Recently, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) was developed, which generalizes the concepts of standard Brownian motion (Bm), fractional Brownian motion (fBm) and G-Brownian motion (GBm) under the framework of sublinear expectation. The fGBm can exhibit long-range dependence and feature the volatility uncertainty of financial markets simultaneously. Thus it can be a better alternative stochastic process in the financial applications. In this paper, some financial markets driven by the fGBm were considered and the corresponding arbitrage opportunities were discussed. Specifically, the financial market M that consists a risk free asset and a risky asset will admit some arbitrage opportunity if the stochastic integrals with respect to the fGBm were established in the path-wise sense, and the arbitrage opportunity can be excluded if the stochastic integrals were established in the Wick calculus sense. What is more, the arbitrage opportunity for the financial market ( M , C ) , consisting of the original market M and some contingent claim C were also investigated in the general case, and some interval of free arbitrage prices for the claim were derived by applying the generalized fractional G-Girsanov theorem and fractional G-Clark-Ocone theorem. This study generalized the well-known existing results of arbitrage theory in financial markets driven by the standard Bm, fBm and GBm.

The primary objective of this article is to investigate the convergence of higher-order moments of the diffusion processes within the state space R . This analysis will subsequently demonstrate weak convergence forms of the diffusion processes. The methodology employed in this paper is relatively straightforward, involving the Poisson equation associated with the infinitesimal generator as presented in Depauw and Derrien (Variance limite d'une marche aléatoire réversible en milieu aléatoire sur Z , C. R. Acad. Sci. Paris, Ser. I 347 (2009), pp. 401–406).

This paper introduces a short rate model in continuous time that adds one or more memory (delay) components to the Merton model [R.C. Merton, A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm, Working Paper 497, Sloan School of Management, MIT, Cambridge, 1970; R.C. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci. 4 (1973), pp. 141–183.] or the Vasiček model [Vasiček, An equilibrium characterization of the term structure, J. Financ. Econ. 5 (1977), pp. 177–188.] for the short rate. The distribution of the short rate in this model is normal, with the mean depending on past values of the short rate, and a limiting distribution exists for certain values of the parameters. The zero coupon bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. This system can be solved analytically, obtaining a closed formula. An analytical expression for the instantaneous forward rate is given: it satisfies the risk neutral dynamics of the Heath-Jarrow-Morton model. Formulae for both forward looking and backward looking caplets on overnight risk free rates are presented. Finally, the proposed model is calibrated against forward looking caplets on SONIA rates and the United States yield curve.

Iksanov and Pilipenko (2023) defined a skew stable Lévy process as a scaling limit of a sequence of perturbed at 0 symmetric stable Lévy processes (continuous-time processes). Here, we provide a simpler construction of the skew stable Lévy process as a scaling limit of a sequence of perturbed at 0 standard random walks (random sequences).

We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP). It is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed.

Given a semi-Markov law, using an additional parameter, we consider a family of stochastic flows corresponding to that law. Then we suitably select a particular flow, for which we obtain expressions of the meeting and merging probabilities of a pair of semi-Markov processes, solving the same equation but having two different initial conditions. A set of sufficient conditions are also obtained under which any two solutions of the flow eventually merge with probability one.

This article studies the problem of evaluating the information that a Principal lacks when establishing an incentive contract with an Agent whose effort is not observable. The Principal (‘she’) pays a continuous rent to the Agent (‘he’), while the latter gives a best response characterized by his effort, until a terminal date decided by the Principal when she stops the contract and gives compensation to the Agent. The output process of the project is a diffusion process driven by a Brownian motion whose drift is impacted by the Agent's effort. The first part of the paper investigates the optimal stochastic control problem when the Principal and the Agent share the same information. This situation, known as the first-best case, is solved by tackling the Lagrangian problem. In the second part, the Principal observes the output process but she may not observe the drift and the Brownian motion separately. This situation is known as the second-best case. We derive the best response of the Agent, then we solve the mixed optimal stopping/stochastic control problem of the Principal under a fixed probability and on the filtration generated by the Brownian motion, which is larger than the one generated by the output process (that corresponds to the information available for the Principal). Under some regularity conditions, the Principal value function is characterized by solving the associated Hamilton Jacobi Bellman Variational Inequality. At the optimum, we prove that the two filtrations coincide. Finally, we compute the value of the information for the Principal provided by the observation of the Agent's effort. It is defined as the difference between the principal value function in the first-best and second-best cases.

Branching processes in a varying environment encompass a wide range of stochastic demographic models, and their complete understanding in terms of limit behaviour poses a formidable research challenge. In this paper, we conduct a thorough investigation of such processes within a continuous-time framework, assuming that the reproduction law of individuals adheres to a specific parametric form for the probability generating function. Our six clear-cut limit theorems support the notion of recognizing five distinct asymptotical regimes for branching in varying environments: supercritical, asymptotically degenerate, critical, strictly subcritical, and loosely subcritical.

We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation. This can be considered a time-space analogue of the classical Kalman filter. The result is illustrated with examples of the filtering problem involving noisy observations of a constant and noisy observations of the Brownian sheet.

This work focuses on the well-posedness of McKean–Vlasov stochastic differential delay equations. Under suitable Lipschitz conditions on the drift and diffusion terms, along with a distribution-dependent Lyapunov condition, this paper shows the existence of a unique solution to the distribution-dependent stochastic differential delay equation.