Jacod, J. (1975) Multivariate point processes: predictable projection, Radon-Nikodym derivative, representation of martingales, Z. fur Wahrscheinlichkeitstheorie verw. Gebiete (Probab. Theory Appl.) 31, 235--253.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....N(dt Theta L) is non explosive (recall that by Lemma 1, the left hand side of the previous equation is an F t stochastic intensity of NL ) In order to conclude from this bound on the stochastic intensity of NL that NL = 0 a.s. it is enough to appeal to the following Lemma from Jacod [13] (see also Lemma 1 in [4] Lemma 3 . Let N be a simple point process on R, admitting an F N t predictable intensity (t) v(t; N) on R . Then for all t 2 R [ f 1g the conditional probability P(N( 0; t] 0 j F N 0 ) equals exp Gamma R t 0 v(s; S 0 N Gamma )ds. Indeed, this ....

....Q(D) 1, the process fZ t g associated with g(z) 1D (z) is regular. Remark 2 . Strong regularity of a point process N on R Theta L obviously implies its weak regularity. Both notions depend on the measure Q, and coincide when L is a finite set. If we take the usual definition (see e.g. [13]) of a simple point process on R as the random measure N associated to a a collection of positive random variables T n such that T n T n 1 , and T n T n 1 on fT n 1g through N(C) X n 0 1C (T n )1 Tn 1 then the associated process Z t = N(0; t] is regular in the sense of the above ....

Jacod, J. (1975), Multivariate point processes: Predictable projections, Radon-Nikodym derivatives, representation of martingales, Z. Wahrs. 31, 235-253. 24

....next coupon date. 5 Martingale representation According to Remark 2.7, the gain process G given by (2.15) for the defaultable contingent claim (C; Z) may be regarded as an asset, and the discounted gain process is an fH t g martingale. Using general martingale representation results due to Jacod [21] (see Hugonnier [20] for their application here) one can show that this martingale has a stochastic integral representation. Rather than appeal to these general results, we work out this representation from rst principles and identify the integrand for the martingale M . This martingale ....

Jacod, J. (1975) Multivariate point processes: predictable projection, Radon-Nikodym derivative, representation of martingales, Zeitschrift fur Wahrscheinlichkeitstheorie verwandte Gebiete (Probab. Theory and Appl.) 31, 235-253. 39

....appropriate (predictable) versions. We shall use conditional failure intensities to formally introduce our class of repairable systems. The following fundamental result exhibits a version of a failure intensity with respect to the internal (natural) history F N t : oe(N(s) s t) see e.g. Jacod (1975). Theorem 2.1 Assume that the conditional distribution P (T n 1 Gamma T n xjT 1 ; T n ) n 0, is absolutely continuous with respect to the Lebesgue measure, i.e. P (T n 1 Gamma T n xjT 1 ; T n ) Z t 0 f (n 1) u)du; where f (n 1) u) is a random function, B Omega ....

....; Z ; V 0 ; T 1 ; Z 1 ; T n ; Z n ) t 0; 2.2) is a conditional failure intensity of N with respect to the internal history F t = oe(Z ; V 0 ) oe( Phi( 0; s] Theta B) s t; B 2 B) t 2 IR : 2.3) This is an extension of Theorem 2. 1 to the context of marked failure processes, see Jacod (1975) and Last and Brandt (1995) 2.2 A model of repairable systems In this section we introduce the model in detail. A new item operates according to a life time distribution function F such that F(0 ) 0. We assume throughout the existence of the generic failure rate function r(t) which satisfies ....

Jacod, J. (1975) Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales, Z. Wahrsch. verw. Geb. 31, 235--253.

....0 1f t Phi 2 Bgdt to obtain a stationary probability measure Q on NE , see Baccelli and Br emaud (1994, p. 27) for more details about this construction. From ( and the inverse construction we see that ( hold for n 1, see Br emaud and Massouli e (1994) for a similar argument) By Jacod s (1975) formula for compensators of marked point processes and taking into account stationarity of Q and shift consistency of the repair kernel D we obtain for all predictable f : NE Theta R Theta ( Gamma1; 1] R by a limiting argument that EQ Z f(t; z) Psi(d(t; z) EQ Z f(t; z) d(t; z) ....

