Using Ito’s Lemma to Derive a Discounted Geometric Brownian Motion

Summary

The document derives the dynamics of a geometric Brownian motion after multiplying the asset process by a deterministic exponential discount factor. The key observation is that the discount factor is continuously differentiable and therefore has finite variation. In applying Ito’s lemma to the product, there is no additional quadratic variation term from that factor, so the ordinary product rule gives the relevant differential.

For the stated process, the drift contribution from the asset’s growth rate is canceled by the differential of the discount factor. The resulting discounted process has only the volatility-driven Brownian term, scaled by the discount factor and the asset level. This is a compact derivation for the specific choice of discount rate shown; it does not discuss other discounting conventions, stochastic rates, or extensions to more general processes.

Key ideas

• A continuously differentiable discount factor has finite variation.
• The product rule applies when combining geometric Brownian motion with a deterministic discount factor.
• The discount factor’s differential can cancel the asset process’s drift term.
• The discounted process in the example retains a volatility-driven Brownian component.

Tags

# Deriving Law of Motion by Ito's Lemma


# Deriving Law of Motion by Ito's Lemma












I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below:

I've managed to take the partial derivative such that I can substitute them into the Ito's Lemma form as shown below:

From this I'm able to simplify down to:

The above is possible because:

However, I am struggling to simplify any further towards an answer for deriving the law of motion. I wonder whether I've miscalculated the partial derivatives or perhaps an error in my simplification thus far, but I can't seem to eliminate the St's which seems necessary. Could I perhaps take the total differential of dUt/Ut of the most simplified version thus far? Thanks for any help in advance.

## Answer by R. Rayl (score 3, accepted)

https://quant.stackexchange.com/a/64147

Since the process $e^{-\mu t}$ is continuously differntiable, then it has finite variation. Thus, Ito's lemma essentially implies the 'normal' product rule:

\begin{align} dU_t &= d(S_te^{-\mu t}) \\ &= e^{-\mu t}dS_t + S_td(e^{-\mu t}) \\ &= e^{-\mu t}\mu S_t dt + e^{-\mu t}\sigma S_t dw_t - \mu e^{-\mu t} S_t dt \\ &= e^{-\mu t}\sigma S_t dw_t \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.