# Do All Stochastic Processes have underlying Probability Distributions?

I am trying to understand how a Stochastic Process can be created based on real world data. Specifically, do all Stochastic Processes have underlying Probability Distributions? If this is the case, I think it might be possible to create a valid mathematical likelihood function based on this underlying Probability Distribution Function - and then estimate the required parameters using some estimation technique (e.g. Maximum Likelihood Estimation).

Now, I will add some more context to my question.

In a previous question (Simulating a Function (that is naturally contained) Within an Interval $a,b$), I tried to define a Stochastic Process that "naturally" only exists between points $$(a,b)$$. This process was based on a modified version of the Ornstein-Uhlenbeck process (whereas the Ornstein-Uhlenbeck process itself is based on the Weiner Process).

Part 1: Definitions

1. Wiener Process: As I understand, the Wiener Process $$W_t$$ can be thought of as successive differences between a Brownian Motion. Here are some standard properties about to Brownian Motion:

• $$W_0 = 0$$ almost surely.
• $$W$$ has independent increments: for every $$t > 0$$, the future increments $$W_{t+u} - W_t$$, $$u \geq 0$$, are independent of the past values $$W_s$$, $$s < t$$.
• $$W$$ has Gaussian increments: $$W_{t+u} - W_t$$ is normally distributed with mean 0 and variance $$u$$, $$W_{t+u} - W_t \sim N(0, u)$$.
2. Ornstein-Uhlenbeck Process: The Ornstein-Uhlenbeck process $$X_t$$ is defined by the following stochastic differential equation:

$$dx_t = \theta (\mu - x_t) \, dt + \sigma \, dW_t$$

where:

• $$\theta > 0$$ and $$\sigma > 0$$ are parameters
• $$W_t$$ denotes the Wiener process.
• $$\mu$$ is a drift constant.
1. Modifying the Ornstein-Uhlenbeck process to be constrained between two points $$(a,b)$$:

In this answer (https://math.stackexchange.com/a/4828166/791334), I learned that the Ornstein-Uhlenbeck process can transformed into a new process $$Y_t$$ that will now be contained between $$(0,1)$$:

$$Y_t = \frac{e^{X_t}}{1 + e^{X_t}}$$

By scaling $$Y_t$$, I think we should now be able to define it between two points $$(a,b)$$:

$$Y_t = a + \left(\frac{e^{X_t}}{1 + e^{X_t}}\right) \cdot (b - a)$$

My Question: In this post here (https://stats.stackexchange.com/questions/605530/estimate-parameters-in-brownian-motion-with-drift-dx-t-mu-dt-sigma-dw-t), an approach is outlined as to how we estimate the parameters of a Brownian Motion using a Likelihood based approach (i.e. consecutive increments in the Brownian Motion are iid Normally Distributed, i.e. Wiener Process - thus, estimating the parameters of a Brownian Motion should correspond to estimating the parameters of a Normal Distribution via Maximum Likelihood Estimation):

It is well known that (note that $$\{W_t\}$$ by definition is a Gaussian process) for $$0 < t_1 < \cdots < t_k$$, the joint density of $$(W_{t_1}, > \ldots, W_{t_k})$$ is (where $$t_0 = w_0 = 0$$) \begin{align} > f_{t_1\cdots t_k}(w_1, \ldots, w_k) = \prod_{i = > 1}^k\frac{1}{\sqrt{2\pi(t_i - t_{i - 1})}} \exp\left[-\frac{(w_i - > w_{i - 1})^2}{2(t_i - t_{i - 1})}\right]. \end{align} Since the transformation $$\mathbf{X} = \mu\mathbf{t} + \sigma\mathbf{W}$$ is affine (where $$\mathbf{W} = (W_{t_1}, \ldots, W_{t_k})$$, $$\mathbf{X} > = (X_{t_1}, \ldots, X_{t_k})$$, $$\mathbf{t} = (t_1, \ldots, t_k)$$), the joint density of $$(X_{t_1}, \ldots, X_{t_k})$$ is then given by (where $$t_0 = x_0 = 0$$): \begin{align} & g_{t_1\cdots t_k}(x_1, > \ldots, x_k) \\ > =& \frac{1}{\sigma^k} \prod_{i = 1}^k\frac{1}{\sqrt{2\pi(t_i - t_{i - 1})}} \exp\left[-\frac{((\sigma^{-1}(x_i - \mu t_i) - \sigma^{-1}(x_{i > - 1} - \mu t_{i - 1}))^2}{2(t_i - t_{i - 1})}\right] \\ > =& \frac{1}{\sigma^k} \prod_{i = 1}^k\frac{1}{\sqrt{2\pi(t_i - t_{i - 1})}} \exp\left[-\frac{(x_i - x_{i - 1} - \mu(t_i - t_{i - > 1}))^2}{2\sigma^2(t_i - t_{i - 1})}\right]. \end{align}

