# Continuous-time models and stochastic calculus Part II

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## Outline

• The following topics will be covered in this lecture:
• Ordinary differential equations
• Stochastic differential equations
• Modes of convergence
• The Fokker-Planck Equations

## Ordinary differential equations

• In our last session, we introduced the notion of the stochastic integral, with two standard forms, the Itô and Stratonovich forms.

• Particularly, we discussed some of the ways that the stochastic integral extends, and is different from the standard deterministic integral.

• Despite the differences, we are able to formally manipulate these equations with e.g., Itô's lemmas.

• In particular, these concepts allow us to derive what is known as a stochastic differential equation as an extension of the ordinary differential equation.

• Giving an intuition on this extension, and how we will use this formalism to sample a target density, will be the focus of this lecture.

### Ordinary differential equations

• A general ordinary differential equation (ODE) is written as

\begin{align} & \frac{\mathrm{d}}{\mathrm{d}t} \pmb{x} := \pmb{f}(t, \pmb{x}) \\ \Leftrightarrow & \mathrm{d}\pmb{x} := \pmb{f}(t,\pmb{x})\mathrm{d}t. \end{align}

• When $$\pmb{f}$$ satisfies a regularity condition, this equation will have a unique solution given some initial data.

Lipshitz Continuity
The function $$\pmb{f}:\mathbb{R}^{N_x} \rightarrow \mathbb{R}^{N_x}$$ is said to be Lipshitz continuous at a point $$\pmb{x}_0$$ if for all $$\pmb{x}_1$$, in a sufficiently small neighborhood of $$\pmb{x}_0$$, \begin{align} \parallel \pmb{f}(\pmb{x}_0) - \pmb{f}(\pmb{x}_1) \parallel \leq K \parallel \pmb{x}_0 - \pmb{x}_1\parallel \end{align} for a fixed constant $$K\in \mathbb{R}$$.
• Lipshitz continuity above is stronger than regular continuity, but weaker than differentiability.

• In particular, if $$\pmb{f}\in \mathcal{C}^1(\mathbb{R}^{N_x})$$, $$\pmb{f}$$ satisfies Lipshitz continuity.
• More generally, a function that is Lipshitz continuous can be shown to be differentiable except on a set of measure zero,
• i.e., with probability one, you will select a point in a bounded interval at which $$\pmb{f}$$ is differentiable.
• For this reason, we consider Lipshitz functions to be differentiable “almost everywhere”, where the number of non-differentiable spikes is limited.

### Ordinary differential equations

• Recall the differential equation on the last slide

\begin{align} \mathrm{d}\pmb{x} := \pmb{f}(t,\pmb{x})\mathrm{d}t. \end{align}

• Provided that $$\pmb{f}$$ is Lipshitz in its components at some initial condition, in the state variable and time, it can be shown that there is a unique solution defined for this initial data.

Picard-Lindelölf theorem
Suppose that $$\pmb{f}$$ satisfies the Lipshitz condition at a point $$(0,\pmb{x}_0)$$ as previously discussed. Then there is a unique solution $$\pmb{x}(t)$$ defined on some time interval $$[ -\epsilon, \epsilon]$$ for which:
• $$\pmb{x}(0)= \pmb{x}_0$$,
• $$\frac{\mathrm{d}}{\mathrm{d}t}|_{t=t_0}\pmb{x} = \pmb{f}(t_0, \pmb{x}(t_0))$$, and
• where we formally write \begin{align} \pmb{x}(t) = \int_0^t \pmb{f}(s, \pmb{x})\mathrm{d}s + \pmb{x}_0. \end{align}
• Notice that this only defines a solution within a local neighborhood, depending on a range of time around the initial condition.

• This known as an initial value problem, as previously discussed in the context of Markov models.

### Ordinary differential equations

• Particularly, suppose that we have an initial prior on the state vector $$\pmb{x}_0$$ and $$\frac{\mathrm{d}}{\mathrm{d}t} \pmb{x}=\pmb{f}$$ is known to satisfy the Lipshitz condition in the support of $$p(\pmb{x}_0)$$.

• This actually defines a deterministic Markov model, but where there is uncertainty in the initial value.

