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We consider the stochastic control problem of the shallow lake and continue the work of Kossioris, Loulakis, and Souganidis. First, we generalise the characterisation of the value function as the viscosity solution of a well-posed problem to include more general recycling rates. Then, we prove approximate optimality under bounded controls and we establish quantitative estimates. Finally, we implement a convergent and stable numerical scheme for the computation of the value function to investigate properties of the optimally controlled stochastic shallow lake. This approach permits to derive tails asymptotics for the invariant distribution and to extend results of Grass, Kiseleva and Wagener. ] beyond the small noise limit. [ABSTRACT FROM AUTHOR]

In this paper, we consider doubly reflected backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients. The existence of solutions can be obtained by a monotone convergence argument, a linearization method, a penalization method and the method of Picard iteration. [ABSTRACT FROM AUTHOR]

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô's formula and leads to pth moment and pathwise exponential stability for p ≥ 2 with explicit Lyapunov exponents. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically. [ABSTRACT FROM AUTHOR]

We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type. [ABSTRACT FROM AUTHOR]

We study first- and second-order linear transport equations, as well as flows for ordinary and stochastic differential equations, with irregular velocity fields satisfying a one-sided Lipschitz condition. Depending on the time direction, the flows are either compressive or expansive. In the compressive regime, we characterize the stable continuous distributional solutions of both the first and second-order nonconservative transport equations as the unique viscosity solution, and we also provide new observations and characterizations for the dual, conservative equations. Our results in the expansive regime complement the theory of Bouchut et al. (Ann Sc Norm Super Pisa Cl Sci (5) 4:1–25, 2005), and we develop a complete theory for both the conservative and nonconservative equations in Lebesgue spaces, as well as proving the existence, uniqueness, and stability of the regular Lagrangian flow for the associated ordinary differential equation. We also provide analogous results in this context for second order equations with degenerate noise coefficients that are constant in the spatial variable, as well as for the related stochastic differential equation flows. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 풪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball's idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time. [ABSTRACT FROM AUTHOR]

This paper concerns the dynamics of a stochastic regime-switching four-species food chain model with distributed delays, Lévy jumps and harvesting in a polluted environment. First, under some basic assumptions, criterions about stochastic persistence and extinction of each species are established. Then, sufficient and necessary conditions for the existence of optimal harvesting policy (OHP) are obtained, as well as the expressions for the optimal harvesting effort (OHE) and the maximum of expectation of sustainable yield (MESY). The effects of both environmental noises and delays on the stochastic persistence, extinction and OHP of the model are revealed. Some results on food chain models without telephone noises or distributed delays are extended and improved. [ABSTRACT FROM AUTHOR]

We consider stochastic differential equations (SDEs) with (distributional) drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, such as continuity with respect to the drift and the link with the Fokker–Planck equation. We also show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. In the second part we identify the dynamics of the solution of the martingale problem by describing the proper associated SDE. Under suitable assumptions we show equivalence with the solution to the martingale problem. [ABSTRACT FROM AUTHOR]

We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by G-Brownian motion (G-BSDEs) with diagonal generators. Two methods, including the penalization method and the Picard iteration argument, are provided to prove the existence and uniqueness of the solutions. We also study its connection with the obstacle problem of a system of fully nonlinear PDEs. [ABSTRACT FROM AUTHOR]

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ℝd for d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝd. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model. [ABSTRACT FROM AUTHOR]

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature and provide a foundation for the use of semimartingale liquidation strategies and the use of portfolio processes of unbounded variation. Our convergence results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the relationship between the equilibrium reinsurance strategy and the tail of the distribution of the risk. Since Mean Residual Life (MRL) has a close relationship with the tail of the distribution, we consider two classes of risk distributions, Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) distributions, which can be used to classify light-tailed and heavy-tailed distributions, respectively. We assume that the underlying risk process is modelled by the classical Cramér-Lundberg model process. Under the mean-variance criterion, by solving the extended Hamilton-Jacobi-Bellman equation, we derive the equilibrium reinsurance strategy for the insurer and the reinsurer under DMRL and IMRL, respectively. Furthermore, we analyze how to choose the reinsurance premium to make the insurer and the reinsurer agree with the same reinsurance strategy. We find that under the case of DMRL, if the distribution and the risk aversions satisfy certain conditions, the insurer and the reinsurer can adopt a reinsurance premium to agree on a reinsurance strategy, and under the case of IMRL, the insurer and the reinsurer can only agree with each other that the insurer do not purchase the reinsurance. [ABSTRACT FROM AUTHOR]

