About Stochastic Calculus in Presence of Jumps at Predictable Stopping Times

In this paper, some basic results of stochastic calculus are revised using the following observation: For any semimartingale, the series of jumps at predictable stopping times converges a.s. on any finite time interval, whereas the series of jumps at totally inaccessible stopping times diverges. This implies that when studying random measures generated by jumps of a given semimartingale, it is naturally to define separately a random measure μ generated by the jumps at totally inaccessible stopping times and an other random measure π generated by the jumps at predictable stopping times. Stochastic integrals ( ) p f μ μ ∗ − are well defined for suitable functions f, where p μ is the predictable compensator of μ . Concerning the stochastic integral h π ∗ , it is well defined without any compensating of the integer valued measure π .


Introduction
Stochastic calculus deals with stochastic integrals and stochastic processes constructed by making use of these integrals.
Initially the stochastic integrals were defined with respect to the Wiener process and the Poisson measures by K. Ito (see [1]).An important contribution in the theory of stochastic processes based on stochastic integrals belongs to A. V. Skorokhod [2] (see also I. I. Gihman and A. V. Skorokhod [3]).
The Poisson measures are generated by jumps of stochastically continuous independent increments processes (IIP's).Note that up to subtract a deterministic function, any IIP is a semimartingale.These processes may admit a countable number of small jumps on any finite time interval.For any such process X, the series of jumps Then the stochastic integral is well defined, for a suitable predictable function

( )
, , h h s x ω = . This process possesses the properties: ( ) is a local martingale; when the stochastic integral exists.
Multiple applications of the stochastic calculus have needed an extension of random measures and stochastic integrals, in particular, to consider the integer-valued measures generated by semimartingales.
A general class of random measures suitable for construction of stochastic integrals was studied by J. Jacod [4], R. Liptser and A. Shiryaev [5] (see also Jacod J. and Shiryaev A. [6]).Without loss of generality, we consider random measures generated by jumps of càdlàg semimartingales.
Let µ be an integer-valued measure generated by jumps of a semimartingale X, i.e.
( ) Similarly to case of the Poisson measure, the stochastic integral of kind h µ * does not exist (except a particular case).For this reason, in [5] [7] for a suitable functions h, a stochastic integral ( ) p h µ µ * − is defined, where p µ is a predictable compensator of the measure µ .The properties of this integral are different of those of the above integral with respect to the Poisson measure.In particular,

(
) ( We propose an alternative approach defining stochastic integrals with respect to random measures generated by jumps of semimartingales.
For any semimartingale X, there exist sequences of totally inaccessible and predictable, respectively, stopping times (s.t.'s) which absorb all jumps of X.The graphs of all n S and n T are disjoint (see [1]).The important property of jumps of X at predictable s.t.'s is that, for any t < ∞ , the series This result implies that one can define a stochastic integral with respect to the integer-valued measure generated by the jumps at predictable s.t.'s without making use of the predictable compensator.
In the paper we consider the integer-valued measures µ and π generated by jumps of a semimartingale X at totally inaccessible and predictable, respectively, s.t.'s, and define stochastic integrals ( ) p g µ µ * − and h π * .Note that the second integral is a local martingale or a semimartingale according to properties of the function h.For this second integral, we give necessary and sufficient conditions on the function h for which the process h π * is a semimartingale.Such result was not considered earlier.
Concerning the our integral with respect to the measure p µ µ − it is the same as in [4] [5] if the measure µ there has been generated only by the jumps at totally inaccessible s.t.'s, that is the process generating the measure µ has not the jumps at predictable s.t.'s.
It should be clarified the difference in results of applying the construction of stochastic integrals with respect to the measure given in [4] [5] and that proposed in this paper for the measure π .It turns out that the first construction leads to addition and subtraction of the term example, in the exponential semimartingale (see (29) and Proposition 4).In some other applications the first construction leads to addition and subtraction of the integral with respect to the compensator, p t f π * , as in the Ito formula.In our construction such a kind of addition and subtraction of some terms is not used.
As application, we revise some basic results of stochastic calculus by making use of this construction of stochastic integrals.
One of consequences of this approach is the following innovation representation of any semimartingale (see Theorem 11 and the formula (71)): where , v m are continuous processes, v is of finite variation, m is a local martingale, µ is an integer valued measure with continuous compensator p µ .Note that the innovation representation is important in statistics of random processes.It was used in nonlinear filtering of diffusion processes (see R. Liptser and A. Shiryaev [8]).
The representation is similar to that of IIP's.This representation implies that any semimartingale X can be presented as , The paper is organized as follows.
In Section 2, we give some necessary general notions.In Section 3, the convergence of series of semimartingale jumps at predictable s.t.'s is proved and some direct applications are discussed.Section 4 contains the construction of stochastic integrals with respect to the measures p µ µ − and π generated by a semi- martingale X.Sections 5-6 contain the innovation presentation of semimartingales and the Ito formula, respectively, revised by using the given construction of stochastic integrals.

