We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C 1, C 2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C 1, C 2,…); here ‘∨’ denotes the maximum operator.

We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.

Continuous-time discrete-state random Markov chains generated by a random linear differential equation with a random tridiagonal matrix are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses comparison theorems for Carathéodory random differential equations and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself with respect to the Hilbert projective metric. It does not involve probabilistic properties of the sample path and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transition probabilities, in which case the attractor is a periodic path.

Our aim in this paper is to identify the limit behavior of the solutions of random degenerate equations of the form −div Aε(x′,∇Uε)+ρεω(x′)Uε=F with mixed boundary conditions on Ωε whenever ε→0, where Ωε is an N-dimensional thin domain with a small thickness h(ε), ρεω(x′)=ρω(x′/ε), where ρω is the realization of a random function ρ(ω) , and Aε(x′,ξ)=a(Tx′ /εω,ξ) , the map a(ω,ξ)being measurable in ω and satisfying degenerated structure conditions with weight ρ in ξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the condition h(ε)/ε→0 , the sequence of solutions of them converges to a limit u0, where u0 is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.

Let M=(M)n∈ℕ be a discrete-time normal martingale satisfying some mild requirements. In this paper we show that through the full Wiener integral introduced by Wang et al. (‘An alternative approach to Privault’s discrete-time chaotic calculus’, J. Math. Anal. Appl.373 (2011), 643–654), one can define a multiplication-type operation on square integrable functionals of M, which we call the convolution. We examine algebraic and analytical properties of the convolution and, in particular, we prove that the convolution can be used to represent a certain family of conditional expectation operators associated with M. We also present an example of a discrete-time normal martingale to show that the corresponding convolution has an integral representation.

Multivariate Lévy-driven mixed moving average (MMA) processes of the type X t = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

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 V 0 define the generalised Ornstein–Uhlenbeck (GOU) process V t :=eξ t (V 0+∫t 0 e-ξ s- dηs ), t≥0, and the associated stochastic integral process Z t :=∫0 t e-ξ s- dηs , t≥0. Let T z :=inf{t>0: V t <0|V 0=z} and ψ(z):=P(T z <∞) 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 T z 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, {R i }i≥1, and {C, C i }i≥1 are independent nonnegative random variables, the {C, C i }i≥1 are identically distributed, and the {R i }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[C α] = 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[C α] < 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.