi }^2(x)\ast {\nu }^{\mathbb{G}};g\ast \mu -g\ast {\nu }^{\mathbb{G}},g\in {\mathcal{C}}^{+}(\mathbb{R}),$$

$$\mathbb{E}(M^{\mathbb{G}}{)}_{{\tau }_n\wedge \sigma }^2=\mathbb{E}(C^{\mathbb{G}}+{\chi }^2\ast {\nu }^{\mathbb{G}}{)}_{{\tau }_n\wedge \sigma },n=1,2,\dots ,$$

$$\mathbb{E}(C^{\mathbb{G}}+{\chi }^2\ast {\nu }^{\mathbb{G}}{)}_{{\tau }_n\wedge \sigma }\le \mathbb{E}(C^{\mathbb{G}}+{\chi }^2\ast {\nu }^{\mathbb{G}}{)}_T=C_T^{\mathbb{G}}+({\chi }^2\ast {\nu }^{\mathbb{G}}{)}_T<\infty ,$$

$$\mathbb{E}\Big{(}(M^{\mathbb{G}})^{2}_{\sigma}-C^{\mathbb{G}}_{\sigma}-(\chi^{2}*\nu^{\mathbb{G}})_{\sigma}\Big{)}=0.$$

$$\mathbb{E}(g\ast {\mu }^X{)}_{\sigma \wedge {\tau }_n}=\mathbb{E}(g\ast {\nu }^{\mathbb{G}}{)}_{\sigma \wedge {\tau }_n},n=1,2,\dots ,$$

$$\mathbb{E}(g\ast \mu {)}_{\sigma }=\mathbb{E}(g\ast {\nu }^{\mathbb{G}}{)}_{\sigma }\le (g\ast {\nu }^{\mathbb{G}}{)}_T<\infty .$$

$$\mathbb{E}\big{(}\int_{0}^{t}|b^{\mathbb{G}}_{u}|a^{\mathbb{G}}_{u}du\big{)}<\infty,$$

$$M{=}^{o,\mathbb{F}}(H\cdot A)-{(}^{o,\mathbb{F}}H)\cdot A$$

$$M{=}^{o,\mathbb{F}}(H\cdot A)-(H\cdot A{)}^{o,\mathbb{F}}$$

$$(H\cdot A{)}^{o,\mathbb{F}}={(}^{o,\mathbb{F}}H)\cdot A,$$

$$M_t^B={}^{o,\mathbb{F}}{B}_t^{\mathbb{G}}-{\int }_0^t{}^{o,\mathbb{F}}(b^{\mathbb{G}}{a}^{\mathbb{G}}{)}_{u}du,t\ge 0,$$

$$L_t=a_t^{\mathbb{G}}{b}_t^{\mathbb{G}}\text{mathbbm}{1}_{\{t\le T\}},t\ge 0.$$

$$N_t:={}^{o,\mathbb{F}}(L\cdot A{)}_t-({}^{o,\mathbb{F}}L\cdot A{)}_t,t\ge 0,$$

$${}^{o,\mathbb{F}}(L\cdot A{)}_t={}^{o,\mathbb{F}}((H\cdot A{)}^T{)}_t={}^{o,\mathbb{F}}(H\cdot A{)}_t.$$

$${}^{o,\mathbb{F}}L={}^{o,\mathbb{F}}(L{1}_{[0,T]})=1_{[0,T]}{}^{o,\mathbb{F}}L=1_{[0,T]}{}^{o,\mathbb{F}}(L^T)=1_{[0,T]}{}^{o,\mathbb{F}}(H^T)=1_{[0,T]}{}^{o,\mathbb{F}}H,$$

$${}^{o,\mathbb{F}}L\cdot A=(1_{[0,T]}{}^{o,\mathbb{F}}H)\cdot A=({}^{o,\mathbb{F}}H\cdot A{)}^T.$$

Full text

The purpose of this work is to study the following two problems:

• A.

If $X$ is $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of $X$ in terms of the $\mathbb{G}$-semimartingale characteristics of $X$.

• B.

If $X$ is a special $\mathbb{G}$-semimartingale but not $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of $\mathbb{F}$-optional projection of $X$ in terms of the $\mathbb{G}$-canonical decomposition and $\mathbb{G}$-semimartingale characteristics of $X$.

