A Longstaff and Schwartz Approach to the Early Election Problem
Develop initial estimate for the value function (Detail Section 4.6)
Establish an estimate for the value function at time across all poll states .
Simulation of the poll process (Detail Section 4.2)
Generate trajectories of the SDE (4.6) by simulating solutions. Denote each of the N simulated poll process as .
Simulation of electoral outcome if calling an election (Detail Section 4.4)
At each decision point , and poll state , we establish the expectation of the value function from a decision to call an election. The electoral result arises from: (i) Continue the diffusion to time (ii) Simulate a sampling error for the poll state (iii) Simulate the randomness in relationship in Section 3.3 for the imperfect relationship between nationwide polling and the distribution of seats.
Simulation of electoral outcome if not calling an election
The alternative to the step 2 is to NOT call an election. In that case, the polls will diffuse to the next timestep and the decision experiment is repeated.
Regress over all N simulations the value function conditioned upon the poll state under the scenario that an election IS called (step 2). Regress over all N simulations the value function conditioned upon the poll state under the scenario that an election IS NOT called (step 3).
Establish max and strategy
Establish the maximum arising from steps 3 or 4; if 3 then assign the optimal strategy as CALL, else CONTINUE. Assign the value function to each of the points , , from the maximum between the two regressions.
Repeat over all timesteps
Repeat over all time steps stepping backwards from to .
Update initial value estimate
When time zero is reached, the value function will disagree with the estimate from step 0. Update with the derived initial value and recommence from step 0 until convergence of the initial estimate is achieved.
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