arXiv subjects

Saul Jacka

Publications and source records attributed to Saul Jacka.

At least 19 recordsLinked to original sources

arXiv2024

Dynamic minimisation of the commute time for a one-dimensional diffusion

Motivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel to $1$ and then return to the origin (the so-called commute time from 0 to 1). Substantially extending results in a previous paper, we consider a dynamic version of this problem where the control mechanism is related to the diffusion's drift via the corresponding scale function. We are only able to choose the drift at each point at the time of first visiting that point and the drift is constrained on a set of the form $[0,\ell)\cup(i,1]$. This leads to a type of stochastic control problem with infinite dimensional state.

math.PR↗

arXiv2022

A generalisation of the Burkholder-Davis-Gundy inequalities

{Consider a càdlàg local martingale $M$ with square brackets $[M]$. In this paper, we provide upper and lower bounds for expectations of the type ${\mathbb E} [M]^{q/2}_τ$, for any stopping time $τ$ and $q\ge 2$, in terms of predictable processes. This result can be thought of as a Burkholder-Davis-Gundy type inequality in the sense that it can be used to relate the expectation of the running maximum $|M^*|^q$ to the expectation of the dual previsible projections of the relevant powers of the associated jumps of $M$. The case for a class of moderate functions is also discussed.

math.PR↗

arXiv2021

The Support and Resistance Line Method: An Analysis via Optimal Stopping

We study a mathematical model motivated by the support/resistance line method in technical analysis where the underlying stock price transitions between three states of nature in a path-dependent manner. For optimal stopping problems with respect to a general class of reward functions and dynamics, using probabilistic methods, we show that the value function is $C^1$ (with respect to the corresponding scale function) and solves a general free boundary problem. Moreover, for a range of utilities, we prove that the best times to buy and sell the stock are obtained by solving free boundary problems corresponding to two linked optimal stopping problems. We use this to compute optimal trading strategies for several types of dynamics and varying degrees of relative risk aversion.

q-fin.TR↗

arXiv2020

Pricing and Hedging the No-Negative-Equity Guarantee in Equity-Release Mortgages

We provide a practical superhedging strategy for the pricing and hedging of the No-Negative-Equity-Guarantee (NNEG) found in Equity-Release Mortgages (ERMs), or reverse mortgages, using a discrete-time model. In contrast to many papers on the NNEG and industry practice we work in an incomplete market setting so that deaths and property prices are not independent under most pricing measures. We give theoretical results and numerical illustrations to show that the assumption of market completeness leads to a considerable undervaluation of the NNEG. By introducing an Excess-of-Loss reinsurance asset, we show that it is possible to reduce the cost of the superhedge for a portfolio of ERMs with the average cost decreasing rapidly as the number of lives in the portfolio increases. All the hedging assets, with the exception of cash, have a term of one year making the availability of a property hedging asset from over-the-counter derivative providers more realistic. We outline how a practical multi-period ERM pricing and hedging model can be built. Although the prices identified by this model will be higher than prices under the completeness assumption, they are considerably lower than those under the Equivalent Value Test mandated by the UK's Prudential Regulatory Authority.

q-fin.RM↗

arXiv2019

Martingale approach to control for general jump processes

We provide verification theorems (at different levels of generality) for infinite horizon stochastic control problems in continuous time for semimartingales. The control framework is given as an abstract "martingale formulation", which encompasses a broad range of standard control problems. Under appropriate conditions we show that the set of admissible controls gives rise to a certain class of controlled special semimartingales. Our results generalise both the standard controlled Itô- and Lévy-diffusion settings as we allow ourselves to locally control not only the drift and diffusion coefficients, but also the jump intensity measure of the jumps. As an illustration, we present a few examples with explicit solutions.

math.PR↗

arXiv2018

Minimising the expected commute time

Motivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel from the origin to $1$ and then return (the so-called commute time from 0 to 1). We consider the static and dynamic versions of this problem where the control mechanism is related to the diffusion\rq{}s drift via the corresponding scale function. In the static version the diffusion's drift can be chosen at each point in [0,1], whereas in the dynamic version, we are only able to choose the drift at each point at the time of first visiting that point. The dynamic version leads to a novel type of stochastic control problem.

math.PR↗

arXiv2018

On the informational structure in optimal dynamic stochastic control

We formulate a very general framework for optimal dynamic stochastic control problems which allows for a control-dependent informational structure. The issue of informational consistency is investigated. Bellman's principle is formulated and proved. In a series of related results, we expound on the informational structure in the context of (completed) natural filtrations of stochastic processes.

math.PR↗

arXiv2018

On representing and hedging claims for coherent risk measures

We provide a dual characterisation of the weak$^*$-closure of a finite sum of cones in $L^\infty$ adapted to a discrete time filtration $\mathcal{F}_t$: the $t^{th}$ cone in the sum contains bounded random variables that are $\mathcal{F}_t$-measurable. Hence we obtain a generalisation of Delbaen's m-stability condition for the problem of reserving in a collection of numéraires $\mathbf{V}$, called $\mathbf{V}$-m-stability, provided these cones arise from acceptance sets of a dynamic coherent measure of risk. We also prove that $\mathbf{V}$-m-stability is equivalent to time-consistency when reserving in portfolios of $\mathbf{V}$, which is of particular interest to insurers.