In this paper problem B is solved under the assumption that the filtration $\mathbb{F}$ is immersed in $\mathbb{G}$. Beyond the obvious mathematical interest, our study is motivated by important practical applications in areas such as finance and insurance (cf. [BJN19]).

Keywords: Semimartingale, special semimartingale, filtration shrinkage, semimartingale characteristics

Mathematics Subjects Classification (2010): 60G99, 60H99

1 Introduction

This paper is meant to initiate a systematic study of the change of properties of semimartingales under shrinkage of filtrations and, when appropriate, under respective projections. The paper does not aim at a complete and comprehensive study of the topic. Nevertheless, our study contributes, we believe, to understanding of these problems and to giving, in the some specific cases, explicit solutions.

We consider a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, which is endowed with two filtrations, $\mathbb{G}$ and $\mathbb{F}$, assumed to satisfy the usual conditions and such that $\mathbb{F}\subset\mathbb{G}$. On this probability space we consider a real valued $\mathbb{G}$-semimartingale $X$. The results can be generalized to the case of $\mathbb{R}^{n}$ valued semimartingales, in a straightforward manner. We fix a truncation function with respect to which the semimartingale characteristics are computed.

The purpose of this work is to study the following two problems:

• A.

If $X$ is $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of $X$ in terms of the $\mathbb{G}$-semimartingale characteristics of $X$.

• B.

If $X$ is a special $\mathbb{G}$-semimartingale but not $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of the $\mathbb{F}$-optional projection of $X$ in terms of the $\mathbb{G}$-canonical decomposition and $\mathbb{G}$-semimartingale characteristics of $X$.

Note that $\mathbb{G}$-semimartingale characteristics of $X$ are unique up to equivalence, even though they may not determine the law of $X$ uniquely. Thus, the above two problems are well posed.

So, in a sense, we study problems, which are complementary to problems that arise when one studies what happens to a semimartingale under enlargement of filtration, where the main object of interest is study how martingales in a given filtration behave when they are considered in a larger filtration. The goal there is to give, under adequate conditions on the filtration enlargement, their semimartingale decomposition in this larger filtration. The literature regarding enlargement of filtrations is quite abundant (see, e.g., the recent monograph [AJ17] and the references therein). On the contrary, the literature regarding the shrinkage of filtration and its effect on the properties of a semimartingale is essentially non-existent. One can quote the seminal paper of Stricker [Str77] who establishes that a ${\mathbb{G}}$-semimartingale which is ${\mathbb{F}}$-adapted (with ${\mathbb{F}}\subset{\mathbb{G}}$) is an ${\mathbb{F}}$-semimartingale, emphasizing that a ${\mathbb{G}}$-local martingale which is ${\mathbb{F}}$-adapted may fail to be an ${\mathbb{F}}$-local martingale. But the problem of how the semimartingale characteristics change under shrinkage of the filtration is not addressed there. Two notable exceptions are Chapter 4. §6 in [LS89] and Section IX.2 in [Jac79], that feature partial versions of some of our results. Related study is also done in [BY78] where, however, a different, from our special semimartingales, class of processes was investigated (called semi-martingales there). Special cases of our Lemma 3.4 are present in the literature in the context of the filtering theory; see for example Lemma 8.4 in [LS01]. It needs to be stressed that, in general, the problems that we study with regard to shrinkage of filtration are, in general, different from problems studied by the theory of filtering, where, due to the noise in the observation, the observation filtration is not included in the signal filtration.

Also, contrary to the theory of the enlargement of the filtrations, where only initial and progressive enlargements are studied, here we do not make any specific restrictions regarding relation between the filtrations $\mathbb{G}$ and $\mathbb{F}$, except for the inclusion condition $\mathbb{F}\subset\mathbb{G}$, and perhaps some additional conditions, such as the immersion condition in Section 4.

