Abstract
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.
Keywords:
Finance; Uncertainty theory; Uncertain process; Uncertain calculus; Uncertain differential equationReview
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 [1] 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 [2] 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 [3] as a sequence of uncertain variables indexed by time. Besides, the concept of uncertain integral was also proposed by Liu [3] in order to integrate an uncertain process with respect to a canonical process. Furthermore, Liu [4] 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 [3] 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 [5] under linear growth condition and Lipschitz continuous condition. The theorem was verified again by Gao [6] under local linear growth condition and local Lipschitz continuous condition. In order to solve uncertain differential equations, Chen and Liu [5] obtained an analytic solution to linear uncertain differential equations. In addition, Liu [7] presented a spectrum of analytic methods to solve some special classes of nonlinear uncertain differential equations. More importantly, Yao and Chen [8] 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 [8] for solving general uncertain differential equations. Furthermore, Yao [9] 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 [4] in which an uncertain stock model was proposed and European option price formulas were documented. Besides, Chen [10] derived American option price formulas for this type of uncertain stock model. In addition, Peng and Yao [11] presented a different uncertain stock model and obtained the corresponding option price formulas, and Yu [12] proposed an uncertain stock model with jumps. Uncertain differential equations were also employed to model uncertain currency markets by Liu and Chen [13] in which an uncertain currency model was proposed. Uncertain differential equations were used to simulate interest rate by Chen and Gao [14], 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 [15] 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.
A paradox of stochastic finance theory
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 [16] and stochastic differential equation [17], stochastic finance theory was well developed among others by Samuelson [18], Black and Scholes [19], and Merton [20] during the 1960s and 1970s.
Traditionally, stochastic finance theory presumes that the stock price (including currency exchange rate and interest rate) follows an Ito’s stochastic differential equation. Is it really reasonable? In fact, this widely accepted presumption was continuously challenged by many scholars. Let us assume that the stock price X_{t} follows the stochastic differential equation
where e is the log‐drift, σ is the log‐diffusion, and W_{t} is a Wiener process. Let us see what will happen with such an assumption. It follows from the stochastic differential equation (1) that X_{t} is a geometric Wiener process, i.e.,
from which we derive
whose increment is
Write
Note that the real stock price X_{t} is actually a step function of time with a finite number of jumps although it looks like a curve. During a fixed period, without loss of generality, we assume that X_{t} is observed to have 100 jumps. Now we divide the period into 10,000 equal intervals. Then we may observe 10,000 samples of X_{t}. It follows from Equation 4 that ΔW_{t} has 10,000 samples that consist of 9,900 A’s and 100 other numbers:
Nobody can believe that those 10,000 samples follow a normal probability distribution with expected value 0 and variance Δt. See Figure 1. This fact is in contradiction with the property of Wiener process that the increment ΔW_{t} is a normal random variable with expected value 0 and variance Δt. Therefore, the stock price X_{t} does not follow the stochastic differential equation.
Figure 1. Normal distribution vs real frequency.
Perhaps some people think that the stock price does behave like a geometric Wiener process (or Ornstein‐Uhlenbeck process) in macroscopy although they recognize the paradox in microscopy. However, as the very core of stochastic finance theory, Ito’s calculus is just built on the microscopic structure (i.e., the differential dW_{t}) of Wiener process rather than macroscopic structure. More precisely, Ito’s calculus is dependent on the presumption that dW_{t} is a normal random variable with expected value 0 and variance dt. This unreasonable presumption is what causes the second order term in Ito’s formula,
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.
What is uncertainty theory?
