By 苏剑林 | June 09, 2016
This chapter applies path integrals to stochastic differential equations (SDEs) and obtains the same results as for asymmetric random walks, thereby proving the equivalence with that model. The idea of using path integrals for the study of stochastic differential equations has a long history. Feynman, in his work [5], already established the relationship between path integrals and linear stochastic differential equations. For non-linear cases, there have been numerous studies, but they are somewhat confusing; for example, reference [8] even provided incorrect results. Starting from the discrete concept of path integrals, this article clearly establishes the Jacobian determinant relationship between two path integral differentials, thereby establishing path integrals for non-linear stochastic differential equations as well. The results of this article are consistent with those in reference [9].
The study in this article is limited to stochastic ordinary differential equations. Their difference from general ordinary differential equations lies in the introduction of a Brownian motion term. A common class of stochastic differential equations is:
$$dx(t)=p(x(t),t)dt + \sqrt{\alpha} dW_t.\tag{48}$$Where $W_t$ represents a standard Brownian motion. Due to the introduction of the stochastic term, the solution $x(t)$ is no longer deterministic but has a certain probability distribution. In the field of stochastic differential equations, there are many quantities of interest, such as the expectation, variance, or stability of a certain quantity related to $x$. There are various analytical techniques in the field of SDEs, but obviously, directly finding the probability distribution of $x(t)$ and then studying that distribution is the most ideal and easiest approach. Path integrals provide exactly a method for finding the probability distribution.
We start with the following linear stochastic differential equation:
$$dx(t)=x(t)dt + \sqrt{\alpha} dW_t,\tag{49}$$We wish to solve this equation. Since our focus is on the probability distribution of $x$, the goal of the solution is to find the probability of starting from $x_a$ at time $t_a$ and arriving at $x_b$ at time $t_b$, which is the Propagator (Green's function). To construct the path integral, we need to find the probability functional $P[x(t)]$ of reaching $(t_b, x_b)$ from $(t_a, x_a)$ via the path $x(t)$. We can start from the probability functional of Brownian motion. Since $x(t)$ can be uniquely calculated given $W(t)$, the probability of a given $x(t)$ is the same as that of a given $W(t)$, i.e.,
$$P[x(t)]\mathscr{D}x(t)= P[W(t)]\mathscr{D}W(t).\tag{50}$$We already know the probability density functional of $W(t)$ in Brownian motion is:
$$P[W(t)]=\exp\left(-\int_{t_a}^{t_b} \frac{1}{2}\dot{W}^2(t)dt\right).\tag{51}$$Note that we still write this in derivative form; this is only formal (the derivative terms in the definition of path integrals are inherently formal), serving as a notation that is easy to understand and remember. Now we have:
$$\begin{aligned} P [x(t)]\mathscr{D}x(t)=& P [W(t)]\mathscr{D}W(t)\\ =&\exp\left(-\int_{t_a}^{t_b} \frac{1}{2}\dot{W}^2(t)dt\right)\mathscr{D}W(t)\\ =&\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right)\mathscr{D}W(t)\end{aligned},\tag{52}$$Up to this point, what we have done is easy to understand, being merely a change of variables. Now what is missing is the final step: the relationship between $\mathscr{D}x(t)$ and $\mathscr{D}W(t)$. Since $\mathscr{D}x(t)$ and $\mathscr{D}W(t)$ are defined as the limit of $n-1$ fold integrals, and the change of variables in finite-dimensional integrals differs by a Jacobian determinant, the relationship between these two should also differ by a Jacobian determinant:
$$\mathscr{D}W(t)=\mathcal{J}[x(t)]\mathscr{D}x(t),\tag{53}$$Due to the infinite nature of $\mathscr{D}$, the Jacobian determinant here is the determinant of an $\infty \times \infty$ dimensional square matrix. Note that the relationship between $W(t)$ and $x(t)$ is linear. We already know that the Jacobian determinant of a linear transformation is a constant, and this case is no exception. Therefore, we can ignore this constant for now and determine it through normalization after the result is obtained. In the case where they differ only by a constant factor, we have:
$$\begin{aligned} P [x(t)]\mathscr{D}x(t)=&\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right)\mathscr{D}W(t)\\ =&\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right)\mathscr{D}x(t)\end{aligned},\tag{54}$$Which is to say:
$$P[x(t)]=\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right).\tag{55}$$With this result, we can use path integrals to calculate the propagator:
$$\begin{aligned}\int P [x(t)]\mathscr{D}x(t)=&\int\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}^2(t)+x^2(t)-2\dot{x}(t) x(t)\right] dt\right)\mathscr{D}x(t)\\ =&\exp\left(\frac{1}{2\alpha}\left(x_b^2-x_a^2\right)\right)\times\\ &\qquad\int\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}^2(t)+x^2(t)\right] dt\right)\mathscr{D}x(t)\end{aligned}.\tag{56}$$The path integral part:
$$K(x_b,t_b;x_a,t_a)=\int\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}^2(t)+x^2(t)\right] dt\right)\mathscr{D}x(t)\tag{57}$$is a basic Gaussian-type path integral, which we can solve exactly.