Some General Notions
≥ Ω  P be a filtered probability space with P -completed right-continuous filtration We denote Let X be a semimartingale, X ∈  .We denote c X the continuous martingale component of X and [X,X] the optional quadratic variation:

Optional and Predictable Projections
Let X be a bounded or positive -adapted process.There exists an ( ) ( ) a.s.for any s.t.T (resp. a.s.for any predictable s.t.S).
The process o X (resp.p X ) is called the optional (resp.the predictable) projection of X on the optional (resp.predictable) σ-field.Each of these projections is unique to within modification on a P-null set (see [9]).

Random Measures
We begin this subsection with some notions and results about random measures (see the book by J. Jacod [4] for details).
Let ( ) , E  be the Lusin space with the borelian σ-algebra (really, we use the case when A random measure µ is called to be integer-valued if

Dual Predictable Projection of a Random Measure
Now we give a basic result on existence of a dual predictable projection (a predictable compensator) of a random measure.
Theorem 1.Let µ be a random measure for which there exists   -predictable partition ( ) ( ) , for any n.Then there exists a unique predictable measure p µ (called a pre- dictable compensator of µ ) verifying the property: where ( ) is the dual predictable projection of the process W µ * .If µ is an integer-valued measure generated by a semimartingale X, then for any predictable s.t.S,

Convergence of Series of Semimartingale Jumps at Predictable s.t.'s
where m is a local martingale, loc m ∈  , A is a process of finite variation on any finite interval a.s., There exist the sequences of totally inaccessible and predictable stopping times (s.t.'s), respectively, which absorb all jumps of X.The graphs of all n S and n T are disjoint.
From finiteness of the optional quadratic variation [ ] , X X it follows that, for any t < ∞ , , a.s.
For the jumps at the predictable s.
We consider some particular cases (see [7]).For any 0 s > , 1) The series when , l n → ∞ , where the second equality follows from orthogonality of martingales and convergence to 0 follows from integrability of optional quadratic variation, H . Choosing a subsequence of indexes n we obtain that this series converges a.s.
Hence the process  [10]).The pre- vious particular cases provide, for any k, , we obtain the statement of theorem.

Applications of Theorem 2
We shall give two applications of this result.
The decomposition is unique to within modification on a P -null set.
Proof.The semimartingale X ′ absorbs all jumps of X at predictable s.t.'s.Hence the process qc X X X′ = − is a quasi left continuous semimartingale. The exponential semimartingale.Let X be a semimartingale.It is well-known the exponential semi-martingale (called the Dolean exponential) ( ) where the infinite product converges a.s.for any t < ∞ and it is the process of finite variation.The semimartingale Z is a unique solution of the equation 0 d , 0.
The following result gives an other form of the solution of Equation (30) taking into account the Theorem 2.
Proposition 4. Let X be a semimartingale from (20) and ( ) ( ) n n S T be the sequences of predictable and totally inaccessible, respectively, s.t.'s from (21).Then the exponential semimartingale ( ) ( ) is the solution of the Equation (30), where ∏ converges a.s.for any t < ∞ and it is a semimartingale, the product ( ) is the process of finite variation for any t < ∞ .
In particular, if the semimartingale X has the jumps only at predictable s.t.'s ( ), then the exponential semimartingale t Z  is as follows: ( ) Proof.Due to Theorem 2 and Proposition 3, the Dolean exponential (29) can be presented as Z  in (31).
One has to show only that the product ( ) ∏ converges a.s. and it is a semimartingale.To that end, note that there is a finite number of jumps such that 1 2 n S X ∆ > . Hence the process ( ) is of finite variation for any t < ∞ .Denote For the process lnU one has where 0 1 θ ≤ ≤ .The first series on the right-hand size converges a.s. and it is a semimartingale, due to Theorem 2, and the second one converges absolutely and it is a process of finite variation being bounded by the series Therefore, the process lnU is a semimartingale and by the Ito formula (see Lemma 2), the processes ln e U V

