In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.

Positive T-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval T = [0, 1] and replace random measures by random functions. We specify a large class of such martingales for which we provide a general sufficient condition for almost sure uniform convergence to a nontrivial limit. Such a limit yields new examples of naturally generated multifractal processes that may be of use in multifractal signals modeling.

The familiar cascade measures are sequences of random positive measures obtained on [0, 1] via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time.

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cramér–Lundberg process. The firm has the option of investing part of the surplus in a Black–Scholes financial market. The objective is to find a strategy consisting of both investment and dividend payment policies which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy. We show that the optimal value function is the smallest viscosity solution of the associated second-order integro-differential Hamilton–Jacobi–Bellman equation. We study the regularity of the optimal value function. We show that the optimal dividend payment strategy has a band structure. We find a method to construct a candidate solution and obtain a verification result to check optimality. Finally, we give an example where the optimal dividend strategy is not barrier and the optimal value function is not twice continuously differentiable.

This paper explores various distributional aspects of random variables defined as the ratio of two independent positive random variables where one variable has an α-stable law, for 0 < α < 1, and the other variable has the law defined by polynomially tilting the density of an α-stable random variable by a factor θ > −α. When θ = 0, these variables equate with the ratio investigated by Lamperti [Trans. Amer. Math. Soc. 88 (1958) 380–387] which, remarkably, was shown to have a simple density. This variable arises in a variety of areas and gains importance from a close connection to the stable laws. This rationale, and connection to the PD (α, θ) distribution, motivates the investigations of its generalizations which we refer to as Lamperti-type laws. We identify and exploit links to random variables that commonly appear in a variety of applications. Namely Linnik, generalized Pareto and z-distributions. In each case we obtain new results that are of potential interest. As some highlights, we then use these results to (i) obtain integral representations and other identities for a class of generalized Mittag–Leffler functions, (ii) identify explicitly the Lévy density of the semigroup of stable continuous state branching processes (CSBP) and hence corresponding limiting distributions derived in Slack and in Zolotarev [Z. Wahrsch. Verw. Gebiete 9 (1968) 139–145, Teor. Veroyatn. Primen. 2 (1957) 256–266], which are related to the recent work by Berestycki, Berestycki and Schweinsberg, and Bertoin and LeGall [Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 214–238, Illinois J. Math. 50 (2006) 147–181] on beta coalescents. (iii) We obtain explicit results for the occupation time of generalized Bessel bridges and some interesting stochastic equations for PD (α, θ)-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension 2 − 2α.

In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676–713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process.

In this paper, we show that this dual approach can actually be used for both deriving a candidate solution and verification in Merton’s problem with logarithmic utility and proportional transaction costs. In particular, we determine the shadow price process.

We study repeated games where players use an exponential learning scheme in order to adapt to an ever-changing environment. If the game’s payoffs are subject to random perturbations, this scheme leads to a new stochastic version of the replicator dynamics that is quite different from the “aggregate shocks” approach of evolutionary game theory. Irrespective of the perturbations’ magnitude, we find that strategies which are dominated (even iteratively) eventually become extinct and that the game’s strict Nash equilibria are stochastically asymptotically stable. We complement our analysis by illustrating these results in the case of congestion games.

The aim of this paper is to introduce a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations. The main idea is to combine random walk on squares or rectangles methods with importance sampling techniques.

The first interest of this approach is that the weights can be easily computed from the density of the one-dimensional Brownian motion. Compared to the Euler scheme this method allows one to obtain a more accurate approximation of diffusions when one has to consider complex boundary conditions. The method provides also an interesting alternative to performing variance reduction techniques and simulating rare events.

We consider a process X_t, which is observed on a finite time interval [0, T], at discrete times 0, Δ_n, 2Δ_n, …. This process is an Itô semimartingale with stochastic volatility σ_t². Assuming that X has jumps on [0, T], we derive tests to decide whether the volatility process has jumps occurring simultaneously with the jumps of X_t. There are two different families of tests for the two possible null hypotheses (common jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh Δ_n goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use on S&P 500 index data.

In the classical balls-and-bins paradigm, where n balls are placed independently and uniformly in n bins, typically the number of bins with at least two balls in them is Θ(n) and the maximum number of balls in a bin is Θ(log n / log log n). It is well known that when each round offers k independent uniform options for bins, it is possible to typically achieve a constant maximal load if and only if k = Ω(log n). Moreover, it is possible w.h.p. to avoid any collisions between n / 2 balls if k > log₂ n.

In this work, we extend this into the setting where only m bits of memory are available. We establish a tradeoff between the number of choices k and the memory m, dictated by the quantity km / n. Roughly put, we show that for km ≫ n one can achieve a constant maximal load, while for km ≪ n no substantial improvement can be gained over the case k = 1 (i.e., a random allocation).

For any k = Ω(log n) and m = Ω(log² n), one can achieve a constant load w.h.p. if km = Ω(n), yet the load is unbounded if km = o(n). Similarly, if km > Cn then n / 2 balls can be allocated without any collisions w.h.p., whereas for km < ɛn there are typically Ω(n) collisions. Furthermore, we show that the load is w.h.p. at least log(n/m) / (log k+log log(n / m)). In particular, for k ≤ polylog (n), if m = n^1−δ the optimal maximal load is Θ(log n / log log n) (the same as in the case k = 1), while m = 2n suffices to ensure a constant load. Finally, we analyze nonadaptive allocation algorithms and give tight upper and lower bounds for their performance.

We deal with backward stochastic differential equations with time delayed generators. In this new type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function, for instance, of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. We give examples of BSDE with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lipschitz constant. This class includes linear time delayed generators which we study in more detail. We are concerned with different properties of a solution of a BSDE with time delayed generator, including the inheritance of boundedness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.

In the Wiener disorder problem, the drift of a Wiener process changes suddenly at some unknown and unobservable disorder time. The objective is to detect this change as quickly as possible after it happens. Earlier work on the Bayesian formulation of this problem brings optimal (or asymptotically optimal) detection rules assuming that the prior distribution of the change time is given at time zero, and additional information is received by observing the Wiener process only. Here, we consider a different information structure where possible causes of this disorder are observed. More precisely, we assume that we also observe an arrival/counting process representing external shocks. The disorder happens because of these shocks, and the change time coincides with one of the arrival times. Such a formulation arises, for example, from detecting a change in financial data caused by major financial events, or detecting damages in structures caused by earthquakes. In this paper, we formulate the problem in a Bayesian framework assuming that those observable shocks form a Poisson process. We present an optimal detection rule that minimizes a linear Bayes risk, which includes the expected detection delay and the probability of early false alarms. We also give the solution of the “variational formulation” where the objective is to minimize the detection delay over all stopping rules for which the false alarm probability does not exceed a given constant.