arXiv subjects

David Hobson

Publications and source records attributed to David Hobson.

At least 19 recordsLinked to original sources

arXiv2026

A multi-player optimal stopping game based on a spider Brownian bridge

This paper studies a multi-player non-zero-sum optimal stopping game driven by a spider Brownian bridge. We begin by analyzing the two-player setting, in which the underlying process is a skew Brownian bridge, and establish sufficient conditions for the existence of a Nash equilibrium. We derive sufficient conditions for the uniqueness of the Nash equilibrium in the symmetric case, (where the skew Brownian bridge reduces to a standard Brownian bridge and both players have the same payoff structures). We also give examples which show that there can be multiple Nash equilibria. We then extend the framework to the multi-player setting and provide sufficient conditions for the existence of a Nash equilibrium in this more general setting with a spider Brownian bridge.

math.PR↗

arXiv2025

Optimal Investment and Consumption in a Stochastic Factor Model

In this article, we study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem, and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an It\^o diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including -- for the first time -- the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme.

q-fin.MF↗

arXiv2025

Optimal Stopping of a Brownian Excursion and an $\alpha$-dimensional Bessel Bridge

We study the optimal stopping of an $\alpha$-dimensional Bessel bridge for the payoff $\phi(x)=x^n$, where $\alpha,n>0$. As a special case we consider the Brownian excursion with the identity function as the payoff ($\alpha=3,n=1$). For the Brownian excursion we can give an explicit solution but in the general case we provide a complete solution via a power series expansion.

math.PR↗

arXiv2025

Model-independent upper bounds for the prices of Bermudan options with convex payoffs

Suppose $\mu$ and $\nu$ are probability measures on $\mathbb{R}$ satisfying $\mu \leq_{cx} \nu$. Let $a$ and $b$ be convex functions on $\mathbb{R}$ with $a \geq b \geq 0$. We are interested in finding $$\sup_{\mathbf{M}} \sup_{\tau} \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ \tau = 1 \} } + b(Y) I_{ \{ \tau = 2 \} } \right] $$ where the first supremum is taken over consistent models $\mathbf{M}$ (i.e., filtered probability spaces $(\Omega, \mathbf{F}, \mathbb{F}, \mathbb{P})$ such that $Z=(z,Z_1,Z_2)=(\int_{\mathbb{R}} x \mu(dx) = \int_{\mathbb{R}} y \nu(dy), X, Y)$ is a $(\mathbb{F},\mathbb{P})$ martingale, where $X$ has law $\mu$ and $Y$ has law $\nu$ under $\mathbb{P}$) and $\tau$ in the second supremum is a $(\mathbb{F},\mathbb{P})$-stopping time taking values in $\{1,2\}$. Our contributions are first to characterise and simplify the dual problem, and second to completely solve the problem under some structural assumptions on the measures $\mu$ and $\nu$ (namely that $\mu$ and $\nu$ are absolutely continuous probability measures that satisfy the Dispersion Assumption). A key finding is that the canonical set-up in which the filtration is that generated by $Z$ is not rich enough to define an optimal model and additional randomisation is required. This holds even though the marginal laws $\mu$ and $\nu$ are atom-free. The problem has an interpretation of finding the robust, or model-free, no-arbitrage bound on the price of a Bermudan option with two possible exercise dates, given the prices of co-maturing European options.

q-fin.MF↗

arXiv2024

Zero-sum Dynkin games under common and independent Poisson constraints

Zero-sum Dynkin games under Poisson constraints, where players can only stop at the event times of a Poisson process, have been studied widely in the recent literature. The constraint can be modelled in two ways: either both players share the same Poisson process (the common constraint) or each player has their own Poisson process (the independent constraint). In a Markovian framework, where payoffs are functions of an underlying diffusion, we establish necessary and sufficient conditions for the equivalence of the game's solution--comprising the value function and optimal stopping sets--under the common and independent constraints. Specifically, if the stopping sets of the maximiser and minimiser in the game under the common constraint are disjoint, then the solution to the game is the same under both the common and the independent constraint. However, the fact that the stopping sets are disjoint in the game under the independent constraint is not sufficient to guarantee that the solution of the game under the independent constraint is also the solution under the common constraint. To demonstrate the broad applicability of our results, we solve infinite-horizon Dynkin games satisfying the assumptions of our main theorems, using backward stochastic differential equation (BSDE) techniques. This requires extending standard BSDE results from the finite-horizon setting to the infinite-horizon case, allowing for unbounded solutions.

