Showing papers in "Annals of Applied Probability in 2015"
TL;DR: In this article, the authors consider a non-nominated model of a discrete-time financial market where stocks are traded dynamically and options are available for static hedging, and show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures.
Abstract: We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decomposition Theorem.
241 citations
TL;DR: The high-dimensional (large N) behavior of the iterates of F for polynomial functions F is studied, and it is proved that it is universal, i.e. it depends only on the first two moments of the entries of A, under a subgaussian tail condition.
Abstract: We consider a class of nonlinear mappings $\mathsf{F}_{A,N}$ in $\mathbb{R}^{N}$ indexed by symmetric random matrices $A\in\mathbb{R}^{N\times N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333–366]. Within information theory, they are known as “approximate message passing” algorithms. We study the high-dimensional (large $N$) behavior of the iterates of $\mathsf{F}$ for polynomial functions $\mathsf{F}$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.
192 citations
TL;DR: In this article, a weak formulation of stochastic optimal control is used to study mean field games with rank and nearest-neighbor effects, where the data may depend discontinuously on the state variable and its entire history.
Abstract: Mean field games are studied by means of the weak formulation of stochastic optimal control. This approach allows the mean field interactions to enter through both state and control processes and take a form which is general enough to include rank and nearest-neighbor effects. Moreover, the data may depend discontinuously on the state variable, and more generally its entire history. Existence and uniqueness results are proven, along with a procedure for identifying and constructing distributed strategies which provide approximate Nash equlibria for finite-player games. Our results are applied to a new class of multi-agent price impact models and a class of flocking models for which we prove existence of equilibria.
141 citations
TL;DR: In this paper, Andrieu and Roberts showed that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm.
Abstract: We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697–725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic or an independent Metropolis–Hastings or a random-walk Metropolis targeting a super-exponential density with regular contours. Our results on geometric and polynomial convergence rates imply central limit theorems. We also prove that under general conditions, the asymptotic variance of the pseudo-marginal algorithm converges to the asymptotic variance of the marginal algorithm if the accuracy of the estimators is increased.
134 citations
TL;DR: It is argued that it is often possible, at least in principle, to develop local particle filtering algorithms whose approximation error is dimension-free and the key to such developments is the decay of correlations property, which is a spatial counterpart of the much better understood stability property of nonlinear filters.
Abstract: The discovery of particle filtering methods has enabled the use of nonlinear filtering in a wide array of applications. Unfortunately, the approximation error of particle filters typically grows exponentially in the dimension of the underlying model. This phenomenon has rendered particle filters of limited use in complex data assimilation problems. In this paper, we argue that it is often possible, at least in principle, to develop local particle filtering algorithms whose approximation error is dimension-free. The key to such developments is the decay of correlations property, which is a spatial counterpart of the much better understood stability property of nonlinear filters. For the simplest possible algorithm of this type, our results provide under suitable assumptions an approximation error bound that is uniform both in time and in the model dimension. More broadly, our results provide a framework for the investigation of filtering problems and algorithms in high dimension.
133 citations
TL;DR: In this article, the authors studied nearly unstable Hawkes processes for which the stability condition is almost violated and showed that they asymptotically behave like integrated Cox-Ingersoll-Ross models.
Abstract: Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the $L^{1}$ norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox–Ingersoll–Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by Bacry et al. [Quant. Finance 13 (2013) 65–77]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.
131 citations
TL;DR: In this paper, rates of convergence of their finitely truncated Karhunen-Lo-ve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample path generation via fast Fourier transforms on the sphere are indicated.
Abstract: Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Lo\`eve expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample H\"older continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen-Lo\`eve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample path generation via fast Fourier transforms on the sphere are indicated. The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated. Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven.
91 citations
TL;DR: A class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees and a general weak law of large numbers is defined.
Abstract: We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $\tau>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value $\tau=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.
80 citations
TL;DR: In this paper, a decoupling random field and its associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components Y and Z, are studied.
Abstract: In this paper we study the wellposedness of the forward-backward stochastic differential equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to overcome some fundamental difficulties that have been longstanding problems fornon-Markovian FBSDEs. Our main devices are a decoupling random field and its associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components Y and Z. We establish various sufficient conditions under which the characteristic BSDE is wellposed, which leads to the existence of the decoupling random field, and ultimately to the solvability of the original FBSDE. We show that all existing frameworks could be analyzed using our new criteria.
