In this paper we consider BSDEs with Lipschitzcoefficient reflected on two discontinuous (RCLL) barriers. In thiscase, we prove first the existence and uniqueness of the solution,then we also prove the convergence of the solutions of the penalizedequations to the solution of the RBSDE. Since the method used in thecase of continuous barriers (see Cvitanic and Karatzas, Ann. Probab.24 (1996) 2024–2056 and Lepeltier and San Martín, J. Appl. Probab.41 (2004) 162–175) does notwork, we develop a new method, by considering the solutions of thepenalized equations as the solutions of special RBSDEs and usingsome results of Peng and Xu in Annales of I.H.P.41 (2005) 605–630.

In practice, it is well known that hedging a derivative instrumentcan never be perfect. In the case of credit derivatives (e.g.synthetic CDO tranche products), a trader will have to face somespecific difficulties. The first one is the inconsistence betweenmost of the existing pricing models, where the risk is theoccurrence of defaults, and the real hedging strategy, where thetrader will protect his portfolio against small CDS spreadmovements. The second one, which is the main subject of thispaper, is the consequence of a wrong estimation of some parametersspecific to credit derivatives such as recovery rates orcorrelation coefficients. We find here an approximation of thedistribution under the historical probability of the final Profit& Loss of a portfolio hedged with wrong estimations of theseparameters. In particular, it will depend on a ratio between thesquare root of the historical default probability and therisk-neutral default probability. This result is quite general andnot specific to a given pricing model.

Asymmetric or partial information in financial markets may berepresented by different filtrations. We consider the case of alarger filtration F – the natural filtration of the“model world” – and a subfiltration $\hat{\mathcal F}$ thatrepresents the information available to an agent in the “realworld”. Given a price system on the larger filtration that isrepresented by a martingale measure Q and an associated numeraire S, we show that there is a canonical and nontrivial numeraire Ŝ such that the price system generated by(Ŝ,Q, $\hat{\mathcal F}$ ) is consistent, in a sense to be madeprecise, with the price system generated by (S,Q,F).

A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singular and regular perturbation analysis to characterize the variance reduced by martingale control variates. This variance analysis is done in the regime where time scales of associated driving volatility processes are well separated. Numerical results for European, Barrier, and American options are presented to illustrate the effectiveness and robustness of this martingale control variate method in regimes where these time scales are not so well separated.

We consider an infinite system of hard balls in $\xR^d$ undergoing Brownian motionsand submitted to a smooth pair potential.It is modelized by an infinite-dimensional stochastic differential equationwith an infinite-dimensional local time term.Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also showthat Gibbs measures are reversible measures.

We consider an extension of the Kyle and Back's model [Back, Rev. Finance Stud.5 (1992) 387–409; Kyle, Econometrica35 (1985) 1315–1335],meaning a model for the market with a continuous time risky assetand asymmetrical information. There are threefinancial agents: the market maker, an insider trader (who knows a randomvariable V which will be revealed at final time) and a non informed agent. Here we assume thatthe non informed agent is strategic, namely he/she uses a utility function to optimize his/her strategy.Optimal control theory is applied to obtain a pricing ruleand to prove the existenceof an equilibrium price when the insider trader and the non informedagent are risk-neutral. We will show that if such an equilibrium exists, then the non informed agent's optimal strategy is to do nothing, in other words to be non strategic.

Expected suprema of a function f observed along the paths of a nice Markov process define an excessive function, and in fact a potential if f vanishes at the boundary. Conversely, we show under mild regularity conditions that any potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal representing function in terms of probabilistic potential theory. Our results are motivated by the work of El Karoui and Meziou (2006) on the max-plus decomposition of supermartingales, and they provide a singular analogue to the non-linear Riesz representation in El Karoui and Föllmer (2005).

Considering the centered empirical distribution function F n-F asa variable in ${\mathbb L}^p(\mu)$ , we derive non asymptotic upperbounds for the deviation of the ${\mathbb L}^p(\mu)$ -norms ofF n-F as well as central limit theorems for the empirical processindexed by the elements of generalized Sobolev balls. These resultsare valid for a large class of dependent sequences, includingnon-mixing processes and some dynamical systems.

