Theory Probab Research Articles

A BSDE approach to fair bilateral pricing under endogenous collateralization

Nie and Rutkowski (Int. J. Theor. Appl. Finance 18:1550048, 2015; Math. Finance, 2016, to appear) examined fair bilateral pricing in models with funding costs and an exogenously given collateral. The main goal of this work is to extend results from Nie and Rutkowski (Int. J. Theor. Appl. Finance 18:1550048, 2015; Math. Finance, 2016, to appear) to the case of an endogenous margin account depending on the contract’s value for the hedger and/or the counterparty. Comparison theorems for BSDEs from Nie and Rutkowski (Theory Probab. Appl., 2016, forthcoming) are used to derive bounds for unilateral prices and to study the range for fair bilateral prices in a general semimartingale model. The backward stochastic viability property, introduced by Buckdahn et al. (Probab. Theory Relat. Fields 116:485–504, 2000), is employed to examine the bounds for fair bilateral prices for European claims with a negotiated collateral in a diffusion-type model. We also generalize in several respects the option pricing results from Bergman (Rev. Financ. Stud. 8:475–500, 1995), Mercurio (Actuarial Sciences and Quantitative Finance, pp. 65–95, 2015) and Piterbarg (Risk 23(2):97–102, 2010) by considering contracts with cash-flow streams and allowing for idiosyncratic funding costs for risky assets.

Spatial mixing and the connective constant: optimal bounds

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model (which is defined as a weighted sum over all matchings where each matching is given a weight $$\gamma ^{|V| - 2 |M|}$$ in terms of a fixed parameter $$\gamma $$ called the monomer activity) and the hard core model (which is defined as a weighted sum over all independent sets where an independent set I is given a weight $$\lambda ^{|I|}$$ in terms of a fixed parameter $$\lambda $$ called the vertex activity). The connective constant is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdős–Renyi model $$\mathcal {G}(n, d/n)$$ . Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called message approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain fully polynomial time approximation schemes (FPTASs) for the two problems on graphs of bounded connective constant. In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati et al. (Proc. 39th ACM Symp. Theory Comput., pp. 122–127, 2007), and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant $$\varDelta $$ whenever the vertex activity $$\lambda < \lambda _c(\varDelta )$$ , where $$\lambda _c(\varDelta ) :=\frac{\varDelta ^\varDelta }{(\varDelta - 1)^{\varDelta + 1}}$$ ; this result is optimal in the sense that an FPTAS for any $$\lambda > \lambda _c(\varDelta )$$ would imply that NP=RP (Sly and Sun, Ann. Probab. 42(6):2383–2416, 2014). The previous best known result in this direction was in a recent manuscript by a subset of the current authors (Proc. 54th IEEE Symp. Found. Comput. Sci., pp 300–309, 2013), where the result was established under the sub-optimal condition $$\lambda < \lambda _c(\varDelta + 1)$$ . Our techniques also allow us to improve upon known bounds for decay of correlations for the hard core model on various regular lattices, including those obtained by Restrepo et al. (Probab Theory Relat Fields 156(1–2):75–99, 2013) for the special case of $$\mathbb {Z}^2$$ using sophisticated numerically intensive methods tailored to that special case.

A new characterization of quadratic transportation-information inequalities

It is known that a quadratic transportation-information inequality $$\mathrm{W}_2\mathrm{I}$$ interpolates between the Talagrand’s inequality $$\mathrm{W}_2\mathrm{H}$$ and the log-Sobolev inequality (LSI for short). The aim of this paper is threefold: (1) To prove the equivalence of $${\mathrm {W}}_2\mathrm{I}$$ and the Lyapunov condition, which gives a new characterization inspired by Cattiaux et al. (Probab Theory Relat Fields 148(1–2):285–304, 2010). (2) To prove the stability of $${\mathrm {W}}_2\mathrm{I}$$ under bounded perturbations, which gives a transference principle in the sense of Holley–Stroock. (3) To prove $$\mathrm{W}_2\mathrm{H}$$ through a restricted $$\mathrm{W}_2\mathrm{I}$$ , which gives a counterpart of the restricted LSI presented by Gozlan et al. (Ann Probab 39(3):857–880, 2011).

