arXiv subjects

Tiziano De Angelis

Publications and source records attributed to Tiziano De Angelis.

At least 19 recordsLinked to original sources

arXiv2026

A continuous-time dynamic contracting problem with limited liability and finite horizon

We perform a detailed study of a principal--agent problem in a continuous time version of the celebrated Holmström--Milgrom model (Econometrica 55 (2), 1987) where we add limited liability for the Agent. We develop a probabilistic methodology to prove that the Principal's value function is the unique bounded classical solution to a fully nonlinear and fully degenerate partial differential equation (PDE) with Cauchy-Dirichlet boundary conditions on $[0,T]\times[0,\infty)$. Indeed, we also prove infinite continuous differentiability of the solution in the interior of the domain. The strength of our regularity result is such that we can ensure existence of optimal controls in strong form---a rare occurrence in dynamic contracting---and we obtain fine properties of the optimal control map, including a characterisation via a further nonlinear degenerate PDE.

math.OC↗

arXiv2026

On variable annuities with surrender charges

In this paper we provide a theoretical analysis of Variable Annuities (VAs) with a focus on the holder's right to an early termination of the contract. Motivated by risk-management considerations, and in line with existing literature, we assume that the surrender option is optimally exercised from a purely financial perspective (i.e., we consider the worst-case scenario for the insurer). In this context, we rigorously derive the pricing formula for the VA and characterise the optimal surrender time. We also illustrate our theoretical results with extensive numerical experiments. The pricing problem is formulated as an optimal stopping problem with a time-dependent payoff which is discontinuous at the maturity of the contract and non-smooth. This structure leads to non-monotonic optimal stopping boundaries which we prove nevertheless to be continuous and regular in the sense of diffusions for the stopping set. The lack of monotonicity of the boundary makes it impossible to use classical methods from optimal stopping. Also more recent results about Lipschitz continuous boundaries are not applicable in our setup. Thus, we contribute a new methodology for non-monotone stopping boundaries.

q-fin.MF↗

arXiv2026

A probabilistic approach to continuous differentiability of optimal stopping boundaries

We obtain the first probabilistic proof of continuous differentiability of time-dependent optimal boundaries in optimal stopping problems. The underlying stochastic dynamics is a one-dimensional, time-inhomogeneous diffusion. The gain function is also time-inhomogeneous and not necessarily smooth. Moreover, we include time-and-state-dependent discount rate and the time-horizon can be either finite or infinite. Our arguments of proof are of a local nature that allows us to obtain the result under more general conditions than those used in the PDE literature. As a byproduct of our main result we also obtain the first probabilistic proof of the link between the value function of an optimal stopping problem and the solution of the Stefan's problem.

math.PR↗

arXiv2026

Integral equations for the optimal boundary surface of a mean-field game of capacity expansion

We prove that the optimal boundary surface that splits the action and inaction regions in a mean-field game of capacity expansion studied in (Campi et al.,\ Ann.\ Appl.\ Probab.,\ {\bf 32}(5),\, pp.\,3674-3717, 2022) is the unique continuous solution of a nonlinear integral equation of Volterra type. In order to do that, we first establish continuity of the optimal surface. Then we develop an extension of Itô's formula which weakens assumptions required in the existing literature on the first-order time-derivative and/or second-order space derivative of the value function. The paper also provides an algorithm for the numerical solution of the integral equation and we compute optimal controls numerically for the mean-field game.

math.OC↗

arXiv2026

A model of strategic sustainable investment

We study a problem of optimal irreversible investment and emission reduction formulated as a nonzero-sum dynamic game between an investor with environmental preferences and a firm. The game is set in continuous time on an infinite-time horizon. The firm generates profits with a stochastic dynamics and may spend part of its revenues towards emission reduction (e.g., renovating the infrastructure). The firm's objective is to maximize the discounted expectation of a function of its profits. The investor participates in the profits, may decide to invest to support the firm's production capacity and uses a profit function which accounts for both financial and environmental factors. Nash equilibria of the game are obtained via a system of variational inequalities. We formulate a general verification theorem for this system in a diffusive setup and construct an explicit solution in the zero-noise limit. Our explicit results and numerical approximations show that both the investor's and the firm's optimal actions are triggered by moving boundaries that increase with the total amount of emission abatement.

q-fin.MF↗

arXiv2026

A class of stochastic control problems with state constraints

We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set $\mathcal{D}\subseteq [0,T]\times\mathbb{R}^d$, a diffusion $X$ in $\mathbb{R}^d$ must be linearly controlled in order to keep the time-space process $(t,X_t)$ inside the set $\mathcal{C}:=([0,T]\times\mathbb{R}^d)\setminus\mathcal{D}$, while at the same time minimising an expected cost that depends on the state $(t,X_t)$ and is quadratic in the speed of the control exerted. We find a probabilistic representation for the value function and an optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set $\mathcal{D}$. The optimally controlled dynamics is in strong form, in the sense that it is adapted to the filtration generated by the driving Brownian motion. Fully explicit formulae are presented in some relevant examples.