75 citations
TL;DR: The problem of detecting a tight community in a sparse random network is considered, formalized as testing for the existence of a dense random subgraph in a random graph, and information theoretic lower bounds are derived.
Abstract: We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdős–Renyi graph on $N$ vertices and with connection probability $p_{0}$; under the alternative, there is an unknown subgraph on $n$ vertices where the connection probability is $p_{1}>p_{0}$. In Arias-Castro and Verzelen [Ann. Statist. 42 (2014) 940–969], we focused on the asymptotically dense regime where $p_{0}$ is large enough that $np_{0}>(n/N)^{o(1)}$. We consider here the asymptotically sparse regime where $p_{0}$ is small enough that $np_{0} 0$. As before, we derive information theoretic lower bounds, and also establish the performance of various tests. Compared to our previous work [Ann. Statist. 42 (2014) 940–969], the arguments for the lower bounds are based on the same technology, but are substantially more technical in the details; also, the methods we study are different: besides a variant of the scan statistic, we study other tests statistics such as the size of the largest connected component, the number of triangles, and the number of subtrees of a given size. Our detection bounds are sharp, except in the Poisson regime where we were not able to fully characterize the constant arising in the bound.
75 citations
TL;DR: This paper proves first a large deviation principle for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exciting function, and then generalizes it to get the result for sum of exponentials exciting functions.
Abstract: Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other fields. In this paper, we study the large deviations for nonlinear Hawkes processes. The large deviations for linear Hawkes processes has been studied by Bordenave and Torrisi. In this paper, we prove first a large deviation principle for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exciting function, and then generalize it to get the result for sum of exponentials exciting functions. We then provide an alternative proof for the large deviation principle for a linear Hawkes process. Finally, we use an approximation approach to prove the large deviation principle for a special class of nonlinear Hawkes processes with general exciting functions.
TL;DR: In this article, the authors investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution.
Abstract: We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality is of great importance as the resulting system is known to blow-up as this becomes large. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when the coefficient of proportionality is small enough.
TL;DR: In this paper, the authors consider the case where the vertices of simplices are the points of a random point process and the edges and faces are determined according to some deterministic rule.
Abstract: There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a random point process in $\mathbb{R}^{d}$, and the edges and faces are determined according to some deterministic rule, typically leading to Cech and Vietoris–Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either i.i.d. observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction or repulsion, we find phenomena quantitatively different from those observed in the i.i.d. and Poisson cases. From the point of view of topological data analysis, our results seriously impact considerations of model (non)robustness for statistical inference. Our proofs rely on analysis of subgraph and component counts of stationary point processes, which are of independent interest in stochastic geometry.
TL;DR: In this paper, the authors studied the existence of optimal actions in a zero-sum game between a stopper and a controller choosing a probability measure, and showed that the game has a value.
Abstract: We study the existence of optimal actions in a zero-sum game $\inf_{\tau}\sup_{P}E^{P}[X_{\tau}]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_{\tau}\mathcal{E}(X_{\tau})$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ such as the $G$-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope $Y$ and prove that the first hitting time $\inf\{t:Y_{t}=X_{t}\}$ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition. Finally, the results are applied to the subhedging of American options under volatility uncertainty.
TL;DR: In this article, a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation in [12] by means of control randomization and backward Stochastic differential equation with nonpositive jumps.
Abstract: We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation in [12] by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity condition on the controlled diffusion coefficient. Explicit implementable scheme based on Monte-Carlo simulations and empirical regressions, associated error analysis, and numerical experiments are performed in the companion paper [13].
TL;DR: In this article, a quasi-linear PDE associated with a coupled FBSDE is proposed, which weakens a critical constraint imposed by Fahim, Touzi and Warin (2011), especially when the generator of the PDE depends only on the diagonal terms of the Hessian matrix.
Abstract: In this paper we propose a feasible numerical scheme for high-dimensional, fully nonlinear parabolic PDEs, which includes the quasi-linear PDE associated with a coupled FBSDE as a special case. Our paper is strongly motivated by the remarkable work Fahim, Touzi and Warin [Ann. Appl. Probab. 21 (2011) 1322–1364] and stays in the paradigm of monotone schemes initiated by Barles and Souganidis [Asymptot. Anal. 4 (1991) 271–283]. Our scheme weakens a critical constraint imposed by Fahim, Touzi and Warin (2011), especially when the generator of the PDE depends only on the diagonal terms of the Hessian matrix. Several numerical examples, up to dimension 12, are reported.
