The classical random walk can be described as the evolution of the position $X_t$ of a walker for integers $t \geqslant 0$, where $X_0 = 0$ and $X_t = X_{t-1} + V_t$ for $t \geqslant 1$, where the "speed" $V_t$ at each time step is uniformly random $V_t \in_{\mathrm R} \{-1,+1\}$ and independent at each time step. It is well-known that this process yields a position which obeys a symmetric binomial distribution with mean $0$ and variance $t$, and that $t^{-1/2} X_t$ tends to a Gaussian distribution with variance $1$.

I am interested in a variant in which the speed itself increases or decreases by random "boosts", behaving like a classical random walk, and where the position is governed by the speed as it evolves over time. That is, we have $$\begin{aligned} X_0 &= 0 & V_0 &= 0 \\ X_t &= X_{t-1} + V_t & V_t &= V_{t-1} + A_t & A_t \in_{\mathrm R} \{-1,+1\}. \end{aligned}$$ Question. What is the probability distribution of $X_t$ as a function of $t$?

Observation. This process is in effect a discrete-time variant of a second-order stochastic differential equation of a form $$\begin{aligned} \frac{\mathrm dx}{\mathrm dt} &= v, & \frac{\mathrm dv}{\mathrm dt} = \xi_t \end{aligned}$$ where $W_t = \int_{0}^t \mathrm d\tau \, \xi_\tau\,$; though I do not know enough to be able to distil what probability density function one would expect for $x(T)$ from the references I found. Failing an answer for the discrete-time case, I will accept an answer for the probability density function of $x(T)$.

The classical random walk can be described as the evolution of the position $X_t$ of a walker for integers $t \geqslant 0$, where $X_0 = 0$ and $X_t = X_{t-1} + V_t$ for $t \geqslant 1$, where the "speed" $V_t$ at each time step is uniformly random $V_t \in_{\mathrm R} \{-1,+1\}$ and independent at each time step. It is well-known that this process yields a position which obeys a symmetric binomial distribution with mean $0$ and variance $t$, and that $t^{-1/2} X_t$ tends to a Gaussian distribution with variance $1$.

I am interested in a variant in which the speed itself increases or decreases by random "boosts", behaving like a classical random walk, and where the position is governed by the speed as it evolves over time. That is, we have $$\begin{aligned} X_0 &= 0 & V_0 &= 0 \\ X_t &= X_{t-1} + V_t & V_t &= V_{t-1} + A_t & A_t \in_{\mathrm R} \{-1,+1\}. \end{aligned}$$ Question. What is the probability distribution of $X_t$ as a function of $t$?

Remark #1. This process is in effect a discrete-time variant of a second-order stochastic differential equation of a form $$\begin{aligned} \frac{\mathrm dx}{\mathrm dt} &= v, & \frac{\mathrm dv}{\mathrm dt} = \xi_t \end{aligned}$$ where $W_t = \int_{0}^t \mathrm d\tau \, \xi_\tau\,$; though I do not know enough to be able to distil what probability density function one would expect for $x(T)$ from the references I found.

Remark #2. I originally requested an answer regarding the density function of the continuous version $x(t)$ as a fall-back. I have accepted an answer on that question for now; and that answer implicitly provides an answer for the discrete-time version in which the steps are normally distributed instead of being of fixed size. However, I will preferentially accept and reward any answer on the discrete-time version of the problem.