....Am as defined in Davis (1984) and modified in Meyn and Tweedie (1993a) Let F t : oe( Phi( a; b] Theta C) a b t; C 2 B(E) t 0; and define the predictable oe field P as the smallest oe field over Omega Theta R containing the sets B Theta (s; t] for all B 2 F s and s t. By Jacod s (1975) formula, E Z 1ft 0gf(t; z) Psi(d(t; z) E Z 1ft 0gf(t; z) d(t; z) 2.14) for all predictable f : Omega Theta R Theta ( Gamma1; 1] R (f is measurable and P measurable in the first two arguments) where as in ( d(t; z) 1ft T1gr(V (t Gamma) D L (dz; A(t Gamma) V ....

J. Jacod, Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales, Z. Wahrsch. verw. Geb. 31 (1975) 235--253.

....jump; see Yushkevich (1980) This construction is relatively complicated. For example, it transforms a CTJMDP with nite state and action sets into an SMDP with the same state space and Borel action sets. Kitaev (1984) constructed past dependent and randomized strategies in an elegant way by using Jacod s (1975) results on predictable projections of random measures; see also Kitaev and Rykov (1995) The main approach used in this paper is based on a new result that a CTJMDP is equivalent to the corresponding ESMDP. Before we describe our results, we make some remarks concerning terminology used in this ....

....the current state and time. Yushkevich (1980) constructed appropriate stochastic processes and expectations by using reduction of the CTJMDPs to SMDPs with actions being Borel mappings from R to the sets of available functions. Kitaev (1985) gave an equivalent construction using the results by Jacod (1975). In this section we introduce both constructions and explain why they are equivalent. We describe the approach introduced by Yushkevich (1977, 1980) rst. According to this approach, a CTJMDP is an SMDP with the set of actions at each state x 2 X being a set of all Borel functions from R to ....

[Article contains additional citation context not shown here]

Jacod, J. (1975). Multivariate point processes: predictable projections, Radon-Nikodym derivatives, representation of martingales. Z. Wahr. verw. Geb. 31 235-253.

....measures P and Q, let P Q mean that P is absolutely continuous to Q. Lemma 3.3 Let 0 be a point in Theta such that R 1 0 xf t (xj 0 )dx 0 for all t 2 [0; Then for any given fi 0 2 R p we have P Nn rn (fi) P Nn rn (fi 0 ) 0 for all and fi. Proof From the results in Jacod (1975) and lemma 3.1 and lemma 3.2, we can see that P N 1 r 1 (fi) P N 1 r 1 (fi 0 ) 0 and P Nn 1 r n 1 (fi) Delta j n n ) P Nn 1 r n 1 (fi 0 ) 0 ( Delta j n n ) for all n n 2 N n , which completes the proof. Denote by L(N k jN k Gamma1 ) the Radon Nikodym derivative of P N k r k ....

.... Radon Nikodym derivative of P N k r k (fi) DeltajN k Gamma1 ) with respect to P N k r k (fi 0 ) 0 ( DeltajN k Gamma1 ) for all k = 1; 2; Then the simple calculation yields that dP N k r k (fi) dP N k r k (fi 0 ) 0 = k Y i=1 L(N i jN i Gamma1 ) From the result in Jacod (1975), we can write L(N k jN k Gamma1 ) Y t2[0; i Y k (t)dA 0 r k Gamma1 (fi) tjr k ) j DeltaN k (t) Theta Y t2[0; i 1 Gamma Y k (t)dA 0 r k Gamma1 (fi) tjr k ) j 1 Gamma DeltaN k (t) Bayes inference for counting process 15 We have thus established the likelihood ....

Jacod, J. (1975). Multivariate point processes: predictable projection, Radon-Nikodym derivative, representation of martingales. Z. Warsch. verw. Gebiete 31, 235-253.

