Toward uncertain finance theory

Someone sat down in nineteen seventy-three with a stock ticker and a legal pad, recording prices as they came in. Every number they wrote down became raw material for one of the most mathematically sophisticated structures in the history of economics. Built into that structure, at its very foundation, is an assumption about the microscopic behavior of those price changes — an assumption Baoding Liu argues is not just imprecise, but logically impossible. That is the claim at the center of Liu's paper "Toward Uncertain Finance Theory." Not an approximation complaint, but an impossibility proof. If he is right, the implications reach all the way back to Bachelier. The canonical story of quantitative finance begins with Louis Bachelier's nineteen hundred dissertation, runs through Kiyosi Itô's invention of stochastic calculus, and arrives at the Black-Scholes-Merton framework of the nineteen sixties and seventies. The core mathematical object in this story is the Wiener process — a continuous random process whose short-time increments are normally distributed. That means at any small time step, the change in the process is drawn from a bell curve centered at zero, with variance equal to the length of the time step. Itô's calculus, and the famous second-order correction term in Itô's formula, depends entirely on that microscopic hypothesis about those increments. Here is Liu's paradox. Real stock prices, recorded in actual markets, are not smooth continuous curves. At the microstructure level — the level of individual ticks — they are step functions. A price sits at one value, then jumps to another. To make this concrete, Liu offers a simple numerical illustration. Suppose a stock price is observed to make exactly one hundred jumps over some fixed period. Now divide that same period into ten thousand equal intervals and sample the price increment at each one. What you get is nine thousand nine hundred zeros — intervals where nothing happened — and one hundred actual changes. That is your dataset. Now ask whether those ten thousand samples could plausibly have been drawn from a normal distribution with mean zero and variance proportional to the time step. The answer is obvious. They cannot. The distribution is dominated by a single repeated value. No one would accept it as normal. And yet that normality of increments is exactly what the Wiener process requires. The microscopic foundation of Itô calculus fails, and with it the theoretical basis for modeling stock prices as geometric Wiener processes. Liu's response is not to patch stochastic finance. It is to build an alternative from scratch. The alternative is uncertainty theory, a mathematical framework Liu developed for situations where you have no frequency data — no long-run samples to calibrate a probability distribution — and instead must work with expert belief degrees. The fundamental object is an uncertainty space, a triplet consisting of a nonempty set, a collection of events over that set, and an uncertain measure that assigns each event a value between zero and one. Liu's axioms for this measure include normality — the measure of the whole space equals one — duality — an event and its complement sum to one — subadditivity for countable unions, and a product axiom governing how measures compose across independent spaces. The key distinction from probability is this. Probability is calibrated by frequency — you run the experiment many times and observe how often events occur. Uncertainty is calibrated by belief — when no repetition is possible, experts supply a degree of confidence, and the framework handles that input rigorously. An uncertain variable is a measurable function from an uncertainty space to the real numbers, and its uncertainty distribution describes the belief profile of that variable. Peng and Iwamura proved that a function qualifies as an uncertainty distribution exactly when it is monotonically increasing. When that distribution is also continuous and strictly increasing in the interior, it is called regular, and it has a well-defined inverse — the inverse uncertainty distribution — which maps confidence levels to values. Expected values of uncertain variables are computed from these distributions through an integral-based definition. With the framework in place, Liu constructs a calculus for uncertain processes. The central object is the canonical Liu process — the uncertain analogue of Brownian motion. Like Brownian motion, it starts at zero and has stationary, independent increments. But the differences are precise and consequential. Its sample paths are almost all Lipschitz continuous — no wild oscillations, no infinite variation — and its increment over a time interval of length t is a normal uncertain variable with expected value zero and variance t squared. That variance scaling of t squared, rather than t, is a deliberate departure from the Wiener process, and it shapes everything that follows. The uncertain integral is defined as the limit of Riemann-type sums taken against the canonical process. An uncertain differential equation then takes the form: the change in the process equals a drift term times dt plus a diffusion term times the canonical increment dCt. Liu also proves an analogue of Itô's formula — a chain rule for functions of uncertain processes. Here is what it says verbally: the differential of a function h applied to time and the canonical process equals the partial derivative of h with respect to time, times dt, plus the partial derivative of h with respect to its second argument, times dCt. That's it. No second-order term. No correction. The reason ties directly back to the paradox: Itô's correction term exists precisely because the Wiener increment has variance proportional to dt, and when you square it you get something non-negligible. The canonical process is built differently. That correction vanishes. Chen and Liu proved existence and uniqueness results for uncertain differential equations under standard Lipschitz and linear growth conditions, and derived closed-form solutions for the linear case. For the general case, Yao and Chen contributed a powerful numerical tool called the Yao-Chen formula. The idea is elegant: instead of simulating the uncertain process directly, you fix a confidence level alpha between zero and one, and for each alpha you solve a deterministic ordinary differential equation — an alpha-path — where the diffusion term is replaced by the absolute value of the diffusion coefficient multiplied by the inverse uncertainty distribution of the standard normal uncertain variable evaluated at alpha. Run this across many values of alpha, and you assemble the full uncertainty distribution of the solution. Uncertain calculus becomes, computationally, a family of deterministic differential equations. That is the toolkit. Liu then applies it to finance. He proposed an uncertain stock model and derived European option pricing formulas under uncertainty theory. Chen extended this to American options. Peng and Yao developed a different uncertain stock model with its own option pricing results. Yu proposed a jump extension. Liu and Chen built an uncertain currency model, treating the exchange rate as the domestic price of one unit of foreign currency and deriving currency option prices by equating expected returns under uncertainty. Chen and Gao modeled interest rates with an uncertain differential equation driven by positive parameters, and from it produced an explicit formula for zero-coupon bond prices. Zhu applied uncertain differential equations to optimal control, proving a necessary condition called Zhu's equation of optimality. Each of these is a working model. Not a critique wrapped in philosophy, but an actual pricing formula, an actual bond price, an actual option result — computed through the alpha-path machinery, grounded in the uncertainty-theoretic axioms, and free from the Itô assumption that infinitesimal increments are normally distributed random variables with variance proportional to dt. What remains open is substantial. This is a research program, not a finished structure. The questions are sharp: which uncertain differential equations best describe different asset classes? How do you elicit and formalize expert belief degrees when market frequency data exists but its applicability is in question? How do uncertain option prices compare numerically to their Black-Scholes counterparts, and where do they diverge meaningfully? But the deepest question the paper leaves is the one embedded in the paradox itself. When you record a stock price, what kind of object are you actually measuring? If the increments can't be normal random draws — and Liu argues they provably cannot be — then the probabilistic interpretation of price movements, which underpins a half-century of financial mathematics, is not describing reality. The increments behave like uncertain variables, shaped by belief rather than frequency. What that means for how we build, test, and trust financial models is a question this paper opens rather than closes. This lecture was created by ennepō. 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