• We can define the discrete mapping under the continuous time model by the flow map discussed before, where

\begin{align} \boldsymbol{\Phi}(t, \pmb{x}_0) = \pmb{x}(t) = \int_0^t \pmb{f}(s, \pmb{x})\mathrm{d}s + \pmb{x}_0. \end{align}

• In the case that $$\pmb{f}$$ is a linear transformation, $$\boldsymbol{\Phi}\equiv \mathbf{M}_t$$ for some matrix, as with the previously defined Gauss-Markov model.

• In this case, we once again generate a transition kernel as

\begin{align} \mathcal{P}\left(\pmb{x}_t | \pmb{x}_0\right) = \delta_{\mathbf{M}_t \pmb{x}_0} \end{align} with $$\delta_{\pmb{v}}$$ referring to the Dirac measure at $$\pmb{v} \in \mathbb{R}^{N_x}$$.

• The Dirac probability measure is defined by the property,

\begin{align} \int f(x) \boldsymbol{\delta}_{\pmb{v}}\left(\mathrm{d}\pmb{x}\right) = f\left(\pmb{v}\right); \end{align} particularly, the Dirac delta is a singular measure, understood by the integral equation.

### Ordinary differential equations

• Similarly, we will say that the transition “density” is given in terms of \begin{align} p(\pmb{x}_t \vert \pmb{x}_{0} ) \equiv \delta \left\{\pmb{x}_t - \mathbf{M}_t\left(\pmb{x}_{0}\right)\right\} \end{align} where $$\delta$$ represents the Dirac distribution.
• Heuristically, this is known as the “function” which has the property $\pmb{\delta}(\pmb{x}) = \begin{cases} +\infty & \pmb{x} = \pmb{0} \\ 0 & \text{else}\end{cases};$
• This is just a convenient abuse of notations, as the Dirac measure does not have a density with respect to the standard Lebesgue measure.
• Rather, the Dirac distribution is understood through the generalized function theory of distributions as a type of kernel that gives the property, \begin{align} \int f(\pmb{x}_{t}) \delta\left\{\pmb{x}_t- \mathbf{M}_t\left(\pmb{x}_{0}\right)\right\}\mathrm{d}\pmb{x}_{t} = f\left(\mathbf{M}_t\left(\pmb{x}_{0}\right)\right). \end{align}
• This equation is to be interpreted that, given a realization of the initial condition $$\pmb{x}_0 \sim P$$, this defines the probability one of the subsequent realizations $$\mathbf{M}_t \pmb{x}_0$$ at all times $$t$$.
• Therefore, such a model is known as a “perfect” model, as it totally determines the subsequent evolution of the random process in time.
• However, our classic Gauss-Markov model was defined in terms of a perfect model that is perturbed by random shocks, \begin{align} \pmb{x}_{k}:= \mathbf{M}_k \pmb{x}_{k-1} + \pmb{w}_k. \end{align}
• The extension of such a Gauss-Markov model to a system generated with continuous time is derived with the notion of a stochastic differential equation.

## Stochastic differential equations

• A general, scalar stochastic differential equation (SDE) is written as

\begin{align} \mathrm{d}X_t := a(t, X_t)\mathrm{d}t + b(t, X_t)\mathrm{d}W_t \end{align} where $$a$$ is known as the drift function and $$b$$ is known as the diffusion function.

• The above SDE is written in the Itô form, while there exists an equivalent Stratonovich form given as,

\begin{align} \mathrm{d}X_t := \left[a(t, X_t) -\frac{1}{2} b(t, X_t) \partial_x b(t, X_t)\right] \mathrm{d}t + b(t, X_t) \circ \mathrm{d}W_t. \end{align}

• The Itô SDE has a formal solution given by

\begin{align} X(T) - X(0) = \int_0^T a(s, X_t)\mathrm{d}t + \int_0^T b(t, X_t)\mathrm{d}W_t \end{align}

• An immediate difference from the ODE initial value problem is that the evolution of the state is given by a random variable, with a distribution that depends in time on the realization of the Wiener process.

• Particularly, the transition probability is no longer given by a Dirac measure, and instead include uncertainty in the evolution.
• In this case, the drift terms represent the mechanistic laws governing the process, while the diffusion terms represent the random shocks to the system.

• If the drift and diffusion $$a,b$$ are linear functions, this furthermore defines a Gauss-Markov model.

• We will note here a particular scenario that is of special relevance to our discussions.

• When the diffusion term $$b$$ has no dependence on the model state $$X_t$$, then the model is said to be one of additive noise.