....B 2 F s and s t. Let f : NE Theta IR Theta ( Gamma1; 1] IR be predictable, i.e. measurable and P measurable in the first two arguments. From (2.12) and the inverse construction we see that (2.5) 2.6) hold for n 1, see also Br emaud and Massouli e 5 (1994) for a similar argument. By Jacod s (1975) formula for compensators of marked point processes we have EQ Z 1ft T 1 gf(t; z) Psi(d(t; z) EQ Z 1ft T 1 gf(t; z) d(t; z) 2.14) where Psi is the marked point process ( T n ; Z n ) n 2 ZZ) and is the following random measure on IR Theta ( Gamma1; 1] d(t; z) r(V ....

....in Davies (1984) As Meyn and Tweedie (1993a) we in fact use a slightly modified definition of this generator. As in the canonical framework we define the predictable oe field P as the smallest oe field over Omega Theta IR containing the sets B Theta (s; t] for all B 2 F s and s t. By Jacod s (1975) formula, E Z 1ft 0gf(t; z) Psi(d(t; z) E Z 1ft 0gf(t; z) d(t; z) 2.19) for all predictable f : Omega Theta IR Theta ( Gamma1; 1] IR (f is measurable and P measurable in the first two arguments) where as in (2.15) d(t; z) 1ft T1 gr(V (t Gamma) D L (dz; ....

J. Jacod, Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales, Z. Wahrsch. verw. Geb. 31 (1975) 235--253.

....x; v) Y (s; x)1 fx6=0g 1 Gamma a 0 s 1 Gamma a s 1 fx=0g : Hence the representation (2.6) of the full likelihood Z reduces to the partial likelihood Z. For multivariate point processes, this observation is due to Arjas and Haara (1984) Their condition A is contained in (3. 1) They use Jacod s (1975) representation of the full likelihood. The reduction of Z to Z under (3.1) also follows from our factorization. The partial likelihood is a projection. Suppose we are given a parametric model for the partial specification B, but leave the model completely unspecified otherwise. If the full ....

Jacod, J. (1975). Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 235--253.

....TNn =N n m a.s. This implies n=N n m a.s. Proof of Lemma 4. To prove local asymptotic normality, we need a representation of the likelihood ratio in terms of the transition distributions Q and Q nh . It is obtained by specializing the representation for multivariate point processes due to Jacod (1975). Associate with the semi Markov process Y a multivariate point process, i.e. a random measure on (0; 1) Theta E, dt; dy) X i1 (T i ;X i ) dt; dy) If the transition distribution is Q, the random measure has compensator (dt; dy) Q(X t Gamma ; dy; dt Gamma TN t Gamma ) Q 2 i ....

.... point process, i.e. a random measure on (0; 1) Theta E, dt; dy) X i1 (T i ;X i ) dt; dy) If the transition distribution is Q, the random measure has compensator (dt; dy) Q(X t Gamma ; dy; dt Gamma TN t Gamma ) Q 2 i X t Gamma ; t Gamma TN t Gamma ; 1) j : See Jacod (1975, Proposition (3.1) or Jacod and Shiryaev (1987, p. 136, Theorem 1.33) Set a t = ftg Theta E) c (dt; dy) 1(a t = 0) dt; dy) Write nh and a nh t if the transition distribution is Q nh , and define Y nh by nh (dt; dy) Y nh (t; y) dt; dy) Since the initial distribution j of the ....

[Article contains additional citation context not shown here]

Jacod, J. (1975). Multivariate point processes: predictable projection, Radon--Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 235--253.

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Jacod, J. (1975) Multivariate point processes: predictable projection, Radon-Nikodym derivative, representation of martingales, Z. fur Wahrscheinlichkeitstheorie verw. Gebiete (Probab. Theory Appl.) 31, 235--253.

No context found.

J. Jacod, Multivariate point processes: predictable projections, Radon-Nikodym derivatives, representation of martingales, Z. Wahrsch. Verw. Gebiete 31 (1971) 235-253.

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Jacod, J. (1975): Multivariate Point Processes: Predictable Projection, RadonNikodym Derivatives, Representation of Martingales. Z. Wahrsch. verw. Gebiete 31, 235-253.

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