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The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean–Vlasov type. Motivated by the recent interest in mean-field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic controlled systems with mean-field interactions when subject to a common policy.

In this paper, in order to study effects of the human immune system response to spread of COVID-19 virus, we establish a stochastic competition model between immune cells and COVID-19 particles by introducing both white and coloured noise. We first prove the existence and uniqueness of the global positive solution of the system under consideration. Furthermore, the stationary distribution and ergodicity of the system are investigated in order to prove weak persistence in mean. We also obtain the conditions for extinction of the disease. The obtained results are related to basic reproduction number of the corresponding deterministic analogue of the system. Finally, we provide numerical simulations with real life data to support theoretical conclusions obtained in the paper.

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by α, is of great importance as the resulting system is known to blow-up for large values of α. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when α is small enough.

We consider stochastic differential equations of the form dYt = V(Yt)dXt + V0(Yt)dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V0 and V = (V1,..., Vd) satisfy Hörmander's bracket condition, we demonstrate that Yt admits a smooth density for any t ∈ (0, T], provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein–Uhlenbeck process and the Brownian bridge returning after time T.

The main objective of this paper and the accompanying one [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint] is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work [Ann. Probab. (2014) 42 204-236], focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in [Stochastic Process. Appl. (2014) 124 3277-3311]. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the well-posedness results established in [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint]. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path-dependent dynamic programming equations.

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms $\\mathbb{R}^d\\ni x\\quad\\longmapsto\\quad\\phi_{s,t}(x)\\in \\mathbb{R}^d,\\qquad s,t\\in\\mathbb{R}$ for a stochastic differential equation (SDE) of the form $dX_t=b(t,X_t)\\,dt+dB_t,\\qquad s,t\\in\\mathbb{R},X_s=x\\in\\mathbb{R}^d.$ The above SDE is driven by a bounded measurable drift coefficient b: ℝ × ℝd → ℝd and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow ϕs,t(·) of the SDE lives in the space L2(Ω; W1,p (ℝd, w)) for all s, t and all p ∈ (1, ∞), where W1,p (ℝd, w) denotes a weighted Sobolev space with weight w possessing a pth moment with respect to Lebesgue measure on ℝd. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant \"culture\" in these dynamical systems is that the flow \"inherits\" its spatial regularity from that of the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation $\\begin {cases}{\\displaystyle d_tu(t,x)+\\bigl(b(t,x)\\cdot Du(t,x)\\bigr)\\,dt+\\sum_{i=1}^de_i\\cdot Du(t,x)\\circ dB_t^i=0,\\cr u(0,x)=u_0(x),}\\end{cases}$ where b is bounded and measurable, u0 is $C_b^1$ and $\\{e_i\\}_{i=1}^d$ a basis for ℝd. It is well known that the deterministic counterpart of the above equation does not in general have a solution.

In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.

In this article, we investigate the existence of periodic solutions to McKean–Vlasov stochastic differential equations subject to periodic Lyapunov conditions with distributional dependence. The proof is based on the construction of periodic Markov processes on the product space , where is the space of probability measures on . Moreover, we also prove the existence of periodic solutions under Lyapunov conditions, where the Lyapunov functions involve only the spatial components. To illustrate our analysis, we present several concrete examples.

Consider an Itô process X satisfying the stochastic differential equation dX = a(X) dt + b(X) dW where a, b are smooth and W is a multidimensional Brownian motion. Suppose that Wn has smooth sample paths and that Wn converges weakly to W. A central question in stochastic analysis is to understand the limiting behavior of solutions Xn to the ordinary differential equation dXn = a(Xn) dt + b(Xn) dWn. The classical Wong-Zakai theorem gives sufficient conditions under which Xn converges weakly to X provided that the stochastic integral ∫ b(X) dW is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of ∫ b(X) dW depends sensitively on how the smooth approximation Wn is chosen. In applications, a natural class of smooth approximations arise by setting Wn (t) = n-1/2 $\\smallint _0^{nt}\\upsilon o{\\phi _s}ds$ where ɸt is a flow (generated, e.g., by an ordinary differential equation) and υ is a mean zero observable. Under mild conditions on ɸt, we give a definitive answer to the interpretation question for the stochastic integral ∫ b(X) dW. Our theory applies to Anosov or Axiom A flows ɸt, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on ɸt. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman–Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional Itô calculus recently introduced by Dupire [Functional Itô calculus (2009) Preprint].