• Recall then the Stratonovich SDE

\begin{align} \mathrm{d}X_t := \left[a(t, X_t) -\frac{1}{2} b(t) \partial_x b(t)\right] \mathrm{d}t + b(t ) \circ \mathrm{d}W_t. \end{align}

• In particular, $$\partial_x b \equiv 0$$ when $$b$$ is only a function of time, i.e.,

\begin{align} \mathrm{d}X_t := a(t, X_t) \mathrm{d}t + b(t) \circ \mathrm{d}W_t. \end{align}

• Therefore, it can be shown that the Stratonovich SDE and the Itô SDE are the same for additive noise.
• This is a scenario that is frequently studied in data assimilation literature, because of the simplification of the SDE above, and for the way this represents precisely unbiased shocks to governing process laws.

• We will return to such systems when we look at numerical solutions shortly.

## Modes of convergence

• As noted with stochastic processes, and stochastic calculus, there are multiple ways we might consider the existence and uniqueness of a solution to an SDE.
Strong convergence
A strong solution $$X_t$$ of an Itô SDE (or equivalent Stratonovich SDE) has the following properties:
• $$X_T$$ satisfies \begin{align} X(T) - X(0) = \int_0^T a(s, X_t)\mathrm{d}t + \int_0^T b(t, X_t)\mathrm{d}W_t, \end{align} and for all times $$T$$, $$X_T$$ is a function of $$a,b$$ and the realization of $$W_t$$ for all times $$t<T$$; and
• the integrals in the above are well-defined in terms of the proper modes of convergence.
• The important notion here is that if we change the realization of the Wiener process $$W_t$$, then also the strong solution $$X_t$$ changes, but the functional relation between $$X_t$$ and $$W_t$$ remains the same.

• This is in analogy to how we looked at the realization of the function $$A(\omega)\sin(t)$$, and how the evolution in time depends on the outcome for $$A$$.

• Different realizations of the Wiener process $$W_t$$ can thus be thought of generating different sequences of shocks to the governing laws, and a strong solution is thought to depend on the specific sequence of shocks.

• However, if we look at the collection of all possible sequences of shocks that can be generated by $$W_t$$, this gives a (non-singular) probability distribution for $$X_t$$ at all times.

• Particularly, each realization of a strong solution gives a particular sample (path) of the probability distribution for $$X_t$$.

### Modes of convergence

• As with ODEs, Lipshitz continuity of the drift and diffusion functions gives the existence and uniqueness of strong solutions to the SDE.

• However, not all SDEs admit strong solutions, and more generally we may be concerned just with the probabilistic aspects of such a simulation.

• This follows the analogy of almost sure convergence (similar to strong convergence) versus convergence in probability alone.

• We may formally define a solution in which we guarantee only that the forward probability distribution matches that generated by the SDE;

• however, we may not actually guarantee a (point-wise) solution that matches a particular sample path given some realization of $$W_t$$.
• This is loosely what is known as weak convergence, which we will consider more in depth when we study the numerical solutions to these equations.

## Fokker-Planck equations

• While strong solutions of the SDE equation give sample realizations of the probability distribution for $$X_t$$, we may also consider solving for this probability distribution directly.

• Suppose that $$X_0$$ has a density defined as $$p(x_0)$$ then, given the SDE equation, we can also study how this initial prior evolves in time.

• Particularly, the SDE defines a Markov model, and we will denote the transition density as $$p(t,x)$$.

Fokker-Planck equations
For a random process $$X_t$$ with an SDE governing the time evolution, an initial prior and transition density $$p(t,x)$$ as above, the Fokker-Planck equations are defined as \begin{align} \partial_t p + \partial_x (a p) - \frac{1}{2} \partial_x^2 \left(pb^2\right) = 0. \end{align} In particular, the above partial differential equation defines the probability density for $$X_t$$ at all times $$t$$ given the initial prior $$p(x_0)$$.
• The Fokker-Planck equations completely define the solution to all sample paths $$X_t$$, as this provides the entire probability density.

• Realizations of sample paths are thus drawn from this joint density in time.
• However, solving the Fokker-Planck equations becomes computationally unfeasible for any dimension $$N_x> 3$$ in practice, so that this full solution is only theoretical.

• Rather, we will typically consider an ensemble of sample path solutions to uncertain O/S DEs to generate empirical statistics from this theoretical density in practice.