Geometric Brownian Motion Between Two Arbitrary Times

Summary

The document derives the transition formula for geometric Brownian motion over a time interval starting at t₁ and ending at t₂. Applying Itô’s lemma to the logarithm of the process turns the multiplicative stochastic differential equation into a process with constant drift and a Brownian increment. Integrating gives the change in log value, which exponentiates to the stated expression for X at t₂ in terms of X at t₁.

The result uses the drift adjustment by half the variance and the Wiener-process increment across the interval. It is a direct consequence of the assumed GBM model with constant parameters; it does not estimate those parameters, discuss calibration, or address changes in drift or volatility over time.

Key ideas

• Taking the logarithm of a GBM process allows Itô’s lemma to express its dynamics additively.
• The log process accumulates adjusted drift and a scaled Brownian increment between the two times.
• Exponentiating the integrated log equation gives the conditional value at the interval endpoint.
• The formula assumes constant drift and volatility over the interval.

Tags

# Geometric Brownian Motion in a general interval $[t_1,t_2]$


# Geometric Brownian Motion in a general interval $[t_1,t_2]$












I know that the Geomtric Brownian Motion, with the expresion $dX_t = v X_t dt + \sigma X_t dW_t$ has the next solution $$X_t = X_0 e^{\sigma W_t+ (v-\frac{\sigma ^2}{2})t}$$ on the interval [0,t]. But, what would be the solution on a general interval $[t_1,t_2]$?

Would it be, $X_{t_2} = X_{t_1} e^{\sigma (W_{t_2}-W_{t_1})+ (v-\frac{1}{2}\sigma^2)(t_2-t_1)}$?

## Answer by user16651 (score 1)

https://quant.stackexchange.com/a/18666

let $Y_t=\ln X_t$ by application of Ito lemma we have \begin{align} dY_t=\frac{1}{X_t}dX_t-\frac{1}{2X_t^2}d[X,X](t)=(v-\frac{1}{2}\sigma^2)dt+\sigma\,dW_t \end{align} by integration on $[t_1,t_2]$, we have \begin{align} Y_{t_2}=Y_{t_1}+(v-\frac{1}{2}\sigma^2)(t_2-t_1)+\sigma\,(W_{t_2}-W_{t_1}) \end{align} then \begin{align} X_{t_2}=X_{t_1}exp\left((v-\frac{1}{2}\sigma^2)(t_2-t_1)+\sigma\,(W_{t_2}-W_{t_1})\right) \end{align}

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