Arithmetic Brownian Motion: Solution and Transition Density

Summary

The document solves an arithmetic Brownian motion with constant drift and volatility over an interval from the current time to a future time. Directly integrating the stochastic differential equation expresses the future state as the current state, plus the drift accumulated over the interval, plus a volatility-scaled Brownian increment.

Because a Brownian increment over that interval is normally distributed with mean zero and variance equal to the elapsed time, the future state conditional on the current state is normal. Its mean is the current value plus drift times elapsed time; its variance is volatility squared times elapsed time. This gives the transition distribution, from which a density can be written using the normal distribution. The result assumes constant drift and volatility and the specified arithmetic Brownian motion model; it does not address estimation, boundary constraints, or whether this process is suitable for a particular asset.

Key ideas

• Integrating the SDE gives the future state as the current state plus drift and a Brownian increment.
• The Brownian increment is normally distributed with variance equal to elapsed time.
• The conditional future state is normal with drift-adjusted mean and time-scaled variance.
• The result relies on constant drift and volatility in the arithmetic Brownian motion model.

Tags

# Find Arithmetic Brownian Motion's transition density


# Find Arithmetic Brownian Motion's transition density












Consider the following stochastic differential equation, an Arithmetic Brownian Motion: 𝑑𝑆(𝑡) = 𝑟 𝑑𝑡 + 𝜎 𝑑𝑊(𝑡) . Find its solution, integrating from t to T, then find its transition density. Hint: using Itô, one would utilize the substitution 𝑉 = 𝑆(𝑡). However doing so is equivalent to simply integrating the SDE directly anyways.

## Answer by Stéphane (score 1)

https://quant.stackexchange.com/a/53170

\begin{align} S(T) - S(t) := \int_t^T dS(s) = r(T-t) + \sigma (W(T) - W(t)) \end{align}

Since $W(T) - W(t) \sim N(0, T-t)$, we have $S(T) \sim N\left(S(t) + r(T-t),\sigma^2(T-t) \right)$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.