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In this letter, we test a scalar stochastic nonlinear equation used to portray the growth of a population with Allee effects. We first testify that there is a unique dynamical bifurcation point Λ to the equation, and the sign of Λ determines the dynamical properties of the equation: if Λ is negative, then the equation has a unique invariant measure — the Dirac measure concentrated at zero; if Λ is positive, the equation has a unique invariant measure concentrated on (0 , + ∞) , and the density function of the invariant measure can be expressed explicitly. Then we probe the lower-growth rate and the continuity of the solution. Finally, we apply the theoretical results to research the growth of African hunting dogs. [ABSTRACT FROM AUTHOR]

In this paper we propose a general framework for modeling an insurance liability cash flow in continuous time, by generalizing the reduced-form framework for credit risk and life insurance. In particular, we assume a nontrivial dependence structure between the reference filtration and the insurance internal filtration. We apply these results for pricing and hedging non-life insurance liabilities in hybrid financial and insurance markets, while taking into account the role of inflation under the benchmarked risk-minimization approach. This framework offers at the same time a general and flexible structure, and an explicit and treatable pricing-hedging formula. [ABSTRACT FROM AUTHOR]

We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients and unbounded divergence. In the first result, we assume the drift is L 2 ([ 0 , T ] × R d) ∩ L ∞ ([ 0 , T ] × R d) , and the divergence is the locally integrable. In the second result, we show that the smoothing acts as a selection criterion when the drift is in L 2 ([ 0 , T ] × R d) ∩ L ∞ ([ 0 , T ] × R d) without any condition on the divergence. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. They assume that the cost function satisfies a convexity and a weak monotonicity property. They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE for short). They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by [Hu, Y. and Peng, S., Solution of forward-backward stochastic differential equations, Probab. Theory Rel., 103(2), 1995, 273–283]. They apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval. [ABSTRACT FROM AUTHOR]

We consider the problem of pricing perpetual American options written on dividend-paying assets whose price dynamics follow a multidimensional Black and Scholes model. For convex Lipschitz continuous reward functions, we give a probabilistic characterization of the fair price in terms of a reflected BSDE, and an analytical one in terms of an obstacle problem. We also provide the early exercise premium formula. [ABSTRACT FROM AUTHOR]

We investigate the existence of a weak martingale solution for a two-dimensional critical viscoelastic flow of the Oldroyd type driven by pure jump Lévy noise. Due to the viscoelastic nature, noise in the equation modeling stress tensor is considered in the Marcus canonical form. Owing to the lack of dissipation and taking into account of the structure of the non-linear terms, the proof requires higher order estimates. [ABSTRACT FROM AUTHOR]

We study the asymptotic organization among many optimizing individuals interacting in a suitable "moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument. [ABSTRACT FROM AUTHOR]

The paper introduces a new weak approximation algorithm for stochastic differential equations (SDEs) of McKean–Vlasov type. The arbitrary order discretization scheme is available and is given using Malliavin weights, certain polynomial weights of Brownian motion, which play a role as correction of the approximation. The new weak approximation scheme works even if the test function is not smooth. In other words, the expectation of irregular functionals of McKean–Vlasov SDEs such as probability distribution functions are approximated through the proposed scheme. The effectiveness of the higher order scheme is confirmed by numerical examples for McKean–Vlasov SDEs including the Kuramoto model. [ABSTRACT FROM AUTHOR]

We study the transport property of Gaussian measures on Sobolev spaces of periodic functions under the dynamics of the one-dimensional cubic fractional nonlinear Schrödinger equation. For the case of second-order dispersion or greater, we establish an optimal regularity result for the quasi-invariance of these Gaussian measures, following the approach by Debussche and Tsutsumi (2021). Moreover, we obtain an explicit formula for the Radon-Nikodym derivative and, as a corollary, a formula for the two-point function arising in wave turbulence theory. We also obtain improved regularity results in the weakly dispersive case, extending those by the first author and Trenberth (2019). Our proof combines the approach introduced by Planchon, Tzvetkov and Visciglia (2020) and that of Debussche and Tsutsumi (2021). [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of an optimal investment and risk control problem for an insurer. The risky asset process and the insurance liability process are governed by stochastic differential equations with jumps and anticipative parameters. The insurer can only get access to partial information about the financial market and the insurance business to make decisions. Taking into account endogenous and exogenous factors, we assume the time horizon is uncertain. With the aim of expected logarithmic utility maximization, we adopt the forward stochastic calculus and the Malliavin calculus to derive a characterization of the optimal strategy. In some particular cases, we obtain the optimal strategies in closed-form and get some new insights. [ABSTRACT FROM AUTHOR]