Synchronization by noise

We provide sufficient conditions for synchronization by noise, i.e. under these conditions we prove that weak random attractors for random dynamical systems consist of single random points. In the case of SDE with additive noise, these conditions are also essentially necessary. In addition, we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point. As a result, synchronization by noise is proven for a large class of SDE with additive noise. In particular, we prove that the random attractor for an SDE with drift given by a (multidimensional) double-well potential and additive noise consists of a single random point. All examples treated in Tearne (Probab Theory Relat Fields 141(1–2):1–18, 2008) are also included.

Excursion probability of Gaussian random fields on sphere

Let $X=\{X(x): x\in\mathbb{S}^N\}$ be a real-valued, centered Gaussian random field indexed on the $N$-dimensional unit sphere $\mathbb{S}^N$. Approximations to the excursion probability ${\mathbb{P}}\{\sup_{x\in\mathbb{S}^N}X(x)\ge u\}$, as $u\to\infty$, are obtained for two cases: (i) $X$ is locally isotropic and its sample functions are non-smooth and; (ii) $X$ is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367--379) and Chan and Lai (Ann. Probab. 34 (2006) 80--121). It is shown that the asymptotics of ${\mathbb{P}}\{\sup_{x\in\mathbb {S}^N}X(x)\ge u\}$ is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.

Complex Outliers of Hermitian Random Matrices

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in Benaych-Georges and Rochet (Probab Theory Relat Fields, 2015), as well as possible correlations between outliers at macroscopic distance as in Knowles and Yin (Ann Probab 42(5):1980–2031, 2014) and Benaych-Georges and Rochet (2015). We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in Benaych-Georges and Nadakuditi (Adv Math 227(1):494–521, 2011) and Belinschi et al. (Outliers in the spectrum of large deformed unitarily invariant models 2012, arXiv:1207.5443v1 ). In the particular case where the perturbation matrix is Hermitian, our results complete the work of Benaych-Georges et al. (Electron J Probab 16(60):1621–1662, 2011), as we consider fluctuations of outliers lying in “holes” of the limit support, which happen to exhibit surprising correlations.

Discrepancy properties for random regular digraphs

AbstractFor the uniform random regular directed graph we prove concentration inequalities for (1) codegrees and (2) the number of edges passing from one set of vertices to another. As a consequence, we can deduce discrepancy properties for the distribution of edges essentially matching results for Erdős–Rényi digraphs obtained from Chernoff‐type bounds. The proofs make use of the method of exchangeable pairs, developed for concentration of measure by Chatterjee in (Chatterjee, Probab Theory and Relat Fields 138 (2007), 305–321). Exchangeable pairs are constructed using two involutions on the set of regular digraphs: a well‐known “simple switching” operation, as well as a novel “reflection” operation. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 23–58, 2017

From the Editors

A correction to the date of receipt listed in [A. A. Borovkov and A. A. Mogulskii, Large deviation principles for trajectories of compound renewal processes. I, Theory Probab. Appl., 60 (2016), pp. 207--224].

BSDEs Driven by Multidimensional Martingales and Their Applications to Markets with Funding Costs

We establish some well-posedness and comparison results for BSDEs driven by one- and multidimensional martingales. On the one hand, our approach is largely motivated by results and methods developed by Carbone, Ferrario, and Santacroce in [Theory Probab. Appl., 52 (2008), pp. 304--314] and by El Karoui and Huang in [Backward Stochastic Differential Equations, Pitman Res. Notes in Math. Ser. 364, Longman, Harlow, 1997, pp. 27--36]. On the other hand, our results are also motivated by the recent developments in the theory of pricing of financial derivatives under funding costs and collateralization. A new version of the comparison theorem for BSDEs driven by a multidimensional martingale is established and applied to the pricing BSDEs studied by Bielecki and Rutkowski in [SIAM J. Financial Math., 6 (2015), pp. 594--655] and by Nie and Rutkowski in [Int. J. Theor. Appl. Finance, 18 (2015), 1550048], [Fair Bilateral Prices under Funding Costs and Exogenous Collateralization, Working paper, University of Sydney, Sydney, 2014], and [Finance Stoch., 20 (2016), pp. 855--900]. We establish the existence and uniqueness results for unilateral prices and demonstrate the existence of no-arbitrage bounds for a collateralized contract when both agents have nonnegative initial endowments.

