Stochastic systems with memory and jumps

Baños, David R.; Cordoni, Francesco; Nunno, Giulia Di; Persio, Luca Di; Røse, Elin Engen

doi:10.1016/j.jde.2018.10.052

Cited by 19 publications

(21 citation statements)

References 38 publications

(107 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For more information about stochastic functional differential equations, we refer to the seminal work of S.E.A. Mohammed [18] and a recent paper by Banos et al [5].…”

Section: Solvability Of Memory Mean-field Sfdementioning

confidence: 99%

See 1 more Smart Citation

Stochastic Control of Memory Mean-Field Processes

Agram

Øksendal

2017

Appl Math Optim

View full text Add to dashboard Cite

By a memory mean-field process we mean the solution X(·) of a stochastic mean-field equation involving not just the current state X(t) and its law L(X(t)) at time t, but also the state values X(s) and its law L(X(s)) at some previous times s < t. Our purpose is to study stochastic control problems of memory mean-field processes.• We consider the space M of measures on R with the norm || · || M introduced by Agram and Øksendal in [1], and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations.• We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (time-) advanced backward stochastic differential equations, one of them with values in the space of bounded linear functionals on path segment spaces.• As an application of our methods, we solve a memory mean-variance problem as well as a linear-quadratic problem of a memory process.MSC (2010): 60H05, 60H20, 60J75, 93E20, 91G80,91B70.

show abstract

“…For more information about stochastic functional differential equations, we refer to the seminal work of S.E.A. Mohammed [18] and a recent paper by Banos et al [5].…”

Section: Solvability Of Memory Mean-field Sfdementioning

confidence: 99%

“…where we denote by the bold X(t) = δ 0 X(t − s)µ(ds) for some bounded Borel-measure µ. As noted in Agram and Røse [2] and Banos et al [5], we have the following:…”

Section: Introductionmentioning

confidence: 99%

Stochastic Control of Memory Mean-Field Processes

Agram

Øksendal

2017

Appl Math Optim

View full text Add to dashboard Cite

show abstract

“…These SFDEs have already been studied in the pioneering works of [28,29,38] in the Brownian framework. The theory has later been developed including models for jumps in [9]. From another perspective models with memory have been studied via the so-called functional Itô calculus as introduced in [17] and then developed steadily in e.g.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Functional Differential Equations and Sensitivity to Their Initial Path

Baños

Nunno

Haferkorn

et al. 2018

Computation and Combinatorics in Dynamics, Stochastics and Control

Self Cite

View full text Add to dashboard Cite

We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks connected to the choice of these models. In particular we focus on the impact of the initial condition on the evaluations. This problem is known as the analysis of sensitivity to the initial condition and, in the terminology of finance, it is referred to as the Delta. In this work the initial condition is represented by the relevant past history of the stochastic functional differential equation. This naturally leads to the redesign of the definition of Delta. We suggest to define it as a functional directional derivative, this is a natural choice. For this we study a representation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our formula is achieved by studying an appropriate relationship between Malliavin derivative and functional directional derivative. For this we introduce the technique of randomisation of the initial condition.

show abstract

“…norm, in L 2 ([−r, 0]; R) =: L 2 . Note that the space M 2 is a separable Hilbert space, see, e.g., [6]. The Delfour-Mitter space can be generalized to be a separable Banach space if we consider p ∈ (1, ∞), equipped with the appropriate norm.…”

Section: Introductionmentioning

confidence: 99%

“…right-continuous functions with finite left limit, on the interval [−r, 0], D := D ([−r, 0]; R) called Skorokhod space; in particular D is a non separable Banach space if endowed with the sup norm · D = sup t∈[−r,0] | · |. We also have that D ⊂ M 2 with the injection being continuous, see, e.g., [6]. Nevertheless, choosing M 2 as state space we cannot deal with the case of discrete delays, see, e.g., [6, pag.…”

Section: Introductionmentioning

confidence: 99%

A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

Cordoni

Persio

Oliva

2017

Nonlinear Differ. Equ. Appl.

Self Cite

View full text Add to dashboard Cite

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) ,

show abstract

Stochastic systems with memory and jumps

Cited by 19 publications

References 38 publications

Stochastic Control of Memory Mean-Field Processes

Stochastic Control of Memory Mean-Field Processes

Stochastic Functional Differential Equations and Sensitivity to Their Initial Path

A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

Contact Info

Product

Resources

About