We study a general kind of partial information nonzero sum two-player stochastic differential games, where the state variable is governed by a stochastic differential equation and the control domain of each player can be non-convex. Moreover, the control variables of both players can enter the diffusion coefficients of the state equation. We establish Pontryagin's maximum principle for open-loop Nash equilibria of the game. Then, a verification theorem is obtained for Nash equilibria when the control domain is convex. Finally, the theoretical results are applied to studying a linear-quadratic game. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of a bounded random solution for a nonlinear stochastic integrodifferential equation on R + . We also show that the sequence of successive approximations converges to the solution x (t , ω) almost surely and in mean square at each t ∈ R + . An application of our result to a stochastic differential system is also given in Section 4. [ABSTRACT FROM AUTHOR]

In this paper we are interested to solve a class of quadratic BSDEs with jumps (QBSDEJs for short) of the following form: Y t = ξ + ∫ t T H (Y s , Z s , U s (⋅)) d s − ∫ t T Z s d W s − ∫ t T ∫ E U s (e) N ~ (d s , d e) , Herein, the terminal data ξ will be assumed to be square integrable. Our study covers the following cases H (y , z , u (⋅)) = f (y) z 2 + [ u ] f (y) =: H f (y , z , u (⋅))) h y , u (⋅) + c z + H f (y , z , u (⋅))) a + b y + c z + d u (⋅) ν , 1 + H f (y , z , u (⋅)) c z + f (y) z 2 − ∫ E u (e) ν (d e) c z + f (y) z 2 h y , u (⋅) + c z + f (y) z 2 H 0 r , X r + H f (y , z , u (⋅))) , (X r ) r ≥ 0 is a Markov process where f is a measurable and integrable function, u f (⋅) is a functional of the unknown processes Y ⋅ and U ⋅ (⋅) to be defined later and h and H 0 enjoy some classical assumptions. The generators show quadratic growth in the Brownian component and non-linear functional form with respect to the jump term. Existence or uniqueness of solutions as well as a comparison and strict comparison principles are established under no monotonicity condition in the third argument of the generator. Probabilistic representations of solutions to some classes of quadratic PIDE are given by means of solutions of these QBSDEJs. The main idea is to use a phase space transformation to transform our initial QBSDEJ to a standard BSDEJ without quadratic term. [ABSTRACT FROM AUTHOR]

The time-evolution of a moderately dense gas in a vacuum is described in classical mechanics by a particle density function obtained from the Boltzmann–Enskog equation. Based on a McKean–Vlasov equation with jumps, the associated stochastic process was recently constructed by modified Picard iterations with the mean-field interactions, and more generally, by a system of interacting particles. By the introduction of a shifted distance that exactly compensates for the free transport term that accrues in the spatially inhomogeneous setting, we prove in this work an inequality on the Wasserstein distance for any two measure-valued solutions to the Boltzmann–Enskog equation. As a particular consequence, we find sufficient conditions for the uniqueness and continuous-dependence on initial data for solutions to the Boltzmann–Enskog equation applicable to hard and soft potentials without angular cut-off. [ABSTRACT FROM AUTHOR]