A stochastic “Fubini” lemma and an approximation theorem forintegrals on the plane are used to produce a simulation algorithmfor an anisotropic fractional Brownian sheet. The convergence rateis given. These results are valuable for any value of the Hurstparameters $(\alpha_1,\alpha_2)\in ]0,1[^2,\alpha_i\neq\frac{1}{2}.$ Finally, theapproximation processis iterative on the quarter plane $\mathbb {R}_+^2.$ A sample of such simulations can be used to test estimatorsof the parameters αi,i = 1,2.

Let $\tau _{D}(Z) $ be the first exit time ofiterated Brownian motion from a domain $D \subset \mathbb{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau _{D}(Z)>t]$ be itsdistribution. In this paper we establish the exact asymptotics of $P_{z}[\tau _{D}(Z)>t]$ over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116(2006) 905–916], for $z\in D$ 
 $ \displaystyle \lim_{t\to\infty}t^{-1/2}\exp\left(\frac{3}{2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}\right)P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber$ 
where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}\left(\psi(z)\int_{D}\psi(y){\rm d}y\right) ^{2}$ . Here λD is thefirst eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$ inD, and ψ is the eigenfunction corresponding toλD . We also study lifetime asymptotics of Brownian-time Brownian motion, $Z^{1}_{t} = z+X(|Y(t)|)$ , where X t and Y t are independentone-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.

In this paper we study asymptotic behavior of convexrearrangements of Lévy processes. In particular weobtain Glivenko-Cantelli-type strong limit theorems for the convexifications when the corresponding Lévy measureis regularly varying at + with exponent α ∈ (1,2).

We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process betweenthe successive times iΔn for i = 0,1,...,n. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/orcentering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function f. As for the associated central limit theorem,one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.

In many markets, especially in energy markets, electricity markets for instance, the detention of the physical asset is quite difficult. This is also the case for crude oil as treated by Davis (2000). So one can identify a good proxy which is an asset (financial or physical) (one)whose the spot price is significantly correlated with the spot price of the underlying (e.g. electicity or crude oil). Generally, the market could become incomplete. We explicit exact hedging strategies for exponential utilities when the risk premium is bounded. Our result is based upon backward stochastic differential equation (BSDE) and a good choice of admissible strategies which allows us to solve our hedging problem.

Different kinds of renewal equations repeatedly arise in connectionwith renewal risk models and variations. It is often appropriate toutilize bounds instead of the general solution to the renewalequation due to the inherent complexity. For this reason, as a firstapproach to construction of bounds we employ a general Lundberg-typemethodology. Second, we focus specifically on exponential boundswhich have the advantageous feature of being closely connected tothe asymptotic behavior (for large values of the argument) of therenewal function. Finally, the last section of this paper includesseveral applications to risk theory quantities.

The aim of this short note is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant measures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously.

For any given random variable Y with infinitelydivisible distribution in a quadratic natural exponential family we obtain a polynomial expansion ofthe power mixture density of Y.We approach the problem generally, and then consider certain distributionsin greater detail.Various applications are indicated and the results are also appliedto obtain approximations and their error bounds.Estimation of density and goodness-of-fit test are derived.

Let X1,...,Xn1 be arandom sample from a population with mean µ 1 and variance $\sigma_1^2$ , and X1,...,Xn1 be a random sample fromanother population with mean µ 2 and variance $\sigma_2^2$ independent of{Xi,1 ≤ i ≤ n1 }.Consider the twosample t-statistic $ T={{\bar X-\bar Y-(\mu_1-\mu_2)} \over\sqrt{s_1^2/n_1+s_2^2/n_2}}$ . This paper shows thatln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2) satisfying x → ∞, x = o(n1 + n2)1/2 as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, inaddition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T \geq x)}{1-\Phi(x)} \to 1 $ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

Assessing the number of clusters of a statistical population is one of the essential issues of unsupervised learning. Given n independent observations X1,...,Xn drawn from an unknown multivariate probability density f, we propose a new approach to estimate the number of connected components, or clusters, of the t-level set $\mathcal L(t)=\{x:f(x) \geq t\}$ . The basic idea is to form a rough skeleton of the set $\mathcal L(t)$ using any preliminary estimator of f, and to count the number of connected components of the resulting graph. Under mild analytic conditions on f, and using tools from differential geometry, we establish the consistency of our method.

We investigate the optimal alignment of two independent random sequences of length n. We providea polynomial lower bound for the probability of the optimal alignmentto be macroscopically non-unique. We furthermore establisha connection between the transversal fluctuation andmacroscopic non-uniqueness.