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 [2]: Axiom 1. (Normality axiom) for the universal set Γ; Axiom 2. (Duality axiom) for any event Λ; Axiom 3. (Subadditivity axiom) For every countable sequence of events Λ_{1},Λ_{2},⋯, we have
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 [4], thus producing the fourth axiom of uncertainty theory: Axiom 4. (Product axiom) Let be uncertainty spaces for k=1,2,⋯ The product uncertain measure is an uncertain measure satisfying
where Λ_{k} are arbitrarily chosen events from for k=1,2,⋯, respectively.
An uncertain variable is defined by Liu [2] as a measurable function ξ from an uncertainty space to the set of real numbers, i.e., for any Borel set B of real numbers, the set
is an event. In order to describe an uncertain variable in practice, the concept of uncertainty distribution is defined by Liu [2] as
Peng and Iwamura [21] 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.
An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0<Φ(x)<1, and
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[22].
It is easy to verify that Φ^{−1}(α) is a continuous and strictly increasing function with respect to α∈(0,1). Conversely, suppose Φ^{−1}(α) is a continuous and strictly increasing function on (0,1). Define
It follows that Φ(x) is an uncertainty distribution of some uncertain variable ξ. Then for each α∈(0,1), we have
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.
The expected value of an uncertain variable ξ is defined by Liu [2] as an average value of the uncertain variable in the sense of uncertain measure, i.e.,
provided that at least one of the two integrals is finite. If ξ has an uncertainty distribution Φ, then the expected value may be calculated by
Independence is an extremely important concept in uncertainty theory. The uncertain variables ξ_{1},ξ_{2},⋯,ξ_{n} are said to be independent [4] if
for any Borel sets B_{1},B_{2},⋯,B_{n} of real numbers. Equivalently, those uncertain variables are independent if and only if
Let ξ_{1},ξ_{2},⋯,ξ_{n} be independent uncertain variables with uncertainty distributions Φ_{1},Φ_{2},⋯,Φ_{n}, respectively. If the function f(x_{1},x_{2},⋯,x_{n}) is strictly increasing with respect to x_{1},x_{2},⋯, x_{m} and strictly decreasing with respect to x_{m+1},x_{m+2},⋯,x_{n}, then ξ=f(ξ_{1},ξ_{2},⋯,ξ_{n}) is an uncertain variable with inverse uncertainty distribution
Then Liu and Ha [23] proved that the uncertain variable ξ=f(ξ_{1},ξ_{2},⋯,ξ_{n}) has an expected value
For exploring the details of uncertainty theory, the readers may consult Liu [24].
Uncertain process
Let T be a totally ordered set (that is usually “time”), and let be an uncertainty space. An uncertain process is defined by Liu [3] as a measurable function from to the set of real numbers, i.e., for each t∈T and any Borel set B of real numbers, the set
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
Then X_{t} is an uncertain process and has the uncertainty distribution Φ_{t}(x). Thus, a function is an uncertainty distribution of uncertain process if and only if at each time t, it is a monotone increasing function except Φ_{t}(x)≡0 and Φ_{t}(x)≡1.
An uncertainty distribution Φ_{t}(x) is said to be regular if at each time t, it is a continuous and strictly increasing function with respect to x at which 0<Φ_{t}(x)<1, and
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
Then X_{t} is an uncertain process and has the inverse uncertainty distribution . Hence, a function is an inverse uncertainty distribution of uncertain process if and only if at each time t, it is a continuous and strictly increasing function with respect to α.
An uncertain process X_{t} is said to have independent increments if
are independent uncertain variables where t_{0} is the initial time and t_{1},t_{2},⋯,t_{k} are any times with t_{0}<t_{1}<⋯<t_{k}. That is, an independent increment process means that its increments are independent uncertain variables whenever the time intervals do not overlap. Let X_{t} be a sample‐continuous independent increment process with an uncertainty distribution Φ_{t}(x) at each time t. When f is a strictly increasing function, Liu [25] proved that the supremum
has an uncertainty distribution
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.
Let X_{t} be a stationary independent increment process with a crisp initial value X_{0}. Liu [22] showed that there exist two real numbers a and b such that the expected value
for any time t≥0. Furthermore, Chen [26] verified that there exists a real number c such that the variance
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 [4] 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}.
