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Absolutely Continuous Compensators U.U.D.M. Report 2010:5 Absolutely Continuous Compensators Svante Janson, Sokhna M’Baye, and Philip Protter Department of Mathematics Uppsala University Absolutely Continuous Compensators Svante Janson∗ and Sokhna M'Bayey and Philip Protterz May 12, 2010 Abstract We give sufficient conditions on the underlying filtration such that all to- tally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a represen- tation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all com- pensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However C¸inlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and slightly of space, have such a representation. 1 Introduction The celebrated Doob-Meyer Decomposition Theorem states that if X is a sub- martingale, then it can be written in the form X = M + A where X is a local martingale and A is a unique, c`adl`agincreasing predictably measurable pro- cess with A0 = 0. (See, for example, [26].) In the case of a point process of N = (Nt)t≥0 it is trivially a submartingale, and hence we know there exists a process A such that N − A is a local martingale. A special case of interest in the theory of Credit Risk is the case 1ft≥Rg − At = a martingale (1) The process A in (1) is known as the compensator of the stopping time R, by an abuse of language. It is common in applications to assume, often without ∗Uppsala University, Department of Mathematics, P.O. Box 480, SE-751 06 Uppsala, Swe- den yD´epartement de de Math´ematiques, Ecole´ Normale Sup´erieurede Cachan, 61 Avenue du Pr´esident Wilson, 94235 Cachan Cedex, France zSchool of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853; Supported by NSF Grant DMS-0906995 1 mention, that A has absolutely continuous paths. That is, one often assumes a priori that (1) is of the form Z t 1ft≥Rg − λsds = a martingale (2) 0 for some adapted process λ. The process λ is often referred to as the hazard rate and has intuitive content as the instantaneous likelihood of the stopping time R occurring in the next infinitesimal time interval. Of course this is not true in general, and for example in the theory of credit risk K. Giesecke and L. Goldberg have given a natural example where it does not hold [11, C3, p. 7]. The goal of this paper is to give simple and natural conditions on the generating underlying filtration to show when the compensators of all of the totally inaccessible stopping times are absolutely continuous; that is, to give sufficient conditions on the filtration such that they all have hazard rates. 2 Prior Results Previous work has been restricted to giving conditions on a given stopping time in relation to the underlying filtration that ensures the compensator is absolutely continuous. Perhaps the most well known of these conditions is that of S. Ethier and T.G. Kurtz [8], which we restate here. Theorem 1 (Ethier{Kurtz Criterion) Let G = (Gt)t≥0 be a given filtration satisfying the usual hypotheses (see [26] for the \usual hypotheses.") Let A be an increasing (not necessarily adapted) and integrable c´adl´agprocess, with A0 = 0. Let A~ be the G compensator of A. If there is a constant K such that for all 0 ≤ s ≤ t EfAt − AsjGsg ≤ K(t − s) a.s. (3) then the compensator of A has absolutely continuous paths, a.s. That is, it is of ~ R t the form At = 0 λsds. An extension of Theorem 1 to necessary and sufficient conditions for the G compensator to have such an intensity process is given in the Cornell PhD thesis of Yan Zeng [28]. A trivial extension is to replace the constant K with an increasing predictable process (Kt)t≥0, and then the inequality (3) becomes: EfAt − AsjGsg ≤ Ks(t − s) a.s. (4) and of course the conclusion in Theorem 1 still holds. Zeng [28, p. 14] did a little better: Theorem 2 (Yan Zeng) Let A be an increasing (not necessarily adapted) and integrable measurable process, with A0 = 0. Let A~ be the compensator of A. Then dA~t dt if and only if there exists an increasing and integrable measurable process D with D0 = 0, such that dD~ t dt and for all t ≥ 0; h ≥ 0, EfAt+h − AtjGtg ≤ EfDt+h − DtjGtg (5) and if equality holds, then we have A~ = D~. 