Nonexistence of Markovian time dynamics for graphical models of correlated default

Filiz et al. (in arXiv:0809.1393 (2008)) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph G=(V,E), where the vertices V correspond to the firms and the edges E describe the network of interdependencies between the firms, a parameter for each vertex that captures the individual propensity of that firm to default, and a parameter for each edge that captures the joint propensity of the two connected firms to default. The correlated default model can be rewritten as a standard Ising model on the graph by identifying the set of defaulting firms in the default model with the set of sites in the Ising model for which the {±1}-valued spin is +1. We ask whether there is a suitable continuous-time Markov chain (X _t ) _t??0 taking values in the subsets of V such that X ₀=?, X _r ?X _s for r??s (that is, once a firm defaults, it stays in default), the distribution of X _T for some fixed time T is the one given by the default model, and the distribution of X _t for other times t is described by a probability distribution in the same family as the default model. In terms of the equivalent Ising model, this corresponds to asking if it is possible to begin at time 0 with a configuration in which every spin is ?1 and then flip spins one at a time from ?1 to +1 according to Markovian dynamics so that the configuration of spins at each time is described by some Ising model and at time T the configuration is distributed according to the prescribed Ising model. We show for three simple but financially natural special cases that this is not possible outside of the trivial case where there is complete independence between the firms.     

The capture time of a graph

We consider the game of Cops and Robbers played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to a new parameter, the capture time of a graph. While the capture time of a cop-win graph on n vertices is bounded above by n−3, half the number of vertices is sufficient for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have capture time within a small additive constant of the number of vertices. We consider the ratio of the capture time to the number of vertices, and extend this notion of capture time density to infinite graphs. For the infinite random graph, the capture time density can be any real number in [0,1]. We also consider the capture time when more than one cop is required to win. While the capture time can be calculated by a polynomial algorithm if the number k of cops is fixed, it is NP-complete to decide whether k cops can capture the robber in no more than t moves for every fixed t.     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

On the service system MX/G/∞

In the present paper we consider the service system M^X/G/∞ characterized by an infinite number of servers anda general service time distribution. The customers arrive at the system in groups of size X, which is a random variable, the time between group arrivals being exponentially distributed. Using simple probability arguments, we obtain probability generating functions (p.g.f.'s) of the number of busy servers at time t and the number that depart by time t. Several other properties of these random variables are also discussed.     

Single item lot-sizing problem for a warm/cold process with immediate lost sales

We consider the dynamic lot-sizing problem with finite capacity and possible lost sales for a process that could be kept warm at a unit variable cost for the next period t + 1 only if more than a threshold value Q_t has been produced and would be cold, otherwise. Production with a cold process incurs a fixed positive setup cost, K_t and setup time, S_t, which may be positive. Setup costs and times for a warm process are negligible. We develop a dynamic programming formulation of the problem, establish theoretical results on the structure of the optimal production plan in the presence of zero and positive setup times with Wagner–Whitin-type cost structures. We also show that the solution to the dynamic lot-sizing problem with lost sales are generated from the full commitment production series improved via lost sales decisions in the presence of a warm/cold process.     

Continuous time random walks and the Cauchy problem for the heat equation

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ?ⁿ. A particle moves in ?ⁿ in such a way that the probability density function u(·, t) of finding it in region Ω of ?ⁿ is given by ∫_Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation

$$u\left( {x,t} \right) = \left[ {\left( {J - \delta } \right)*u} \right]\left( {x,t} \right)$$

, where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.

     

Optimal selling time in stock market over a finite time horizon

In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T ], where the stock price is modelled by a geometric Brownian motion and the ’closeness’ is measured by the relative error of the stock price to its highest price over [0, T ]. More precisely, we want to optimize the expression: where (V t ) t≥0 is a geometric Brownian motion with constant drift α and constant volatility σ > 0, M t = max Vs is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤τ≤ T adapted to the natural filtration (F t ) t≥0 of the stock price. The above problem has been considered by Shiryaev, Xu and Zhou (2008) and Du Toit and Peskir (2009). In this paper we provide an independent proof that when α = 1 2 σ 2 , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the moment when the stock price hits its maximum price so far. Besides, when α > 1 2 σ 2 , selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping ρτ of a stopping time τ to the optimal stopping strategy arisen in the classical "Secretary Problem".     

Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

We consider a model composed of a signal process X given by a classic stochastic differential equation and an observation process Y, which is supposed to be correlated to the signal process. We assume that process Y is observed from time 0 to s>0 at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process X crosses a fixed barrier after a given time t>0. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities.     

Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f₀(ξ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f(ξ,t)∈H^∞(R³) for all t>0. Furthermore, for any t₀>0 and s?0 the Hs norm of f(ξ,t) is bounded for t?t₀. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition.     

Operating characteristic analysis on the M/G/1 system with a variant vacation policy and balking

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a variant vacation policy, where the server leaves for a vacation as soon as the system is empty. The server takes at most J vacations repeatedly until at least one customer is found waiting in the queue when the server returns from a vacation. If the server is busy or on vacation, an arriving batch balks (refuses to join) the system with probability 1 − b. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Finally, important system characteristics are derived along with some numerical illustration.     

Approximation and estimation of some compound distributions

The distribution of the total amount claimed up to time t can often be written in the form of a compound distribution G_t(x) = Σp_n(t)F⁽ⁿ⁾(x) where p_n(t) is the probability of exactly n claims while F is the distribution of a single claim. In the actuarial literature one often finds approximations of G_t(x) when the time t is large. It seems more natural to take t fixed and to look for approximations for x large. This paper contains a number of such results for a Poisson process and for a Pascal process. Different hypotheses on the tail behaviour of F(t) yield different expressions to estimate 1 - G_t(x). The results obtained should prove to have wider applicability than suggested by the insurance context. Within it, however, applications to premium calculation principles are immediate.     

Detailing Losses in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1/1 Transient Loss System

This paper is aimed at investigating the transient losses in the M/M/1/1 Erlang loss system. We evaluate the explicit form of the probability distribution of the number of losses in the time interval [0, t) and provide two alternative representations: one based on the iterated derivatives of hyperbolic sinus and cosine and the other on the spherical modified Bessel function of the second kind. The mathematical structures of the transient loss rate and of the transient probability of losing all customers are described and several analytical properties are derived.     

Markov Chains with finite convergence time

We study the properties of finite ergodic Markov Chains whose transition probability matrix P is singular. The results establish bounds on the convergence time of P^m to a matrix where all the rows are equal to the stationary distribution of P. The results suggest a simple rule for identifying the singular matrices which do not have a finite convergence time. We next study finite convergence to the stationary distribution independent of the initial distribution. The results establish the connection between the convergence time and the order of the minimal polynomial of the transition probability matrix. A queuing problem and a maintenance Markovian decision problem which possess the property of rapid convergence are presented.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems

We consider initial value/boundary value problems for fractional diffusion-wave equation: , where 0<α?2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. Second for α∈(0,1), we apply the eigenfunction expansions and prove (i) stability in the backward problem in time, (ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as t→∞, (iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over (0,T).     

Optimal policies for inventory models with some specified markets and finite time horizon

The paper presents the deterministic finite time horizon inventory lot size model, without backlogs and with no lead time, for a single commodity, with some specified markets. The specified markets are represented by the family b(t)=kt^r of demand functions where k>0, r>?2 are known parameters and t stands for time, 0<t₀?t?T. The strict positivity of t₀. compared to the restrictive condition t₀=0 which has been already solved, is crucial and implies entirely different analytical techniques. An important special case is the affine function (r = 1) partly treated already by Donaldson [3]. The problem is to find the optimal schedule of replenishments, i.e., the number and timings of orders.The problem is completely resolved (compared to a recent heuristic by Silver [8]) and the solution is given in a closed form and is proven to be unique. Numerical examples are provided.     

Graph evolution by stochastic additions of points and lines

We define the graph G₁ recursively from G_t?1 by adding a point or a line with probability ? and q, respectively, if G_t?1 is not complete; if G_t?1 is complete we always add a point. By using recursions we investigate the probability distribution of the order and size of G_? of the minimum and maximum orders for a fixed size, and of the minimum and maximum sizes for a fixed order. Expected values and generating functions are determined.     

Besov regularity for the generalized local time of the indefinite Skorohod integral

Let (t∈[0,1]) be the indefinite Skorohod integral on the canonical probability space (Ω,F,P), and let Lt(x) (t∈[0,1], x∈R) be its the generalized local time introduced by Tudor in [C.A. Tudor, Martingale-type stochastic calculus for anticipating integral processes, Bernoulli 10 (2004) 313-325]. We prove that the generalized local time, as function of x, has the same Besov regularity as the Brownian motion, as function of t, under some conditions imposed on the anticipating integrand u.