It is easy to verify that the canonical process C_{t} is a normal uncertain variable with expected value 0 and variance t^{2} and has an uncertainty distribution
at each time t>0. In addition, for each time t>0, the ratio C_{t}/t is a normal uncertain variable with expected value 0 and variance 1. That is,
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
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.
Let X_{t} be an uncertain process and let C_{t} be a canonical process. For any partition of closed interval [a,b] with a=t_{1}<t_{2}<⋯<t_{k+1}=b, the mesh is written as
Then the uncertain integral of X_{t} with respect to C_{t} is defined by Liu [4] as
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.
Let Z_{t} be an uncertain process. If there exist two uncertain processes μ_{t} and σ_{t} such that
for any t≥0, then we say Z_{t} has an uncertain differential
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.
Let h(t,c) be a continuously differentiable function. Liu [4] showed that the uncertain process Z_{t}=h(t,C_{t}) has an uncertain differential
This result is called the fundamental theorem of uncertain calculus.
Example 1
Let us calculate the uncertain differential of tC_{t}. In this case, we have h(t,c)=tc whose partial derivatives are
It follows from the fundamental theorem of uncertain calculus that
Example 2
Let us calculate the uncertain differential of . In this case, we have h(t,c)=c^{2} whose partial derivatives are
It follows from the fundamental theorem of uncertain calculus that
Example 3
Let f(c) be a continuously differentiable function. Then we have
It follows from the fundamental theorem of uncertain calculus that the uncertain process f(C_{t}) has an uncertain differential
This formula is also called the chain rule of uncertain calculus.
As supplements to uncertain integral, Liu and Yao [27] suggested an uncertain integral with respect to multiple canonical processes. More generally, Chen and Ralescu [28] presented an uncertain integral with respect to the general Liu process.
Uncertain differential equation
The study of uncertain differential equation was pioneered by Liu [3]. Nowadays, uncertain differential equation has achieved fruitful results in both theory and practice. Let f and g be two functions. Then
is called an uncertain differential equation. A solution is an uncertain process X_{t} that satisfies Equation 36 identically in t.
Some analytic methods have been proposed for solving uncertain differential equations. For example, Chen and Liu [5] showed that the linear uncertain differential equation
has a solution
where
In addition, Liu [7] verified that the nonlinear uncertain differential equation like
has a solution
where
and Z_{t} is the solution of uncertain differential equation
with initial value Z_{0}=X_{0}.
An important contribution to uncertain differential equation is the existence and uniqueness theorem by Chen and Liu [5]. An uncertain differential equation has a unique solution if the coefficients f(t,x) and g(t,x) satisfy linear growth condition
and Lipschitz condition
for some constant L. Moreover, the solution is sample‐continuous.
The concept of stability was given by Liu [4]. An uncertain differential equation is said to be stable if for any two solutions X_{t} and Y_{t}, we have
for any given number ϵ>0. Yao et al. [29] proved that the uncertain differential equation is stable if the coefficients f(t,x) and g(t,x) satisfy linear growth condition
for some constant K and strong Lipschitz condition
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 [30], Ge and Zhu [31], and Liu and Fei [32]. In addition, uncertain differential equation with jumps was suggested by Yao [33], and backward uncertain differential equation was discussed by Ge and Zhu [34].
Numerical method
Let α be a number with 0<α<1. An uncertain differential equation
is said to have an α‐path if it solves the corresponding ordinary differential equation
where Φ^{−1}(α) is the inverse uncertainty distribution of standard normal uncertain variable, i.e.,
Then
This result is called the Yao‐Chen formula[8]. In addition, at each time t, the solution X_{t} has an inverse uncertainty distribution
Furthermore, for any monotone (increasing or decreasing) function J, we have
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.
It is almost impossible to find analytic solutions for general uncertain differential equations. This fact provides a motivation to design a numerical method to solve general uncertain differential equation
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 [8] designed the following algorithm: Step 1. Fix α on (0,1). Step 2. Solve by any method of ordinary differential equation and obtain the α‐path , for example, by using the recursion formula
