By 苏剑林 | June 09, 2016
The path integral method provides a new perspective for solving certain stochastic problems.
An Example: Stock Price Model
Consider a risky asset (such as a stock) with a price $S_t$ at time $t$. We consider the time interval $[0,T]$, where 0 is the initial time and $T$ is the maturity date. $S_t$ is treated as a continuous-time variable that follows the stochastic differential equation:
\begin{equation}
dS_t^0=rS_t^0 dt; \quad dS_t=S_t(\mu dt+\sigma dW_t). \tag{70}
\end{equation}
Where $\mu$ and $\sigma$ are two constants, and $W_t$ is a standard Brownian motion.
The equation for $S_t$ is a stochastic differential equation (SDE), which is generally solved using stochastic calculus. Stochastic calculus differs from ordinary calculus in that, when performing a first-order expansion, the $dS_t^2$ term cannot be ignored because $dW_t^2=dt$. For example, let $S_t=e^{x_t}$, then $x_t=\ln S_t$:
\begin{align}
dx_t &= \ln(S_t+dS_t)-\ln S_t = \frac{dS_t}{S_t} - \frac{dS_t^2}{2S_t^2} \nonumber \\
&= \frac{S_t(\mu dt+\sigma dW_t)}{S_t} - \frac{[S_t(\mu dt+\sigma dW_t)]^2}{2S_t^2} \nonumber \\
&= \mu dt+\sigma dW_t - \frac{1}{2}\sigma^2 dW_t^2 \quad (\text{all other terms are of lower order than } dt) \nonumber \\
&= \left(\mu-\frac{1}{2}\sigma^2\right) dt + \sigma dW_t, \tag{71}
\end{align}
This transforms it into the form of equation $(48)$. According to previous research, this is equivalent to an asymmetric random walk model, which we can simulate numerically; or we can write the equivalent partial differential equation according to $(12)$:
\begin{equation}
\frac{\partial P}{\partial t} = \frac{\sigma^2}{2}\frac{\partial^2 P}{\partial x^2} + \left(\mu-\frac{1}{2}\sigma^2\right)\frac{\partial P}{\partial x}, \tag{72}
\end{equation}
Or equivalently as a path integral:
\begin{equation}
\int_{x_a}^{x_b} \exp\left\{ -\frac{1}{2\sigma} \int_{t_a}^{t_b} \left[ \dot{x} - \left(\mu-\frac{1}{2}\sigma^2\right) \right]^2 dt \right\} \mathscr{D}x(t). \tag{73}
\end{equation}
Since the path integral for this problem is quadratic, it can be solved exactly. The result is:
\begin{align}
& \exp\left[ -\frac{1}{2\sigma}\frac{(x_b-x_a)^2}{t_b-t_a} - \frac{1}{2\sigma}\left(\mu-\frac{1}{2}\sigma^2\right)^2(t_b-t_a) + \frac{1}{\sigma}\left(\mu-\frac{1}{2}\sigma^2\right)(x_b-x_a) \right] \nonumber \\
= & \exp\left\{ -\frac{t_b-t_a}{2\sigma} \left[ \frac{x_b-x_a}{t_b-t_a} - \left(\mu - \frac{1}{2}\sigma^2 \right) \right]^2 \right\}, \tag{74}
\end{align}
As can be seen, it only depends on the relative values $T=t_b-t_a$ and $\Delta x = x_b-x_a$:
\begin{equation}
\exp\left\{ -\frac{T}{2\sigma} \left[ \frac{\Delta x}{T} - \left(\mu - \frac{1}{2}\sigma^2 \right) \right]^2 \right\}, \tag{75}
\end{equation}
As previously emphasized, this result is up to a normalization factor. By normalizing, the complete result is obtained as:
\begin{equation}
P(\Delta x) = \frac{1}{\sqrt{2\pi \sigma T}} \exp\left\{ -\frac{T}{2\sigma} \left[ \frac{\Delta x}{T} - \left(\mu - \frac{1}{2}\sigma^2 \right) \right]^2 \right\}. \tag{76}
\end{equation}
Note that this is the distribution of $x_t$. We want to analyze the distribution of $S_t$. By substituting variables $\Delta x = \ln S_b - \ln S_a = \ln(S_b/S_a)$, we get:
\begin{equation}
P(S_b) = \frac{1}{S_b\sqrt{2\pi \sigma T}} \exp\left\{ -\frac{T}{2\sigma} \left[ \frac{\ln (S_b / S_a)}{T} - \left(\mu - \frac{1}{2}\sigma^2 \right) \right]^2 \right\}. \tag{77}
\end{equation}
This is a log-normal distribution. It tells us that if the current value of the stock is $S_a$, then after time $T$, the probability that the value is $S_b$ is $P(S_b)$.
Many financial problems can be described by stochastic differential equations, and stochastic differential equations can be transformed into corresponding partial differential equations or path integrals. Path integrals originated in quantum mechanics; therefore, in recent years, the combination of both has given rise to an emerging field called Quantum Finance, or Quantum Economics. In reality, it involves taking conclusions from the field of quantum mechanics and applying them to the field of finance using the path integral method. From a mathematical perspective, it does not offer much that is substantially new; however, from a practical perspective, it saves research time and costs and is highly significant. A book regarding this subject is "Quantum Finance" [10].
Summary
Through some explanations and examples, this article demonstrates the application of the path integral method to a large class of stochastic problems.
However, the discussion in this article is not exhaustive. Regarding the application of the path integral method in this area, there are still many directions worth studying:
1. For higher-order non-linear stochastic field differential equations, find the corresponding path integrals and find models similar to asymmetric random walks for them;
2. Study the path integrals corresponding to stochastic partial differential equations (SPDEs);
3. Search for models similar to asymmetric random walks for stochastic partial differential equations.
The author believes that these works would be quite meaningful.