math.OC↗

arXiv2024

Portfolio Optimization under Transaction Costs with Recursive Preferences

The Merton investment-consumption problem is fundamental, both in the field of finance, and in stochastic control. An important extension of the problem adds transaction costs, which is highly relevant from a financial perspective but also challenging from a control perspective because the solution now involves singular control. A further significant extension takes us from additive utility to stochastic differential utility (SDU), which allows time preferences and risk preferences to be disentangled. In this paper, we study this extended version of the Merton problem with proportional transaction costs and Epstein-Zin SDU. We fully characterise all parameter combinations for which the problem is well posed (which may depend on the level of transaction costs) and provide a full verification argument that relies on no additional technical assumptions and uses primal methods only. The case with SDU requires new mathematical techniques as duality methods break down. Even in the special case of (additive) power utility, our arguments are significantly simpler, more elegant and more far-reaching than the ones in the extant literature. This means that we can easily analyse aspects of the problem which previously have been very challenging, including comparative statics, boundary cases which heretofore have required separate treatment and the situation beyond the small transaction cost regime. A key and novel idea is to parametrise consumption and the value function in terms of the shadow fraction of wealth, which may be of much wider applicability.

econ.GN↗

arXiv2023

An injective martingale coupling

We give an injective martingale coupling; in particular, given measures $\mu$ and $\nu$ in convex order on $\mathbb R$ such that $\nu$ is continuous, we construct a martingale transport such that for each $y$ in the support of the target law $\nu$ there is a {\em unique} $x$ in {a support of} the initial law $\mu$ such that (some of) the mass at $x$ is transported to $y$. Then $\pi$ has disintegration $\pi(dx,dy) = \nu(dy) \delta_{\theta(y)}(dx)$ for some function $\theta$. More precisely we construct a martingale coupling $\pi$ of the measures $\mu$ and $\nu$ such that there is a set $\Gamma_\mu$ such that $\mu(\Gamma_\mu)=1$ and a disintegration $(\pi_x)_{x \in \Gamma_\mu}$ of $\pi$ of the form $\pi(dx,dy) = \pi_x(dy) \mu(dx)$ such that, with $\Gamma_{\pi_x}$ a support of $\pi_x$, we have $\# \{ x \in \Gamma_\mu : y \in \Gamma_{\pi_x} \} \in \{ 0,1 \}$ for all $y$ and $\{ y : \# \{ x \in \Gamma_\mu : y \in \Gamma_{\pi_x} \} = 1 \} = supp(\nu)$. Moreover, if $\mu$ is continuous we may take $\Gamma_{\pi_x} = supp(\pi_x)$ for each $x$. However, we cannot also insist that $\Gamma_\mu = supp (\mu)$.

math.PR↗

arXiv2021

Proper solutions for Epstein-Zin Stochastic Differential Utility

In this article, we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein--Zin stochastic differential utility (EZ-SDU) who invests in a constant-parameter Black-Scholes-Merton market over the infinite horizon. The parameter combinations that we consider in this paper are such that the risk aversion parameter $R$ and the elasticity of intertemporal complementarity $S$ satisfy $\theta=\frac{1-R}{1-S}>1$. In this sense, this paper is complementary to Herdegen, Hobson and Jerome [arXiv:2107.06593]. The main novelty of the case $\theta>1$ (as opposed to $\theta\in(0,1)$) is that there is an infinite family of utility processes associated to every nonzero consumption stream. To deal with this issue, we introduce the economically motivated notion of a proper utility process, where, roughly speaking, a utility process is proper if it is nonzero whenever future consumption is nonzero. We then proceed to show that for a very wide class of consumption streams $C$, there exists a proper utility process $V$ associated to $C$. Furthermore, for a wide class of consumption streams $C$, the proper utility process $V$ is unique. Finally, we solve the optimal investment-consumption problem in a constant parameter financial market, where we optimise over the right-continuous attainable consumption streams that have a unique proper utility process associated to them.

