Example 1 - convert continuous SDE to discrete soln without drift term

The continuous stochastic differential equation (SDE) is given by:

$$dx = \epsilon \sqrt{dt} \quad // \text{cont eqn}$$

where $\epsilon \sim \mathcal{N}(0,1)$.

To convert this continuous SDE to a discrete solution, we consider the increments:

$$X_{t_{i+1}} - X_{t_i} = \sqrt{t_{i+1} - t_i} \cdot \epsilon$$

Assuming $t$ were not discrete, we can express it as:

$$X_{t_{i+1}} = X_{t_i} + \sqrt{t_{i+1} - t_i} \cdot \epsilon$$

Finally, for a discrete solution, we have:

$$x_{i+1} = x_i + \epsilon, \quad \epsilon \sim \mathcal{N}(0,1) \quad // \text{discrete soln}$$

Example 2 - convert continuous SDE to discrete soln with drift term

Let’s look at another example of converting a discrete equation to a continuous form, this time with a drift term:

  1. Starting with the continuous SDE: $$dx = -\frac{\alpha}{2}xdt + \beta dB_t$$
    where $\alpha$ is the drift term and $\beta$ is the diffusion term.

  2. To discretize this equation, we:

    • Replace $dt$ with finite time step $\Delta t = t_{i+1} - t_i$
    • Replace $dB_t$ with $\epsilon_i \sim \mathcal{N}(0,I)$ scaled by $\sqrt{\Delta t}$

  3. This gives us: $$x_{t_{i+1}} - x_{t_i} = -\frac{\alpha}{2}x_{t_i}(t_{i+1}-t_i) + \beta\sqrt{t_{i+1}-t_i}\epsilon_i$$

  4. Rearranging terms: $$x_{t_{i+1}} = x_{t_i}\left(1-\frac{\alpha}{2}(t_{i+1}-t_i)\right) + \beta\sqrt{t_{i+1}-t_i}\epsilon_i$$

  5. For unit time steps where $t_{i+1}-t_i = 1$, we get: $$x_{i+1} = x_i\left(1-\frac{\alpha}{2}\right) + \beta\epsilon_i \quad \text{where} \quad \epsilon_i \sim \mathcal{N}(0,I)$$

This example shows how a continuous SDE with both drift and diffusion terms can be discretized into an iterative update rule.