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. The time interval considered is [0, T], where 0 represents the initial time and T represents the maturity date. S_t is treated as a continuous-time variable changing over time and follows the following stochastic differential equations: dS_t^0 = rS_t^0 dt; \quad dS_t = S_t(\mu dt + \sigma dW_t). \tag{70} 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). The general approach to solving it is through 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{aligned} dx_t &= \ln(S_t + dS_t) - \ln S_t = \frac{dS_t}{S_t} - \frac{dS_t^2}{2S_t^2} \\ &= \frac{S_t(\mu dt + \sigma dW_t)}{S_t} - \frac{[S_t(\mu dt + \sigma dW_t)]^2}{2S_t^2} \\ &= \mu dt + \sigma dW_t - \frac{1}{2}\sigma^2 dW_t^2 \quad (\text{other terms are of order lower than } dt) \\ &= \left(\mu - \frac{1}{2}\sigma^2\right) dt + \sigma dW_t \end{aligned}, \tag{71} This transforms it into the form of (48). According to previous research, it is equivalent to an asymmetric random walk model, which allows for numerical simulation; or, according to (12), we can write the equivalent partial differential equation: \sigma \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} or equivalently, the path integral: \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}
The path integral for this problem is quadratic and can therefore be solved exactly. The result is: \begin{aligned} &\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) \right. \\ &\qquad\qquad\qquad \left. + \frac{1}{\sigma} \left(\mu - \frac{1}{2}\sigma^2\right) (x_b - x_a) \right] \\ &= \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\} \end{aligned}, \tag{74} It can be seen that it only depends on the relative values T = t_b - t_a and \Delta x = x_b - x_a: \exp \left\{ -\frac{T}{2\sigma} \left[ \frac{\Delta x}{T} - \left(\mu - \frac{1}{2}\sigma^2\right) \right]^2 \right\}, \tag{75} As previously emphasized, this result is up to a normalization factor. After normalization, the complete result is: 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} Note that this is the distribution of x_t. We want to analyze the distribution of S_t. By the change of variables \Delta x = \ln S_b - \ln S_a = \ln(S_b/S_a), we obtain: 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} 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 of the value being S_b is P(S_b).
Many financial problems can be described by stochastic differential equations, which can be converted into corresponding partial differential equations or path integrals. Path integrals originated in quantum mechanics; thus, the combination of the two has recently given rise to an emerging field—Quantum Finance, or Quantum Economics. In reality, it involves applying conclusions from the field of quantum mechanics to finance using path integral methods. From a mathematical perspective, there is no substantially new content; however, from a practical perspective, it saves research time and costs, which is very meaningful. A work in this area is Quantum Finance [10].
Review and Summary
Through several explanations and examples, this article demonstrates the application of path integral methods in a large class of stochastic problems.
However, the discussion in this article is not exhaustive. Regarding the application of path integral methods in this area, there are still many directions worth researching:
1. For high-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. Find models similar to asymmetric random walks for stochastic partial differential equations.
I believe that all the above-mentioned work will be quite meaningful.
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