The investigation of the interrelationships among different oceanic and atmospheric circulation patterns is crucial for future climate projections in the current century. This paper presents the transition matrix approach of the stochastic Markov chain process to investigate the state/event based relationship between the new index of the Interdecadal Pacific Oscillation (IPO) named as the IPO Tripole index (TPI) and different sea surface temperature anomalies (SSTA) based El Nino-Southern Oscillation (ENSO) indices such as Niño 1.2, Niño 3, Niño 3.4, Niño 4 and the Multivariate ENSO Index (MEI) for the period of 120 years (1900–2019) respectively. Several Markovian descriptors like state dependency, temporal stationarity, expected number of state visits and entropy are derived from the estimated transition matrix. These descriptors are helpful in establishing the validity of Markov chain method and useful to characterize the dynamical properties of a time series like persistence, randomness and behaviour of cycles. Through the Markov chain analysis and by derived descriptors, this study finds similar self-communication (periodic) pattern between the transition states, resemblance in expected number of visits from one transition state to another, asymmetric and truncated cyclic nature of the data sequence and the existence of randomness in the transition states. Finally, a strong 2-dimensional correlation values endorses the existence of strong relations between selected indices datasets. This analysis approach may be helpful in understanding the role of the IPO and ENSO in modulating future climate variability and to formulate effective predictive models at the climatic state. [ABSTRACT FROM AUTHOR]

This paper studies an asset-liability management (ALM) problem under mean-variance criterion with inside information. The asset-liability manager is allowed to invest in a financial market composed of a bond and a stock. The stock price process is governed by a diffusion process with random parameters. The uncontrolled liability process is described by a general diffusion process with hedgeable risks and unhedgeable risks. We model the inside information by a general random variable related to the future values of financial assets and liabilities. By using the Donsker Delta functional technique and the BSDE method, we derive the analytic expressions of efficient strategy and efficient frontier. To illustrate the general result, an example is provided in which the efficient strategy and efficient frontier are obtained in closed form. The comparison of efficient frontiers with and without inside information demonstrates that taking advantage of inside information can improve the efficient frontier. [ABSTRACT FROM AUTHOR]

We study in this paper weak approximations in Wasserstein-1 distance to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations, and obtain uniform error bounds. Our approach is via Malliavin calculus and a refined Lindeberg principle. [ABSTRACT FROM AUTHOR]

In this work, we provide a general mathematical formalism to study the optimal control of an epidemic, such as the COVID-19 pandemic, via incentives to lockdown and testing. In particular, we model the interplay between the government and the population as a principal–agent problem with moral hazard, à la Cvitanić et al. (Finance Stoch 22(1):1–37, 2018), while an epidemic is spreading according to dynamics given by compartmental stochastic SIS or SIR models, as proposed respectively by Gray et al. (SIAM J Appl Math 71(3):876–902, 2011) and Tornatore et al. (Phys A Stat Mech Appl 354(15):111–126, 2005). More precisely, to limit the spread of a virus, the population can decrease the transmission rate of the disease by reducing interactions between individuals. However, this effort—which cannot be perfectly monitored by the government—comes at social and monetary cost for the population. To mitigate this cost, and thus encourage the lockdown of the population, the government can put in place an incentive policy, in the form of a tax or subsidy. In addition, the government may also implement a testing policy in order to know more precisely the spread of the epidemic within the country, and to isolate infected individuals. In terms of technical results, we demonstrate the optimal form of the tax, indexed on the proportion of infected individuals, as well as the optimal effort of the population, namely the transmission rate chosen in response to this tax. The government’s optimisation problems then boils down to solving an Hamilton–Jacobi–Bellman equation. Numerical results confirm that if a tax policy is implemented, the population is encouraged to significantly reduce its interactions. If the government also adjusts its testing policy, less effort is required on the population side, individuals can interact almost as usual, and the epidemic is largely contained by the targeted isolation of positively-tested individuals. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study backward stochastic differential equation driven by a class of normal martingale. We prove the existence and uniqueness results of the solution under the stochastic Lipschitz condition by mean the martingale representation theorem. In addition, we show that the solution of these equations provides a viscosity solution of the associated system with partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a martingale representation theorem in the progressive enlargement G by a random time τ of the filtration F L generated by a Lévy process. The assumptions on the random time are that F L is immersed in G and that τ avoids F L stopping times. We also study the multiplicity of a progressively enlarged filtration. [ABSTRACT FROM AUTHOR]

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus. In an influential paper, Lebowitz et al. (J Stat Phys 50(3–4):657–687, 1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (Ann Inst H Poincaré Anal Non Linéaire 31(6):1267–1288, 2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz et al. (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schrödinger equation on the one-dimensional torus. [ABSTRACT FROM AUTHOR]