TL;DR: The authors investigated the maximal domain of the moment generating function of affine processes in the sense of Duffie, Filipovic and Schachermayer, and showed the validity of the affine transform formula that connects exponential moments with the solution of a generalized Riccati differential equation.
Abstract: We investigate the maximal domain of the moment generating function of affine processes in the sense of Duffie, Filipovic and Schachermayer [Ann. Appl. Probab. 13 (2003) 984–1053], and we show the validity of the affine transform formula that connects exponential moments with the solution of a generalized Riccati differential equation. Our result extends and unifies those preceding it (e.g., Glasserman and Kim [Math. Finance 20 (2010) 1–33], Filipovic and Mayerhofer [Radon Ser. Comput. Appl. Math. 8 (2009) 1–40] and Kallsen and Muhle-Karbe [Stochastic Process Appl. 120 (2010) 163–181]) in that it allows processes with very general jump behavior, applies to any convex state space and provides both sufficient and necessary conditions for finiteness of exponential moments.
TL;DR: In this paper, the parametrix method is used for constructions of fundamental solutions as a general method based on semigroups and difference of generators, which leads to a probabilistic interpretation of the parametric method that is amenable to Monte Carlo simulation.
Abstract: In this article, we introduce the parametrix method for constructions of fundamental solutions as a general method based on semigroups and difference of generators. This leads to a probabilistic interpretation of the parametrix method that are amenable to Monte Carlo simulation. We consider the explicit examples of continuous diffusions and jump driven stochastic differential equations with Ho lder continuous coefficients.
TL;DR: In this article, the authors considered an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative Levy process (in the absence of dividend payments).
Abstract: This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative Levy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin, which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber–Shiu function. A number of concrete examples are analyzed.
TL;DR: In this paper, the authors deal with asset price bubbles modeled by strict local martingales and determine the "default term" apparent in risk-neutral option prices if the underlying stock exhibits a bubble modeled by a strict local Martingale.
Abstract: This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term” apparent in risk-neutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given.
TL;DR: In this paper, the error analysis of the time discretization of systems of Forward Backward Stochastic Dierential Equations with drivers having polynomial growth and that are also monotone in the state variable is performed.
Abstract: In this paper we undertake the error analysis of the time discretization of systems of ForwardBackward Stochastic Dierential Equations (FBSDEs) with drivers having polynomial growth and that are also monotone in the state variable. We show with a counter-example that the natural explicit Euler scheme may diverge, unlike in the canonical Lipschitz driver case. This is due to the lack of a certain stability property of the Euler scheme which is essential to obtain convergence. However, a thorough analysis of the family of -schemes reveals that this required stability property can be recovered if the scheme is suciently implicit. As a by-product of our analysis we shed some light on higher order approximation schemes for FBSDEs under non-Lipschitz condition. We then return to fully explicit schemes and show that an appropriately tamed version of the explicit Euler scheme enjoys the required stability property and as a consequence converges. In order to establish convergence of the several discretizations we extend the canonical pathand first order variational regularity results to FBSDEs with polynomial growth drivers which are also monotone. These results are of independent interest for the theory of FBSDEs.
TL;DR: A mathematically tractable model is formulated for theensation phenomenon in social networks such as Twitter where one “superstar” vertex gains a positive fraction of the edges, while the remaining empirical degree distribution still exhibits a power law tail.
Abstract: Condensation phenomenon is often observed in social networks such as Twitter where one “superstar” vertex gains a positive fraction of the edges, while the remaining empirical degree distribution still exhibits a power law tail. We formulate a mathematically tractable model for this phenomenon that provides a better fit to empirical data than the standard preferential attachment model across an array of networks observed in Twitter. Using embeddings in an equivalent continuous time version of the process, and adapting techniques from the stable age-distribution theory of branching processes, we prove limit results for the proportion of edges that condense around the superstar, the degree distribution of the remaining vertices, maximal nonsuperstar degree asymptotics and height of these random trees in the large network limit.
TL;DR: In this paper, the authors considered Brownian motions with one-sided collisions, where each particle is reflected at its right neighbour and the joint distribution of the positions of a subset of par-ticles is expressed as a Fredholm determinant with a kernel defininga signed determinantal point process.