An important motivation behind the study originated in this paper is coming from the theory of stochastic structures that has been under works in recent years (cf. [BJN19]). One of the problems arising in this theory can be summarized as follows: Suppose that $S=(S^{1},\ldots,S^{n})$ is a multivariate semimartingale. Suppose that $(B^{i},C^{i},\nu^{i})$ are the semimartingale characteristics of the semimartingale $S^{i}$ in the natural filtration of $S$. The problem is to find the semimartingale characteristics of $S^{i}$ in the filtration of a sub-group of coordinates $S^{i_{1}},\ldots,S^{i_{k}}$, $i_{1},\ldots,i_{k}\in\{1,\ldots,n\},$ of $S$, in terms of $(B^{i},C^{i},\nu^{i})$. Once it is understood how to do this, then one can proceed with construction of semimartingale structures, which, by definition, are multivariate semimartingales whose components are semimartingales with predetermined marginal characteristics in their own filtrations. In a sense, this corresponds to what is being done in the realm of finite dimensional probability distributions via the classical copula theory, where a multivariate distribution is constructed with given margins. This allows for modeling dependence between components of a multivariate random variable, with preservation of the predetermined marginal distributions. Semimartingale structures find applications in areas such as finance and insurance. For example, in insurance, when designing claim policies for a group of claimants it is important to model dependence between multiple claim processes, subject to idiosyncratic statistical properties of these claim processes. In finance, semimartingale structures come in handy for traders who trade basket derivatives, as well as the individual constituents of these derivatives, and need to make sure that respective models for evolution of the basket price process and the price processes of individual constituents are calibrated in a consistent way. In this regard, quite importantly, semimartingale structures allow for separation of estimating (calibrating) individual (idiosyncratic) characteristics of the components of the structure, from estimation (calibration) of the stochastic dependence between the components of the structure. We refer to [BJN19] for more applications of stochastic structures.

The paper is organized as follows. In Section 2 we formulate the mathematical set-up for our study and we recall some useful concepts and results. In Section 3 we study problem A. In Section 4 we study problem B. In Section 5 we provide several examples illustrating and complementing our theoretical developments. The complexity of the examples varies. But all of them are meant to illustrate our theoretical developments, even though results presented in some of the examples might possibly be obtained directly.

Finally, in Section 6 we formulate some non-trivial open problems, solution of which will require more in-depth understanding of subject matters discussed in this paper.

2 Preliminaries

We begin with recalling the concept of characteristics of a semimartingale. These characteristics depend on the choice of filtration and the choice of so called truncation function. In what follows, we will use the standard truncation function $\chi(x)=x\mathbbm{1}_{\left|x\right|\leq 1}$. Given the truncation function $\chi$, the $\mathbb{G}$-characteristic triple $(B^{\mathbb{G}},C^{\mathbb{G}},\nu^{\mathbb{G}})$ of a $\mathbb{G}$-semimartingale $X$ is given in the following way. First we define the process $X(\chi)$ by

[TABLE]

Since $X(\chi)$ has bounded jumps it is a special $\mathbb{G}$-semimartingale. Thus, it admits a unique canonical decomposition

[TABLE]

where $M^{\mathbb{G}}$ is a $\mathbb{G}$-local martingale such that $M^{\mathbb{G}}_{0}=0$, and $B^{\mathbb{G}}$ is a $\mathbb{G}$-predictable process with finite variation and $B^{\mathbb{G}}_{0}=0$. The process $B^{\mathbb{G}}$ is called the first characteristic of $X$, and this is the only characteristic that depends on the truncation function.

The $\mathbb{G}$-local martingale $M^{\mathbb{G}}$ can be decomposed uniquely into the sum of two orthogonal martingales $M^{\mathbb{G}}=M^{c,\mathbb{G}}+M^{d,\mathbb{G}}$, where $M^{c,\mathbb{G}}$ is a continuous $\mathbb{G}$-local martingale and $M^{d,\mathbb{G}}$ is a purely discontinuous $\mathbb{G}$-local martingale. It can be shown that $M^{c,\mathbb{G}}$ does not depend on the choice of truncation function and it is called the continuous $\mathbb{G}$-martingale part of $X$ and is denoted by $X^{c,\mathbb{G}}$. Then, the second characteristic of $X$ is defined as $C^{\mathbb{G}}=\langle X^{c,\mathbb{G}}\rangle$, where $\langle X^{c,\mathbb{G}}\rangle$ is the predictable quadratic variation process of $X^{c,\mathbb{G}}$. Finally, the third characteristic of $X$ is denoted as $\nu^{\mathbb{G}}$ and is defined as the $\mathbb{G}$-predictable measure which is the $\mathbb{G}$-compensator of $\mu$ - the jump measure of $X$ as defined in Proposition II.1.16 in [JS03]. Clearly, $\nu^{\mathbb{G}}$ does not depend on a choice of truncation function. It is clear that, given a truncation function, the $\mathbb{G}$-characteristic triple $(B^{\mathbb{G}},C^{\mathbb{G}},\nu^{\mathbb{G}})$ is unique (up to equivalence).