In Friz et al. [Precise asymptotics for robust stochastic volatility models. Ann. Appl. Probab, 2021, 31(2), 896–940], we introduce a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small-noise formulae for option prices, using the framework [Bayer et al., A regularity structure for rough volatility. Math. Finance, 2020, 30(3), 782–832]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence. [ABSTRACT FROM AUTHOR]

We study the two-dimensional surface quasi-geostrophic equations and the three-dimensional Hall-magnetohydrodynamics system forced by degenerate noise. In comparison with a vorticity formulation of the Navier–Stokes equations, the non-linear term of the surface quasi-geostrophic equations is more singular by one derivative. In comparison with the magnetohydrodynamics system, the Hall term of the Hall-magnetohydrodynamics system is also more singular by one derivative. We prove the existence and uniqueness of an invariant measure for the Galerkin approximations of the surface quasi-geostrophic equations and the Hall-magnetohydrodynamics system, both forced by degenerate noise which consists of only a few modes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a specific kind of doubly reflected backward stochastic differential equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki's condition, we show the existence of the solution (in consideration of the driver's nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of Itô's formula due to Gal'chouk and Lenglart we provide a suitable a priori estimates which immediately implies the uniqueness of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients in (y, z). The existence and uniqueness of solutions are obtained via a method of Picard iteration, a linearization method and a monotone convergence argument. Furthermore, we establish the corresponding comparison theorem and related nonlinear Feynman–Kac formula. [ABSTRACT FROM AUTHOR]

In this paper, we study a numerical approximation scheme for reflected stochastic differential equations (SDEs) with non-Lipschitzian coefficients in a bounded convex domain. It is shown, under some mild conditions, that the approximation scheme converges in uniform L 2 to the solution of reflected SDEs. Moreover, we move from local to global monotonicity conditions and consider the rate of convergence for our approximation scheme to reflected SDEs with coefficients which have at most polynomial growth. [ABSTRACT FROM AUTHOR]

Superspreaders (individuals with a high propensity for disease spread) have played a pivotal role in recent emerging and re-emerging diseases. In disease outbreak studies, host heterogeneity based on demographic (e.g. age, sex, vaccination status) and environmental (e.g. climate, urban/rural residence, clinics) factors are critical for the spread of infectious diseases, such as Ebola and Middle East Respiratory Syndrome (MERS). Transmission rates can vary as demographic and environmental factors are altered naturally or due to modified behaviors in response to the implementation of public health strategies. In this work, we develop stochastic models to explore the effects of demographic and environmental variability on human-to-human disease transmission rates among superspreaders in the case of Ebola and MERS. We show that the addition of environmental variability results in reduced probability of outbreak occurrence, however the severity of outbreaks that do occur increases. These observations have implications for public health strategies that aim to control environmental variables.

This paper studies optimal investment and risk control strategies for an insurer who owns insider information. The insurance risk process is governed by a general jump diffusion process with random parameters and is correlated with the risky asset process in the financial market. We model the inside information by a general random variable related to the insurance risk process and the risky asset process. Under the criterion of expected utility maximization of the terminal wealth, we adopt white noise calculus and BSDE approach to analyze the problem for various utility functions. [ABSTRACT FROM AUTHOR]

This paper is to establish a sufficient maximum principle for one kind of stochastic optimal control problem with three types of delays: a discrete delay, a moving-average delay and a noisy memory. The main features of this research include the introduction of a unified adjoint equation and a simple method to get the adjoint process. One kind of optimal consumption problem and its special cases are studied as illustrative examples, for which the adjoint equations are solved with two different approaches and the optimal consumption strategies are obtained. [ABSTRACT FROM AUTHOR]

Usually Fokker–Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form of a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process. [ABSTRACT FROM AUTHOR]

In this paper we consider a stochastic Keller–Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work. [ABSTRACT FROM AUTHOR]

In this paper, we derive the existence and uniqueness of the solution for a new class of mean-field backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. In addition, we study the generalized mean-field backward stochastic differential equations in a Markovian framework, that is, it is associated with a reflected McKean-Vlasov stochastic differential equation. This allows us to give a probabilistic formula for viscosity solution of a nonlocal partial differential equation with a nonlinear Neumann boundary condition. [ABSTRACT FROM AUTHOR]