The usual theory of asset pricing in finance assumes that the financial strategies, i.e. the quantity of risky assets to invest, are real-valued so that they are not integer-valued in general, see the Black and Scholes model for instance. This is clearly contrary to what it is possible to do in the real world. Surprisingly, it seems that there are not many contributions in that direction in the literature, except for a finite number of states. In this paper, for arbitrary Ω , we show that, in discrete-time, it is possible to evaluate the minimal super-hedging price when we restrict ourselves to integer-valued strategies. To do so, we only consider terminal claims that are continuous piecewise affine functions of the underlying asset. We formulate a dynamic programming principle that can be directly implemented on historical data and which also provides the optimal integer-valued strategy. The problem with general payoffs remains open but should be solved with the same approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the differentiability of solutions of neutral stochastic differential equations driven by G-Brownian motion with respect to parameter. Under suitable assumptions, we show that solutions are differentiable with respect to the parameter which appears in the initial data. In addition, the stochastic differential equation of the derivative is given and the existence-uniqueness of solution is proved. Moreover, an example to illustrate the theoretically obtained results is presented. [ABSTRACT FROM AUTHOR]

A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

The main aim of this article is to demonstrate the collocation method based on the barycentric rational interpolation function to solve nonlinear stochastic differential equations driven by fractional Brownian motion. First of all, the corresponding integral form of the nonlinear stochastic differential equations driven by fractional Brownian motion is introduced. Then, collocation points followed by the Gauss-quadrature formula and Simpson’s quadrature method are used to reduce them into a system of algebraic equations. Finally, the approximate solution is obtained using Newton’s method. The rigorous convergence and error analysis of the presented method has been discussed in detail. The proposed method has been applied to some well-known stochastic models, such as the stock model and a few other examples, to demonstrate the applicability and plausibility of the discussed method. Also, the numerical results of the collocation method based on the barycentric rational interpolation function and barycentric Lagrange interpolation function get compared with an exact solution. [ABSTRACT FROM AUTHOR]

We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options in the case where the investor is required to liquidate her position. Our main result is establishing a non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. We compute the limit of the corresponding utility indifference prices and find explicitly a family of portfolios which are asymptotically optimal. [ABSTRACT FROM AUTHOR]

We introduce a new formulation of reflected backward stochastic differential equations (BSDEs) and doubly reflected BSDEs associated with irregular obstacles. In the first part of the paper, we consider an extension of the classical optimal stopping problem over a larger set of stopping systems than the set of stopping times (namely, the set of split stopping times), where the payoff process ξ is irregular and in the case of a general filtration. Split stopping times are a powerful tool for modeling financial contracts and derivatives that depend on multiple conditions or triggers, and for incorporating stochastic processes with jumps and other types of discontinuities. We show that the value family can be aggregated by an optional process v, which is characterized as the Snell envelope of the reward process ξ over split stopping times. Using this, we prove the existence and uniqueness of a solution Y to irregular reflected BSDEs. In the second part of the paper, motivated by the classical Dynkin game with completely irregular rewards considered by Grigorova et al. (Electron J Probab 23:1–38, 2018), we generalize the previous equations to the case of two reflecting barrier processes. [ABSTRACT FROM AUTHOR]

In this article, we consider a two-player zero-sum stochastic differential game with regime-switching. Different from the results in existing literature on stochastic differential games with regime-switching, we consider a game between a Markov chain and a state process which are two fully coupled stochastic processes. The payoff function is given by an integral with random terminal horizon. We first study the continuity of the lower and upper value functions under some additional conditions, based on which we establish the dynamic programming principle. We further prove that the lower and upper value functions are unique viscosity solutions of the associated lower and upper Hamilton–Jacobi–Bellman–Isaacs equations with regime-switching, respectively. These two value functions coincide under the Isaacs condition, which implies that the game admits a value. We finally apply our results to an example. [ABSTRACT FROM AUTHOR]

