Arithmetic Brownian Motion Process and SDEs

We discuss various things related to the Arithmetic Brownian Motion Process- these include solution of the SDEs, derivation of its Characteristic Function and Moment Generating Function, derivation of the mean, variance, and covariance, and explanation of the calibration and simulation of the process.

Solution of Arithmetic Brownian SDE

Let’s recall the Arithmetic Brownian motion SDE,

$$dX_t=\mu \,dt+\sigma \,dB_t$$

Solving this equation is relatively straightforward as all the variables are nicely separated, so we can just integrate (we are interested in the process value at time T, so we integrate from 0 to T).

$$\int_0^T{d X_t} =\int_0^T{\mu dt}+\int_0^T{\sigma \,dB_t}$$

We can take the constants out of the integrals,

$$\int_0^T{d X_t} =\mu \int_0^T{ dt}+\sigma \int_0^T{dB_t}$$

And then evaluate the integrals,

$$X_T-X_0=\mu \left( T -0\right) +\sigma \left( B_T-B_0\right)$$

We just need to rearrange and simplify the equation to get the solution,

$$X_T=X_0 + \mu \, T +\sigma \, B_T$$

The solution, as one would expect, is stochastic, so even though we know the solution that gives the process value, this process value is a random quantity. But we can still give it meaningful characterisation using familiar concepts from statistics such as 1) we can calculate its moments - e.g., mean, variance, 2) we can determine its probability distribution, and 3) we can simulate its values. These are covered in the next few tabs.