Abstract
Conditional expectations (like, e.g., discounted prices in financial applications) are martingales under an appropriate filtration and probability measure. When the information flow arrives in a punctual way, a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates. Providing a way to find and construct piecewise constant martingales evolving in a connected subset of mathbb {R} is the purpose of this paper. After a brief review of possible standard techniques, we propose a construction scheme based on the sampling of latent martingales tilde {Z} with lazy clocksθ. These θ are time-change processes staying in arrears of the true time but that can synchronize at random times to the real (calendar) clock. This specific choice makes the resulting time-changed process Z_{t}=tilde {Z}_{theta _{t}} a martingale (called a lazy martingale) without any assumption on tilde {Z}, and in most cases, the lazy clock θ is adapted to the filtration of the lazy martingale Z, so that sample paths of Z on [0,T] only requires sample paths of left (theta, tilde {Z}right) up to T. This would not be the case if the stochastic clock θ could be ahead of the real clock, as is typically the case using standard time-change processes. The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on (interval of) mathbb {R}.
Highlights
Martingales play a central role in probability theory, and in many applications
Definition 3 The stochastic process θ : R+ → R+, t → θt is a F-lazy clock if it satisfies the following properties i) it is an F-time change process: in particular, it is grounded (θ0 = 0), cadlag and non-decreasing; ii) it has piecewise constant sample paths : θt = s≤t θs; iii) it can jump at any time and, when it does, it synchronizes to the calendar clock, i.e. there is the equality {s > 0; θs = θs− } = {s > 0; θs = s}
In some specific cases like when we work under partial information or when market a b c d quotes arrive in a scarce way, it is more realistic to assume that conditional expectations move in a more piecewise constant fashion
Summary
Martingales play a central role in probability theory, and in many applications. In contrast with the common setup, it is more realistic to represent R as a martingale whose trajectories remain constant for long period of times, but “changes” only occasionally, upon arrival of related information (e.g., when a dealer updates its view to specialized data providers) Such types of martingales could be used to model discounted price processes of financial instruments, observed under partial (punctual) information, e.g., at some random times, and to represent price processes of illiquid products. Our approach allows one to build martingales that evolve in a bounded interval, a problem that received little attention so far and which relevance is stressed with the above recovery example, but could be of interest for modeling stochastic probabilities or correlations This is achieved by introducing a new class of time-change processes called lazy clocks. As our objective is to provide a workable methodology, we derive the analytical expression for the distributions and moments in some particular cases, and provide efficient sampling algorithms for the simulations of such martingales
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