Multilayers in a modulated stochastic game

We are concerned with an antagonistic stochastic game between two players A and B which finds applications in economics and warfare. The actions of the players are manifested by a series of strikes of random magnitudes at random times exerted by each player against his opponent. Each of the assaults inflicts a random damage to enemy's vital areas. In contrast with traditional games, in our setting, each player can endure multiple strikes before perishing. Predicting the ruin time (exit) of player A, along with the total amount of casualties to both players at the exit is a main objective of this work. In contrast to the time sensitive analysis (earlier developed to refine the information on the game) we insert auxiliary control levels, which both players will cross in due game before the ruin of A. This gives A (and also B) an additional opportunity to reevaluate his strategy and change the course of the game. We formalize such a game and also allow the real time information about the game to be randomly delayed. The delayed exit time, cumulative casualties to both players, and prior crossings are all obtained in a closed-form joint functional.     

On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process  = {τ₀,τ₁,…} marked by a multivariate random walk (,) and its behavior about fixed levels to be crossed by one of the components of (,). We derive the joint distribution of first passage time τ_ρ, pre-exit time τ_ρ?1 (i.e., the instant one phase prior to the first passage time), and the respective values of (,) at τ_ρ and τ_ρ?1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk ((Π),) embedded in Π over . Processes like these can model various phenomena including stock market and option trading.

One of the central issues in the investigation of ((Π),) is to obtain the information about Π at any moment of time in random vicinities of τ_ρ and τ_ρ?1 previously available only upon . The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.     

On Strategic Defense in Stochastic Networks

This paper deals with the detection and prediction of losses due to cyber attacks waged on vital networks. The accumulation of losses to a network during a series of attacks is modeled by a 2-dimensional monotone random walk process as observed by an independent delayed renewal process. The first component of the process is associated with the number of nodes (such as routers or operational sites) incapacitated by successive attacks. Each node has a weight associated with its incapacitation (such as loss of operational capacity or financial cost associated with repair), and the second component models the cumulative weight associated with the nodes lost. Each component has a fixed threshold, and crossing of a threshold by either component represents the network entering a critical condition. Results are given as joint functionals of the predicted time of the first observed threshold crossing along with the values of each component upon this time.     

Lowest priority waiting time distribution in an accumulating priority Lévy queue

We derive the waiting time distribution of the lowest class in an accumulating priority (AP) queue with positive Lévy input. The priority of an infinitesimal customer (particle) is a function of their class and waiting time in the system, and the particles with the highest AP are the next to be processed. To this end we introduce a new method that relies on the construction of a workload overtaking process and solving a first-passage problem using an appropriate stopping time.     

Itô’s excursion theory and its applications

A presentation of It?’s excursion theory for general Markov processes is given, with several applications to Brownian motion and related processes.     

Stochastic applications of media theory: Random walks on weak orders or partial orders

This paper presents the axioms of a real time random walk on the set of states of a medium and some of their consequences, such as the asymptotic probabilities of the states. The states of the random walk coincide with those of the medium, and the transitions of the random walk are governed by a probability distribution on the set of token-events, together with a Poisson process regulating the arrivals of such events. We examine two special cases. The first is the medium on strict weak orders on a set of three elements, the second the medium of strict partial orders on the same set. Thus, in each of these cases, a state of the medium is a binary relation. We also consider tune in-and-out extensions of these two special cases. We review applications of these models to opinion poll data pertaining to the 1992 United States presidential election. Each strict weak order or strict partial order is interpreted as being the implicit or explicit opinion of some individual regarding the three major candidates in that election, namely, Bush, Clinton and Perot. In particular, the strict partial order applications illustrate our notion of a response function that provides the link between theory and data in situations where, in contrast to previous papers, the permissible responses do not span the entire set of permissible states of the medium.     

Random walks in cones: The case of nonzero drift

We consider multidimensional discrete valued random walks with nonzero drift killed when leaving general cones of the euclidean space. We find the asymptotics for the exit time from the cone and study weak convergence of the process conditioned on not leaving the cone. We get quasistationarity of its limiting distribution. Finally we construct a version of the random walk conditioned to never leave the cone.     

Aging in two-dimensional Bouchaud's model

Let E _x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on ℤ² is a Markov chain X(t) whose transition rates are given by w _xy = ν exp (−βE _x ) if x, y are neighbours in ℤ². We study the behaviour of two correlation functions: ℙ[X(t _w +t) = X(t _w )] and ℙ[X(t') = X(t _w ) ∀ t'∈ [t _w , t _w + t]]. We prove the (sub)aging behaviour of these functions when β > 1.     

General tax structures for a Lévy insurance risk process under the Cramér condition

We investigate the Lévy insurance risk model with tax under Cramér’s condition. A direct analogue of Cramér’s estimate for the probability of ruin in this model is obtained, together with the asymptotic distribution, conditional on ruin occurring, of several variables of interest related to ruin including the surplus immediately prior to ruin (undershoot) and shortfall at ruin (overshoot). We also compute the present value of all tax paid conditional on ruin occurring. The proof involves first transferring results from the model with no tax to the reflected process, and from there to the model with tax.     

Fluctuations of Omega-killed spectrally negative Lévy processes

In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases $\omega \left(x\right)=q$ and $\omega \left(x\right)=q{\mathbf{1}}_{(a,b)}\left(x\right)$, we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy probability in the so-called Omega model, where bankruptcy occurs at rate $\omega \left(x\right)$ when the Lévy surplus process is at level $x<0$. Finally, we apply these results to obtain some exit identities for spectrally positive self-similar Markov processes. The main method throughout all the proofs relies on the classical fluctuation identities for Lévy processes, the Markov property and some basic properties of a Poisson process.     

The Upper Envelope of Positive Self-Similar Markov Processes

We establish integral tests in connection with laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation and on the study of the upper envelope of the future infimum due to the author (see Pardo in Stoch. Stoch. Rep. 78:123–155, [2006]). These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdős (Proceedings of the Second Berkeley Symposium, [1951]) and stable Lévy processes with no positive jumps conditioned to stay positive due to Bertoin (Stoch. Process. Appl. 55:91–100, [1995]). Research supported by a grant from CONACYT (Mexico).     

Weak limits of random coefficient autoregressive processes and their application in ruin theory

We prove that a large class of discrete-time insurance surplus processes converge weakly to a generalized Ornstein–Uhlenbeck process, under a suitable re-normalization and when the time-step goes to 0. Motivated by ruin theory, we use this result to obtain approximations for the moments, the ultimate ruin probability and the discounted penalty function of the discrete-time process.     

American Options in the Heston Model with Stochastic Interest Rate and Its Generalizations

Abstract

We consider the Heston model with the stochastic interest rate of Cox–Ingersoll–Ross (CIR) type and more general models with stochastic volatility and interest rates depending on two CIR-factors; the price, volatility and interest rate may correlate. Time-derivative and infinitesimal generator of the process for factors that determine the dynamics of the interest rate and/or volatility are discretized. The result is a sequence of embedded perpetual options arising in the time discretization of a Markov-modulated Lévy model. Options in this sequence are solved using an iteration method based on the Wiener–Hopf factorization. Typical shapes of the early exercise boundary are shown, and good agreement of option prices with prices calculated with the Longstaff–Schwartz method and Medvedev–Scaillet asymptotic method is demonstrated.