Q: $X_1,\cdots,X_N$ are recursively defined random variables, $$ X_n = \underset{\color{blue}{\text{Taylor series like part}}}{\underbrace{X_{n-1} + f(X_{n-1})\delta+ (X_n-X_{n-1})f'(X_{n-1})\delta}}+\underset{\color{red}{\text{Noise}}}{\underbrace{\;\;W_n\;\;}} $$ where $X_0=0$ is a constant, $\delta$ and $\beta$ are also constants and $W_1,\cdots,W_N$ are iid $N(0,\beta\delta)$. Show that the joint density of $(X_1,\cdots, X_N)$ is given by, $$ C\exp\left(\sum_{n=1}^NV_n\right) $$ where $C$ is a normalizing constant and $V_n$ are defined as, $$ V_n = \frac{\left(x_n-x_{n-1} -\delta f(x_{n-1}) -(x_n-x_{n-1})\delta f'(x_{n-1})\right)^2}{2\beta\delta} $$ with $x_0 = 0$.
Attempt: $$ X_1 = \frac{f(0)\delta + W_1}{1-\delta f'(0)} $$ is a normal random variable with mean $\frac{f(0)\delta}{1-\delta f'(0)}$ and variance $\frac{\beta\delta}{(1-\delta f'(0))^2}$ and therefore, $$ V_1 = \frac{\left(x_1 -\frac{\delta f(0)}{1-\delta f'(0)}\right)^2}{\frac{2\beta\delta}{(1-\delta f'(0))^2}} = \frac{\left(x_1-x_{0} -\delta f(x_{0}) -(x_1-x_{0})\delta f'(x_{0})\right)^2}{2\beta\delta} $$ The numerator of $V_n$ mimics the Taylor series like part. But I don't know how to proceed to the density of $(X_1, X_2)$ or use induction.
Source: Implicit sampling for particle filters - Chorin and Tu (page - $1$, section: sampling by interpolation and iteration)
Let's consider the case when $X_1=x_1$. Let $\varepsilon$ be a small positive number and $f_X$ denote probability density of $X$.
\begin{align} X_2 &= \frac{x_1+f(x_1)\delta+x_1f'(x_1)\delta + W_2}{1-\delta f'(x_1)}\\ \implies& p(X_2\in B(x_2, \varepsilon)|X_1=x_1) = p(W_2\in B(x_2-x_1-f(x_1)\delta-(x_2-x_1)f'(x_1)\delta, \varepsilon) )\\ \implies& \int_{x_2-\varepsilon}^{x_2+\varepsilon}f_{X_2|X_1}(\tau|x_1)\,d\tau=\int_{x_2-\varepsilon}^{x_2+\varepsilon}C'\exp\left(-\frac{(\tau-x_1-f(x_1)\delta-(\tau-x_1)f'(x_1)\delta )^2}{2\beta\delta}\right)\,d\tau\\ \implies& f_{X_2|X_1}(x_2|x_1)\times2\varepsilon+O(\varepsilon^2) = C'\exp\left(-\frac{(x_2-x_1-f(x_1)\delta-(x_2-x_1)f'(x_1)\delta )^2}{2\beta\delta}\right)\times2\varepsilon+O(\varepsilon^2)\\ \implies& f_{X_2|X_1}(x_2|x_1) = C'\exp\left(-V_2\right)\\ \implies& f_{X_2,X_1}(x_2,x_1) = f_{X_2|X_1}(x_2|x_1)f_{X_1}(x_1)=C''\exp(-(V_1+V_2)) \end{align} And similarly we can progress to $f_{X_n,\cdots,X_1}(x_n, \cdots, x_1)$.