We consider a Heston local-stochastic volatility (HLSV) model to study the optimal investment, consumption, and work effort strategies for an executive with equity incentives and stochastic salary. The executive's work behavior can affect the stock price of the company, and after retirement, he can choose whether to accept re-employment. Our goal is to maximize the expected discounted utility of consumption and terminal wealth and reduce the loss utility generated by work. In addition, we establish the Hamilton–Jacobi–Bellman (HJB) equation through the dynamic programming principle and solve complex nonlinear partial differential equations using a perturbation method to obtain asymptotic solutions for the optimal strategies and value function under the HLSV model. We also consider a general case in which the price process of risky asset follows the stochastic volatility (SV) model, which can be compared with the HLSV model. Finally, we provide a numerical example to illustrate the impact of some important parameters on the optimal strategies and value function. [ABSTRACT FROM AUTHOR]

Releasing Wolbachia-infected male mosquitoes into the wild to sterilize wild female mosquitoes is a bio-safe and effective strategy to block the transmission of the dengue virus. Since the spread of the dengue virus is often subject to some randomness, stochastic differential equations can be used to describe the transmission difference of dengue affected by environmental changes. Considering the latent period of virus transmission, we established a delay stochastic differential system model based on the release of Wolbachia-infected males, which has received relatively little attention. By constructing suitable Lyapunov functions, we obtain sufficient conditions for the extinction of the dengue virus and the existence of a unique stationary distribution. To study the efficacy of the releases Wolbachia-infected males, we find two threshold values r 1 ∗ and r 2 ∗ of the release ratio r , where r 1 ∗ > r 2 ∗ . If r > r 1 ∗ , then the dengue virus is extinct. If r < r 2 ∗ , a unique stationary distribution exists, which indicates that the virus is transmitted between humans and mosquitoes. We present some numerical examples to illustrate the impact of environmental noise on the virus dynamics, and we also simulate the dynamics of viruses under different release ratios. [ABSTRACT FROM AUTHOR]

The greater the persistence in a financial time series, the more predictable it becomes, allowing for the development of more effective investment strategies. Desirable attributes for financial portfolios include persistence, smoothness, long memory, and higher auto-correlation. We argue that these properties can be achieved by adjusting the composition weights of the portfolio. Considering the fractal nature of typical financial time series, the fractal dimension emerges as a natural metric to gauge the smoothness of the portfolio trajectory. Specifically, the Hurst exponent is designed for measuring the persistence of time series. In this paper, we introduce an optimization method inspired by the Hurst exponent and signal processing to mitigate the irregularities in the portfolio trajectory. We illustrate the effectiveness of this approach using real data from an S &P100 dataset. [ABSTRACT FROM AUTHOR]

We consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of ℝd for d = 1 , 2 , 3. We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup. [ABSTRACT FROM AUTHOR]

The dynamical equation of the boundary vorticity has been obtained, which shows that the viscosity at a solid wall is doubled as if the fluid became more viscous at the boundary. For certain viscous flows the boundary vorticity can be determined via the dynamical equation up to bounded errors for all time, without the need of knowing the details of the main stream flows. We then validate the dynamical equation by carrying out stochastic direct numerical simulations (i.e. the random vortex method for wall-bounded incompressible viscous flows) by two different means of updating the boundary vorticity, one using mollifiers of the Biot–Savart singular integral kernel, another using the dynamical equations. [ABSTRACT FROM AUTHOR]

Abstract.This article investigates the robust time-consistent investment strategy for a DC pension plan with model uncertainty under the mean-variance optimization objective. For the avoidance of reductions in investment returns due to inflation risk and premature death of members, an inflation-indexed bond is introduced into the financial market and a return of premium clause is incorporated into the model. By establishing extended Hamilton-Jacobi-Bellman (HJB) equations, the explicit solutions of both the robust precommitment strategy and the robust time-consistent strategy are presented by virtue of the robust optimal control theory and the stochastic dynamic programming approach. Furthermore, two degradation cases are derived in detail. Finally, a numerical example demonstrates the analysis of the obtained results. [ABSTRACT FROM AUTHOR]

