Fake Exponential Brownian Motion

We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.Comment: 8 page

The left-curtain martingale coupling in the presence of atoms

Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the left-curtain martingale coupling of probability measures μ and ν, and proved that, when the initial law μ is continuous, it is supported by the graphs of two functions. We extend the later result by constructing the generalised left-curtain martingale coupling and show that for an arbitrary starting law μ it is characterised by two appropriately defined lower and upper functions. As an application of this result, we derive the model-independent upper bound of an American put option. This extends recent results of Hobson and Norgilas (2017) on the atom-free case

Integrability of solutions of the Skorokhod embedding problem for diffusions

Suppose X is a time-homogeneous diffusion on an interval IX⊆R and let μ be a probability measure on IX. Then τ is a solution of the Skorokhod embedding problem (SEP) for μ in X if τ is a stopping time and Xτ∼μ. There are well-known conditions which determine whether there exists a solution of the SEP for μ in X. We give necessary and sufficient conditions for there to exist an integrable solution. Further, if there exists a solution ofthe SEP then there exists a minimal solution. We show that every minimal solution of the SEP has the same first moment. When X is Brownian motion, there exists an integrable embedding of μ if and only if μ is centred and in L2. Further,every integrable embedding is minimal. When X is a general time-homogeneous diffusion the situation is more subtle. The case with drift can be reduced to the local martingale case by a change of scale. If Y is a diffusion in natural scale, and if the target law is centred, then as in the Brownian case, there is an integrable embedding if the target law satisfies an integral condition. However, unlike in the Brownian case, there exist integrable embeddings of target laws which are not centred. Further, there exist integrable embeddings which are not minimal. Instead, if there exists an integrable embedding, then the set of minimal embeddings is the set of embeddings such that the mean equals a certain quantity, which we identify

Optimal consumption and sale strategies for a risk averse agent

In this article we consider a special case of an optimal consumption/optimal portfolio problem first studied by Constantinides and Magill and by Davis and Norman, in which an agent with constant relative risk aversion seeks to maximise expected discounted utility of consumption over the infinite horizon, in a model comprising a risk-free asset and a risky asset with proportional transaction costs. The special case that we consider is that the cost of purchases of the risky asset is infinite, or equivalently the risky asset can only be sold and not bought. In this special setting new solution techniques are available, and we can make considerable progress towards an analytical solution. This means we are able to consider the comparative statics of the problem. There are some surprising conclusions, such as consumption rates are not monotone increasing in the return of the asset, nor are the certainty equivalent values of the risky positions monotone in the risk aversion

The shape of the value function under Poisson optimal stopping

In a classical problem for the stopping of a diffusion process , where the goal is to maximise the expected discounted value of a function of the stopped process , maximisation takes place over all stopping times . 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 (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate ) inherits monotonicity and convexity properties from . It turns out that monotonicity (respectively convexity) of in depends on the monotonicity (respectively convexity) of the quantity rather than . Our main technique is stochastic coupling

Randomized strategies and prospect theory in a dynamic context

When prospect theory (PT) is applied in a dynamic context, the probability weighting com- ponent brings new challenges. We study PT agents facing optimal timing decisions and consider the impact of allowing them to follow randomized strategies. In a continuous-time model of gam- bling and optimal stopping, Ebert and Strack (2015) show that a naive PT investor with access only to pure strategies never stops. We show that allowing randomization can signi cantly alter the predictions of their model, and can result in voluntary cessation of gambling

Randomised rules for stopping problems

In a classical, continuous-time, optimal stopping problem, the agent chooses the best time to stop a stochastic process in order to maximise the expected discounted return. The agent can choose when to stop, and if at any moment they decide to stop, stopping occurs immediately with probability one. However, in many settings this is an idealistic oversimplification. Following Strack and Viefers we consider a modification of the problem in which stopping occurs at a rate which depends on the relative values of stopping and continuing: there are several different solutions depending on how the value of continuing is calculated. Initially we consider the case where stopping opportunities are constrained to be event times of an independent Poisson process. Motivated by the limiting case as the rate of the Poisson process increases to infinity, we also propose a continuous-time formulation of the problem where stopping can occur at any instant

The potential of the shadow measure

It is well known that given two probability measures μ and ν on 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. Beiglböck 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

The infinite horizon investment-consumption problem for Epstein-Zin stochastic differential utility. II : Existence, uniqueness and verification for ϑ∈(0,1)

In this article, we consider the optimal investment–consumption problem for an agent with preferences governed by Epstein–Zin (EZ) stochastic differential utility (SDU) over an infinite horizon. In a companion paper Herdegen et al. (Finance Stoch. 27:127–158, 2023), we argued that it is best to work with an aggregator in discounted form and that the coefficients R of relative risk aversion and S of elasticity of intertemporal complementarity (the reciprocal of the coefficient of elasticity of intertemporal substitution) must lie on the same side of unity for the problem to be well founded. This can be equivalently expressed as ϑ:=1−R1−S>0. In this paper, we focus on the case ϑ∈(0,1). The paper has three main contributions: first, to prove existence of infinite-horizon EZ SDU for a wide class of consumption streams and then (by generalising the definition of SDU) to extend this existence result to any consumption stream; second, to prove uniqueness of infinite-horizon EZ SDU for all consumption streams; and third, to verify the optimality of an explicit candidate solution to the investment–consumption problem in the setting of a Black–Scholes–Merton financial market