This means that given data $$x_1, \ldots, x_k$$ observed at $$0 < t_1 < > \cdots < t_k$$, the log-likelihood function of $$(\mu, \sigma)$$ is \begin{align} > -k\log\sigma - \frac{1}{2}\sum_{i = 1}^k\log(2\pi(t_i - t_{i - 1})) - \frac{1}{2\sigma^2}\sum_{i = 1}^k((x_i - x_{i - 1} - \mu(t_i - t_{i - > 1}))^2. \tag{1} \end{align}

From $$(1)$$ it is easy to determine the MLE of $$\mu$$ and $$\sigma$$.

In my modified process $$Y_t$$, I have an additional parameter $$\theta$$ (note that $$a$$ and $$b$$ are pre-defined and do not need to be estimated).

• Differences between consecutive increments of the Brownian Motion are i.i.d Normal - thus making it possible to create a valid likelihood function. However, I am not sure if differences between consecutive increments of my process $$Y_t$$ are i.i.d. Normal, thus allowing for the existence and construction of a valid likelihood function?
• Is it somehow possible to still create a valid mathematical likelihood function corresponding to $$Y_t$$, such that all parameters can be estimated via Maximum Likelihood Estimation?

Thanks!

Note that the transformation $$f(x) = \frac{e^x}{1 + e^x}$$ is invertible. Given any realization $$Y_t$$, you can back out what $$X_t$$ must have been.

Now $$dX_t = \theta(\mu - X_t) dt + \sigma dW_t$$. So the increment, given $$X_t$$ is normally distributed with mean $$\theta(\mu - X_t) dt$$ and variance $$\sigma^2 dt$$.

This suggests a simple inference scheme. Choose a small $$\Delta t$$ and divide up your sample into $$\Delta X_t$$ increments. Create the list of tuples $$(X_{t_i}, \Delta X_{t_i})$$ for samples $$i \in 1 \ldots N$$. Use least squares to fit the model $$\min_{A, B} \sum_{i} ||A - B X_{t_i} - \Delta X_{t_i}||^2$$.

Then $$\hat \theta = B$$ and $$\hat \mu = A/B$$.

Then the estimator $$\hat \sigma^2 = \frac{1}{N} \sum_i||A - B X_{t_i} - \Delta X_{t_i}||^2$$ which is just the residual variance.

I believe this will approximate the MLE estimator given the data, but I haven't done the math to confirm this. The approximation comes in because the $$E[\Delta X_t | X_t]$$ is actually $$X_t (e^{- \theta \Delta t}-1) + \mu(1 - e^{-\theta \Delta t})$$ (this formula is given in the Wikipedia article) and for small $$\Delta t$$, this is $$\approx -X_t \theta \Delta + \mu \theta \Delta t$$ and a similar effect with the variances.

• @ Mark: thank you so much for your answer! I am trying to learn more about "mean reversion" in general. Here, I posted a question trying to understand the following point: what makes a stochastic process "mean reverting"? Can you please take a look at it if you have time? thank you so much for all your help over the past few weeks - I really appreciate it! math.stackexchange.com/questions/4835878/… Commented Dec 30, 2023 at 4:48