math.OC↗

arXiv2026

A quickest detection problem with false negatives

We formulate and solve a variant of the quickest detection problem which features false negatives. A standard Brownian motion acquires a drift at an independent exponential random time which is not directly observable. Based on the observation in continuous time of the sample path of the process, an optimizer must detect the drift as quickly as possible after it has appeared. The optimizer can inspect the system multiple times upon payment of a fixed cost per inspection. If a test is performed on the system before the drift has appeared then, naturally, the test will return a negative outcome. However, if a test is performed after the drift has appeared, then the test may fail to detect it and return a false negative with probability $ε\in(0,1)$. The optimisation ends when the drift is eventually detected. The problem is formulated mathematically as an optimal multiple stopping problem, and it is shown to be equivalent to a recursive optimal stopping problem. Exploiting such connection and free boundary methods we find explicit formulae for the expected cost and the optimal strategy. We also show that when $ε= 0$ our expected cost is an affine transformation of the one in Shiryaev's classical optimal detection problem with a rescaled model parameter.

math.OC↗

arXiv2025

Martingale theory for Dynkin games with asymmetric information

This paper provides necessary and sufficient conditions for a pair of randomised stopping times to form a saddle point of a zero-sum Dynkin game with partial and/or asymmetric information across players. The framework is non-Markovian and covers essentially any information structure. Our methodology relies on the identification of suitable super and submartingales involving players' equilibrium payoffs. Saddle point strategies are characterised in terms of the dynamics of those equilibrium payoffs and are related to their Doob-Meyer decompositions.

math.PR↗

arXiv2025

Optimal Annuitization with stochastic mortality: Piecewise Deterministic Mortality Force

This paper addresses the problem of determining the optimal time for an individual to convert retirement savings into a lifetime annuity. The individual invests their wealth into a dividend-paying fund that follows the dynamics of a geometric Brownian motion, exposing them to market risk. At the same time, they face an uncertain lifespan influenced by a stochastic mortality force. The latter is modelled as a piecewise deterministic Markov process (PDMP), which captures sudden and unpredictable changes in the individual's mortality force. The individual aims to maximise expected lifetime linear utility from consumption and bequest, balancing market risk and longevity risk under an irreversible, all-or-nothing annuitization decision. The problem is formulated as a three-dimensional optimal stopping problem and, by exploiting the PDMP structure, it is reduced to a sequence of nested one-dimensional problems. We solve the optimal stopping problem and find a rich structure for the optimal annuitization rule, which cover all parameter specifications. Our theoretical analysis is complemented by a numerical example illustrating the impact of a single health shock on annuitization timing, along with a sensitivity analysis of key model parameters.

q-fin.MF↗

arXiv2025

Nash equilibria for dividend distribution with competition

We construct Nash equilibria in feedback form for a class of two-person stochastic games of singular control with absorption, arising from a stylized model for corporate finance. More precisely, the paper focusses on a strategic dynamic game in which two financially-constrained firms operate in the same market. The firms distribute dividends and are faced with default risk. The strategic interaction arises from the fact that if one firm defaults, the other one becomes a monopolist and increases its profitability. The firms choose their dividend distribution policies from a class of randomised strategies and we identify two types of equilibria, depending on the firms' initial endowments. In both situations the optimal strategies and the equilibrium payoffs are found explicitly.

math.OC↗

arXiv2025

Finite-time horizon, stopper vs. singular-controller games on the half-line

We prove existence of a value for two-player zero-sum stopper vs. singular-controller games on finite-time horizon, when the underlying dynamics is one-dimensional, diffusive and bound to evolve in $[0,\infty)$. We show that the value is the maximal solution of a variational inequality with both obstacle and gradient constraint and satisfying a Dirichlet boundary condition at $[0,T)\times\{0\}$. Moreover, we obtain an optimal strategy for the stopper. In order to achieve our goals, we rely on new probabilistic methods, yielding gradient bounds and equi-continuity for the solutions of penalised partial differential equations that approximate the variational inequality.

math.OC↗

arXiv2024

On the saddle point of a zero-sum stopper vs. singular-controller game

We construct a saddle point in a class of zero-sum games between a stopper and a singular-controller. The underlying dynamics is a one-dimensional, time-homogeneous, singularly controlled diffusion taking values either on $\mathbb{R}$ or on $[0,\infty)$. The games are set on a finite-time horizon, thus leading to analytical problems in the form of parabolic variational inequalities with gradient and obstacle constraints. The saddle point is characterised in terms of two moving boundaries: an optimal stopping boundary and an optimal control boundary. These boundaries allow us to construct an optimal stopping time for the stopper and an optimal control for the singular-controller. Our method relies on a new link between the value function of the game and the value function of an auxiliary optimal stopping problem with absorption. We show that the smooth-fit condition at the stopper's optimal boundary (in the game), translates into an absorption condition in the auxiliary problem. This is somewhat in contrast with results obtained in problems of singular control with absorption and it highlights the key role of smooth-fit in this context.

