Stochastic Models: Growth, Mean Reversion and the Assumptions Between Them

A stochastic differential equation specifies how a state responds to predictable forces and random shocks. Choosing that equation determines whether a model permits negative values, how uncertainty grows and whether deviations tend to persist or dissipate. Those choices are economic assumptions expressed mathematically.

This reference compares geometric growth and mean reversion, explains what existence and uniqueness establish, and shows how exact transition formulas improve simulation and estimation. It contains model illustrations rather than market predictions.

The equation and its solution are different objects

An Ito stochastic differential equation has the form

$$dX_t=a(X_t,t)dt+b(X_t,t)dW_t,\qquad X_0=x_0.$$

Here $a$ is drift, $b$ is diffusion, $W$ is Brownian motion and $x_0$ is the initial state. Its integral meaning is

$$X_t=x_0+\int_0^t a(X_s,s)ds+\int_0^t b(X_s,s)dW_s.$$

The first integral accumulates finite-variation drift. The second accumulates shocks under a non-anticipating convention. The initial condition, probability space and information filtration are part of a complete model specification.

A strong solution is constructed using the given Brownian motion and initial information. A weak solution permits the probability space and Brownian driver to be part of the construction. “Strong” does not mean empirically better; it describes the mathematical relationship between the solution and the noise.

Pathwise uniqueness means two solutions with the same driver and initial condition are indistinguishable. Uniqueness in law means they have the same distribution. These concepts answer different implementation questions and should not be treated as synonyms.

Conditions that prevent ambiguous or explosive behaviour

A standard sufficient set of conditions combines global Lipschitz continuity and linear growth. In simplified notation, constants $L,K$ bound

$$|a(x,t)-a(y,t)|+|b(x,t)-b(y,t)|\leq L|x-y|$$

and

$$|a(x,t)|^2+|b(x,t)|^2\leq K(1+|x|^2).$$

Together with appropriate time measurability and an integrable initial state, these conditions yield a unique nonexplosive strong solution over a finite horizon. They are sufficient, not necessary. Failure to satisfy a global Lipschitz condition does not itself prove that a model has no solution.

Local conditions can establish uniqueness up to an explosion time. Additional growth or Lyapunov arguments may rule out explosion. Square-root diffusions are an important finance example where more specialised results apply despite a diffusion coefficient that is not globally Lipschitz at zero.

Existence and uniqueness do not establish that the model fits observed markets, that a numerical scheme is stable, or that parameter estimates are reliable. They establish that the mathematical specification identifies a well-defined process under its assumptions.

Geometric Brownian motion models proportional growth

For an asset level $S_t$, geometric Brownian motion is

$$dS_t=\mu S_tdt+\sigma S_tdW_t.$$

The drift $\mu$ is expected proportional growth and $\sigma$ is proportional volatility. With $S_0>0$, the exact solution remains strictly positive:

$$S_t=S_0\exp[(\mu-\tfrac12\sigma^2)t+\sigma W_t].$$

Its moments are

$$\mathbb E[S_t]=S_0e^{\mu t},\qquad \operatorname{Var}(S_t)=S_0^2e^{2\mu t}(e^{\sigma^2t}-1).$$

In words: The average level reflects growth and elapsed time, while its spread also depends on the shock size.

The logarithm is Gaussian; the level is lognormal. Constant volatility and independent log-return increments make the model convenient, but they exclude volatility clustering and jumps. Positivity is a mathematical property of this process, not evidence that it is suitable for every positive financial variable.

Illustrative geometric-growth inputs and resulting levels

The difference comes from the skewed distribution. These values demonstrate the formulas and do not represent a security forecast.

Under a pricing measure for a non-dividend-paying asset, the drift is replaced by the appropriate risk-free financing rate in the basic model. A historical estimate of $\mu$ should not be inserted into a risk-neutral option valuation merely because it came from the same underlying data.

Exact transitions avoid an unnecessary approximation

Over a step $h$, an exact geometric Brownian transition is

$$S_{t+h}=S_t\exp[(\mu-\tfrac12\sigma^2)h+\sigma\sqrt h\,Z],$$

where $Z$ is an independent standard normal draw. An Euler approximation instead uses $S_{t+h}\approx S_t+\mu S_th+\sigma S_t\sqrt h Z$. For sufficiently adverse draws or large steps, that approximation can produce a negative level even though the exact model cannot.

When the exact transition is available, using it removes time-discretisation error for the sampled states. It does not remove discretisation error in continuously monitored barriers or other path functionals between those states.

Ornstein–Uhlenbeck dynamics model a restoring force

An Ornstein–Uhlenbeck process follows

$$dX_t=\kappa(\theta-X_t)dt+\sigma dW_t,$$

where $\kappa>0$ is the speed of mean reversion, $\theta$ is the long-run level and $\sigma$ is the scale of additive shocks. A state above $\theta$ has negative drift; one below it has positive drift. New shocks can still move the process farther away.

Its conditional mean is

$$\mathbb E[X_{t+h}\mid X_t]=\theta+(X_t-\theta)e^{-\kappa h},$$

and its conditional variance is

$$\operatorname{Var}(X_{t+h}\mid X_t) =\frac{\sigma^2}{2\kappa}(1-e^{-2\kappa h}).$$

In words: Uncertainty grows from zero toward a limit determined by shock size and how strongly the state is pulled back.

These formulas separate three effects: the equilibrium level, the decay of the initial displacement and the uncertainty introduced by subsequent shocks.

