This paper is concerned with existence and uniqueness of solution for the the optimal control problem governed by the stochastic FitzHugh-Nagumo equation driven by a Gaussian noise. First order conditions of optimality are also obtained.
We consider the pricing problem related to payoffs that can have discontinuities of polynomial growth. The asset price dynamic is modeled within the Black and Scholes framework characterized by a stochastic volatility term driven by a fractional Ornstein-Uhlenbeck process. In order to solve the aforementioned problem, we consider three approaches. The first one consists in a suitable transformation of the initial value of the asset price, in order to eliminate possible discontinuities. Then we discretize both the Wiener process and the fractional Brownian motion and estimate the rate of convergence of the related discretized price to its real value, the latter one being impossible to be evaluated analytically. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Then we derive a closed formula which involves only integral functional depending on the fBm trajectory, to evaluate the price; finally we discretize the fBm and estimate the rate of convergence of the associated numerical scheme to the option price. In both cases the rate of convergence is the same and equals n −rH , where n is a number of the points of discretization, H is the Hurst index of fBm, and r is the Hölder exponent of volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus also providing the option price in terms of the associated probability density.
We consider a symmetric n-player nonzero-sum stochastic differential game with jump-diffusion dynamics and mean-field type interaction among the players. Under the assumption of existence of a regular Markovian solution for the corresponding limiting mean-field game, we construct an approximate Nash equilibrium for the n-player game for n large enough, and provide the rate of convergence. This extends to a class of games with jumps classical results in mean-field game literature. This paper complements our previous work [2] on the existence of solutions of mean-field games for jump-diffusions.
In the present paper we derive the existence and uniqueness of a solution for the optimal control problem determined by a stochastic FitzHugh-Nagumo equation with recovery variable. In particular due the cubic non-linearity in the drift coefficients, standard techniques cannot be applied so that the Ekeland's variational principle has to be exploited.
Stochastic systems with memory naturally appear in life science, economy, and finance. We take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps. Our results concern existence and uniqueness of strong solutions, estimates for the moments and the fundamental tools of calculus, such as the Itô formula. We study the robustness of the solution to the change of noises. Specifically, we consider the noises with infinite activity jumps versus an adequately corrected Gaussian noise. The study is presented in two different frameworks: we work with random variables in infinite dimensions, where the values are considered either in an appropriate L p -type space or in the space of càdlàg paths. The choice of the value space is crucial from the modelling point of view as the different settings allow for the treatment of different models of memory or delay. Our techniques involve tools of infinite dimensional calculus and the stochastic calculus via regularisation.
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