q-fin.MF↗

arXiv2021

Callable convertible bonds under liquidity constraints and hybrid priorities

This paper investigates the callable convertible bond problem in the presence of a liquidity constraint modelled by Poisson signals. We assume that neither the bondholder nor the firm has absolute priority when they stop the game simultaneously, but instead, a proportion $m\in[0,1]$ of the bond is converted to the firm's stock and the rest is called by the firm. The paper thus generalizes the special case studied in [Liang and Sun, Dynkin games with Poisson random intervention times, SIAM Journal on Control and Optimization, 57 (2019), 2962-2991] where the bondholder has priority ($m=1$), and presents a complete solution to the callable convertible bond problem with liquidity constraint. The callable convertible bond is an example of a Dynkin game, but falls outside the standard paradigm since the payoffs do not depend in an ordered way upon which agent stops the game. We show how to deal with this non-ordered situation by introducing a new technique which may be of interest in its own right, and then apply it to the bond problem.

q-fin.MF↗

arXiv2021

The Infinite Horizon Investment-Consumption Problem for Epstein-Zin Stochastic Differential Utility

In this article we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein-Zin stochastic differential utility who invests in a constant-parameter Black-Scholes-Merton market. The paper has three main goals: first, to provide a detailed introduction to infinite-horizon Epstein-Zin stochastic differential utility, including a discussion of which parameter combinations lead to a well-formulated problem; second, to prove existence and uniqueness of infinite horizon Epstein-Zin stochastic differential utility under a restriction on the parameters governing the agent's risk aversion and temporal variance aversion; and third, to provide a verification argument for the candidate optimal solution to the investment-consumption problem among all admissible consumption streams. To achieve these goals, we introduce a slightly different formulation of Epstein-Zin stochastic differential utility to that which is traditionally used in the literature. This formulation highlights the necessity and appropriateness of certain restrictions on the parameters governing the stochastic differential utility function.

q-fin.MF↗

arXiv2021

A construction of the left-curtain coupling

In a martingale optimal transport (MOT) problem mass distributed according to the law $\mu$ is transported to the law $\nu$ in such a way that the martingale property is respected. Beiglb\"ock and Juillet (On a problem of optimal transport under marginal martingale constraints, Annals of Probability, 44(1):42-106, 2016) introduced a solution to the MOT problem which they baptised the left-curtain coupling. The left-curtain coupling has been widely studied and shown to have many applications, including to martingale inequalities and the model-independent pricing of American options. Beiglb\"ock and Juillet proved existence and uniqueness, proved optimality for a family of cost functions, and proved that when $\mu$ is a continuous distribution, mass at $x$ is mapped to one of at most two points, giving lower and upper functions. Henry-Labord\`ere and Touzi (An explicit martingale version of Brenier`s theorem, Finance and Stochastics, 20:635-668, 2016) showed that the left-curtain coupling is optimal for an extended family of cost functions and gave a construction of the upper and lower functions under an assumption that $\mu$ and $\nu$ are continuous, together with further simplifying assumptions of a technical nature. In this article we construct these upper and lower functions in the general case of arbitrary centred measures in convex order, and thereby give a complete construction of the left-curtain coupling. In the case where $\mu$ has atoms these upper and lower functions are to be interpreted in the sense of a lifted martingale.

math.PR↗

arXiv2020

The potential of the shadow measure

It is well known that given two probability measures $\mu$ and $\nu$ on $\mathbb{R}$ in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglb\"{o}ck and Juillet (Ann. Probab. 44 (2016) 42-106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.

math.PR↗

arXiv2020

An elementary approach to the Merton problem

In this article we consider the infinite-horizon Merton investment-consumption problem in a constant-parameter Black - Scholes - Merton market for an agent with constant relative risk aversion R. The classical primal approach is to write down a candidate value function and to use a verification argument to prove that this is the solution to the problem. However, features of the problem take it outside the standard settings of stochastic control, and the existing primal verification proofs rely on parameter restrictions (especially, but not only, R<1), restrictions on the space of admissible strategies, or intricate approximation arguments. The purpose of this paper is to show that these complications can be overcome using a simple and elegant argument involving a stochastic perturbation of the utility function.

