arXiv subjects

Nacira Agram

Publications and source records attributed to Nacira Agram.

At least 37 records · Page 2Linked to original sources

arXiv2024

Optimal control of SPDEs driven by time-space Brownian motion

In this paper we study a Pontryagin type stochastic maximum principle for the optimal control of a system, where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). We first discuss some properties of a Brownian sheet driven linear SPDE which models the growth of an ecosystem. Further, applying time-space white noise calculus we derive sufficient conditions and necessary conditions of optimality of the control. Finally, we illustrate our results by solving a linear quadratic control problem and an optimal harvesting problem in the plane. We also study possible applications to machine learning.

math.OC↗

arXiv2023

The Donsker delta function and local time for McKean-Vlasov processes and applications

The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean-Vlasov (mean-field) stochastic differential equation. If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon-Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process. For some particular McKean-Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.

math.PR↗

arXiv2023

Stochastic Fokker-Planck PIDE for conditional McKean-Vlasov jump diffusions and applications to optimal control

The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate an Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of conditional McKean-Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker-Planck equation reduces to a stochastic partial differential equation (SPDE) for the Radon-Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the following problems: -Linear-quadratic optimal control of conditional stochastic McKean-Vlasov jump diffusions. -Optimal consumption from a cash flow modelled as a conditional stochastic McKean-Vlasov differential equation with jumps.

math.PR↗

arXiv2023

Impulse control of conditional McKean-Vlasov jump diffusions

This paper establishes a verification theorem for impulse control problems involving conditional McKean-Vlasov jump diffusions. We obtain a Markovian system by combining the state equation of the problem with the stochastic Fokker-Planck equation for the conditional probability law of the state. We derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control which satisfy the verification theorem.

math.OC↗

arXiv2022

Optimal stopping of conditional McKean-Vlasov jump diffusions

We study the problem of optimal stopping of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we combine the state equation for the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck equation for the conditional law of the solution of the state. This gives us a Markovian system which can be handled by using a version of the Dynkin formula. We illustrate our result by solving explicitly two optimal stopping problems for conditional McKean-Vlasov jump diffusions. More specifically, we first find the optimal time to sell in a market with common noise and jumps, and, next, we find the stopping time to quit a project whose state is modelled by a jump diffusion, when the performance functional involves the conditional mean of the state.

math.OC↗

arXiv2021

SPDEs with space interactions and application to population modelling

We consider optimal control of a new type of non-local stochastic partial differential equations (SPDEs). The SPDEs have space interactions, in the sense that the dynamics of the system at time $t$ and position in space x also depend on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population growth studies and epidemiology. We prove the existence and uniqueness of solutions of a class of SPDEs with space interactions, and we show that, under some conditions, the solutions are positive for all times if the initial values are. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an optimal vaccine strategy problem for an epidemic by modelling the population density as a space-mean stochastic reaction-diffusion equation.

math.OC↗

arXiv2021

Stochastic Maximum Principle with Default

In this paper, we derive sufficient and necessary maximum principles for a stochastic optimal control problem where the system state is given by a controlled stochastic differential equation with default. We prove existence of a unique solution to the controlled default stochastic differential equation. Furthermore, we prove existence and uniqueness of solution to the adjoint backward stochastic differential equation which appears in connection to the maximum principles. Finally, we apply the maximum principles to solve a utility maximisation problem with logarithmic utility functions.

math.OC↗

arXiv2021

Optimal stopping, randomized stopping and singular control with partial information flow

The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a filtration with no a priori assumed relation to the filtration of the process. We call these problems optimal stopping and randomized stopping with general information. Following an idea of Krylov [K] we introduce a special singular stochastic control problem with general information and show that this is also equivalent to the partial information optimal stopping and randomized stopping problems. Then we show that the solution of this singular control problem can be expressed in terms of partial information variational inequalities.

math.OC↗

arXiv2021

Pricing of European options in incomplete jump diffusion markets

We study option prices in financial markets where the risky asset prices are modelled by jump diffusions. It was proposed by Schweizer (1996) in a general semimartingale setting, following earlier works by Föllmer and Sondermann (1986) and Bouleau and Lamberton (1989), that the right price of such an option is the initial wealth needed to make it possible to generate by a self-financing portfolio a terminal wealth which is as close as possible to the payoff F in the sense of variance. Schweizer calls this price the approximation price and he investigates interesting general properties of this price and its corresponding optimal portfolio. -However, neither of these authors compute explicitly this price in concrete cases. This is the motivation for the current paper: We apply stochastic control methods to compute this price in the setting of markets with assets described by jump diffusions. Our method involves Stackelberg games and a suitably modified stochastic maximum principle. We show that such an optimal initial wealth, denoted by z^, with corresponding optimal portfolio π^ exist and are unique. - If the coefficients of the risky asset prices are deterministic, we show that z^=E_{Q*}[F], for a specific equivalent martingale measure (EMM) Q*. This shows in particular that the minimal variance price is free from arbitrage. -Then for the general case we apply a suitable maximum principle of optimal stochastic control to relate the minimal variance price z^=p_{mv}(F) to the Hamiltonian and its adjoint processes, and we show that, under some conditions, z^=E_{Q_0}[F] for any Q_0 in a family M_0 of EMMs, described by the set of solutions of a system of linear equations. -Finally, we illustrate our results by looking at specific examples.