We prove the Perelman type W-entropy formula for the geodesic flow on the L 2 -Wasserstein space over a complete Riemannian manifold equipped with Otto's infinite dimensional Riemannian metric. To better understand the similarity between the W-entropy formula for the geodesic flow on the Wasserstein space and the W-entropy formula for the heat flow of the Witten Laplacian on the underlying manifold, we introduce the Langevin deformation of flows on the Wasserstein space over a Riemannian manifold, which interpolates the gradient flow and the geodesic flow on the Wasserstein space over a Riemannian manifold, and can be regarded as the potential flow of the compressible Euler equation with damping on a Riemannian manifold. We prove the existence, uniqueness and regularity of the Langevin deformation on the Wasserstein space over the Euclidean space and a compact Riemannian manifold, and prove the convergence of the Langevin deformation for c → 0 and c → ∞ respectively. Moreover, we prove the W-entropy-information formula along the Langevin deformation on the Wasserstein space on Riemannian manifolds. The rigidity theorems are proved for the W-entropy for the geodesic flow and the Langevin deformation on the Wasserstein space over complete Riemannian manifolds with the CD(0, m)-condition. Our results are new even in the case of Euclidean spaces and complete Riemannian manifolds with non-negative Ricci curvature. [ABSTRACT FROM AUTHOR]

In this paper, we first derive the existence and uniqueness theorems for solutions to a class of generalized mean-field delay stochastic differential equations and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-field delay control problem. Since the state equation is distribution-depending, we define the adjoint equation as a MFABSDE in which all the derivatives of the coefficients are in Lions' sense. We also provide a sufficient condition for the optimality of the control. [ABSTRACT FROM AUTHOR]

We study the response of the optimal investment problem to small changes of the stock price dynamics. Starting with a multidimensional semimartingale setting of an incomplete market, we suppose that the perturbation process is also a general semimartingale. We obtain second-order expansions of the value functions, first-order corrections to the optimisers, and provide the adjustments to the optimal control that match the objective function up to the second order. We also give a characterisation in terms of the risk-tolerance wealth process, if it exists, by reducing the problem to the Kunita–Watanabe decomposition under a change of measure and numéraire. Finally, we illustrate the results by examples of base models that allow closed-form solutions, but where this structure is lost under perturbations of the model where our results allow an approximate solution. [ABSTRACT FROM AUTHOR]

We solve the superhedging problem for European options in an illiquid extension of the Black–Scholes model, in which transactions have transient price impact and the costs and strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring nonnegativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black–Scholes model with a relative price impact proportional to the volume of shares traded, where the transience for impact on log-prices is modelled like in Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) for nominal prices. More generally, we allow nonlinear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. The pricing equations under illiquidity extend no-arbitrage pricing à la Black–Scholes for complete markets in a non-paradoxical way (cf. Çetin et al. (Finance Stoch. 14:317–341, 2010)) even without additional frictions, and can recover it in base cases. [ABSTRACT FROM AUTHOR]

A limited participation economy models the real-world phenomenon that some economic agents have access to more of the financial market than others. We prove the global existence of a Radner equilibrium with limited participation, where the agents have exponential preferences and derive utility from both running consumption and terminal wealth. Our analysis centers around a coupled quadratic backward stochastic differential equation (BSDE) system whose equations describe the economic agents' stochastic control solutions and equilibrium prices. We define a candidate equilibrium in terms of the BSDE system solution and prove through a verification argument that the candidate is a Radner equilibrium with limited participation. Finally, we prove that the BSDE system has a unique solution in S ∞ × bmo . This work generalises the model of Basak and Cuoco (Rev. Financ. Stud. 11:309–341, 1998) to allow a stock with a general dividend stream and agents with stochastic income streams and exponential preferences. We also provide an explicit example. [ABSTRACT FROM AUTHOR]

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

We study the mean-field type stochastic control problem where the dynamics is governed by a general Lévy process with moments of all orders. For this, we introduce the power jump processes and the related Teugels martingales and give the Malliavin derivative with respect to Teugels martingales. We derive necessary and sufficient conditions for optimality of our control problem in the form of a mean-field stochastic maximum principle. [ABSTRACT FROM AUTHOR]