We propose a fully backward representation of semilinear PDEs with application to stochastic control. Based on this, we develop a fully backward Monte-Carlo scheme allowing to generate the regression grid, backwardly in time, as the value function is computed. This offers two key advantages in terms of computational efficiency and memory. First, the grid is generated adaptively in the areas of interest, and second, there is no need to store the entire grid. The performances of this technique are compared in simulations to the traditional Monte-Carlo forward-backward approach on a control problem of thermostatic loads. [ABSTRACT FROM AUTHOR]

The classical Method of Successive Approximations (MSA) is an iterative method for solving stochastic control problems and is derived from Pontryagin's optimality principle. It is known that the MSA may fail to converge. Using careful estimates for the backward stochastic differential equation (BSDE) this paper suggests a modification to the MSA algorithm. This modified MSA is shown to converge for general stochastic control problems with control in both the drift and diffusion coefficients. Under some additional assumptions the rate of convergence is shown. The results are valid without restrictions on the time horizon of the control problem, in contrast to iterative methods based on the theory of forward-backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of Green's functions of this process. To that purpose we establish a new kernel functional equation connecting moment generating functions of Green's functions inside the quadrant and on its edges. This is reminiscent of the recurrent case where a functional equation derives from the basic adjoint relationship which characterizes the stationary distribution. This equation leads us to a non-homogeneous Carleman boundary value problem. Its resolution provides a formula for the moment generating function in terms of contour integrals and a conformal mapping. [ABSTRACT FROM AUTHOR]

In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process by X = { X t , t ≥ 0 } and set τ l = inf { t ≥ 0 , X t ≤ l } and τ l , u = inf { t ≥ 0 , X t ∉ (l , u) } with l < u . We first establish the existence and uniqueness of strong solutions of the stochastic differential equation (SDE) satisfied by X. Then we investigate the Laplace transforms associated with τ l and τ l , u . It turns out that the explicit expressions for those Laplace transforms can be expressed in terms of exponential functions. [ABSTRACT FROM AUTHOR]

In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator ∂ t - K , where K is the following nonlocal kinetic operator: K f (x , v) = p. v. ∫ R d (f (x , v + w) - f (x , v)) κ (x , v , w) | w | d + α d w + v · ∇ x f (x , v) + b (x , v) · ∇ v f (x , v). Here κ 0 - 1 ⩽ κ (x , v , w) ⩽ κ 0 belongs to C b ∞ (R 3 d) and is symmetric in w, p.v. stands for the Cauchy principal value, and b ∈ C b ∞ (R 2 d ; R d) . [ABSTRACT FROM AUTHOR]

Photovoltaic power system is taking a significant percentage of power system and the demands for accurate forecasting of the power outputs is surging. In prior works, the forecasting problem was formulated as a regression problem, however, which most cannot guarantee that the forecasted outputs is nonnegative. To solve this problem, we proposed a novel probabilistic model by using nonlinear regression and Bayesian learning method. In the paper, we present the detailed theoretical derivations and interpretations. The simulation results show the validity and feasibility of the proposed algorithm by comparing with the traditional SVM algorithm.

The mean time to finish a project is studied where the rate of work varies randomly. It is proved that the mean time to finish is bounded below by the time taken to complete the project if the work is performed at a fixed average rate, specifically, at the average work rate of the corresponding unrestricted or non-exit problem. Several examples illustrate the usefulness of the result in a variety of areas including population dynamics, engineering, finance, and philosophy. [ABSTRACT FROM AUTHOR]

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. In this paper, we focus on stationary distributions for sticky Brownian motions. Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions. The kernel method, copula concept and extreme value theory are the main tools used in our analysis. [ABSTRACT FROM AUTHOR]

This article deals with the reflected and doubly reflected generalized backward stochastic differential equations when the noise is given by Brownian motion and Teugels martingales associated with an independent pure jump Lévy process. We prove the existence and the uniqueness of the solution for these equations with monotone generators and right continuous left limited obstacles. [ABSTRACT FROM AUTHOR]