where Φ is the standard normal uncertainty distribution and h is the step length. Step 3. The inverse uncertainty distribution of the solution X_{t} is determined by
Uncertain stock model
Uncertain differential equations were first introduced into finance by Liu [4] in which an uncertain stock model was proposed,
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.
A European call option is a contract that gives the holder the right to buy a stock at an expiration time s for a strike price K. The payoff from a European call option is (Y_{s}−K)^{+} since the option is rationally exercised if and only if Y_{s}>K. Considering the time value of money resulted from the bond, the present value of this payoff is exp(−rs)(Y_{s}−K)^{+}. Hence, the European call option price should be the expected present value of the payoff, i.e.,
Liu [4] proved that
A European put option is a contract that gives the holder the right to sell a stock at an expiration time s for a strike price K. The payoff from a European put option is (K−Y_{s})^{+} since the option is rationally exercised if and only if Y_{s}<K. Considering the time value of money resulted from the bond, the present value of this payoff is exp(−rs)(K−Y_{s})^{+}. Hence, the European put option price should be the expected present value of the payoff, i.e.,
Liu [4] proved that
An American call option is a contract that gives the holder the right to buy a stock at any time prior to an expiration time s for a strike price K. It is clear that the payoff from an American call option is the supremum of (Y_{t}−K)^{+} over the time interval [0,s]. Considering the time value of money resulted from the bond, the present value of this payoff is
Hence, the American call option price should be the expected present value of the payoff, i.e.,
Chen [10] proved that
An American put option is a contract that gives the holder the right to sell a stock at any time prior to an expiration time s for a strike price K. It is clear that the payoff from an American put option is the supremum of (K−Y_{t})^{+} over the time interval [0,s]. Considering the time value of money resulted from the bond, the present value of this payoff is
Hence, the American put option price should be the expected present value of the payoff, i.e.,
Chen [10] proved that
It is emphasized that other stock models were also actively investigated by Peng and Yao [11], Yu [12], and Chen et al. [35], among others.
Uncertain currency model
Liu and Chen [13] assumed that the exchange rate follows an uncertain differential equation and then proposed an uncertain currency model,
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.
A currency option is a contract that gives the holder the right to exchange one unit of foreign currency at an expiration time s for K units of domestic currency. Suppose that the price of this contract is f in domestic currency. Then the investor pays f for buying the contract at time 0 and receives (Z_{s}−K)^{+} in domestic currency at the expiration time s. Thus, the expected return of the investor is
On the other hand, the bank receives f for selling the contract at time 0 and pays (1−K/Z_{s})^{+} in foreign currency at the expiration time s. Thus, the expected return of the bank is
The fair price of this contract should make the investor and the bank have an identical expected return, i.e.,
Thus, the currency option price is
Liu and Chen [13] proved that
Uncertain interest rate model
Real interest rates do not remain unchanged. Chen and Gao [14] assumed that the interest rate X_{t} follows an uncertain differential equation,
where m,a, and σ are positive numbers, and C_{t} is a canonical process.
A zero‐coupon bond is a bond bought at a price lower than its face value, that is, the amount it promises to pay at the maturity date. For simplicity, we assume that the face value is always US$1. Then the price of a zero‐coupon bond with a maturity date s is
Chen and Gao [14] proved that
where
Uncertain finance theory
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.
If we say that the classical finance theory is a methodology dealing with financial markets by using probability theory, then uncertain finance theory is a methodology dealing with financial markets by using uncertainty theory. In addition to the uncertain stock models shown above, we may also accept other uncertain differential equations, for example,
Conclusions
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.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This work was supported by the National Natural Science Foundation of China, grant no. 61273044.
References

Liu, B: Why is there a need for uncertainty theory?. J. Uncertain Syst. 6(1), 3–10 (2012)

Liu, B: Fuzzy process, hybrid process and uncertain process. J. Uncertain Syst. 2(1), 3–16 (2008)

Liu, B: Some research problems in uncertainty theory. J. Uncertain Syst. 3(1), 3–10 (2009)

Chen, XW, Liu, B: Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization Decis. Mak. 9(1), 69–81 (2010). Publisher Full Text

Gao, Y: Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition. J. Uncertain Syst. 6(3), 223–232 (2012)

Liu, YH: An analytic method for solving uncertain differential equations. J. Uncertain Syst. 6(4), 244–249 (2012)

Yao, K, Chen, XW: A numerical method for solving uncertain differential equations. J. Intell Fuzzy Syst. (2013, in press) (2013)

Yao, K: Extreme values and integral of solution of uncertain differential equation. J. Uncertainty Anal. Appl. 1, Article 3 (2013)

Chen, XW: American option pricing formula for uncertain financial market. Int. J. Oper. Res. 8(2), 32–37 (2011)

Peng, J, Yao: A new option pricing model for stocks in uncertainty markets. Int J. Oper. Res. 8(2), 18–26 (2011)

Yu, XC: A stock model with jumps for uncertain markets. Int. J. Uncertainty Fuzziness Knowledge‐Based Syst. 20(3), 421–432 (2012). Publisher Full Text

Liu, YH, Chen, XW: Uncertain currency model and currency option pricing http://orsc.edu.cn/online/091010.pdf webcite (2013) (2013)

Chen, XW, Gao, J: Uncertain term structure model of interest rate. Soft Comput.. 17(4), 597–604 (2013). Publisher Full Text

Zhu, Y: Uncertain optimal control with application to a portfolio selection model. Cybern. Syst. 41(7), 535–547 (2010). Publisher Full Text

Ito, K: Stochastic integral. Proc. Jpn. Acad. Ser. A. 20(8), 519–524 (1944). Publisher Full Text

Ito, K: On stochastic differential equations. Mem. Am. Math. Soc. 4, 1–51 (1951)

Samuelson, PA: Rational theory of warrant pricing. Ind. Manage. Rev. 6, 13–31 (1965)

Black, F, Scholes, M: The pricing of option and corporate liabilities. J. Pol. Econ. 81, 637–654 (1973). Publisher Full Text

Merton, RC: Theory of rational option pricing. Bell J. Econ. Manage. Sci. 4, 141–183 (1973). Publisher Full Text

Peng, ZX, Iwamura, K: A sufficient and necessary condition of uncertainty distribution. J. Interdiscip. Math. 13(3), 277–285 (2010)

Liu, B: Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Berlin: Springer (2010)

Liu, YH, Ha, MH: Expected value of function of uncertain variables. J. Uncertain Syst. 4(3), 181–186 (2010)

Liu, B: Uncertainty Theory http://orsc.edu.cn/liu/ut.pdf webcite (2013) (2013)

Liu, B: Extreme value theorems of uncertain process with application to insurance risk model. Soft Comput. 17(4), 549–556 (2013). Publisher Full Text

Chen, XW: Variation analysis of uncertain stationary independent increment process. Eur. J. Oper. Res. 222(2), 312–316 (2012). Publisher Full Text

Liu, B, Yao, K: Uncertain integral with respect to multiple canonical processes. J. Uncertain Syst. 6(4), 250–255 (2012)

Chen, XW, Ralescu, DA: Liu process and uncertain calculus. J. Uncertainty Anal. Appl. 1, Article 2 (2013)

Yao, K, Gao, J, Gao, Y: Some stability theorems of uncertain differential equation. Fuzzy Optimization Decis. Mak. 12(1), 3–13 (2013). Publisher Full Text

Barbacioru, IC: Uncertainty functional differential equations for finance. Surv Math. Appl. 5, 275–284 (2010)

Ge, XT, Zhu, Y: Existence and uniqueness theorem for uncertain delay differential equations. J. Comput. Inf. Syst. 8(20), 8341–8347 (2012)

Liu, HJ, Fei, WY: Neutral uncertain delay differential equations. Inf. Int. Interdiscip. J. 16(2), 1225–1232 (2013)

Yao, K: Uncertain calculus with renewal process. Fuzzy Optimization Decis. Mak. 11(3), 285–297 (2012). Publisher Full Text

Ge, XT, Zhu, Y: A necessary condition of optimality for uncertain optimal control problem. Fuzzy Optimization Decis. Mak. 12, (2013)

Chen, XW, Liu, YH, Ralescu, DA: Uncertain stock model with periodic dividends. Fuzzy Optimization Decis. Mak. 12(1), 111–123 (2013). Publisher Full Text