In view of Proposition II.2.9 in [JS03], there exists a $\mathbb{G}$-predictable, locally integrable increasing process, say $A^{\mathbb{G}}$, such that

[TABLE]

where

• i.

$b^{\mathbb{G}}$ is an $\mathbb{R}$-valued and $\mathbb{G}$-predictable process,

• ii.

$c^{\mathbb{G}}$ is an $\mathbb{R}_{+}$-valued and $\mathbb{G}$-predictable process,

• iii.

$K^{\mathbb{G}}_{t}(\omega,dx)$ is a transition kernel from $(\Omega\times\mathbb{R}_{+},{\mathcal{P}}_{\mathbb{G}})$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, satisfying condition analogous to condition II.2.11 in [JS03], and where ${\mathcal{P}}_{\mathbb{G}}$ is the $\mathbb{G}$-predictable $\sigma$ -field on $\Omega\times\mathbb{R}_{+}$,

and where $\cdot$ denotes the stochastic or Stieltjes integral, wherever appropriate.

We will assume that

[TABLE]

where $a^{\mathbb{G}}$ is a $\mathbb{G}$-progressively measurable process. This assumption will be satisfied in examples studied in Section 5.

In what follows we use the following notions and notation:

For a given process $Z$, we denote by ${}^{o,\mathbb{F}}Z$ the optional projection of $Z$ on $\mathbb{F}$ defined in the sense of He et al. [HWY92], i.e., the unique $\mathbb{F}$-optional, finite valued process such that for every $\mathbb{F}$-stopping time $\tau$ we have

[TABLE]

Note that by Theorem 5.1 in [HWY92] this optional projection exists if $Z$ is a measurable process such that $Z_{\tau}\mathbbm{1}_{\tau<\infty}$ is $\sigma$-integrable with respect to $\mathcal{F}_{\tau}$ for every $\mathbb{F}$-stopping time $\tau$. That is, there exists a sequence of sets $(A_{n})_{n=1}^{\infty}$ such that $A_{n}\in\mathcal{F}_{\tau}$, $A_{n}\uparrow\Omega$ and $\mathbb{E}(Z_{\tau}\mathbbm{1}_{\tau<\infty}\mathbbm{1}_{A_{n}})<\infty$ for $n=1,2,\ldots$. 2. 2.

For a given process $Z$, we denote by ${}^{p,\mathbb{F}}Z$ the predictable projection of $Z$ on $\mathbb{F}$ defined in the sense of He et al. [HWY92], i.e., the unique $\mathbb{F}$-predictable, finite valued process such that for every $\mathbb{F}$-predictable stopping time $\tau$ we have

[TABLE]

Note that by Theorem 5.2 in [HWY92] this predictable projection exists if $Z$ is a measurable process such that $Z_{\tau}\mathbbm{1}_{\tau<\infty}$ is $\sigma$-integrable with respect to $\mathcal{F}_{\tau-}$ for every predictable $\mathbb{F}$-stopping time $\tau$. That is, there exists a sequence of sets $(A_{n})_{n=1}^{\infty}$ such that $A_{n}\in\mathcal{F}_{\tau-}$, $A_{n}\uparrow\Omega$ and $\mathbb{E}(Z_{\tau}\mathbbm{1}_{\tau<\infty}\mathbbm{1}_{A_{n}})<\infty$ for $n=1,2,\ldots$. 3. 3.

