Our focus in this work is to investigate an efficient state estimation scheme for a singularly perturbed stochastic hybrid system. As stochastic hybrid systems have been used recently in diverse areas, the importance of correct and efficient estimation of such systems cannot be overemphasized. The framework of nonlinear filtering provides a suitable ground for on-line estimation. With the help of intrinsic multiscale properties of a system, we obtain an efficient estimation scheme for a stochastic hybrid system.

In this note we re-examine the analysis of the paper ‘On the martingale property of stochastic exponentials’ by Wong and Heyde (2004). Some counterexamples are presented and alternative formulations are discussed.

The limit behaviour in probability of realised quadratic variation is discussed under a relatively simple ambit process setting. The relation of this to the underlying volatility/intermittency field is in focus, especially as concerns the question of no volatility/intermittency memory.

For a bivariate Lévy process (ξt,ηt)t≥ 0 and initial value V0 define the generalised Ornstein–Uhlenbeck (GOU) process Vt:=eξt (V0+∫t0 e-ξs-dηs), t≥0, and the associated stochastic integral process Zt:=∫0t e-ξs-dηs, t≥0. Let Tz:=inf{t>0: Vt<0|V0=z} and ψ(z):=P(Tz<∞) for z≥0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for ψ(z) and the distribution of Tz as z→∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.

We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spectrally one-sided Lévy process. We show that this SDE has some connections with queueing and storage models, and we use this observation to obtain the invariant distribution.

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

Let (Ω,ℱ,ℙ) be a probability space and Z=(ZK)k∈ℕ a Bernoulli noise on (Ω,ℱ,ℙ) which has the chaotic representation property. In this paper, we investigate a special family of functionals of Z, which we call the coherent states. First, with the help of Z, we construct a mapping ϕ from l2(ℕ) to ℒ2(Ω,ℱ,ℙ) which is called the coherent mapping. We prove that ϕ has the continuity property and other properties of operation. We then define functionals of the form ϕ(f)with f∈l2 (ℕ)as the coherent states and prove that all the coherent states are total in ℒ2 (Ω,ℱ,ℙ) . We also show that ϕ can be used to factorize ℒ2 (Ω,ℱ,ℙ) . Finally we give an application of the coherent states to calculus of quantum Bernoulli noise.

Within an anticipative stochastic calculus framework, we study a market game with asymmetric information and feedback effects. We derive necessary and sufficient criteria for the existence of Nash equilibria and study how general welfare is affected by the level of information. In particular, we show that, under certain conditions in a competitive environment, an increased level of information may in fact lower the level of general welfare, leading to the so-called Hirshleifer effect (see Hirshleifer (1971)). Finally, we determine equilibrium prices for particular pieces of information, by extending our market game with a pre-stage, in which information is traded.

We establish an upper bound on the tails of a random variable that arises as a solution of a stochastic difference equation. In the nonnegative case our bound is similar to a lower bound obtained in Goldie and Grübel (1996).

In this paper we consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google's PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form where N, Q, {Ri}i≥1, and {C, Ci}i≥1 are independent nonnegative random variables, the {C, Ci}i≥1 are identically distributed, and the {Ri}i≥1 are independent copies of stands for equality in distribution. We study the asymptotic properties of the distribution of R that, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found to be useful in the analysis of many other algorithms. Our first main result shows that if ENE[Cα] = 1, α > 0, and Q, N satisfy additional moment conditions, then R has a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie's (1991) implicit renewal theorem. Furthermore, when N is regularly varying of index α > 1, ENE[Cα] < 1, and Q, C have higher moments than α, then the distributions of R and N are tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution of R is determined by the tail of Q. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.

We consider a portfolio optimization problem in a defaultable market. The investor can dynamically choose a consumption rate and allocate his/her wealth among three financial securities: a defaultable perpetual bond, a default-free risky asset, and a money market account. Both the default risk premium and the default intensity of the defaultable bond are assumed to rely on some stochastic factor which is described by a diffusion process. The goal is to maximize the infinite-horizon expected discounted log utility of consumption. We apply the dynamic programming principle to deduce a Hamilton-Jacobi-Bellman equation. Then an optimal Markov control policy and the optimal value function is explicitly presented in a verification theorem. Finally, a numerical analysis is presented for illustration.

A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.

We present conditions that imply the conditional full support (CFS) property, introduced in Guasoni, Rásonyi and Schachermayer (2008), for processes Z := H + ∫K dW, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case, under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.

We examine the long-time behavior of forward rates in the framework of Heath-Jarrow-Morton-Musiela models with infinite-dimensional Lévy noise. We give an explicit condition under which the rates have a mean reversion property. In a special case we show that this condition is fulfilled for any Lévy process with variance smaller than a given constant, depending only on the state space and the volatility.

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

In this paper, we deal with a class of reflected backward stochastic differential equations (RBSDEs) corresponding to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness of the solution for RBSDEs by means of the penalization method. As an application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions.

Using key tools such as Itô's formula for general semimartingales, Kunita's moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable.

An existence and uniqueness theorem for mild solutions of stochastic evolution equations is presented and proved. The diffusion coefficient is handled in a unified way which allows a unified theorem to be formulated for different cases, in particular, of multiplicative space–time white noise and trace-class noise.

Let 𝒩* be a Hilbert inductive limit and X a Banach space. In this paper, we obtain a necessary and sufficient condition for an analytic mapping Ψ:𝒩*↦X to have a factorization of the form Ψ=T∘ℰ, where ℰ is the exponential mapping on 𝒩* and T:Γ(𝒩*)↦X is a continuous linear operator, where Γ(𝒩*) denotes the Boson Fock space over 𝒩*. To prove this result, we establish some kernel theorems for multilinear mappings defined on multifold Cartesian products of a Hilbert space and valued in a Banach space, which are of interest in their own right. We also apply the above factorization result to white noise theory and get a characterization theorem for white noise testing functionals.