q-fin.MF↗

arXiv2017

Multi-currency reserving for coherent risk measures

We examine the problem of dynamic reserving for risk in multiple currencies under a general coherent risk measure. The reserver requires to hedge risk in a time-consistent manner by trading in baskets of currencies. We show that reserving portfolios in multiple currencies $\mathbf{V}$ are time-consistent when (and only when) a generalisation of Delbaen's m-stability condition \cite{D06}, termed optional $\V$-m-stability, holds. We prove a version of the Fundamental Theorem of Asset Pricing in this context. We show that this problem is equivalent to dynamic trading across baskets of currencies (rather than just pairwise trades) in a market with proportional transaction costs and with a frictionless final period.

q-fin.MF↗

arXiv2015

Markov chain approximations to scale functions of Lévy processes

We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with its (nonnegative) coefficients given explicitly in terms of the Lévy triplet of $X$. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of $X$ and its scale functions, not unlike the one-dimensional Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.

math.PR↗

arXiv2014

Monotonicity of the value function for a two-dimensional optimal stopping problem

We consider a pair $(X,Y)$ of stochastic processes satisfying the equation $dX=a(X)Y\,dB$ driven by a Brownian motion and study the monotonicity and continuity in $y$ of the value function $v(x,y)=\sup_τE_{x,y}[e^{-qτ}g(X_τ)]$, where the supremum is taken over stopping times with respect to the filtration generated by $(X,Y)$. Our results can successfully be applied to pricing American options where $X$ is the discounted price of an asset while $Y$ is given by a stochastic volatility model such as those proposed by Heston or Hull and White. The main method of proof is based on time-change and coupling.

math.PR↗

arXiv2013

Markov chain approximations for transition densities of Lévy processes

We consider the convergence of a continuous-time Markov chain approximation X^h, h>0, to an R^d-valued Levy process X. The state space of X^h is an equidistant lattice and its Q-matrix is chosen to approximate the generator of X. In dimension one (d=1), and then under a general sufficient condition for the existence of transition densities of X, we establish sharp convergence rates of the normalised probability mass function of X^h to the probability density function of X. In higher dimensions (d>1), rates of convergence are obtained under a technical condition, which is satisfied when the diffusion matrix is non-degenerate.

math.PR↗

arXiv2011

Minimizing the time to a decision

Suppose we have three independent copies of a regular diffusion on $[0,1]$ with absorbing boundaries. Of these diffusions, either at least two are absorbed at the upper boundary or at least two at the lower boundary. In this way, they determine a majority decision between 0 and 1. We show that the strategy that always runs the diffusion whose value is currently between the other two reveals the majority decision whilst minimizing the total time spent running the processes.

math.PR↗

arXiv2009

Markov chains conditioned never to wait too long at the origin

Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by $τ$ the first time that the chain, $X$, waits for at least one unit of time at the origin, we consider conditioning the chain on the event $(τ>T)$. We show there is a weak limit as $T\to \infty$ in the cases where either the statespace is finite or $X$ is transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than $τ$ and $τ$ is subexponential.

math.PR↗

arXiv2008

The noisy veto-voter model: a Recursive Distributional Equation on [0,1]

We study a particular example of a recursive distributional equation (RDE) on the unit interval. We identify all invariant distributions, the corresponding "basins of attraction" and address the issue of endogeny for the associated tree-indexed problem, making use of an extension of a recent result of Warren.

math.PR↗

arXiv2008

No-arbitrage and closure results for trading cones with transaction costs

The paper considers trading with proportional transaction costs. We give a necessary and sufficient condition for A, the cone of claims attainable from zero endowment, to be closed, and show, in general, how to represent its closure in such a way that it is the cone of claims attainable for zero endowment, for a different set of trading prices. The new representation obeys the Fundamental Theorem of Asset Pricing. We then show how to represent claims and in a final section show how any such setup corresponds to a coherent risk measure.

math.PR↗

arXiv2007

On representing claims for coherent risk measures

We consider the problem of representing claims for coherent risk measures. For this purpose we introduce the concept of (weak and strong) time-consistency with respect to a portfolio of assets, generalizing the one defined by Delbaen. In a similar way we extend the notion of m-stability, by introducing weak and strong versions. We then prove that the two concepts of m-stability and time-consistency are still equivalent, thus giving necessary and sufficient conditions for a coherent risk measure to be represented by a market with proportional transaction costs. We go on to deduce that, under a separability assumption, any coherent risk measure is strongly time-consistent with respect to a suitably chosen countable portfolio, and show the converse: that any market with proportional transaction costs is equivalent to a market priced by a coherent risk measure, essentially establishing the equivalence of the two concepts.

math.PR↗

arXiv2007

On the density of properly maximal claims in financial markets with transaction costs

We consider trading in a financial market with proportional transaction costs. In the frictionless case, claims are maximal if and only if they are priced by a consistent price process--the equivalent of an equivalent martingale measure. This result fails in the presence of transaction costs. A properly maximal claim is one which does have this property. We show that the properly maximal claims are dense in the set of maximal claims (with the topology of convergence in probability).

math.PR↗