Filtration and Novikov Condition for a Vasicek Interest Rate Model

Summary

The document poses two mathematical questions in a model combining a money account, a Vasicek short rate, and a stock with constant drift and volatility. The money account grows at the short rate, which follows mean-reverting stochastic dynamics driven by one Brownian motion; the stock is driven by an independent Brownian motion. Initial values and model parameters are assumed known.

First, it asks whether the short rate can be recovered as an adapted process from the augmented natural filtration generated by the two asset prices. Second, it asks whether the exponential integrability requirement known as Novikov’s condition holds over a finite time horizon for the market-price-of-risk term involving the short rate. The document provides the model and the conditions to investigate, but no solution, proof, or parameter-based conclusion. Its value is as a stochastic-calculus problem setup rather than a worked result, and the validity of the integrability condition is left unresolved.

Key ideas

• The short rate follows a mean-reverting Vasicek process and drives the money account.
• The stock has constant drift and volatility and is driven by a Brownian motion independent of the rate process.
• The first question concerns whether the short rate is observable from the asset-price filtration.
• The second question asks whether the market-price-of-risk process satisfies Novikov’s integrability condition over a finite horizon.
• The document states the problem but does not provide a proof or answer.

Tags

# Novikov condition for Vasicek process


# Novikov condition for Vasicek process












Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \end{align} Moreover, suppose that there is a stock $S^{(1)}$ such that \begin{align} d S^{(1)}_{t} = \alpha S^{(1)}_{t} \, dt + \sigma S^{(1)}_{t}dW_{t}^{(1)}. \end{align} Assume that $a,b,\sigma_{r},\alpha,\sigma>0$ and that the initial values of the processes are known constants and that $W^{(0)}$ and $W^{(1)}$ are independent.

First question: Is $r_{t}$ adapted to the (augmented) natural filtration of $S^{(0)}$ and $S^{(1)}$?

Second question: Given $T>0$, does the Novikov condition hold, i.e. \begin{align} \mathbb{E}\left[\text{e}^{\frac{1}{2}\int_{0}^{T}\left(\frac{\alpha-r_{t}}{\sigma}\right)^{2}dt}\right]<\infty? \end{align}

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