The half-life of a conditional-mean displacement is

$$h_{1/2}=\frac{\log 2}{\kappa}.$$

For an illustrative $\kappa=0.5$ per year, the half-life is about 1.39 years. This is a statement about the model’s conditional mean. It is not a promise that every observed spread returns halfway within that period.

Stationarity does not mean a path settles down

The invariant distribution of the OU process is normal with mean $\theta$ and variance $\sigma^2/(2\kappa)$. If the process starts in that distribution, its marginal distribution is unchanged over time. From a fixed starting point, its distribution approaches that invariant law.

An individual realised path continues to fluctuate. It does not converge to $\theta$. Confusing convergence in distribution with pathwise convergence turns a model of persistent random variation into an unjustified convergence trade.

Gaussian support also allows negative values. OU dynamics can be sensible for a spread or rate factor that may cross zero. A strictly positive variance level may require a different model or a transformation, such as modelling its logarithm. A transformation changes the distribution and economic interpretation and should be documented.

Discrete observations imply an autoregression

At regular spacing $h$, the exact OU transition can be written

$$X_{n+1}=c+\phi X_n+\varepsilon_n,$$

with $\phi=e^{-\kappa h}$, $c=\theta(1-\phi)$, and innovation variance $\sigma^2(1-\phi^2)/(2\kappa)$. An estimated autoregression therefore maps into continuous-time parameters through

$$\kappa=-\frac{\log\phi}{h},\qquad \theta=\frac{c}{1-\phi}.$$

When $\phi$ is close to one, small estimation errors produce large changes in the implied half-life and equilibrium mean. A short sample may appear mean reverting without identifying the speed reliably. Structural breaks can make a single equilibrium estimate misleading even when the regression fit looks acceptable.

Residual analysis should check changing variance, nonlinear drift, outliers and regime shifts. Estimation under the physical measure supports forecasting; pricing-measure parameters need calibration or a documented market-price-of-risk relationship.

An exact mean-reversion simulation

The OU conditional distribution gives an exact transition over spacing $h$:

$$X_{t+h}=\theta+(X_t-\theta)e^{-\kappa h} +\sigma\sqrt{\frac{1-e^{-2\kappa h}}{2\kappa}}Z,$$

In words: The next state combines the long-run level, a fading part of its current deviation and a new random shock.

with independent standard normal $Z$. This preserves the correct conditional mean and variance for the sampled states. A large Euler step can instead exaggerate or reverse persistence because it replaces $e^{-\kappa h}$ with $1-\kappa h$.

For an illustrative annual step with $\kappa=2$, the exact persistence is about 0.1353, whereas Euler gives minus one. The negative coefficient creates alternating numerical behaviour absent from the exact conditional mean. Reducing the time step can resolve the approximation problem, but using the available exact transition avoids it directly.

Even the exact transition does not guarantee correct continuous-time extrema, occupation times or hitting events. Those quantities depend on what happens between sampled states, so the relevant path treatment remains part of the simulation design.

Model parameters carry units

In an additive OU process, $\sigma$ has units of the state variable per square root of time. In geometric Brownian motion it has units of inverse square root time because it multiplies the state. Mean-reversion speed has units of inverse time.

An annual estimate cannot be inserted unchanged into a daily-step formula if the formula treats one step as one time unit. For a model measured in years, a daily step is a fraction of a year; drift scales with that fraction and diffusion with its square root.

Rate data recorded as percentages and as decimals differ by a factor of 100. Additive volatility must change by the same factor, while a square-root diffusion coefficient transforms differently because the state appears under a square root. Consistent units should be verified before comparing parameter magnitudes across models.

Calibration fit can conceal weak identification

A slow mean-reverting process observed over a short sample can resemble a random walk. Several combinations of equilibrium level and speed may produce similar likelihoods. A precise optimisation result can therefore conceal a broad region of statistically plausible parameters.

Confidence intervals or sensitivity ranges should accompany the estimated half-life. The transformation $-\log\phi/h$ is nonlinear, so uncertainty in an autoregressive coefficient does not translate symmetrically into uncertainty in mean-reversion speed. Near $\phi=1$, the long-run mean $c/(1-\phi)$ is particularly unstable.

Testing an investment interpretation requires more than finding a positive fitted reversion speed. The state may contain structural breaks, selection bias or a changing equilibrium. A spread built from estimated hedge weights introduces another layer of estimation error. These issues can produce an attractive in-sample path without an implementable out-of-sample strategy.

Distinguish a state model from a tradable asset model

A mean-reverting state can be a useful latent factor without itself being a directly tradable asset. Specifying its physical drift does not automatically identify derivative prices. The connection to traded instruments, pricing measure and any market price of risk must be supplied.

Conversely, treating a directly tradable price as forced toward a known deterministic level under a frictionless model can have implications for arbitrage and admissible strategies. Model design should therefore state whether the variable is a price, a spread, a short rate, a volatility factor or an unobserved economic state.

Checking the chosen approximation

Euler–Maruyama approximates the drift and diffusion at the beginning of each step. Under suitable regularity, its strong error is commonly of order one-half and its weak error of order one. Strong error concerns coupled sample paths; weak error concerns expectations of sufficiently regular functions.

Milstein adds a term involving the diffusion derivative and the squared Brownian increment. In a one-dimensional sufficiently smooth setting it can improve strong convergence. That benefit is not universal for nonsmooth payoffs, multidimensional noncommutative noise or poorly behaved coefficients.

Match the validation check to the modelling question

The model choice should follow the required behaviour: proportional growth for a positive multiplicative state, mean reversion for a restoring force, and specialised dynamics where boundaries or jumps matter. A reliable implementation then preserves those properties while exposing the uncertainty in both parameters and numerical approximations.