2 Another observation is perhaps useful to make. Once a stopping time has an absolutely continuous compensator in a given filtration, say G, then if it is also a stopping time for a smaller filtration it also has an absolutely continuous compensator in the smaller filtration. Actually one can obtain a more precise result, which is established in the book of Martin Jacobsen [13]. We provide here an original and elementary proof of this result, and inter alia we extend the result a little. R t Theorem 3 Let R be a G stopping time with compensator given by 0 λsdc(s) for some G adapted process λ, where G satisfies the usual hypotheses. Here s 7! c(s) is non random, continuous, and non-decreasing. Let F be a subfiltration of G also satisfying the usual hypotheses, and suppose R is also an F stopping R t o 1 time. Then the F compensator of R is given by 0 λsdc(s). That is we have Z t If 1ft≥Rg − λsdc(s) = a martingale in G 0 Z t o then 1ft≥Rg − λsdc(s) = a martingale in F: (6) 0 Proof. Let Z t Mt = 1ft≥Rg − λsdc(s): (7) 0 o Then M is a G martingale. Since λs ≥ 0, the optional projection λs exists o o with 0 ≤ λs ≤ 1. For every s, E λs = Eλs, and thus, by Fubini's theorem, R 1 o R 1 R t o E 0 λs dc(s) = E 0 λs dc(s) = E1fR<1g < 1. Thus, At = 0 λs dc(s) is o an integrable increasing continuous adapted process; in particular, λs < 1 for a.e. s a.s. We define Z t o Lt = 1ft≥Rg − At = 1ft≥Rg − λs dc(s): (8) 0 If 0 ≤ s ≤ t and H is bounded and Fs measurable, then, by Fubini's theorem o and the fact that for fixed r, λr = EfλrjFrg a.s., Z t E H(Lt − Ls) = E H1ft≥R>sg − E HE(λrjFr) dc(r) s Z t (9) = E H1ft≥R>sg − E(Hλr) dc(r) s = E H(Mt − Ms) = 0: Hence the uniformly integrable process Lt is an F martingale. This gives the F canonical decomposition of the F submartingale 1ft≥Rg as 1ft≥Rg = Lt + At, and thus A is the F compensator of 1ft≥Rg. We include for emphasis the following obvious but important (and well known) corollary: 1 o Note that for fixed s, λs = EfλsjFsg a.s.; the optional projection gives a method to define the projection via conditional expectation for all s ≥ 0 simultaneously. Since λ is positive, the optional projection exists. See [26] for more details. 3 R t Corollary 4 Let R be a G stopping time with compensator given by 0 λsds for some G adapted process λ, where G satisfies the usual hypotheses. Let F be a subfiltration of G also satisfying the usual hypotheses, and suppose R is also an R t o F stopping time. Then the F compensator of R is given by 0 λsds. That is we have Z t If 1ft≥Rg − λsds = a martingale in G 0 Z t o then 1ft≥Rg − λsds = a martingale in F: (10) 0 Theorem 3 and its Corollary 4 show that once there is a filtration H such that a stopping time R is totally inaccessible, if R has an AC compensator in H, then it has an AC compensator in any smaller filtration G as well. In particular Dellacherie's result (Theorem 5 below) implies that the law of R is absolutely continuous (ie, has a density) as well. We recall Dellacherie's result here for the reader's convenience. A proof can be found in [26, p. 120]. Theorem 5 (Dellacherie's Theorem) Let R be a nonnegative random vari- able with P (R = 0) = 0;P (R > t) > 0 for each t > 0. Let Ft = σ(t ^ R), the minimal filtration which renders R a stopping time. Let F denote the law of R. That is, F (x) = P (R ≤ x) for x ≥ 0. Then the compensator A = (At)t≥0 of the process 1fR≥tg is given by Z t 1 At = dF (u): 0 1 − F (u−) If F is continuous, then A is continuous, R is totally inaccessible, and At = − ln(1 − F (R ^ t)). One may ask if that, once a compensator of a stopping time R is a.s. sin- gular with respect to Lebesgue measure, does that propagate down to smaller filtrations, and in particular does it imply that the law of the stopping time is singular as well? The next example shows that this is not true in general. Example 6 Let B be a standard one dimensional Brownian motion with nat- ural filtration F and with a local time at zero L = (Lt)t≥0. Define the change of time τt = inffs > 0 : Ls > tg: Then (τt)t≥0 is a family of F stopping times.
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