Now we derive the path integral for the general form of a stochastic differential equation:
$$dx(t)=p(x(t), t)dt + \sqrt{\alpha} dW_t ,\tag{58}$$As in the previous section, it is easy to obtain:
$$\begin{aligned} P [x(t)]\mathscr{D}x(t)=& P [W(t)]\mathscr{D}W(t)\\ =&\exp\left(-\int_{t_a}^{t_b} \frac{1}{2}\dot{W}^2 dt\right)\mathscr{D}W(t)\\ =&\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}-p(x,t)\right]^2 dt\right)\mathscr{D}W(t)\end{aligned}.\tag{59}$$The key is the relationship between $\mathscr{D}x(t)$ and $\mathscr{D}W(t)$. In the linear case, we know it is a constant and can be ignored until the final normalization; however, in the non-linear case, it cannot be bypassed, and we must find a way to calculate its determinant. Discretize the equation as:
$$(x_k - x_{k-1}) - \frac{1}{2}\epsilon\left[p(x_k,t_k)+p(x_{k-1},t_{k-1})\right]=\sqrt{\alpha} (W_k-W_{k-1}),\tag{60}$$where $k=0,1,2,\dots,n$, while $W_0\equiv W(t_a), W_n\equiv W(t_b), x_0\equiv x(t_a), x_n\equiv x(t_b)$ are pre-given initial and boundary conditions, $\epsilon=\frac{t_b-t_a}{n}$ is the time interval, and $x_k\equiv x\left(t_a + k\epsilon\right)$. Note that we have taken the average for $p(x,t)$. This is necessary; failing to take the average leads to incorrect results. Because $\frac{x(t)-x(t-\epsilon)}{\epsilon}$ as an approximation of $\dot{x}(t)$ only has zeroth-order accuracy, while in fact it is closest to $\dot{x}(t-\epsilon/2)$, where it has first-order $\epsilon$ accuracy. In other words, when we write $(x_k-x_{k-1})/\epsilon$, it actually represents the derivative at the midpoint between $x_k$ and $x_{k-1}$. Therefore, the discretization of $p(x,t)$ must also take the midpoint part; one can use either $\frac{1}{2}\left[p(x_k,t_k)+p(x_{k-1},t_{k-1})\right]$ or $p\left(\frac{x_k+x_{k-1}}{2},\frac{t_k+t_{k-1}}{2}\right)$, both of which have the same accuracy and yield the same result (in integral approximation, we can use the trapezoid or midpoint rectangle to approximate the area of a curved trapezoid, which has a similar meaning). We write:
$$\left\{\begin{aligned}&\mathscr{D}x(t)\approx dx_1 dx_2\dots dx_{n-1}\\ &\mathscr{D}W(t)\approx dW_1 dW_2\dots dW_{n-1}\end{aligned}\right. .\tag{61}$$Then we use finite-dimensional calculus to find the (approximate) Jacobian determinant. Note that the Jacobian matrix from $W_k$ to $x_k$ is a triangular matrix, so the determinant is the product of the diagonal elements, and the diagonal elements are:
$$\frac{\partial W_k}{\partial x_k}=\frac{1}{\sqrt{\alpha}}\left(1-\frac{1}{2}\epsilon\frac{\partial p(x_k,t_k)}{\partial x_k}\right)\approx \frac{1}{\sqrt{\alpha}}\exp\left(-\frac{1}{2}\epsilon\frac{\partial p(x_k,t_k)}{\partial x_k}\right),\tag{62}$$Hence, up to a constant, we have:
$$\mathcal{J}[x(t)]\approx \prod_{k=1}^{n-1} \left(1-\frac{1}{2}\epsilon\frac{\partial p(x_k,t_k)}{\partial x_k}\right)\approx \exp\left[-\frac{1}{2}\left(\sum_{k=1}^{n-1}\frac{\partial p(x_k,t_k)}{\partial x_k}\right)\epsilon\right],\tag{63}$$Taking the limit $n\to\infty$, we have:
$$\mathcal{J}[x(t)]=\exp\left(-\frac{1}{2}\int_{t_a}^{t_b}\frac{\partial p(x,t)}{\partial x}dt\right),\tag{64}$$Substituting back, we get:
$$\begin{aligned} P [x(t)]\mathscr{D}x(t)=&\exp\left\{-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}-p(x,t)\right]^2 dt\right\}\\ &\qquad\qquad\exp\left(-\frac{1}{2}\int_{t_a}^{t_b}\frac{\partial p(x,t)}{\partial x}dt\right)\mathscr{D}x(t)\\ =&\exp\left\{-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\left(\dot{x}-p(x,t)\right)^2 + \alpha\frac{\partial p(x,t)}{\partial x} \right]dt\right\}\mathscr{D}x(t) \end{aligned},\tag{65}$$Thus,
$$\begin{aligned} P [x(t)]=& \exp\left\{-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\left(\dot{x}-p(x,t)\right)^2 + \alpha\frac{\partial p(x,t)}{\partial x} \right]dt\right\}\\ =&\exp\left\{-\frac{1}{\alpha}\int_{t_a}^{t_b}\left[\frac{1}{2}\dot{x}^2-\dot{x}p(x,t)+\frac{1}{2}p^2(x,t)+\frac{1}{2}\alpha\frac{\partial p(x,t)}{\partial x}\right]dt\right\} \end{aligned}.\tag{66}$$Note that:
$$\begin{aligned}\frac{d}{dt}\int p(x,t)dx =& \dot{x}\frac{\partial}{\partial x}\int p(x,t)dx+\frac{\partial}{\partial t}\int p(x,t)dx\\ =&\dot{x}p(x,t)+\int \frac{\partial p(x,t)}{\partial t}dx \end{aligned},\tag{67}$$Using this formula, the above expression can be rewritten as:
$$P [x(t)]=\exp\left[-\frac{1}{\alpha}\int_{t_a}^{t_b}\left(\frac{1}{2}\dot{x}^2-V(x,t)\right)dt\right]\exp\left(\frac{1}{\alpha}\int_{x_a}^{x_b}p(x,t)dx\right).\tag{68}$$Where
$$V(x,t)=-\frac{1}{2}\left(\alpha\frac{\partial p}{\partial x}+p^2\right)-\int \frac{\partial p}{\partial t}dx.\tag{69}$$Comparing these with the results of equations $(16)\sim(19)$, we can find that the stochastic differential equation $(48)$ yields identical results to the asymmetric random walk model.
Now that we have proven the equivalence between stochastic differential equations and the asymmetric random walk model, and in the first chapter we analyzed the partial differential equations satisfied by the asymmetric random walk model, it means that we can immediately write down the partial differential equation corresponding to the stochastic differential equation. Thus, we have achieved the mutual transformation of the three, with the path integral method acting as the connecting link. This also demonstrates the power of the path integral method.