Likelihood Ratio Martingales and Change of Measure

Why Change Measure by Comparing Intensities?

An event-history model specifies how quickly events occur given the information available just before each time. Changing the intensity changes the probability assigned to event paths: paths with events in times or types favoured by the new intensity gain relative weight, while paths with long event-free intervals may lose or gain weight according to the cumulative rates. A likelihood-ratio process makes this reweighting explicit at every time, not only at the end of follow-up.

This is useful for comparing event-process models, expressing a new event law relative to a reference law, and proving that a proposed change of intensity is mathematically valid. The key caution is that the pathwise likelihood formula alone does not guarantee a probability measure: its density must have expectation one.

Definition and Conditions

Here each intensity is the rate of jumps conditional on the information immediately before time \(t\); if an at-risk indicator is used, it is included in the intensity. Assume the two laws use the same initial distribution, share the same filtration and event-history structure, and have no simultaneous jumps across the counted event types. The rates must be nonnegative, predictable, and locally integrable, and the processes must be nonexplosive on the interval. These conditions allow the usual event-history likelihood construction.

For \(Q\) to be absolutely continuous with respect to \(P\), a path possible under \(Q\) must not require a jump where \(P\) assigns zero intensity. In particular, \(\mu_j(t)=0\) wherever \(\lambda_j(t)=0\), up to the relevant predictable measure. If both are zero, that event type contributes no jump or cumulative-rate difference there. When \(\lambda_j>0\), the pathwise likelihood ratio is

The product over jumps multiplies the relative rates at observed events. The exponential term accounts for the different event-free probabilities between jumps. Equivalently, where the ratios are defined, the process solves

This stochastic differential equation identifies \(L\) as a nonnegative local martingale under \(P\). To use \(L_\tau\) as a probability density, one needs the stronger property \(E_P(L_\tau)=1\). A nonnegative local martingale has expectation at most its initial value; it need not preserve expectation. One sufficient condition on a finite interval is that the intensities are bounded and the ratios \(\mu_j/\lambda_j\) are bounded on the region where \(\lambda_j>0\), with \(\mu_j=0\) where \(\lambda_j=0\). These bounds make the stochastic-exponential integrands bounded and their predictable quadratic variation finite. More generally, other integrability conditions can establish the same unit-expectation property.

Why the Formula Has This Form

For a path with jumps at specified times, the event-history likelihood under an intensity is the product of the intensity at each jump times the exponential of minus the integrated intensity. This is the multiplicative intensity likelihood. Taking the likelihood under the proposed rates and dividing by the likelihood under the reference rates gives the displayed formula: jump contributions divide to \(\mu_j/\lambda_j\), and the integrated-rate terms divide to \(\exp\{-\int(\mu_j-\lambda_j)\}\). The shared initial distribution cancels.

A short stochastic-calculus check gives the same result. At a jump of type \(j\), the product in \(L_t\) changes by the factor \(\mu_j/\lambda_j\). Between jumps, its derivative is \(-L_{t-}\sum_j(\mu_j-\lambda_j)\). Thus

The equality follows because \(\lambda_j(\mu_j/\lambda_j-1)=\mu_j-\lambda_j\). The final expression is a stochastic integral against the compensated counting processes under \(P\), hence a local martingale under the stated local conditions. If it is a true martingale with \(E_P(L_\tau)=1\), define \(Q\) on \(\mathcal{F}_\tau\) by \(Q(A)=E_P[L_\tau I(A)]\). The density process is then \(L_t=E_P(L_\tau\mid\mathcal{F}_t)\).

Under this change of measure, the compensated processes have the proposed rates: \(N_j(t)-\int_0^t\mu_j(s)\,ds\) is a local martingale under \(Q\). One way to verify the claim is to apply the product rule to \(L_t\) and this proposed compensated process. The drift terms cancel because the change in intensity is \(\mu_j-\lambda_j\); the remaining terms are local-martingale terms. This is the counting-process change-of-measure result. It requires the density and support conditions above, not merely a formal substitution of rates.

Worked Path Calculations

Assumptions, Diagnostics, and Alternatives

A practical warning sign is a computed density with an expected value below one: it may be a strict local martingale rather than a probability density. A ratio that becomes unbounded, a process that can accumulate infinitely many jumps in finite time, or incompatible event support also calls for closer justification. These are mathematical checks on the change of measure, not merely numerical diagnostics of model fit.

The path-likelihood ratio is often the clearest approach when observed paths and intensities are explicit. The stochastic-exponential form is more useful for martingale arguments and predictable integrands. A score or log-likelihood calculation instead studies local changes in model parameters; it does not by itself establish a global probability measure for a proposed alternative intensity. In all approaches, the likelihood formula needs the same support and normalization scrutiny.

A common mistake is to see that \(L_t\) is a nonnegative local martingale and conclude immediately that \(E_P(L_t)=1\). Nonnegative local martingales are supermartingales, so only \(E_P(L_t)\leq L_0=1\) follows in general. The correction is to establish the true-martingale property on the time interval of interest, for example with suitable boundedness conditions, before using \(L_t\) as a density. A second error is to use \(\mu_j/\lambda_j\) at a point where \(\lambda_j=0\); the ratio is not defined there, and support compatibility must be checked rather than repaired by algebra.