math.OC↗

arXiv2024

Stopper vs. singular-controller games with degenerate diffusions

We study zero-sum stochastic games between a singular controller and a stopper when the (state-dependent) diffusion matrix of the underlying controlled diffusion process is degenerate. In particular, we show the existence of a value for the game and determine an optimal strategy for the stopper. The degeneracy of the dynamics prevents the use of analytical methods based on solution in Sobolev spaces of suitable variational problems. Therefore we adopt a probabilistic approach based on a perturbation of the underlying diffusion modulated by a parameter $γ>0$. For each $γ>0$ the approximating game is non-degenerate and admits a value $u^γ$ and an optimal strategy $τ^γ_*$ for the stopper. Letting $γ\to 0$ we prove convergence of $u^γ$ to a function $v$, which identifies the value of the original game. We also construct explicitly optimal stopping times $θ^γ_*$ for $u^γ$, related but not equal to $τ^γ_*$, which converge almost surely to an optimal stopping time $θ_*$ for the game with degenerate dynamics.

math.OC↗

arXiv2024

The American put with finite-time maturity and stochastic interest rate

In this paper we study pricing of American put options on the Black and Scholes market with a stochastic interest rate and finite-time maturity. We prove that the option value is a $C^1$ function of the initial time, interest rate and stock price. By means of Ito calculus we rigorously derive the option value's early exercise premium formula and the associated hedging portfolio. We prove the existence of an optimal exercise boundary splitting the state space into continuation and stopping region. The boundary has a parametrisation as a jointly continuous function of time and stock price, and it is the unique solution to an integral equation which we compute numerically. Our results hold for a large class of interest rate models including CIR and Vasicek models. We show a numerical study of the option price and the optimal exercise boundary for Vasicek model.

q-fin.MF↗

arXiv2024

Zero-sum stopper vs. singular-controller games with constrained control directions

We consider a class of zero-sum stopper vs. singular-controller games in which the controller can only act on a subset $d_0<d$ of the $d$ coordinates of a controlled diffusion. Due to the constraint on the control directions these games fall outside the framework of recently studied variational methods. In this paper we develop an approximation procedure, based on $L^1$-stability estimates for the controlled diffusion process and almost sure convergence of suitable stopping times. That allows us to prove existence of the game's value and to obtain an optimal strategy for the stopper, under continuity and growth conditions on the payoff functions. This class of games is a natural extension of (single-agent) singular control problems, studied in the literature, with similar constraints on the admissible controls.

math.OC↗

arXiv2023

The maximality principle in singular control with absorption and its applications to the dividend problem

Motivated by a new formulation of the classical dividend problem, we show that Peskir's maximality principle can be transferred to singular stochastic control problems with 2-dimensional degenerate dynamics and absorption along the diagonal of the state space. We construct an optimal control as a Skorokhod reflection along a moving barrier, where the barrier can be computed analytically as the smallest solution to a certain non-linear ordinary differential equation. Contrarily to the classical 1-dimensional formulation of the dividend problem, our framework produces a non-trivial solution when the firm's (pre-dividend) equity capital evolves as a geometric Brownian motion. Such solution is also qualitatively different from the one traditionally obtained for the arithmetic Brownian motion.

math.OC↗

arXiv2023

Variational inequalities on unbounded domains for zero-sum singular-controller vs. stopper games

We study a class of zero-sum games between a singular-controller and a stopper over finite-time horizon. The underlying process is a multi-dimensional (locally non-degenerate) controlled stochastic differential equation (SDE) evolving in an unbounded domain. We prove that such games admit a value and provide an optimal strategy for the stopper. The value of the game is shown to be the maximal solution, in a suitable Sobolev class, of a variational inequality of `min-max' type with obstacle constraint and gradient constraint. Although the variational inequality and the game are solved on an unbounded domain we do not require boundedness of either the coefficients of the controlled SDE or of the cost functions in the game.

math.OC↗

arXiv2023

A change of variable formula with applications to multi-dimensional optimal stopping problems

We derive a change of variable formula for $C^1$ functions $U:\R_+\times\R^m\to\R$ whose second order spatial derivatives may explode and not be integrable in the neighbourhood of a surface $b:\R_+\times\R^{m-1}\to \R$ that splits the state space into two sets $\cC$ and $\cD$. The formula is tailored for applications in problems of optimal stopping where it is generally very hard to control the second order derivatives of the value function near the optimal stopping boundary. Differently to other existing papers on similar topics we only require that the surface $b$ be monotonic in each variable and we formally obtain the same expression as the classical Itô's formula.

math.PR↗