Maximum Likelihood Estimation for Correlated SDE Systems

Summary

The document asks how to estimate parameters for three synchronously sampled processes: one geometric Brownian motion and two arithmetic Brownian motions. Their driving Brownian motions have specified correlations with the first process. The proposed route is to work with discrete observations, account for correlated innovations through a covariance matrix, and potentially use a Cholesky factorization with a multivariate normal likelihood.

It is a problem statement rather than a worked solution: it gives no likelihood derivation, estimator, data example, or empirical evidence. Any implementation would need to clarify which drift and volatility terms are constant or time-varying, how the correlation between the second and third processes is handled, and whether the sampling interval supports the assumed diffusion approximation. For the geometric process, log price increments are typically the natural Gaussian observations under constant parameters; the arithmetic processes use level increments. The document motivates a useful estimation question but does not resolve it.

Key ideas

• The model combines one multiplicative diffusion with two additive diffusions.
• Synchronous observations allow the increments to be modeled jointly across processes.
• A multivariate normal likelihood can represent correlated innovations through their covariance matrix.
• The document proposes Cholesky decomposition as a possible computational method but does not derive an estimator.
• Parameter constancy and the full correlation structure require clarification before fitting the model.

Tags

# Maximum likelihood estimation of system of correlated SDEs


# Maximum likelihood estimation of system of correlated SDEs












I have the following system of SDEs (which you can think of as 3 different stocks) $$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$ $$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$ $$dX_t^3 = \mu_3 dt + \sigma_3 dW_t^3$$ where $dW_t$ is a standard Brownian motion and correlated such that $dW_t^1 dW_t^2 = \rho_2 dt$ and $dW_t^1 dW_t^3 = \rho_3 dt$. I have discrete time series data on $X_t^1$, $X_t^2$ and $X_t^3$, and they are all sampled at the same time. How do I find maximum likelihood estimators in this model?

My initial thought was to rewrite the setup to a system in terms of $X_t^i$ instead of $dX_t^i$, apply Cholesky decomposition and then use MLE for multivariate normal variables. But I am unsure how to do this specifically, and if that is the best approach to estimate parameters in a system of correlated SDEs?

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