Deriving the Exponential Martingale in a Girsanov Example

Summary

The document asks how a stochastic exponential in a Girsanov theorem example reduces to a formula involving time, a constant market-price-of-risk term, and Brownian motion. The general expression contains integrals of a process θ over time and against Brownian motion. The question concerns the substitution and simplification that produce the stated outcome in the cited example.

This is a conceptual question about applying a change of measure in a stochastic-process setting, relevant to mathematical finance. The document gives the general exponential-martingale expression and the resulting expression, but does not explain the assumptions that make θ constant or work through the integrals. It offers no separate derivation, numerical example, or discussion of conditions required for the exponential process to define a valid measure change, so those details cannot be inferred from the question alone.

Key ideas

• The document asks how a Girsanov exponential martingale is simplified in a worked example.
• The general expression contains a time integral of the squared process and a Brownian stochastic integral.
• The stated result uses a constant involving the interest rate, drift, and volatility.
• The document poses the derivation question but does not provide the assumptions or steps needed to answer it.

Tags

# Girsanov Theorem example


# Girsanov Theorem example












My reference is here : https://arxiv.org/pdf/1504.05309.pdf

My question is related to the example 2.6.1 : page 21-22; 2.6 Girsanov Theorem

It said in equation (2.8) $Z_t = exp(-\frac{1}{2}∫^t_0θ_s^2ds+∫^t_0θ_sdW_s)$.

And there is outcome for the exponential martingale, $Z_t = exp(-\frac{t}{2}(\frac{(r-μ)}{σ})+\frac{(r-μ)}{σ}W_t)$.

But I don't understand the logic behind this derivation. I want to the procedure how $θ_s$ term changed to $Z_t = exp(-\frac{t}{2}(\frac{(r-μ)}{σ})+\frac{(r-μ)}{σ}W_t)$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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