Toward uncertain finance theory
© Liu; licensee Springer. 2013
Received: 12 February 2013
Accepted: 18 February 2013
Published: 24 April 2013
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© Liu; licensee Springer. 2013
Received: 12 February 2013
Accepted: 18 February 2013
Published: 24 April 2013
This paper first introduces a paradox of stochastic finance theory that shows the real stock price is impossible to follow any Ito’s stochastic differential equation. After a survey on uncertainty theory, uncertain process, uncertain calculus, and uncertain differential equation, this paper discusses some possible applications of uncertain differential equations to financial markets. Finally, it is suggested that a new uncertain finance theory should be developed based on uncertainty theory and uncertain differential equation.
When no samples are available to estimate a probability distribution, we have to invite some domain experts to evaluate their belief degree that each event will occur. Perhaps some people think that personal belief degree is subjective probability or fuzzy concept. However, Liu  declared that it is inappropriate because both probability theory and fuzzy set theory may lead to counterintuitive results in this case. In order to rationally deal with the belief degree, an uncertainty theory was founded by Liu  and subsequently studied by many scholars. Nowadays, uncertainty theory has become a branch of axiomatic mathematics for modeling human uncertainty.
Based on uncertainty theory, the concept of uncertain process was given by Liu  as a sequence of uncertain variables indexed by time. Besides, the concept of uncertain integral was also proposed by Liu  in order to integrate an uncertain process with respect to a canonical process. Furthermore, Liu  recast his work via the fundamental theorem of uncertain calculus and thus produced the techniques of chain rule, change of variables, and integration by parts. Since then, the theory of uncertain calculus was well developed.
After uncertain differential equation was proposed by Liu  as a differential equation involving uncertain process, an existence and uniqueness theorem of a solution of uncertain differential equation was proved by Chen and Liu  under linear growth condition and Lipschitz continuous condition. The theorem was verified again by Gao  under local linear growth condition and local Lipschitz continuous condition. In order to solve uncertain differential equations, Chen and Liu  obtained an analytic solution to linear uncertain differential equations. In addition, Liu  presented a spectrum of analytic methods to solve some special classes of nonlinear uncertain differential equations. More importantly, Yao and Chen  showed that the solution of an uncertain differential equation can be represented by a family of solutions of ordinary differential equations, thus relating uncertain differential equations and ordinary differential equations. On the basis of the Yao‐Chen formula, a numerical method was also designed by Yao and Chen  for solving general uncertain differential equations. Furthermore, Yao  presented some formulas to calculate the extreme values, first hitting time and integral of solution of uncertain differential equation.
Uncertain differential equations were first introduced into finance by Liu  in which an uncertain stock model was proposed and European option price formulas were documented. Besides, Chen  derived American option price formulas for this type of uncertain stock model. In addition, Peng and Yao  presented a different uncertain stock model and obtained the corresponding option price formulas, and Yu  proposed an uncertain stock model with jumps. Uncertain differential equations were also employed to model uncertain currency markets by Liu and Chen  in which an uncertain currency model was proposed. Uncertain differential equations were used to simulate interest rate by Chen and Gao , and an uncertain interest rate model was presented. On the basis of this model, the price of zero‐coupon bond was also produced. Uncertain differential equations were applied to optimal control by Zhu  in which Zhu’s equation of optimality is proved to be a necessary condition for extremum of uncertain optimal control model.
This paper first introduces a paradox of stochastic finance theory. After a survey on uncertainty theory, uncertain process, uncertain calculus, and uncertain differential equation, this paper shows some possible applications of uncertain differential equations to financial markets. Finally, this paper suggests to develop an uncertain finance theory by using uncertainty theory and uncertain differential equation.
The origin of stochastic finance theory can be traced to Louis Bachelier’s doctoral dissertation Théorie de la Speculation in 1900. However, Bachelier’s work had little impact for more than a half century. After Kiyosi Ito invented stochastic calculus  and stochastic differential equation , stochastic finance theory was well developed among others by Samuelson , Black and Scholes , and Merton  during the 1960s and 1970s.
In fact, the increment of stock price is impossible to follow any continuous probability distribution. On the basis of the above paradox, personally, I do not think Ito’s calculus can play the essential tool of finance theory because Ito’s stochastic differential equation is impossible to model real stock price.
Let Γ be a nonempty set, and a σ‐algebra over Γ. Each element Λ in is called an event. A set function from to [0,1] is called an uncertain measure if it satisfies the following axioms :
Axiom 1. (Normality axiom) for the universal set Γ;
Axiom 2. (Duality axiom) for any event Λ;
The triplet is called an uncertainty space. In order to obtain an uncertain measure of compound event, a product uncertain measure was defined by Liu , thus producing the fourth axiom of uncertainty theory:
where Λ k are arbitrarily chosen events from for k=1,2,⋯, respectively.
Peng and Iwamura  proved that a function is an uncertainty distribution if and only if it is a monotone increasing function except Φ(x)≡0 and Φ(x)≡1.
Let ξ be an uncertain variable with regular uncertainty distribution Φ(x). Then the inverse function Φ −1(α) is called the inverse uncertainty distribution of ξ.
Thus, Φ −1(α) is just the inverse uncertainty distribution of the uncertain variable ξ. Hence, we have a sufficient and necessary condition of inverse uncertainty distribution: A function is an inverse uncertainty distribution if and only if it is a continuous and strictly increasing function with respect to α.