Abstract: We consider Brownian motions with one-sided collisions, meaningthat each particle is reflected at its right neighbour. For a finite num-ber of particles a Schu¨tz-type formula is derived for the transitionprobability. We investigate an infinite system with periodic initialconfiguration, i.e., particles are located at the integer lattice at timezero. The joint distribution of the positions of a finite subset of par-ticles is expressed as a Fredholm determinant with a kernel defininga signed determinantal point process. In the appropriate large timescaling limit, the fluctuations in the particle positions are describedby the Airy 1 process. 1 Introduction A widely studied model of interacting Brownian motions is governed by thecoupled stochastic differential equationsdx j =V ′ (x j+1 −x j )− V ′ (x j − x j−1 )dt+√2dB j (t), (1.1)j = 1,...,N, written here for the case where particles diffuse in one dimen-sion. Hence x j (t) ∈ Rand {B j (t),j = 1,...,N} is a collection of N indepen-dent standard Brownian motions. The boundary terms V
TL;DR: In this paper, a central limit theorem of Lindeberg-Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme was proved.
Abstract: This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607–617] which is significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg–Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [Ann. Probab. 26 (1998) 267–307], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. A complexity of the multilevel Monte Carlo algorithm is carried out.
TL;DR: In this article, the authors studied optimal stochastic control problems for non-Markovian SDEs where the drift, diffusion coefficients and gain functionals are path-dependent, and they did not make any ellipticity assumptions on the SDE.
Abstract: We study optimal stochastic control problems for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients and gain functionals are path-dependent, and importantly we do not make any ellipticity assumptions on the SDE. We develop a control randomization approach and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent version of the Hamilton–Jacobi–Bellman equation, and an extension to $G$-expectation.
TL;DR: In this paper, a continuous-time model for a large investor who trades with a finite number of market makers at their utility indifference prices is developed from basic economic principles, where the market makers compete with their quotes for the investor's orders and trade among themselves to attain Pareto optimal allocations.
Abstract: We develop from basic economic principles a continuous-time model for a large investor who trades with a finite number of market makers at their utility indifference prices. In this model, the market makers compete with their quotes for the investor’s orders and trade among themselves to attain Pareto optimal allocations. We first consider the case of simple strategies and then, in analogy to the construction of stochastic integrals, investigate the transition to general continuous dynamics. As a result, we show that the model’s evolution can be described by a nonlinear stochastic differential equation for the market makers’ expected utilities.
TL;DR: In this paper, the correlation kernel for the determinantal process of the Aztec diamond tiling of the diamond was studied. And the authors showed that at the northern boundary, the southern domino process converges to a thinned Airy point process, which is a multiple point process.
Abstract: We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.
TL;DR: Results about the distribution of the first time when two neutral mutations have accumulated in some cell in dimensions d ≥ 2 are proved, extending work done by Komarova for d = 1.
Abstract: We consider a multistage cancer model in which cells are arranged in a d-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated in some cell in dimensions d ≥ 2, extending work done by Komarova [12] for d = 1.
TL;DR: In this paper, a handy integral equation for the freeboundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion X was derived.
Abstract: In this paper, we derive a new handy integral equation for the freeboundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion X. The new integral equation allows to explicitly find the freeboundary b(.) in some so far unsolved cases, as when the operating profit function is not multiplicatively separable and X is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we first show that b(X (t)) = l* (t), with l* the unique optional solution of a representation problem in the spirit of Bank El Karoui [Ann. Probab. 32 (2004) 1030-1067]; then, thanks to such an identification and the fact that l* uniquely solves a backward stochastic equation, we find the integral problem for the free-boundary.
TL;DR: In this paper, the authors discuss importance sampling schemes for the estimation of finite time exit probabilities of small noise diffusions that involve escape from an equilibrium, where rest points are included in the domain of interest.
Abstract: We discuss importance sampling schemes for the estimation of finite time exit probabilities of small noise diffusions that involve escape from an equilibrium. A factor that complicates the analysis is that rest points are included in the domain of interest. We build importance sampling schemes with provably good performance both pre-asymptotically, that is, for fixed size of the noise, and asymptotically, that is, as the size of the noise goes to zero, and that do not degrade as the time horizon gets large. Simulation studies demonstrate the theoretical results.