Normal approximation for the net flux through a random conductor

We consider solutions of an elliptic partial differential equation in \({\mathbb R}^d\) with a stationary, random conductivity coefficient. The boundary condition on a square domain of width L is chosen so that the solution has a macroscopic unit gradient. We then consider the average flux through the domain. It is known that in the limit \(L \rightarrow \infty \), this quantity converges to a deterministic constant, almost surely. Our main result is about normal approximation for this flux when L is large: we give an estimate of the Kantorovich–Wasserstein distance between the law of this random variable and that of a normal random variable. This extends a previous result of the author (Probab Theory Relat Fields, 2013. doi:10.​1007/​s00440-013-0517-9) to a much larger class of random conductivity coefficients. The analysis relies on elliptic regularity, on bounds for the Green’s function, and on a normal approximation method developed by Chatterjee (Ann Probab 36:1584–1610, 2008) based on Stein’s method.

Erratum to: Asymptotic behaviour of first passage time distributions for Lévy processes

In this Erratum, we correct an error in our paper “Asymptotic behaviour of first passage time distributions for Levy processes” published in Probab Theory Relat Fields, 157(1–2):1–45, 2013. Let X be a real valued Levy process with law P, characteristic exponent and characteristic triplet (a, σ, ). We assume that X is in the domain of attraction of a stable distribution without centering, that is there exists a deterministic function c : (0,∞) → (0,∞) such that

Erratum to: An $$L_{p}$$ L p -theory of Stochastic PDEs of Divergence Form on Lipschitz Domains

The purpose of this erratum is to correct the assumptions in Theorem 2.10 of [2] (Kim in J Theor Probab 22:220–238, 2009).

Measure valued solutions to the stochastic Euler equations in $$\mathbb {R}^d$$ R d

We establish the existence of measure-valued solutions of the stochastic Euler equations in \(\mathbb {R}^d, \, d \ge 2.\) Our definition of a measure-valued solution is based on that for the deterministic Euler equations. Our first result is the existence over a given probability space in the spirit of Bensoussan and Temam (J Funct Anal 13:195–222, 1973). The main tool is measurable selection established in Stroock and Varadhan (Multidimensional diffusion processes, 1997). As a necessary step for this result, we also present a new existence result on the stochastic Navier–Stokes equations. Our second result is the existence of a martingale measure-valued solution. The main tool is the generalized Skorokhod representation theorem due to Jakubowski (Theory Probab Appl, 42:167–174, 1998).

On distribution of the norm of deviation of a sub-Gaussian random process in Orlicz spaces

Abstract The paper is devoted to the study of sub-Gaussian random variables and stochastic processes. Recall that along with centered Gaussian random variables the space Sub(Ω) of sub-Gaussian random variables contains all bounded zero-mean random variables and all zero-mean random variables whose distribution tails decrease no slower than the tails of the distribution of a Gaussian random variable. Here we study a square deviation of a sub-Gaussian random process from a constant and derive an upper estimate for the exponential moment of the deviation. The obtained result allows to estimate the distribution of deviation of a sub-Gaussian random process from some measurable function in the norm of Lp (𝕋) and in the norm of Orlicz space. The paper generalizes results of [Theory Probab. Math. Statist. 58 (1999), 51–66] for the norm of a sub-Gaussian random process in Orlicz space. As an example we apply the obtained estimates to a sub-Gaussian Wiener process deviated from a linear and a square root functions.

Generic properties for random repeated quantum iterations

We denote by $$M^n$$ the set of n by n complex matrices. Given a fixed density matrix $$\beta :\mathbb {C}^n \rightarrow \mathbb {C}^n$$ and a fixed unitary operator $$U : \mathbb {C}^n \otimes \mathbb {C}^n \rightarrow \mathbb {C}^n \otimes \mathbb {C}^n$$ , the transformation $$\Phi : M^n \rightarrow M^n$$ $$\begin{aligned} Q \rightarrow \Phi (Q) = \text {Tr}_2 (U ( Q \otimes \beta ) U^*)\\ \end{aligned}$$ describes the interaction of Q with the external source $$\beta $$ . The result of this operation is $$\Phi (Q)$$ . If Q is a density operator then $$\Phi (Q)$$ is also a density operator. The main interest is to know what happens when we repeat several times the action of $$\Phi $$ in an initial fixed density operator $$Q_0$$ . This procedure is known as random repeated quantum iterations and is of course related to the existence of one or more fixed points for $$\Phi $$ . In Nechita and Pellegrini (Probab Theory Relat Fields 52:299–320, 2012), among other things, the authors show that for a fixed $$\beta $$ , there exists a set of full probability for the Haar measure such that the unitary operator U satisfies the property that for the associated $$\Phi $$ there is a unique fixed point $$ Q_\Phi $$ . Moreover, there exists convergence of the iterates $$\Phi ^n (Q_0) \rightarrow Q_\Phi $$ , when $$n \rightarrow \infty $$ , for any given initial $$Q_0$$ . We show here that there is an open and dense set of unitary operators $$U: \mathbb {C}^n \otimes \mathbb {C}^n \rightarrow \mathbb {C}^n \otimes \mathbb {C}^n $$ such that the associated $$\Phi $$ has a unique fixed point. We will also consider a detailed analysis of the case when $$n=2$$ . We will be able to show explicit results. We consider the $$C^0$$ topology on the coefficients of U. In this case, we will exhibit the explicit expression on the coefficients of U which assures the existence of a unique fixed point for $$\Phi $$ . Moreover, we present the explicit expression of the fixed point $$Q_\Phi $$ .

Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation

We study the nonlinear fractional stochastic heat equation in the spatial domain \({\mathbb {R}}\) driven by space-time white noise. The initial condition is taken to be a measure on \({\mathbb {R}}\), such as the Dirac delta function, but this measure may also have non-compact support. Existence and uniqueness, as well as upper and lower bounds on all pth moments \((p\ge 2)\), are obtained. These bounds are uniform in the spatial variable, which answers an open problem mentioned in Conus and Khoshnevisan (Probab Theory Relat Fields 152(3–4):681–701, 2012). We improve the weak intermittency statement by Foondun and Khoshnevisan (Electron J Probab 14(21):548–568, 2009), and we show that the growth indices (of linear type) introduced in Conus and Khoshnevisan (Probab Theory Relat Fields 152(3–4):681–701, 2012) are infinite. We introduce the notion of “growth indices of exponential type” in order to characterize the manner in which high peaks propagate away from the origin, and we show that the presence of a fractional differential operator leads to significantly different behavior compared with the standard stochastic heat equation.

A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu (J Theor Probab 27:220–248, 2014). Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used.

Shock Fluctuations in Flat TASEP Under Critical Scaling

We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate \(1\), while particles to the right have jump rate \(\alpha \). When \(\alpha <1\) there is a formation of a shock where the density jumps to \((1-\alpha )/2\). For \(\alpha <1\) fixed, the statistics of the associated height functions around the shock is asymptotically (as time \(t\rightarrow \infty \)) a maximum of two independent random variables as shown in Ferrari and Nejjar (Probab Theory Rel Fields 161:61–109, 2015). In this paper we consider the critical scaling when \(1-\alpha =a t^{-1/3}\), where \(t\gg 1\) is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of \(a\). We see that the convergence to \(F_\mathrm{GOE}^2\) occurs quite rapidly as \(a\) increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes (Baik et al. in Ann Probab 33:1643–1697, 2006).

Identifying Berlin’s land value map using adaptive weights smoothing

We use adaptive weights smoothing (AWS) of Polzehl and Spokoiny (J R Stat Soc Ser B 62:335–354, 2000; Ann Stat 31:30–57, 2003; Probab Theory Relat Fields 135:335–362, 2006) to estimate a map of land values for Berlin, Germany. Our data are prices of undeveloped land that was transacted between 1996 and 2009. Even though the observed land price is an indicator of the respective land value, it is influenced by transaction noise. The iterative AWS applies piecewise constant regression to reduce this noise and tests at each location for constancy at the margin. If not rejected, further observations are included in the local regression. The estimated land value map conforms overall well with expert-based land values. Our application suggests that the transparent AWS could prove a useful tool for researchers and real estate practitioners alike.

Letter to the Editors

Previous article Letter to the EditorsV. R. FatalovV. R. Fatalovhttps://doi.org/10.1137/S0040585X97T987168PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this letter, the author provides erratum to his paper [On the Laplace method for Gaussian measures in a Banach space, Theory Probab. Appl., 58 (2014), pp. 216--251].1. V. R. Fatalov and On Laplace method for Gaussian measures in a Banach space , Theory Probab. Appl. , 58 ( 2014 ), pp. 216 -- 251 . LinkGoogle Scholar2. A. S. Mishchenko and A. T. Fomenko, A Course of Differential Geometry and Topology, Mir Publishers, Moscow, 1988.Google Scholar Previous article FiguresRelatedReferencesCited ByDetails Volume 59, Issue 2| 2015Theory of Probability & Its Applications History Published online:09 June 2015 Information© 2015, Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X97T987168Article page range:pp. 357-358ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics