Distribution of Return Transition for Bohm-Vigier Stochastic Mechanics in Stock Market

Abstract

The Bohm-Vigier stochastic model is assumed as a natural generalization of the Black-Scholes model in stock market. The behavioral factor of stock market recognizes as a hidden sector in Bohmian mechanics. A Fokker-Planck equation description for the Bohm-Vigier stochastic model is presented. We find the familiar Boltzmann distribution is a stationary solution of the Fokker-Planck equation for the Bohm-Vigier model. The return transition distribution of stock market, which corresponds with a time-dependent solution of the Fokker-Planck equation, is obtained.

1. Introduction

The Bohmian mechanics is also called the pilot-wave model [1]. It provides the quantum mechanical causal interpretation. The Bohmian mechanics is often called a hidden variables interpretation of quantum mechanics. In quantum mechanics, one cannot say anything certainly about moment of the individual particle. Prior to measurement, it is impossible to assume that a particle exists equally. The particle becomes whatever it is measured to be once the measurement has been made. According to this interpretation of quantum theory, a particle is actually in all of its potential states up until a measurement or precise observation is made of it. Everything in the Copenhagen Model of quantum physics is based on probabilities. In the late 1920s and the first half of the 1930s, the Copenhagen Model reached its height.Bohr, Schrödinger, Heisenberg, and Pauli are pioneers of the Copenhagen Model. Schrödinger suggested the so-called Schrödinger equation to explain the likelihood of a specific particle state,

$$i\hslash \frac{\partial \Psi (q,t)}{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\Psi (q,t)}{\partial q^2}+V\left(q\right)\Psi (q,t),$$

(1)

where q is the position of particle at time t, m is the mass of the particle, ℏ denotes the Planck constant, and ${|\Psi (q,t)|}^2$ describes the probability of the particle appears at position q and time t. One of the biggest critics of the Copenhagen Model of quantum physics comes from Einstein. Einstein was unable to accept the idea that a single particle might exist in multiple universes. The classic “Schrodinger’s cat” puzzle is an excellent illustration of the Copenhagen Model’s perspective on reality. A cat is placed into a dark box. A bottle of cyanide may have cracked and killed the cat if it had been placed in that box. It could also have survived unharmed and saved the cat. The Copenhagen Model of quantum theory would assume that the cat is simultaneously in both states, that is, both alive and dead, as long as this is unknown. In Bohmian mechanics, one can partially explain a system by looking at its changing wave function, according to Schrödinger’s equation. According to Bohm, the wave function only fully describes the system. The actual positions of each particle are specified, completing this description. In Bohmian quantum mechanics, ${\Phi }^2$ and ${\rm Y}$ satisfy the equations

$${\displaystyle \frac{\partial {\Phi }^2(q,t)}{\partial t}+\frac{1}{m}\frac{\partial }{\partial q}\left({\Phi }^2(q,t)\frac{\partial {\rm Y}}{\partial q}\right)=0,}$$

(2)

$${\displaystyle \frac{\partial {\rm Y}(q,t)}{\partial t}+\frac{1}{2m}{\left(\frac{\partial {\rm Y}(q,t)}{\partial q}\right)}^2+\left(V-U\right)=0,}$$

(3)

where ${\displaystyle U\equiv \frac{{\hslash }^2}{2m\Phi }\frac{{\partial }^2\Phi }{\partial M^2}}$ is the so called Bohm quantum potential. $\Phi$ and ${\rm Y}$ are related with Schrödinger’s wave function $\Psi (q,t)$ as

$$\Psi (q,t)=\Phi (q,t)exp\left(i\frac{{\rm Y}}{\hslash }\right).$$

(4)

At the case of ${\displaystyle \frac{{\hslash }^2}{2m}<<1}$, the Bohm quantum potential U can be neglected. And the Equation (3) reduces to the Hamilton-Jacobi equation in classical dynamics

$${\displaystyle \frac{\partial {\rm Y}}{\partial t}+\frac{1}{2m}{\left(\frac{\partial {\rm Y}}{\partial q}\right)}^2+V=0.}$$

(5)

In classical dynamics, it is well known that the dynamics of particles are represented by the Hamilton-Jacobi equation,

$$m\frac{dq}{dt}=\frac{\partial {\rm Y}}{\partial q}.$$

(6)

Bohm proposed that in quantum level the dynamics behaves the same way but added with the quantum potential U,

$$m\frac{d^{2}q}{d{t}^2}=-\frac{\partial V}{\partial q}-\frac{\partial U}{\partial q}.$$

(7)

It should be noticed that the Bohm quantum potential U is itself driven by the Schrödinger Equation (1).

The Bohmian mechanics offers the opportunity to explain a particle’s quantum-level trajectory $q\left(t\right)$. Standard quantum mechanics does not permit such a route. However, the Bohmmian mechanics is nonlocal. One particle would affect others with large distances via the pilot wave field. It has been proposed that these Bohmian mechanics drawbacks will turn into benefits when they are applied to the financial market [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].

The efficient market hypothesis states that for a highly liquid market, the prices of the financial assets are efficient representing all information that is currently accessible, and the Black-Scholes model may accurately capture a financial market’s dynamics [19,20]. A price estimate for options in the European style is provided by the Black-Scholes formula [21]. Regardless of the security’s risk or anticipated return, the option has a fixed price. A stock price is typically assumed to move in accordance with a geometric Brownian motion described by the stochastic differential equation $dM=M(\sigma dB+\mu dt)$, where B is a Wiener process. Ito’s lemma provides if the value of an option at time t is $f(t,M)$

$${\displaystyle df(t,M)=\left(\frac{\partial f}{\partial t}+\frac{1}{2}{\left(M\sigma \right)}^2\frac{{\partial }^{2}f}{\partial M^2}\right)dt+\frac{\partial f}{\partial M}dM.}$$

(8)

The expression $\frac{\partial f}{\partial M}dM$ denotes the change in value over time dt of the trading strategy that entails keeping an amount of the stock equal to $\frac{\partial f}{\partial M}$. If this trading strategy is used, the total value V of this portfolio satisfies the stochastic differential equation, and any cash held is expected to grow at the risk-free rate r

$${\displaystyle dV=r\left(V-\frac{\partial f}{\partial M}M\right)dt+\frac{\partial f}{\partial M}dM.}$$

(9)

If $V=f(t,M)$, this approach duplicates the option in the European style. We obtain the well-known Black-Scholes equation by combining these equations

$${\displaystyle \frac{\partial f}{\partial t}+\frac{{\sigma }^2{M}^2}{2}\frac{{\partial }^{2}f}{\partial M^2}+rM\frac{\partial f}{\partial M}-rf=0.}$$

(10)

The underlying asset’s average future volatility serves as the sole input for the Black-Scholes calculation. In the financial market, the parameter cannot be immediately monitored. The Black-Scholes model’s presumptions have been broadened and loosened in numerous ways. As a result, the pricing of derivatives and risk management today use a variety of methods (see [22,23,24,25,26,27,28,29,30,31]).

A geometric Brownian motion is frequently utilized to explain the stochastic dynamics of stock prices. A process M is said to follow a geometric Brownian motion if it meets the stochastic differential equation $dM=M(\sigma dB+\mu dt)$ for a Brownian motion B. Here, $\sigma$ represent the constant volatility and $\mu$ is the constant drift. The result of using Ito’s lemma with $f\left(M\right)=log\left(M\right)$ is

$${\displaystyle \begin{array}{ccc}\hfill dlog\left(M\right)& =& f^{\prime }\left(M\right)dM+\frac{1}{2}{f}^{\prime \prime }\left(M\right)M^2{\sigma }^{2}dt\hfill \\ \hfill & =& \frac{1}{M}\left(\sigma MdB+\mu Mdt\right)-\frac{1}{2}{\sigma }^{2}dt\hfill \\ \hfill & =& \sigma dB+\left(\mu -\frac{{\sigma }^2}{2}\right)dt.\hfill \end{array}}$$

(11)

The well-known log-normal probability distribution for stock price movements is produced by the Brownian motion’s geometric solution. The tails of the distribution, however, degrade more slowly than the log-normal distribution, according to actual data on the financial markets (see [32,33,34]). The possible origins of the anomaly may be perfect investor rationality contradict the efficient market hypothesis. Market participants in an efficient market evaluate stock values logically based on all current and foreseeable internal and external elements. The buying and selling of stocks on the actual financial market may be influenced by psychological and sociological variables. Substantial market anomalies like bubbles and deep recessions are also related with the psychological and social factors. Investors and portfolio managers are interested in understanding behavioral finance trends. To take account behavioral factors of real market and enhance the geometric Brownian motion assumption that underpins the Black-Scholes model, several models have been put forth (see [35,36,37,38,39]). It has been proposed that, the Black-Scholes model assumes volatility of an empirical financial market to be a constant, however this is incorrect (see [40,41,42,43,44,45,46,47,48,49]). Models like ARCH and GARCH are proposed by scholars to modify the constant variance assumptions (see [50,51,52,53,54]). Researchers utilized these models [55] to represent financial data sets that have time-varying volatility.

In mathematical finance, the fundamental theorem of asset pricing has been presented (see [56,57]). The fundamental theorem of asset pricing states that for a stochastic process the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. In the financial modeling with no-arbitrage pricing theory, there is a so called the efficient market hypothesis. The efficient market hypothesis states that for a highly liquid market, the prices of the financial assets are efficient representing all information that is currently accessible.

Investors in the stock market have relatively typical and self-controlling tendencies but are not entirely rational and self-aware. Decision-makings for invest usually rely on the investor’s temporary mental as well as physical condition. It is well-known that the stock prices usually adjust quickly to new information. The stock price behavior heavily relies on the investors’ confidence. An event announcement makes market’s quick reaction. Information security breach will affect the long-run confidence of investors [58,59]. From many different angles, behavioral finance can be examined. Recently, Bohmian quantum mechanics description of behavioral financial factors has been proposed [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. The Bohmian interpretation of quantum physics is frequently referred to as the hidden variables interpretation. The hidden sector of Bohmian mechanics should be recognized as behavioral sector in the financial market.

In this paper, we assume the Bohm-Vigier model as a natural generalization of the Black-Scholes model in stock market. The Bohm-Vigier model is described using the Fokker-Planck equation. We find the Boltzmann distribution is a stationary solution of the Fokker-Planck equation for the Bohm-Vigier model. The return transition of stock market, which corresponds with a time-dependent solution of the Fokker-Planck equation, is obtained.

The structure of the paper is as follows. In Section 2, we briefly review the Fokker-Planck formalism of the geometric Brownian motion and present the return transition distribution for the Black-Scholes model. In Section 3, the Bohm-Vigier stochastic model is introduced as a natural generalization of the Black-Scholes model in stock market. In Section 4, we set up the Fokker-Planck equation for the Bohm-Vigier stochastic model. The familiar Boltzmann distribution obtained as a stationary solution of the Fokker-Planck equation. We emphasis that the empirical return transition distribution of stock market corresponds with the time-dependent solutions of the Fokker-Planck equation. Conclusions are provided in Section 5.

2. Geometric Brownian Motion and Normal Return Transition Distribution

We use the Langevin equation to describe a Gaussian noise. A stochastic differential equation called the Langevin equation describes the temporal evolution of a subset of degrees of freedom. In contrast to the other system variables, these degrees of freedom often have collective variables that change gradually. The stochastic aspect of the Langevin equation is caused by the fast variables. The time-dependent stochastic differential equation of first order is what we refer to as the Langevin equation

$$\frac{dM\left(t\right)}{dt}=\mu M+\sigma Mn\left(t\right),$$

(12)

where the stochastic function $n\left(t\right)$ is referred to as the noise. A functional probability distribution, $\left[dP\right(n\left)\right]$ can be used to describe the noise. In relation to Markov’s processes, the Gaussian noise probability distribution has the following form:

$$\left[dP\left(n\right)\right]=\left[dn\right]e^{-\frac{1}{2}\int n^2\left(t\right)dt}.$$

(13)

The Gaussian noise correlations at 1 and 2 points are

$$\begin{array}{ccc}\hfill \langle n\left(t\right)\rangle & =& 0,\hfill \\ \hfill \langle n\left(t\right)n\left(t^{\prime }\right)\rangle & =& \delta (t-t^{\prime }),\hfill \end{array}$$

(14)

where $\delta (t-t^{\prime })$ denotes the Dirac’s delta function. Given the value of $M\left(t\right)$ at the beginning time $t_0$, $M\left(t_0\right)=M_0$.

We may rephrase (12) in terms of the log-return $r\left(t\right)=lnM\left(t\right)/M\left(0\right)$ by using Ito’s formula as

$$\frac{dr\left(t\right)}{dt}=\mu -\frac{{\sigma }^2}{2}+\sigma n\left(t\right).$$

(15)

It should be noted that the formula $y\left(t\right)=r\left(t\right)-\mu t$ measures the log-return in relation to growth rate $\mu$. The Langevin equation takes the form when viewed in terms of the relative log-return

$$\frac{dy\left(t\right)}{dt}=-\frac{{\sigma }^2}{2}+\sigma n\left(t\right).$$

(16)

For the stochastic variable $y\left(t\right)$, the Langevin equation produces a time-dependent probability distribution $P(y,t)$, which is formally denoted as

$$P(y,t)=\langle \delta \left[y\right(t)-y]\rangle .$$

(17)

Average over noise is indicated by the brackets. The argument of the probability distribution $P(y,t)$ is the variable y, which is unrelated to the function $y\left(t\right)$.

The probability distribution $P(y,t)$’s argument is variable y, which is unrelated to the function $y\left(t\right)$. Assuming $f\left(y\right)$ to be arbitrary, it can be obtained

$$\langle f\left(y\right)\rangle =\int P(y,t)f\left(y\right)dy.$$

(18)

Differentiating Equation (17) with respect to time and utilizing Equation (16), then we have

$$\frac{\partial P(y,t)}{\partial t}=\langle \left[-\frac{{\sigma }^2}{2}+\sigma n\left(t\right)\right]\frac{\partial }{\partial y\left(t\right)}\delta \left(y\left(t\right)-y\right)\rangle .$$

(19)

The differentiation ${\displaystyle \frac{\partial }{\partial y\left(t\right)}}$ can be replaced by ${\displaystyle -\frac{\partial }{\partial y}}$, which can be subtracted from the average, thanks to the symmetrical character of the $\delta$-function. Hence, by substituting y for $y\left(t\right)$, we may use the $\delta$-function and get

$$\frac{\partial P(y,t)}{\partial t}=\frac{\partial }{\partial y}\left[\frac{{\sigma }^2}{2}P(y,t)-\sigma \langle n\left(t\right)\delta \left(y\left(t\right)-y\right)\rangle \right].$$

(20)

Since we integrate over the entire y space, we discover that the equation, which is unaffected by the noise distribution, entails the possibility’s conservation:

$$\int dy\frac{dP(y,t)}{dt}=0.$$

(21)

We first offer a lemma to create an equation for the partial differential of the probability $P(y,t)$.

Lemma 1. 

The algebraic identity is satisfied by every arbitrary functional $F\left(n\right)$ of $n\left(t\right)$ 

$${\displaystyle \langle F\left(n\right)n\left(\tau \right)\rangle =\langle \frac{\Delta F\left(n\right)}{\Delta n\left(\tau \right)}\rangle .}$$

(22)

Here $\frac{\Delta F\left(n\right)}{\Delta n\left(\tau \right)}$ denotes the functional derivative of $F\left(n\right)$ with respect to $n\left(\tau \right)$. The Fokker-Planck equation’s theorem is now ready to be stated.

Theorem 1. 

The following equation is satisfied by the time-dependent probability distribution that corresponds to the Langevin Equation (16) 

$$\frac{\partial P(y,t)}{\partial t}=\frac{1}{2}\frac{\partial }{\partial y}\left[\sigma \frac{\partial P(y,t)}{\partial y}+{\sigma }^{2}P(y,t)\right].$$

(23)

This is the description of Fokker-Planck equation.

We are attempting to obtain a time-dependent solution to the Fokker-Planck Equation (23). To solve the Fokker-Planck equation, we formally rewrite (23) as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial P(y,t)}{\partial t}}& =& {\mathcal{L}}_{FP}P(y,t),\hfill \\ \hfill {\mathcal{L}}_{FP}& \equiv & {\displaystyle \frac{1}{2}\frac{\partial }{\partial y}\left[\sigma \frac{\partial }{\partial y}+{\sigma }^2\right].}\hfill \end{array}$$

(24)

The Fokker-Planck operator is represented by ${\mathcal{L}}_{FP}$.

In accordance with Mantegna and Stanley [33,60], we compute the probability density function of the index changes to analyze the financial markets

$$Z_{\Delta t}\equiv ln\left[M(t+\Delta t)/M\left(0\right)\right]-ln\left[M\left(t\right)/M\left(0\right)\right].$$

(25)

In this paper, we focus on the high frequency time series of stock market. We notice that

$$\begin{array}{ccc}\hfill Z\left(t\right)& =& lnM(t+\Delta t)-lnM\left(t\right)\hfill \\ \hfill & =& {\displaystyle \frac{ln\left(M\right(t+\Delta t)}{M\left(t\right))}}\hfill \\ \hfill & =& {\displaystyle ln\left(1+\frac{M(t+\Delta t)-M\left(t\right)}{M\left(t\right)}\right)}\hfill \\ \hfill & =& ln(1+r(t\left)\right).\hfill \end{array}$$

(26)

For high-frequency data, $M(t+\Delta t)-M\left(t\right)$ is small and the return $r\left(t\right)$ is less far than 1. Hence one has

$$Z\left(t\right)=ln(1+r(t\left)\right)\approx r\left(t\right).$$

(27)

Thus, in the high-frequency regime, where we focus on, the two definitions for return is equivalent approximately.

To examine the index price changes, we can construct a conditional probability density as the probability density of the stochastic process at time $t+\Delta t$ ($y(t+\Delta t)$) if the stochastic process at time t has the sharp value $y\left(t\right)$:

$${P(y(t+\Delta t),t+\Delta t|y\left(t\right),t)=\langle \delta (y-y(t+\Delta t))\rangle |}_{y=y\left(t\right)}.$$

(28)

As a result, the Fokker-Planck Equation (24) must also be obeyed for the transition probability $P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)$,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)}{\partial t}}& =& {\mathcal{L}}_{FP}P(y(t+\Delta t),t+\Delta t|y\left(t\right),t),\hfill \\ \hfill P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)& =& \delta \left(y\right(t+\Delta t)-y(t\left)\right).\hfill \end{array}$$

(29)

The equations’ formal solution can be read as

$$P(y(t+\Delta t),t+\Delta t|y\left(t\right),t)=e^{{\mathcal{L}}_{FP}\Delta t}\delta (y(t+\Delta t)-y\left(t\right)).$$

(30)

Using iteration [61], we are able to determine the index changes probability density distribution:

$$\begin{array}{ccc}\hfill P\left(Z_{\Delta t}\right)& =& {\displaystyle \int \frac{1}{\sqrt{2\pi \sigma \Delta t}}exp\left(-\frac{{[Z_{\Delta t}+\frac{{\sigma }^2}{2}\Delta t]}^2}{2\sigma \Delta t}\right)\frac{1}{N}exp\left(-\sigma y\right)dy}\hfill \\ \hfill & =& {\displaystyle \frac{1}{\sqrt{2\pi \sigma \Delta t}}exp\left(-\frac{{[Z_{\Delta t}+\frac{{\sigma }^2}{2}\Delta t]}^2}{2\sigma \Delta t}\right).}\hfill \end{array}$$

(31)

3. Bohm-Vigier Stochastic Model and Behavioral Factor in Stock market

Olga Choustova first proposed the Bohm-Vigier model to use real financial data [10]. Consider a stock market with a huge number of agents interacted with one another. The price at which financial assets are bought or sold depends on economic, political, social as well as meteorological conditions. We denote the price vector as

$$M=(M_1,M_2,\cdots ,M_n),$$

(32)

where $M_i$ is price of a share of the i-th corporation. The price dynamics are described by the trajectory $M\left(t\right)=(M_1\left(t\right),M_2\left(t\right),\cdots ,M_n\left(t\right))$. The log-return is defined as $r_i\left(t\right)=ln{M}_i\left(t\right)/M_i\left(0\right)$. Remember that $y_i\left(t\right)=r_i\left(t\right)-\mu t$ measures the log-return in relation to growth rate $\mu$. In line with Mantegna and Stanley [33] and better understand the financial markets, we need to analyze the return transition

$$Z_{i,\Delta t}\equiv ln\left[M_i(t+\Delta t)/M_i\left(0\right)\right]-ln\left[M_i\left(t\right)/M_i\left(0\right)\right].$$

(33)

We introduce m as the case shares of a stock, which can be analogue of mass. Then, the market capitalization of the i-th trader is equal to $T_i=m_i{M}_i$. The financial energy can be written as

$$H(M,t)=\frac{1}{2}\sum _{i=1}^n{m}_i{\left(y_i{M}_i\right)}^2+V(M_1,M_2,\cdots ,M_n).$$

(34)

The interactions between investors are presented by the potential financial energy V. Both external political and economic conditions should be considered. The Bohm-Vigier stochastic model assume that the relative log-return satisfies the Langevin equation,

$$\frac{dy\left(t\right)}{dt}=\left(-\frac{{\sigma }^2}{2}+\frac{1}{m}\frac{\partial {\rm Y}}{\partial M}\right)+\sigma n\left(t\right),$$

(35)

where ${\rm Y}$ is the phase factor of solution $\Psi =\Phi e^{i\frac{{\rm Y}}{\hslash }}$ for the Schrödinger equation

$$i\hslash \frac{\partial \Psi (M,t)}{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\Psi (M,t)}{\partial M^2}+V\left(M\right)\Psi (M,t).$$

(36)

In Bohmian quantum mechanics, ${\Phi }^2$ and ${\rm Y}$ satisfy the equations

$$\begin{array}{c}{\displaystyle \frac{\partial {\Phi }^2}{\partial t}+\frac{1}{m}\frac{\partial }{\partial M}\left({\Phi }^2\frac{\partial {\rm Y}}{\partial M}\right)=0,}\\ {\displaystyle \frac{\partial {\rm Y}}{\partial t}+\frac{1}{2m}{\left(\frac{\partial {\rm Y}}{\partial M}\right)}^2+\left(V-U\right)=0.}\end{array}$$

(37)

where ${\displaystyle U\equiv \frac{{\hslash }^2}{2m\Phi }\frac{{\partial }^2\Phi }{\partial M^2}}$ is the so called Bohm quantum potential.

4. Distribution of Return Transition for the Bohm-Vigier Model

Consider the Bohm-Vigier model (35) and assume a positive definite, radially unbounded, twice continuously differentiable function $\Xi \left(y\right)$ that causes the infinitesimal generator to be constant [62,63,64]

$${\mathcal{L}}_{BG}\Xi \left(y\right)=\frac{\partial \Xi }{\partial y}\left(-\frac{{\sigma }^2}{2}+\frac{1}{m}\frac{\partial {\rm Y}}{\partial M}\right)+\frac{1}{2}{\sigma }^2\frac{{\partial }^2\Xi }{{\partial }^{2}y},$$

(38)

is negative definite up to a constant. If so, (35)’s equilibrium of $y=0$ of is probabilistically globally asymptotically stable.

By introducing the virtual tracking error

$$z\left(t\right)=-\frac{{\sigma }^2}{2}+\frac{1}{m}\frac{\partial {\rm Y}}{\partial M}+\theta y\left(t\right),\theta >0,$$

(39)

we can reformulate the Bohm-Vigier model (35) as

$$\frac{dy\left(t\right)}{dt}=\left(-\theta y\left(t\right)+z\left(t\right)\right)+\sigma n\left(t\right).$$

(40)

We have derived the Itô’s lemma, which gives

$$\frac{dz\left(t\right)}{dt}=\left(-\theta y\left(t\right)+z\left(t\right)\right)\frac{\partial }{\partial y}\left(\frac{1}{m}\frac{\partial {\rm Y}}{\partial M}\right)+\frac{{\sigma }^2}{2}\frac{{\partial }^2}{\partial y^2}\left(\frac{1}{m}\frac{\partial {\rm Y}}{\partial M}\right)+\sigma \frac{\partial }{\partial y}\left(\frac{1}{m}\frac{\partial {\rm Y}}{\partial M}\right)n\left(t\right).$$

(41)

A convenient Lyapunov function is of the form

$$\Xi =\frac{1}{2}{y}^2\left(t\right)+\frac{1}{4}{z}^4\left(t\right).$$

(42)

Then, the bound of the infinitesimal generator is of the form [65],

$${\mathcal{L}}_{BG}\Xi \left(y\right)\le \left(-\theta +\frac{1}{2}\right)y^2\left(t\right)-\kappa z^4\left(t\right)+\frac{{\sigma }^2+2}{2}.$$

(43)

Thus, for the case of $\theta >1/2$ and $\kappa >0$ the equilibrium $y=0$ of the Bohm-Vigier model (35) is globally asymptotically stable in probability. The Bohm-Vigier model can be described as the Ornstein-Uhlenbeck process,

$$\frac{dy\left(t\right)}{dt}=-\theta y\left(t\right)+\sigma n\left(t\right).$$

(44)

We are ready to present the Fokker-Planck equation for the Bohm-Vigier stochastic mechanics.

Theorem 2. 

The following equation is true for time-dependent probability distribution corresponding to the Bohm-Vigier model (35) 

$$\frac{\partial P(y,t)}{\partial t}=\frac{\partial }{\partial y}\left[\frac{\sigma }{2}\frac{\partial P(y,t)}{\partial y}+\theta yP(y,t)\right].$$

(45)

The stationary Fokker-Planck equation is satisfied by the stationary solution, $P_0\left(y\right)$.

$$\frac{\partial }{\partial y}\left[\frac{\sigma }{2}\frac{\partial P_0\left(y\right)}{\partial y}+\theta y{P}_0\left(y\right)\right]=0.$$

(46)

Once this equation has been integrated, we can have

$$\frac{\sigma }{2}\frac{\partial P_0\left(y\right)}{\partial y}+\theta y{P}_0\left(y\right)=C,$$

(47)

Here C represent the constant of integration. It is compatible with the requirement that the probability current be constant for a stationary process. As the boundary is naturally limited to be $(y_{\min },y_{\max })=(-\infty ,\infty )$, the probability current should equal to zero. Therefore, the above equation can be rewritten as

$$\frac{\partial P_0\left(y\right)}{\partial y}+\frac{2\theta y}{\sigma }{P}_0\left(y\right)=0.$$

(48)

Consequently, we can solve the stationary form of the corresponding Fokker-Planck equation explicitly,

$${\displaystyle P_0\left(y\right)=\sqrt{\frac{\theta }{\sigma \pi }}exp\left(-\frac{\theta }{\sigma }{y}^2\right).}$$

(49)

This just is the familiar Boltzmann distribution.

A time-dependent solution for the Bohm-Vigier model can be found by the Fokker-Planck Equation (45). Here, the Fokker-Planck equation can be rewriten formally (45) as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial P(y,t)}{\partial t}}& =& {\mathcal{L}}_{FP-BG}P(y,t),\hfill \\ \hfill {\mathcal{L}}_{FP-BG}& \equiv & {\displaystyle \frac{\partial }{\partial y}\left[\frac{\sigma }{2}\frac{\partial }{\partial y}+\theta y\right].}\hfill \end{array}$$

(50)

Here ${\mathcal{L}}_{FP-BG}$ represent the Fokker-Planck operator for the Bohm-Vigier model.

In order to examine price changes, we may construct a conditional probability density as the density function of the stochastic process at time $t+\Delta t$ ($y(t+\Delta t)$) if the stochastic process at time t has the sharp value $y\left(t\right)$,

$${P(y(t+\Delta t),t+\Delta t|y\left(t\right),t)=\langle \delta (y-y(t+\Delta t))\rangle |}_{y=y\left(t\right)}.$$

(51)

Therefore, the distribution $P(y,t)$ for the unique initial condition $P(y,t_0)=\delta (y-y\left(t_0\right))$ equals to the transition probability $P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)$. Moreover, this distribution must adhere to the Fokker-Planck Equation (50), that is,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)}{\partial t}}& =& {\mathcal{L}}_{FP-BG}P(y(t+\Delta t),t+\Delta t|y\left(t\right),t),\hfill \\ \hfill P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)& =& \delta \left(y\right(t+\Delta t)-y(t\left)\right).\hfill \end{array}$$

(52)

The formal solution of the equation can be found here

$$P(y(t+\Delta t),t+\Delta t|y\left(t\right),t)=e^{{\mathcal{L}}_{FP-BG}\Delta t}\delta (y(t+\Delta t)-y\left(t\right)).$$

(53)

Using iteration [61], we can have

$$\begin{array}{c}\hfill {\displaystyle P(y(t+\Delta t),t+\Delta t|y\left(t\right),t)=\delta (y(t+\Delta t)-y\left(t\right))+{\int }_t^{t+\Delta t}{\mathcal{L}}_{FP-BG}(y,t_1)d{t}_1\delta (y(t+\Delta t)-M\left(t\right))}\\ \hfill {\displaystyle +{\int }_t^{t+\Delta t}d{t}_1{\int }_t^{t_1}d{t}_2{\mathcal{L}}_{FP}(y,t_1){\mathcal{L}}_{FP-BG}(y,t_2)\delta (y(t+\Delta t)-y\left(t\right))+\cdots }\\ \hfill {\displaystyle =\left[1+\sum _{n=1}^{\infty }{\int }_t^{t+\Delta t}d{t}_1{\int }_t^{t_1}d{t}_2\cdots {\int }_t^{t_{n-1}}d{t}_n{\mathcal{L}}_{FP-BG}(y,t_1)\cdots {\mathcal{L}}_{FP-BG}(y,t_n)\right]\delta (y(t+\Delta t)-y\left(t\right)).}\end{array}$$

(54)

By switching the time-dependent operators so that the ones with longer times stay to the left of those with shorter times, we establish the time-ordering operator T. We may rephrase the transition probability density by using the time-order operator T as

$$\begin{array}{c}P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)\hfill \\ {\displaystyle =T\left[1+\sum _{n=1}^{\infty }\frac{1}{n!}{\int }_t^{t+\Delta t}d{t}_1{\int }_t^{t+\Delta t}d{t}_2\cdots {\int }_t^{t+\Delta t}d{t}_n{\mathcal{L}}_{FP}(y,t_1)\cdots {\mathcal{L}}_{FP-BG}(y,t_n)\right]\delta (y(t+\Delta t)-y\left(t\right))}\hfill \\ =Texp\left[{\int }_t^{t+\Delta t}{\mathcal{L}}_{FP-BG}(y,t^{\prime })d{t}^{\prime }\right]\delta (y(t+\Delta t)-y\left(t\right)).\hfill \end{array}$$

(55)

For short time interval $\Delta t$, we may rephrase the solution as

$$P(y(t+\Delta t),t+\Delta t|y\left(t\right),t)=\left[1+{\mathcal{L}}_{FP-BG}(y,t)\Delta t+O\left({(\Delta t)}^2\right)\right]\delta (y(t+\Delta t)-y\left(t\right)).$$

(56)

Utilizing the $\delta$ function’s integration

$$\delta (y-y^{\prime })=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{e}^{iu(y-y^{\prime })}du,$$

(57)

It can be obtained

$$\begin{array}{c}P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)\hfill \\ {\displaystyle =\left[1+\frac{\sigma }{2}\frac{{\partial }^2}{\partial y{(t+\Delta t)}^2}+\frac{\partial }{\partial y(t+\Delta t)}\left(\theta y\left(t\right)\right)\right]\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{e}^{iu\left(y\right(t+\Delta t)-y(t\left)\right)}du.}\hfill \end{array}$$

(58)

Following [66,67], we replace in the drift and diffusion coefficients $y(t+\Delta t)$ by $y\left(t\right)$. Then, we can get

$$\begin{array}{c}P\left(y\right(t+\Delta t),t+\Delta t|y\left(t\right),t)\hfill \\ {\displaystyle =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }exp\left[-u^2\frac{1}{2}\sigma \Delta t+iu\theta y\Delta t+iu(y(t+\Delta t)-y\left(t\right))\right]du}\hfill \\ {\displaystyle =\frac{1}{\sqrt{2\pi \sigma \Delta t}}exp\left(-\frac{{[Z_{\Delta t}+\theta y\Delta t]}^2}{2\sigma \Delta t}\right).}\hfill \end{array}$$

(59)

Finally, we have obtained the index price changes’ probability density distribution in the following form:

$$\begin{array}{ccc}\hfill P\left(Z_{\Delta t}\right)& =& {\displaystyle \int \frac{1}{\sqrt{2\pi \sigma \Delta t}}exp\left(-\frac{{[Z_{\Delta t}+\theta y\Delta t]}^2}{2\sigma \Delta t}\right)\sqrt{\frac{\theta }{\sigma \pi }}exp\left(-\frac{\theta }{\sigma }{y}^2\right)dy}\hfill \\ \hfill & =& {\displaystyle \frac{1}{\sqrt{(2+\theta \Delta t)\sigma \pi \Delta t}}exp\left(-\frac{\theta \Delta t+1}{(\theta \Delta t+2)\sigma \Delta t}{Z}_{\Delta t}^2\right).}\hfill \end{array}$$

(60)

5. Conclusions and Remarks

The Bohmian mechanics, in which a system is partially described by its changing wave function that follows Schrödinger’s equation, is frequently referred to as a hidden variables explanation of quantum mechanics. According to Bohm, the wave function only partly describes the system. The precise locations of each particle are specified, completing this description. The Bohmian mechanics presents the possibility to describe on quantum level a particle’s trajectory $q\left(t\right)$, which is not permitted by standard quantum mechanics. However, the Bohmmian mechanics is nonlocal. One particle would affect others with large distances via the pilot wave field. It has been proposed that these Bohmian mechanics drawbacks will turn into benefits when they are applied to the financial market.

According to the theory of efficient markets, stock prices are efficiently valued at any given time in a highly liquid market to reflect all the information that is currently available, and the Black-Scholes model can accurately capture the dynamics of a financial market that contains derivatives.The cost of European-style options can be estimated using the Black-Scholes model. Given the security’s risk and predicted return, the option has a specific price, which is typically assumed to move in a geometric Brownian motion. The underlying asset’s average future volatility serves as the sole input for the Black-Scholes calculation, yet this input variable could hardly be observed directly in the financial market. Many efforts have been made to relax the Black-Scholes model’s assumptions, especially in the field of derivative pricing and risk management.

The geometric Brownian motion solution yields a well-known log-normal distribution for financial asset price movements. Yet, empirical evidence indicates that the distribution’s tails drop more slowly than the log-normal distribution. The anomaly’s possible causes include investor irrationality, which contradicts the efficient market concept. In an efficient market, market participants rationally assess stock values based on all current and future internal and external factors. In the real financial market, stock purchases and sales are affected by psychological and sociological factors. Substantial market anomalies like bubbles and deep recessions are also related with the psychological and social factors. Investors and portfolio managers are interested in understanding behavioral finance trends. Scholars have developed various approaches to modify the assumption of geometric Brownian motion that underpins Black-Scholes model in order to account for real-market behavioral characteristics. It has been proposed that rather than a constant an actual financial asset’s volatility should be a stochastic variable like in ARCH and GARCH models. These models are often used to represent financial time series with time-varying volatility.

Rather than fully rational and self-controlling, stock market investors are generally influenced by psychological factors. Decision-makings for invest usually rely on the investor’s temporary mental as well as physical condition. Behavioral finance may be studied from a variety of angles. Recently, Bohmian quantum mechanics description of behavioral financial factors has been proposed. Bohmian mechanics is frequently referred to as a hidden variables interpretation of quantum physics. The hidden sector of Bohmian mechanics should be recognized as behavioral sector in the financial market.

In this paper, we proposed the Bohm-Vigier stochastic model as a natural generalization of the Black-Scholes model in stock market. A Fokker-Planck equation description for the Bohm-Vigier model was presented. For the Bohm-Vigier model, we discovered that the Boltzmann distribution is a stationary solution of the Fokker-Planck equation. This is in agreement with the empirical stock market. The return transition distribution of stock market, which corresponds with a time-dependent solution of the Fokker-Planck equation, was obtained. Of course, the results obtained in the paper should be tested by data fitting for empirical financial market [68]. In fact, the main result (Equation (58)) has been tested initially using the S&P 500 index [69]. In the forthcoming paper, we should investigate high frequency time series for empirical financial market within the framework of the paper. We wish the return transition distribution describe the behavioral factor of stock market.

This paper just presented some preliminary attempts on using Bohmian quantum mechanics to describe behavioral factors in the stock market. The Bohmian interpretation of quantum physics is frequently referred to as the hidden variables interpretation. The relations of hidden sector of Bohmian mechanics with behavioral sector in the financial market is worthing extensive studies in the future.