# Continuous-time versus discrete-time stochastic models

When modeling a dynamic phenomenon, (from a general point of view) people can use two type of models: (1) continuous-time models, (2) discrete-time models.

To be more precise, assume that we try to model a phenomenon observed in a time series form via dynamical systems. If the stochasticity involved in the system, we can use two type of models: continuous-time stochastic processes or discrete-time stochastic processes. Since the time-series data is discrete, it seems very natural to model the processes in discrete-time. However, in some application domains, like stochastic finance, continuous-time models are of great importance and interest.

Here are my questions:

1) Is there an advantage of continuous-time modeling? Since the calculus is more difficult (continuous-time models require Ito calculus), what is the computational or modeling advantage of continuous-time models?

2) If there is no analytical solution of a continuous-time model, we discretize it and solve it numerically as far as I know. So, is it so different from discrete-time modeling?

As George Box said, "All Models are Wrong, some are useful." Continuous time models are useful when the dynamics you are interested in are fine enough to be approximated as a continuous process. Moreover, it is a common misconception that somehow continuous time dynamics are more difficult than discrete time. This is hardly the case and depends on how coarse your model is. There are two cases where continuous space/time models become effective: when space becomes extremely granular and when time becomes extremely granular (or both!). This is parallel to differential equations. If you're measuring how fast a car is going, then you need to know the acceleration and velocity. If you want really fine measurements, then you need to take a lot of them which is why derivatives naturally arise. It's simply easier to encapsulate a derivative. You're right though, when simulating on a computer you discretize your equations when they are explicitly unsolvable. Think of derivatives as a shorthand for discrete numerical work.

Take a simple example such as a stock price $$X$$ being a random variable whose range is between 1 and 2 dollars. If the stock price takes only the values \$1,\$ 2, then clearly a discrete model is what you want. Now instead suppose that the stock can take any value between [1,2] in units of \$1/1000 of a dollar. To make the example concrete, lets suppose that $$X=1+B(n,p)/n$$, where $$B(p,n)$$ is a binomial random variable (taking on values $$0,1,\ldots, n$$), so that $$X$$ lives in $$[1,2]$$ and we set $$n=1000$$. Now ask the question, what's the chance that $$X$$ equals \$1.3? It's simply not instructive to calculate something like $$\binom{1000}{300}p^{300}(1-p)^{700}$$ when you know that $$B(n,p)$$ becomes roughly normally distributed as $$n$$ gets large. In particular you know that $$B(n,p)$$ looks something like $$\mathcal{N}(np,np(1-p))$$. Divide by $$n$$ and it's as if you are sampling a normal distribution which is also a Brownian motion at a specific time.

Another example is that if you have a simple random walk $$S_n=X_1+X_2+\ldots+X_n$$, where $$X_{n+1}$$ equals $$X_{n}\pm 1$$ with probability 1/2 each. Then $$\frac{X_1+\ldots+X_{nt}}{\sqrt{nt}}$$ looks like $$B_t$$, a standard Brownian motion. In this example space is granular but time is becoming continuous. What you need to take home from this example is that even minor complications to $$X_t$$ will result in completely intractable formulas for calculating probabilities of say $$S_{36}=25$$. Again, there is a kind of central limit theorem at work here which is trying to help you understand what's going on with probability.

Combine the two examples above and you can imagine that if you are tracking a stock, then it can go up and down more or less than 1/2 or whatever unit you are using. The point here is that if the time interval is extremely short, then you'll want to make sure that you don't have that big of a jump, which is something that Brownian motion encapsulates. If I fix $$B_0$$, then $$B_t$$ for really small $$t$$ is roughly equal to $$B_0$$ (in fact Brownian motion is continuous). Then if we want to get our hands dirty on a computer for stuff that's unsolvable, we discretize equations and go to work knowing that the finer our discretizations, the more accurate our simulations are to the continuous model which is something we can work with on paper.

• Very nice answer. Thanks you.
– user48547
Apr 2, 2013 at 14:39
• How exactly is time becoming continuous? $X_{nt}$ doesn't make sense if $t$ is continuous. Oct 7, 2018 at 20:16
• @samlaf: You can make it rigorous by considering $[n t]$, or instead assuming $t=c/n$, where $c,n$ are integers and then using this to approximate irrational times. Oct 7, 2018 at 20:39
• Ah so this is Donsker's theorem right? Oct 19, 2018 at 18:20