Multidimensional Itô’s Lemma for Correlated Geometric Brownian Motions

Summary

The document frames a portfolio problem involving N assets, each modeled as a geometric Brownian motion with drift and volatility. It asks how to express the stochastic differential equation for a function of all asset prices, with clear drift and diffusion components. The asset shocks are correlated, and the question specifies that the instantaneous cross-product of two shocks equals their correlation times the time increment.

This setup points to the multidimensional Itô formula: the drift includes first derivatives against asset drifts and second derivatives weighted by the covariance matrix, while the diffusion combines the first derivatives with the asset shocks. The document itself contains no derivation or answer, so it does not establish a particular compact notation or discuss assumptions such as regularity of the function, constant parameters, or time dependence. It is a useful formulation of the modeling issue, but not a complete worked explanation.

Key ideas

• Each asset price is modeled as a geometric Brownian motion with drift and volatility.
• Correlated Brownian shocks create cross-variation terms in the multidimensional Itô expansion.
• The desired SDE separates drift contributions from diffusion contributions for a function of several prices.
• The document poses the derivation problem but does not provide its solution.

Tags

# stochastic calculus and multidimentional itos lemma


# stochastic calculus and multidimentional itos lemma












I am considering a number of assets (N) in a portfolio. each asset follows a geometric Brownian motion process therefore the stochastic differential equation is dS(i) = S(i)μdt + S(i)σdX(i). The price changes are correlated as measured by the linear correlation coefficients rho(ij) . how do i deduce the multi-dimensional Ito's Lemma to write down the SDE for F(S1; S2; : : : ; SN) in the most compact form possible (with clear drift and diffusion terms) including crorrelation between dX(i)dX(j) that is rho(i,j)dt?

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