Stochastic Processes

• Random Walk — the discrete-time process of summing independent steps; the Wiener process is its continuous-time limit.
• Markov Property — the "no memory" property every process in this cluster has: the next state depends only on the current one.
• Wiener Process — continuous-time Brownian motion, \(dW\), the source of randomness every process below is driven by.
• Ito's Lemma — the stochastic chain rule, needed to derive e.g. a bond-price SDE from a short-rate SDE.

• Arithmetic Brownian Motion — additive increments (price + increment=); implemented by arithmetic_gaussian_mixture_model_process.
• Geometric Brownian Motion — multiplicative, log-normal increments (price * exp(log_return)=); implemented by gaussian_mixture_model_process. Both generalise their single-Gaussian textbook form to a K-component Gaussian mixture, reproducing fat tails and volatility clustering a single Normal shock cannot — see Volatility clustering and GARCH models for the companion discrete-time picture of that same fat-tail problem. Currently consumed for FX spot and equity price generation.

Mean reversion is the opposite behaviour to the unbounded wandering of the price processes above: instead of a random shock accumulating without limit, a mean-reverting process has a built-in pull back toward a long-run level whenever it drifts away — the further it strays, the harder it gets pulled back. Nothing stops it wandering away again on the very next tick, so the path still looks jittery moment to moment; what mean reversion changes is the long-run behaviour, not the short-run randomness — a mean-reverting path keeps circling back toward its target level instead of drifting off to arbitrarily large or small values the way an unbounded price process can. This is the natural shape for quantities that are anchored by some external equilibrium force (a central bank's policy target for an interest rate, for instance), whereas a price process is the natural shape for quantities with no such anchor at all.

• Ornstein-Uhlenbeck Process — the general mean-reverting building block, \(dX = \kappa(\theta-X)\,dt + \sigma\, dW\); implemented by ornstein_uhlenbeck_process.
• Vasicek Process — the OU process applied to a short rate with a constant long-run level; implemented by vasicek_process, a thin wrapper composing hull_white_process.
• Hull-White Process — Vasicek generalised to a piecewise-constant, time-varying mean-reversion level \(\theta(t)\), letting the process fit an observed initial term structure exactly; implemented by hull_white_process.
• Cox-Ingersoll-Ross (CIR) Process — Vasicek with a \(\sigma\sqrt{r}\) volatility term that keeps the rate non-negative, at the cost of a non-Gaussian (non-central chi-squared) transition density; implemented by cox_ingersoll_ross_process.

Currently all four are consumed as short-rate models; the underlying maths applies to any quantity that reverts to a long-run level, not just a rate.

Vasicek, Hull-White, and CIR are not three independent models — they are three points on one design axis, all sharing the OU drift shape \(\kappa(\text{target} - X)\):

Vasicek is literally the \(\theta(t) = \text{const}\) special case of Hull-White — vasicek_process composes hull_white_process rather than reimplementing it. CIR cannot be reached from Hull-White by any parameter substitution: its non-negativity comes from a genuinely different volatility term, which is also why its exact transition requires the Poisson-mixture-of-central-chi-squared construction described in its own atomic doc, rather than the closed-form Gaussian update the other three share.

The processes above answer "what is the level now?". A point process answers "when does the next event happen?". It is the third family, and its foundation note defines the conditional intensity that all of them share.

• Point Process and Conditional Intensity — the rate of events given the history, the compensator, and the time-rescaling test.
• Poisson Process — a constant intensity; memoryless exponential gaps.
• Hawkes Process — each event raises the intensity for a while; bursts breed bursts. It reduces to the Poisson process when its branching ratio is zero.

These do not implement i_stochastic_process, because next() there returns a level. Their consumer, and the interface they sit behind, are in Tick arrival.

ores::analytics::quant::service::process_factory constructs the concrete process an ir_curve_generation_config or market_data_generation_config names, behind the common i_stochastic_process interface every process implements — the generation loop that drives next() on whichever process was configured does not need to know which one it got. This is the first, not the only, place these processes are expected to be used in ORE Studio.

The synthetic volatility surfaces use the mean-reverting family for something other than a rate. The forward variances of an SSVI surface are Cox-Ingersoll-Ross processes, because a variance must stay positive, and its skew and curvature are Ornstein-Uhlenbeck processes mapped into their bounds, as the option vol surface section of Synthetic market data generation: approach describes. The FX spot price processes carry a daily GJR-GARCH variance, a discrete-time model outside this cluster, described in Volatility clustering and GARCH models.

• Wikipedia: Stochastic process.