We give a dynamic extension result of the (static) notion of a deviation measure. We also study distribution-invariant deviation measures and show that the only dynamic deviation measure which is law invariant and recursive is the variance.

In this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.

We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations $\mathsf{E}$ and $\mathsf{F}$. We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation

$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$

$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$

$(1+1)$

We present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

We find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows us to reduce the problem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$$(1/4<H\leqslant 1/2)$. Under Hörmander’s condition on the coefficient vector fields, the solution has a smooth density for each fixed time. Using Watanabe’s distributional Malliavin calculus, we obtain a short time full asymptotic expansion of the density under quite natural assumptions. Our main result can be regarded as a “fractional version” of Ben Arous’ famous work on the off-diagonal asymptotics.

We introduce variance-optimal semi-static hedging strategies for a given contingent claim. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy we use a Fourier approach in a multidimensional factor model. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in the Heston model, which is affine, in the 3/2 model, which is not, and in a market model including jumps.

Asset allocation with a derivative security is studied in a hidden, Markovian regime-switching, economy using filtering theory and the martingale approach. A generalized delta-hedged ratio and a generalized elasticity of an option are introduced to accommodate the presence of the information state process and the derivative security. Malliavin calculus is applied to derive a solution for a general utility function which includes an exponential utility, a power utility, and a logarithmic utility. A compact solution is obtained for a logarithmic utility. Some economic implications of the solutions are discussed.

In last passage percolation models lying in the Kardar–Parisi–Zhang (KPZ) universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at $(0,0)\in \mathbb{R}^{2}$, and the other is varied horizontally, over $(z,1)$, $z\in \mathbb{R}$, the polymer weight profile as a function of $z\in \mathbb{R}$ is locally Brownian; indeed, by Hammond [‘Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation’, Preprint (2016), arXiv:1609.02971, Theorem 2.11 and Proposition 2.5], the law of the profile is known to enjoy a very strong comparison to Brownian bridge on a given compact interval, with a Radon–Nikodym derivative in every $L^{p}$ space for $p\in (1,\infty )$, uniformly in the scaling parameter, provided that an affine adjustment is made to the weight profile before the comparison is made. In this article, we generalize this narrow wedge case and study polymer weight profiles begun from a very general initial condition. We prove that the profiles on a compact interval resemble Brownian bridge in a uniform sense: splitting the compact interval into a random but controlled number of patches, the profile in each patch after affine adjustment has a Radon–Nikodym derivative that lies in every $L^{p}$ space for $p\in (1,3)$. This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in Hammond [‘Exponents governing the rarity of disjoint polymers in Brownian last passage percolation’, Preprint (2017a), arXiv:1709.04110] using techniques from Hammond (2016) and [‘Modulus of continuity of polymer weight profiles in Brownian last passage percolation’, Preprint (2017b), arXiv:1709.04115].

We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.

In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We study load balancing under the batch sampling routeing scheme for a network of n servers storing a set of files using the maximum distance separable (MDS) code (cf. Li (2016)). Specifically, each file is stored in equally sized pieces across L servers such that any k pieces can reconstruct the original file. When a request for a file is received, the dispatcher routes the job into the k-shortest queues among the L for which the corresponding server contains a piece of the file being requested. We establish a law of large numbers and a central limit theorem as the system becomes large (i.e. n → ∞), for the setting where all interarrival and service times are exponentially distributed. For the central limit theorem, the limit process take values in ℓ2, the space of square summable sequences. Due to the large size of such systems, a direct analysis of the n-server system is frequently intractable. The law of large numbers and diffusion approximations established in this work provide practical tools with which to perform such analysis. The power-of-d routeing scheme, also known as the supermarket model, is a special case of the model considered here.

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are derived under heavy-tailed conditions allowing both dependence between Q and N and the tails to be of the same order of magnitude. Similar results are derived for a K-class version with applications to multi-type branching processes and busy periods in multi-class queues.

In this paper we consider the asymptotics of logarithmic tails of a perpetuity R=D∑j=1∞Qj∏k=1j-1Mk, where (Mn,Qn)n=1∞ are independent and identically distributed copies of (M,Q), for the case when ℙ(M∈[0,1))=1 and Q has all exponential moments. If M and Q are independent, under regular variation assumptions, we find the precise asymptotics of -logℙ(R>x) as x→∞. Moreover, we deal with the case of dependent M and Q, and give asymptotic bounds for -logℙ(R>x). It turns out that the dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such a phenomenon is not observed in the case of heavy-tailed perpetuities.

We consider a large class of $1+1$-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf–Cole solutions to the KPZ equation.

In this paper we study the Assouad dimension of graphs of certain Lévy processes and functions defined by stochastic integrals. We do this by introducing a convenient condition which guarantees a graph to have full Assouad dimension and then show that graphs of our studied processes satisfy this condition.