Stochastic Integrals with Respect to the Random Measures µ − µ p and π
Let X be a semimartingale with values in E.
On the product space ( ) ( )  , we define two integer-valued random measures where { } ( ) We denote by p µ (resp.p π ) the predictable compensator of µ (resp., of π ).Since X has not a jump at the 2) ( ) For the proof of this result (see J. Jacod [4], Theorem 2.45).

Stochastic Integrals with Respect to the Random Measures µ − µ p .
Let us introduce the functional spaces, for 1, 2 q = , ( ) where  (resp.loc  ) denote the space of processes of integrable (resp.locally integrable) variation.By making use of Lemma 1, we obtain the following results about stochastic integrals with respect to the random measure p µ µ − .This integral is the same that is given in [4] [5], when the predictable compensator p µ is continuous (see Proposition 5).

Theorem 7. Let f be 
 -measurable function.For existence a unique process it is necessary and sufficiently that ( ) The process Z is called to be the stochastic integral , , | 0 a.s.. .

Stochastic Integrals with Respect to the Random Measure π
Now we consider stochastic integrals with respect to the measure π which is a purely discontinuous local martingale.Theorem 9. Let h be   -measurable function.Denote, for 0 t > , ( For existence a unique process it is necessary and sufficiently that The process Z is called to be the stochastic integral Proof.We have to verify only the condition 0 Due to Theorem 1, ( ) The process Z is called to be the stochastic integral ( ) p h π π * − (see [4] [5]).Since the jumps are the same, Z h ∆ =  , we have two different forms of the same process ˆˆ, .

Semimartingale Stochastic Integrals
We have studied stochastic integrals which are local martingales.Now we consider a stochastic integral with respect to the integer-valued measure π that is a semimartingale.
Denote by dp  the space of semimartingales that are purely discontinuous with jumps at predictable s.   ,  , d  ,  , d .
Therefore, Â h ∆ = and ( ) J is the complement of J. Therefore, as ( ) ( ) where , v m are continuous processes, Proof.From the definition of the measure π , one has, for any and due to Theorem 1, the stochastic integral in the right-hand side is a ( ) The process di X is a ( ) X   -measurable semimartigale being the sum of a X  -local martingale and a ( ) X   -measurable process of locally finite variation.
di X absorbs all jumps of X at times n T .Indeed  [11] [12]).
Remark 7. It is known that the semimartigale property is stable with respect to a narrowed filtration (see, for example, [4]).In our case, the result claims that any ( ) X   -measurable process from ( )   belongs also to ( ) X   .

The Ito Formula
Lemma 2. Let ( ) ( ) Taking into account that, for any predictable stopping time S and any totally inaccessible stopping time T,[ ] [ ]

2 .
It follows from Lemma 1.  Remark We have for optional quadratic variation of

Remark 5 .
For the optional quadratic variation of ( ) The Ito formula is well known when the semimartingale (82) has not the last term t h π * .We explain only that the last term in (83) is well defined and it is a semimartingale.Denote


we have to verify the conditions of corollary of theorem 10.One has = , this integral equals the above series of jumps).For this reason, one must use the compensated Poisson measure p − Π with } t.'s we get the following stronger result.
It should be noted that in the exponential (29) the term