q-fin.MF↗

arXiv2020

The shape of the value function under Poisson optimal stopping

In a classical problem for the stopping of a diffusion process $(X_t)_{t \geq 0}$, where the goal is to maximise the expected discounted value of a function of the stopped process ${\mathbb E}^x[e^{-\beta \tau}g(X_\tau)]$, maximisation takes place over all stopping times $\tau$. In a Poisson optimal stopping problem, stopping is restricted to event times of an independent Poisson process. In this article we consider whether the resulting value function $V_\theta(x) = \sup_{\tau \in {\mathcal T}({\mathbb T}^\theta)}{\mathbb E}^x[e^{-\beta \tau}g(X_\tau)]$ (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate $\theta = (\theta(X_t))_{t \geq 0}$) inherits monotonicity and convexity properties from $g$. It turns out that monotonicity (respectively convexity) of $V_\theta$ in $x$ depends on the monotonicity (respectively convexity) of the quantity $\frac{\theta(x) g(x)}{\theta(x) + \beta}$ rather than $g$. Our main technique is stochastic coupling.

math.PR↗

arXiv2019

Constrained Optimal Stopping, Liquidity and Effort

In a classical optimal stopping problem in continuous time, the agent can choose any stopping time without constraint. Dupuis and Wang (Optimal stopping with random intervention times, Advances in Applied Probability, 34, 141--157, 2002) introduced a constraint on the class of admissible stopping times which was that they had to take values in the set of event times of an exogenous, time-homogeneous Poisson process. This can be thought of as a model of finite liquidity. In this article we extend the analysis of Dupuis and Wang to allow the agent to choose the rate of the Poisson process. Choosing a higher rate leads to a higher cost. Even for a simple model for the stopped process and a simple call-style payoff, the problem leads to a rich range of optimal behaviours which depend on the form of the cost function. Often the agent accepts the first offer --- if they are not going to accept an offer then there is no point in putting in effort to generate offers, and thus there may be no offers to accept or decline --- but for some set-ups this is not the case.

math.PR↗

arXiv2018

Robust bounds for the American Put

We consider the problem of finding a model-free upper bound on the price of an American put given the prices of a family of European puts on the same underlying asset. Specifically we assume that the American put must be exercised at either $T_1$ or $T_2$ and that we know the prices of all vanilla European puts with these maturities. In this setting we find a model which is consistent with European put prices and an associated exercise time, for which the price of the American put is maximal. Moreover we derive a cheapest superhedge. The model associated with the highest price of the American put is constructed from the left-curtain martingale transport of Beiglböck and Juillet.

q-fin.MF↗

arXiv2017

Optimal Stopping and the Sufficiency of Randomized Threshold Strategies

In a classical optimal stopping problem the aim is to maximize the expected value of a functional of a diffusion evaluated at a stopping time. This note considers optimal stopping problems beyond this paradigm. We study problems in which the value associated to a stopping rule depends on the law of the stopped process. If this value is quasi-convex on the space of attainable laws then it is a well known result that it is sufficient to restrict attention to the class of threshold strategies. However, if the objective function is not quasi-convex, this may not be the case. We show that, nonetheless, it is sufficient to restrict attention to mixtures of threshold strategies.

math.PR↗

arXiv2016

A multi-asset investment and consumption problem with transaction costs

In this article we study a multi-asset version of the Merton investment and consumption problem with proportional transaction costs. In general it is difficult to make analytical progress towards a solution in such problems, but we specialise to a case where transaction costs are zero except for sales and purchases of a single asset which we call the illiquid asset. Assuming agents have CRRA utilities and asset prices follow exponential Brownian motions we show that the underlying HJB equation can be transformed into a boundary value problem for a first order differential equation. The optimal strategy is to trade the illiquid asset only when the fraction of the total portfolio value invested in this asset falls outside a fixed interval. Important properties of the multi-asset problem (including when the problem is well-posed, ill-posed, or well-posed only for large transaction costs) can be inferred from the behaviours of a quadratic function of a single variable and another algebraic function.

q-fin.MF↗