We will also need a notion of $\mathbb{F}$-optional and $\mathbb{F}$-predictable projections for any function $W:\widetilde{\Omega}\rightarrow\mathbb{R}$, which is measurable with respect to $\widetilde{\mathcal{F}}$, where

[TABLE]

The $\mathbb{F}$-optional projection of such a function $W$ is defined as the jointly measurable function ${}^{o,\mathbb{F}}W$ on $\Omega\times\mathbb{R}_{+}\times\mathbb{R}$, which is such that for all $x\in\mathbb{R}$ the process ${}^{o,\mathbb{F}}W(\cdot,x)$ is the optional projection on $\mathbb{F}$ of the process $W(\cdot,x)$. Similarly, the $\mathbb{F}$-predictable projection of such a function $W$ is defined as the function ${}^{p,\mathbb{F}}W$ on $\Omega\times\mathbb{R}_{+}\times\mathbb{R}$, which is such that for all $x\in\mathbb{R}$ the process ${}^{p,\mathbb{F}}W(\cdot,x)$ is the predictable projection on $\mathbb{F}$ of the process $W(\cdot,x)$. 4. 4.

We denote by $\mathcal{O}_{\mathbb{F}}$ (resp. $\mathcal{P}_{\mathbb{F}}$), the $\mathbb{F}$-optional (resp. the $\mathbb{F}$-predictable) sigma-field on $\Omega\times\mathbb{R}_{+}$ generated by $\mathbb{F}$-adapted càdlàg (resp. continuous) processes. Analogously we introduce the sigma fields $\widetilde{\mathcal{O}}_{\mathbb{F}}$ and $\widetilde{\mathcal{P}}_{\mathbb{F}}$ on $\widetilde{\Omega}$ defined by

[TABLE] 5. 5.

A random measure $\pi$ on $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{B}(\mathbb{R})$ is $\mathbb{F}$-optional (resp. $\mathbb{F}$-predictable) if for any $\widetilde{\mathcal{O}}_{\mathbb{F}}$-measurable (resp. $\widetilde{\mathcal{P}}_{\mathbb{F}}$-measurable) positive real function $W$, the real valued process

[TABLE]

is $\mathbb{F}$-optional (resp. $\mathbb{F}$-predictable); equivalently if for any positive real, measurable function $W$ on $\widetilde{\Omega}$

[TABLE]

where $q=o$ (resp. $q=p$). 6. 6.

We say that a random measure $\pi$ on $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{B}(\mathbb{R})$ is $\mathbb{F}$-optionally (resp. $\mathbb{F}$-predictably), $\sigma$-integrable if the measure $M_{\pi}$ on $\widetilde{\mathcal{F}}$ defined by

[TABLE]

restricted to $\widetilde{\mathcal{O}}_{\mathbb{F}}$ (resp. to $\widetilde{\mathcal{P}}_{\mathbb{F}}$), is a $\sigma$-finite measure. In other words $\pi$ is $\mathbb{F}$-optionally, resp. $\mathbb{F}$-predictable, $\sigma$-integrable if there exist a sequence of sets $(\widetilde{A}_{k})_{k=1}^{\infty}$ such that $\widetilde{A}_{k}\in\widetilde{\mathcal{O}}_{\mathbb{F}}$ (resp. $\widetilde{A}_{k}\in\widetilde{\mathcal{P}}_{\mathbb{F}}$), with $M_{\pi}(\widetilde{A}_{k})<\infty$ for each $k$ and $\widetilde{A}_{k}\uparrow\widetilde{\Omega}$. 7. 7.

For a random measure $\pi$ on $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{B}(\mathbb{R})$ we denote by $\pi^{o,\mathbb{F}}$ the $\mathbb{F}$-dual optional projection of $\pi$ on $\mathbb{F}$, i.e., the unique $\mathbb{F}$-optional measure on $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{B}(\mathbb{R})$ such that it is $\mathbb{F}$-optionally $\sigma$-integrable and for every positive $\widetilde{\mathcal{O}}_{\mathbb{F}}$-measurable function $W$ on $\widetilde{\Omega}$, we have

[TABLE]

The $\mathbb{F}$-dual predictable projection of $\pi$ on $\mathbb{F}$, denoted by $\pi^{p,\mathbb{F}}$, is defined analogously, as the unique $\mathbb{F}$-predictable measure on $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{B}(\mathbb{R})$ such that it is $\mathbb{F}$-predictably $\sigma$-integrable and for every positive $\widetilde{\mathcal{P}}_{\mathbb{F}}$-measurable function $W$ on $\widetilde{\Omega}$, we have

[TABLE]

We note that existence and uniqueness of $\pi^{o,\mathbb{F}}$ (resp. $\pi^{p,\mathbb{F}}(dt,dx)$) holds under assumption that $\pi$ is $\mathbb{F}$-optionally (resp. $\mathbb{F}$-predictably), $\sigma$-integrable (see e.g. [HWY92, Theorem 11.8]). 8. 8.