For exploring the details of uncertainty theory, the readers may consult Liu .
is an event. In other words, an uncertain process is a sequence of uncertain variables indexed by time.
Note that if the index set T becomes a partially ordered set (e.g., time × space, or a surface), then X t is called an uncertain field provided that X t is an uncertain variable at each point t. That is, an uncertain field is a generalization of an uncertain process.
An uncertain process X t is said to have an uncertainty distribution Φ t (x) if at each time t, the uncertain variable X t has the uncertainty distribution Φ t (x). It is easy to prove that Φ t (x) is a monotone increasing function with respect to x and Φ t (x)≢0, Φ t (x)≢1. Conversely, if at each time t, Φ t (x) is a monotone increasing function except Φ t (x)≡0 and Φ t (x)≡1, it follows that there exists an uncertain variable ξ t whose uncertainty distribution is just Φ t (x). Define
Let X t be an uncertain process with regular uncertainty distribution Φ t (x). Then the inverse function is called the inverse uncertainty distribution of X t . It is easy to prove that is a continuous and strictly increasing function with respect to α∈(0,1). Conversely, if is a continuous and strictly increasing function with respect to α∈(0,1), it follows that there exists an uncertain variable ξ t whose inverse uncertainty distribution is just . Define
This result is called the extreme value theorem of uncertain process.
An uncertain process X t is said to have stationary increments if its increments are identically distributed uncertain variables whenever the time intervals have the same length, i.e., for any given t>0, the increments X s+t −X s are identically distributed uncertain variables for all s>0.
for any time t≥0.
As an important type of uncertain process, a canonical process is a stationary independent increment process whose increments are normal uncertain variables. More precisely, an uncertain process C t is called a canonical process by Liu  if (1) C 0=0 and almost all sample paths are Lipschitz continuous, (2) C t has stationary and independent increments, and (3) every increment C s+t −C s is a normal uncertain variable with expected value 0 and variance t 2.
for any t>0.
What is the difference between canonical process and the Wiener process? First, canonical process is an uncertain process while the Wiener process is a stochastic process. Second, almost all sample paths of canonical process are Lipschitz continuous functions while almost all sample paths of the Wiener process are continuous but non‐Lipschitz functions. Third, canonical process has a variance t 2 while the Wiener process has a variance t at each time t.
Uncertain calculus is a branch of mathematics that deals with differentiation and integration of uncertain processes. The key concept in uncertain calculus is the uncertain integral that allows us to integrate an uncertain process (the integrand) with respect to the canonical process (the integrator). The result of the uncertain integral is another uncertain process.
provided that the limit exists almost surely and is finite. Since X t and C t are uncertain variables at each time t, the limit in Equation 29 is also an uncertain variable.
In this case, Z t is called an uncertain process with drift μ t and diffusion σ t . It is clear that uncertain integral and differential are mutually inverse operations. Please also note that an uncertain differential of an uncertain process has two parts, the “ dt” part and the “ dC t ” part.
This result is called the fundamental theorem of uncertain calculus.
Let f(c) be a continuously differentiable function. Then we have
This formula is also called the chain rule of uncertain calculus.
As supplements to uncertain integral, Liu and Yao  suggested an uncertain integral with respect to multiple canonical processes. More generally, Chen and Ralescu  presented an uncertain integral with respect to the general Liu process.
is called an uncertain differential equation. A solution is an uncertain process X t that satisfies Equation 36 identically in t.
with initial value Z 0=X 0.
for some constant L. Moreover, the solution is sample‐continuous.
for some bounded and integrable function L(t) on [0,+∞).
Uncertain differential equation has been extended by many scholars. For example, uncertain delay differential equation was studied among others by Barbacioru , Ge and Zhu , and Liu and Fei . In addition, uncertain differential equation with jumps was suggested by Yao , and backward uncertain differential equation was discussed by Ge and Zhu .
The Yao‐Chen formula relates uncertain differential equations and ordinary differential equations, just like that Feynman‐Kac formula relates stochastic differential equations and partial differential equations.
In order to do so, a key point is to obtain an inverse uncertainty distribution of its solution X t at any given time t. For this purpose, Yao and Chen  designed the following algorithm:
Step 1. Fix α on (0,1).
where Φ is the standard normal uncertainty distribution and h is the step length.
where X t is the bond price, Y t is the stock price, r is the riskless interest rate, e is the log‐drift, σ is the log‐diffusion, and C t is a canonical process.
where X t represents the domestic currency with domestic interest rate u, Y t represents the foreign currency with foreign interest rate v, and Z t represents the exchange rate, that is, the domestic currency price of one unit of foreign currency at time t.
where m,a, and σ are positive numbers, and C t is a canonical process.
At the beginning of this paper, a paradox was proposed to show that the real stock price is impossible to follow an Ito’s stochastic differential equation. It follows from Figure 1 that the increments behave like an uncertain variable rather than a random variable. This fact motives us to model stock prices by uncertain differential equations. Personally, I think uncertain calculus may play a potential mathematical foundation of finance theory.
At first, a paradox of stochastic finance theory was introduced in this paper. After a survey on uncertainty theory, uncertain process, uncertain calculus, and uncertain differential equation, this paper summarized uncertain stock model, uncertain currency model, and uncertain interest model by using the tool of uncertain differential equation. Finally, it was suggested that an uncertain finance theory should be developed based on uncertainty theory.
This work was supported by the National Natural Science Foundation of China, grant no. 61273044.
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