Abstract:We consider a reaction-diffusion equation on a network subjected to dynamic boundary conditions, with time delayed behaviour, also allowing for multiplicative Gaussian noise perturbations. Exploiting semigroup theory, we rewrite the aforementioned stochastic problem as an abstract stochastic partial differential equation taking values in a suitable product Hilbert space, for which we prove the existence and uniqueness of a mild solution. Eventually, a stochastic optimal control application is studied.
“…, n, will denote vertexes. We refer to [10,11,19], for further details The structure of the graph is based on the incidence matrix Φ := Φ + − Φ − , where the sum is intended componentwise and Φ = (φ i,α ) n×m , together with the incoming incidence matrix Φ + = φ + i,α n×m , and the outgoing incidence matrix Φ − = φ − i,α n×m , where…”
Section: A General Framework For Systemic Risk In Financial Networkmentioning
“…, n, will denote vertexes. We refer to [10,11,19], for further details The structure of the graph is based on the incidence matrix Φ := Φ + − Φ − , where the sum is intended componentwise and Φ = (φ i,α ) n×m , together with the incoming incidence matrix Φ + = φ + i,α n×m , and the outgoing incidence matrix Φ − = φ − i,α n×m , where…”
Section: A General Framework For Systemic Risk In Financial Networkmentioning
“…where η is defined as in (19). While, the first four integrals in the right hand side of equation ( 28) can be bounded similarly as done proving Lemma 2.5, see (19), the last two terms can be treated exploiting the Young inequality…”
We consider the existence and first order conditions of optimality for a stochastic optimal control problem inspired by the celebrated FitzHugh-Nagumo model, with nonlinear diffusion term, perturbed by a linear multiplicative Brownian-type noise. The main novelty of the present paper relies on the application of the rescaling method which allows us to reduce the original problem to a random optimal one.
“…The present section closely follows in [35,Section. 7], in particular we will consider weak control problems, we refer to [32] for a general treatment of the present notion of control, or [18,19,50,51]. Let us therefore consider the following R−valued controlled delay equation, dX(t) = (µ(t, X t , X(t)) + F (t, X(t), α(t))) dt+ +σ(t, X(t))dW (t) + R0 γ(t, X(t), z)Ñ (dt, dz) , (X t0 , X(t 0 )) = (x, η) , In what follows we assume µ, σ and γ to satisfy assumptions 2.2, we also require that it exists a constant C σ > 0 such that, for any t ∈ [0, T ] and x ∈ R, |σ −1 (t, x)| ≤ C σ .…”
We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an L 2 −valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non-linear Feynman-Kac representation theorem under mild assumptions of differentiability.Remark 1.1. In what follows we will only consider the 1−dimensional case, the case of a R d −valued stochastic process, perturbed by a general R m − dimensional Wiener process and a R n −dimensional Poisson random measure, with d > 1, m > 1 and n > 1, can be easily obtained from the present one.In order to take into account the delay component, we study the equation (1.2) in the Delfour-Mitter space defined as follows M 2 := L 2 ([−r, 0]; R) × R, endowed with the scalar product (X t , X(t)), (Y t , Y (t)) M2 = X t , Y t L 2 + X(t) · Y (t) ,
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