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We consider the stopping time problems whose costs are given by time average forms, a generalization of the Gittins index. We show the existence of the optimal solutions and give their approximation to these stopping time problems.

Unfortunately, the proof of Proposition 4.8 in [HW13] is not correct. In fact, equation (4.60) whereRθ is rewritten as a stochastic integral is not correct, because the integrand is not adapted to the filtration. Hence in (4.61) we are not allowed to apply the Burkholder-Davies-Gundy inequality. The argument was corrected in [HMW12, Lemma 3.6 and Corollary 3.7] in a more complicated situation. In order to transfer these statement to the situation in [HW13] some smaller changes are needed. First of all the regularity of the linearised process X should be measured in a Holder norm with slightly bigger index α∗ than the solution u. This can be done without further problems. Then the definition of the stopping time in (4.53) becomes. For K > 0 and for an α? ∈ (α, 1/2) we introduce the stopping time

If X is a symmetric Levy process on the line, then there exists a non-decreasing, cadlag process H such that X(H(x)) = x for all x≥ 0 if and only if X is recurrent and has a non-trivial Gaussian component. The minimal such H is a subordinator K. The law of K is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion, K is just the usual ladder times process and this result extends the classical result of Levy that the maximum process has the same law as the local time at 0. Write G t for last point in the range of K prior to t. In a parallel with classical fluctuation theory, the process Z := (X t −X Gt ) t ≥0 is Markov with local time at 0 given by (X Gt ) t ≥0. The transition kernel and excursion measure of Z are identified. A similar programme is outlined for Levy processes on the circle. This leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle.

In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality.

As a first step in the development of a general theory of set-indexed martingales, we define predictability on a general space with respect to a filtration indexed by a lattice of sets. We prove a characterization of the predictable σ-algebra in terms of adapted and “left-continuous” processes without any form of topology for the index set. We then define a stopping set and show that it is a natural generalization of the stopping time; in particular, the predictable σ-algebra can be characterized by various stochastic intervals generated by stopping sets.

LetX be a transient right process for which semipolar sets are polar. We characterize the measures which can arise as the distribution ofX T withT a non-randomized stopping time.

Consider the set $$C$$ of all possible distributions of triples (τ, κ, η), such that τ is a finite stopping time with associated mark κ in some fixed Polish space, while η is the compensator random measure of (τ, κ). We prove that $$C$$ is convex, and that the extreme points of $$C$$ are the distributions obtained when the underlying filtration is the one induced by (τ, κ). Moreover, every element of $$C$$ has a corresponding unique integral representation. The proof is based on the peculiar fact that EV τ, κ=0 for every predictable processV which satisfies a certain moment condition. From this it also follows thatT τ, κ isU(0, 1) wheneverT is a predictable mapping into [0, 1] such that the image of ζ, a suitably discounted version of η, is a.s. bounded by Lebesgue measure. Iterating this, one gets a time change reduction of any simple point process to Poisson, without the usual condition of quasileftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.

Given (μt)t≥0 a transient convolution semigroup over Rd, we characterize the transient semi-groups which are subordinate in the wide sense to the latter, via some quotient relation between the two Levy generators. This is also equivalent to an inclusion between excessive cones. Thanks to a theorem of Hirsch, this allows us in big dimensions to give a probabilistic description of some measures satisfying the classical maximum principle.

We study some features concerning the occupation timeAt of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A1

We study the distance in variation between probability measures defined on a measurable space (Ω, ℱ) with right-continuous filtration (ℱt)t ≦0. To every pair of probability measures P and $$\tilde P$$ an increasing predictable process $$h = h(P,{\text{ }}\tilde P)$$ (called the Hellinger process) is associated. For the variation distance $$\left\| {P_T - \tilde P_T } \right\|$$ between the restrictions of P and $$\tilde P$$ to ℱ T (T is a stopping time), lower and upper bounds are obtained in terms of h. For example, in the case when $$P_0 = \tilde P_0 $$ , $$2(1 - (E{\text{ }}\exp {\text{ }}( - h_T ))^{1/2} ) \leqq \left\| {P_T - \tilde P_T } \right\| \leqq 4(Eh_T )^{1/2} $$ In the cases where P and $$\tilde P$$ are distributions of multivariate point processes, diffusion-type processes or semimartingales h are expressed explicitly in terms of given predictable characteristics.

Summary. We prove that the derivative of a diAerentiable family XtOaU of continuousmartingalesinamanifold M isamartingaleinthetangentspacefor the complete lift of the connection in M, provided that the derivative is bicontinuousin tand a.WeconsiderafilteredprobabilityspaceOX;OFtU0t1; PU such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a7! LOaU of random variable with values in M and such that LO0U x 02 M, there exists an analytic family a7! XOaUof continuous martingales such that X1OaUa LO aU . Forthis,weinvestigatetheconvexityofthetangentspaces T OnU M,andweprove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time T such that POT < 1U is arbitrarilysmall.Weusethisconstructionoffamiliesofmartingalesincomplex analytic manifolds to prove that every F1-measurable random variable with valuesinacompactconvexset V withconvexgeometryinamanifoldwitha C 1 connection is reachable by a V -valued martingale. Mathematics Subject Classification (1991): 60G44

A super-Brownian motion X in ℝ with "hyperbolic" branching rate ρ2 (b) = 1/b2, b ∈ ℝ, is constructed, which symbolically could be described by the formal stochastic equation dXt = ½ ΔXt dt + √2ρ2XtdWt, t ≥ 0, (1) (with a space-time white noise W). If the finite starting measure X0 does not have mass at b = 0, then this superprocess X will never hit the catalytic center: There is Brownian stopping time r strictly smaller than the hitting time of 0 such that Dynkin's stopped measures Xr vanishes except a.s.

Let\(M = \left( {M_t , \mathfrak{F}_t } \right)\) be a continuous BMO-martingale. Then the associated exponential martingale ℰ(M) satisfies the reverse Holder inequality $$\left( {R_p } \right)E[\mathcal{E}\left( M \right)_\infty ^p |\mathfrak{F}_T ] \leqq C_p \mathcal{E}\left( M \right)_T^p $$ for somep>1, whereT is an arbitrary stopping time (see [3, 5]). Our claim is, in a word, that the (Rp) condition bears upon the distance betweenM andL∞ if BMO. Especially, we shall prove thatM belongs to the BMO-closure ofL∞ if and only if ℰ(λM) satisfies all (Rp) for every real number λ. Some related problems are also considered.