For any process $A$ and any (stopping) time $\vartheta$, we denote by $A^{\vartheta}$ the process $A$ stopped at $\vartheta$. 9. 9.

We use the standard notation $[\cdot,\cdot]$ (resp. $[\cdot]$) for the quadratic co-variation (resp. variation) of real-valued semimartingales. We denote by $\langle\cdot,\cdot\rangle$ (resp. $\langle\cdot\rangle$) the $\mathbb{F}$-predictable quadratic co-variation (resp. variation) of real-valued semimartingales. We denote by $\langle\cdot,\cdot\rangle^{\mathbb{G}}$ (resp. $\langle\cdot\rangle^{\mathbb{G}}$) the $\mathbb{G}$-predictable quadratic co-variation (resp. variation) of real-valued semimartingales. 10. 10.

We use the usual convention that $\int_{0}^{t}=\int_{(0,t]}$, for any $t\geq 0$.

In the rest of the paper we shall study the $\mathbb{F}$-characteristics of $X$ in the case when $X$ is $\mathbb{F}$-adapted, and the $\mathbb{F}$-characteristics of the optional projection of $X$ on $\mathbb{F}$ in the case when $X$ is not $\mathbb{F}$-adapted, providing conditions for such optional projection to exist.

3 Study of Problem A: The Case of $X$ adapted to $\mathbb{F}$

In this section, we consider the case where $X$ is a $\mathbb{G}$-semimartingale, which is $\mathbb{F}$-adapted. Thus, it is an $\mathbb{F}$-semimartingale (see [Str77, Theorem 3.1]).

We start with a result regarding semimartingales with deterministic $\mathbb{G}$-characteristics, that is, semimartingales with independent increments (cf. [JS03, Theorem II.4.15]).

Proposition 3.1**.**

If $X$ has deterministic $\mathbb{G}$-characteristics, then they are also $\mathbb{F}$-characteristics of $X$.

Proof.

In the proof we will use the standard notation for integral of $g$ with respect to a measure $\gamma$, that is

[TABLE]

Note that $X(\chi)$ given in (2.1) is an $\mathbb{F}$-adapted process with jumps bounded by $1$. From assumption (2.4) it follows that the process $B^{\mathbb{G}}$ does not have jumps. Given this, from [JS03, Theorem II.2.21] we know that the processes

[TABLE]

are $\mathbb{G}$-local martingales, where the class $\mathcal{C}^{+}(\mathbb{R})$ of functions is defined in [JS03, II.2.20]. These processes are $\mathbb{F}$-adapted processes since the $\mathbb{G}$ characteristics of $X$ are assumed to be deterministic. Thus, by [Str77, Theorem 3.1], $M^{\mathbb{G}}$ is an $\mathbb{F}$-semimartingale. Moreover, since $M^{\mathbb{G}}$ has bounded jumps, it is a special $\mathbb{F}$-semimartingale. So, by [Str77, Theorem 2.6] it is an $\mathbb{F}$-local martingale. In order to analyze the $\mathbb{G}$–local martingale $Y$ in (3.1) let us take a localizing sequence of $\mathbb{G}$-stopping times $(\tau_{n})_{n=1}^{\infty}$. Then we have

[TABLE]

for every bounded $\mathbb{G}$-stopping time $\sigma$. This implies that

[TABLE]

since the $\mathbb{G}$-characteristics of $X$ are deterministic. So, upon letting $n\rightarrow\infty$, we obtain

[TABLE]

Since $\mathbb{F}\subset\mathbb{G}$ the above holds for all bounded $\mathbb{F}$-stopping times. Thus $(M^{\mathbb{G}})^{2}-C^{\mathbb{G}}-\chi^{2}*\nu^{\mathbb{G}}$