This paper investigates the optimal asset-liability management problems for two managers subject to relative performance concerns in the presence of stochastic inflation and stochastic volatility. The objective of the two managers is to maximize the expected utility of their relative terminal surplus with respect to that of their competitor. The problem of finding the optimal investment strategies for both managers is modeled as a non-zero-sum stochastic differential game. Both managers have access to a financial market consisting of a risk-free asset, a risky asset, and an inflation-linked index bond. The risky asset’s price process and uncontrollable random liabilities are not only affected by the inflation risk but also driven by a general class of stochastic volatility models embracing the constant elasticity of variance model, the family of state-of-the-art 4/2 models, and some path-dependent models. By adopting a backward stochastic differential equation (BSDE) approach to overcome the possibly non-Markovian setting, closed-form expressions for the equilibrium investment strategies and the corresponding value functions are derived under power and exponential utility preferences. Moreover, explicit solutions to some special cases of our model are provided. Finally, we perform numerical studies to illustrate the influence of relative performance concerns on the equilibrium strategies and draw some economic interpretations. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of backward stochastic differential equations (BSDEs) with mean reflection formulated by Briand et al. (Ann Appl Probab 28(1):482–510, 2018). We investigate the solvability of a generalized mean reflected BSDE, whose driver also depends on the distribution of solution term Y. Using a fixed-point argument, BMO martingale theory and the θ -method, we establish existence and uniqueness results for such BSDEs in several typical situations, including the case where the driver is quadratic with bounded or unbounded terminal condition. [ABSTRACT FROM AUTHOR]

Fractional Brownian motion with Hurst parameter H < 1 2 is used widely, for instance, to describe 'rough' volatility data in finance. In this paper, we examine a generalised Ait-Sahalia-type model driven by a fractional Brownian motion with H < 1 2 and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions. [ABSTRACT FROM AUTHOR]

This paper studies the weak Euler approximation for solutions to stochastic differential equations (SDEs) driven by point and martingale measures, with Hölder-continuous coefficients. The equation under consideration includes a nondegenerate main part whose jump intensity measure is absolutely continuous with respect to the Lévy measure of a spherically symmetric stable process. It encompasses a broad range of stochastic processes including the nondegenerate diffusions and SDEs driven by Lévy processes. To investigate the dependence of the convergence rate on the regularity of the coefficients and driving processes, the regularity of a solution to the associated backward Kolmogorov equation is considered. In particular, for the first time the Hölder norm of the subordinated part of the corresponding generator is rigorously estimated. [ABSTRACT FROM AUTHOR]

As defaultable options are subject to default risks in over-the-counter (OTC) markets, vulnerable options are one of the financial securities closely linked with the risk caused by the situation that the option issuer may be unable to execute their contractual obligations. In this study, taking account of the changes in stock prices show different degrees of long-term correlation and autocorrelation at different times in the real-world situations of the financial market, contrary to the assumption of Black-Scholes settings, we propose the closed-form pricing formulas for the vulnerable options based on a fractional Brownian environment. First of all, from the model dynamics for the underlying asset under the fractional Brownian motions, we derive partial differential equations (PDEs) for vulnerable option prices, and subsequently derive the closed-form solutions for the options based on the issuer's credit risk via double Mellin transforms. In addition, we evaluate the accuracy of vulnerable option pricing by comparing the solutions obtained using the aforementioned closed formulas with those obtained via Monte Carlo simulations. Finally, we examine the price sensitivities of vulnerable options in fractional Brownian markets with respect to the model parameters. [ABSTRACT FROM AUTHOR]

The representation of functions by artificial neural networks depends on a large number of parameters in a non-linear fashion. Suitable parameters are found by minimizing a ‘loss functional’, typically by stochastic gradient descent (SGD) or an advanced SGD-based algorithm. In a continuous time model for SGD with noise that follows the ‘machine learning scaling’, we show that in a certain noise regime, the optimization algorithm prefers ‘flat’ minima of the objective function in a sense which is different from the flat minimum selection of continuous time SGD with homogeneous noise. [ABSTRACT FROM AUTHOR]

In this article, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and dependent risks, in which the insurer's surplus process is assumed to follow a risk model with two dependent classes of insurance business. The insurer is allowed to purchase proportional reinsurance and invest his surplus in a financial market, which contains one risk-free asset and one risky asset whose price process satisfies a jump-diffusion model. Under the consideration of the performance-related capital inflow or outflow, the insurer's wealth process is modeled by a stochastic differential delay equation. The insurer's aim is to develop the robust optimal reinsurance and investment strategy for the insurer by maximizing the expected exponential utility of the combination of the average historical performance and terminal wealth under the worst-case scenario of the alternative measures. By adopting the stochastic dynamic programming technique, the expressions of the robust optimal strategy and value function are explicitly obtained. Finally, we present some numerical examples to illustrate the effects of some model parameters on the optimal strategy. [ABSTRACT FROM AUTHOR]

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory. [ABSTRACT FROM AUTHOR]