math.OC↗

arXiv2021

Singular optimal control of stochastic Volterra integral equations

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution X^{u,ξ}(t)=X(t) is given by X(t) =ϕ(t)+\int_{0}^{t}}b(t,s,X(s),u(s)) ds+\int_{0}^{t}σ(t,s,X(s),u(s))dB(s) +\int _{0}^{t}\int_{0}^{t}h(t,s) dξ(s). Here dB(s) denotes the Brownian motion Itô type differential and ξdenotes the singular control (singular in time t with respect to Lebesgue measure) and u denotes the regular control (absolutely continuous with respect to Lebesgue measure). Such systems may for example be used to model harvesting of populations with memory, where X(t) represents the population density at time t, and the singular control process ξrepresents the harvesting effort rate. The total income from the harvesting is represented by J(u,ξ) =E[\int _{0}^{T}\int_{0}^{T} f_{0}(t,X(t),u(t))dt+\int _{0}^{T}\int_{0}^{T} f_{1}(t,X(t))dξ(t)+g(X(T))], for given functions f_{0},f_{1} and g, where T>0 is a constant denoting the terminal time of the harvesting. Note that it is important to allow the controls to be singular, because in some cases the optimal controls are of this type. Using Hida-Malliavin calculus, we prove sufficient conditions and necessary conditions of optimality of controls. As a consequence, we obtain a new type of backward stochastic Volterra integral equations with singular drift. Finally, to illustrate our results, we apply them to discuss optimal harvesting problems with possibly density dependent prices.

math.OC↗

arXiv2020

A financial market with singular drift and no arbitrage

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow & Protter for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas & Shreve (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay θ>0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as the delay θ> 0. This implies that there is no arbitrage in the market in that case. However, when θgoes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al and the references therein.

q-fin.MF↗

arXiv2020

Reflected Backward Stochastic Volterra Integral Equations and related time-inconsistent optimal stopping problems

We study solutions of a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We prove existence and uniqueness by a fixed point argument and we derive a comparison result. Moreover, we show how the solution of our problem is related to a time-inconsistent optimal stopping problem and derive an optimal strategy.

math.PR↗

arXiv2019

Mean-Field Stochastic Control with Elephant Memory in Finite and Infinite Time Horizon

Our purpose of this paper is to study stochastic control problem for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case. - In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem. - For infinite horizon, we derive sufficient and necessary maximum principles. As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

math.OC↗

arXiv2019

Mean-field FBSDE and optimal control

We study optimal control for mean-field forward backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem.

math.OC↗

arXiv2019

Singular control of SPDEs with space-mean dynamics

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

math.OC↗

arXiv2019

Introduction to White Noise, Hida-Malliavin Calculus and Applications

The purpose of these lectures is threefold: We first give a short survey of the Hida white noise calculus, and in this context we introduce the Hida-Malliavin derivative as a stochastic gradient with values in the Hida stochastic distribution space $(\mathcal{S}% )^*$. We show that this Hida-Malliavin derivative defined on $L^2(\mathcal{F}_T,P)$ is a natural extension of the classical Malliavin derivative defined on the subspace $\mathbb{D}_{1,2}$ of $L^2(P)$. The Hida-Malliavin calculus allows us to prove new results under weaker assumptions than could be obtained by the classical theory. In particular, we prove the following: (i) A general integration by parts formula and duality theorem for Skorohod integrals, (ii) a generalised fundamental theorem of stochastic calculus, and (iii) a general Clark-Ocone theorem, valid for all $F \in L^2(\mathcal{F}_T,P)$. As applications of the above theory we prove the following: A general representation theorem for backward stochastic differential equations with jumps, in terms of Hida-Malliavin derivatives; a general stochastic maximum principle for optimal control; backward stochastic Volterra integral equations; optimal control of stochastic Volterra integral equations and other stochastic systems.

math.OC↗

arXiv2019

Mean-field backward stochastic differential equations and applications

In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the form dY(t) =-f(t,Y(t),Z(t),K(t, . ),E[φ(Y(t),Z(t),K(t,.))])dt+Z(t)dB(t) +\int_{R_{0}}K(t,ζ)\tilde{N}(dt,dζ), where B is a Brownian motion, \tilde{N} is the compensated Poisson random measure. Under some mild conditions, we prove the existence and uniqueness of the solution triplet (Y,Z,K). It is commonly believed that there is no comparison theorem for general mean-field bsde. However, we prove a comparison theorem for a subclass of these equations. When the mean-field bsde is linear, we give an explicit formula for the first component Y(t) of the solution triplet. Our results are applied to solve a mean-field recursive utility optimization problem in finance.

math.OC↗

arXiv2019

Dynamic risk measure for BSVIE with jumps and semimartingale issues

Risk measure is a fundamental concept in finance and in the insurance industry, it is used to adjust life insurance rates. In this current paper, we will study dynamic risk measures by means of backward stochastic Volterra integral equations (BSVIEs) with jumps. We prove a comparison theorem for such a type of equations. Since the solution of a BSVIEs is not a semimartingale in general, we will discuss some particular semimartingale issues.

math.OC↗