The Value of Insight

An investor may invest in a riskless bank account and in a stock that is a standard Black–Scholes asset with occasional Gaussian jumps of the log price, as proposed by Merton [Merton RC ( 1976 ) Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3(1):125–144.]. It is well known how to solve the standard running consumption problem for this investor, which we take as a benchmark for comparing the performance of two different insiders, one who knows in advance of each jump exactly when the jump will happen, and the other who has information in advance of each jump about the size of the jump but no information about the time. These considerations give rise to two novel and concrete stochastic control problems. For each problem, rigorous verification proofs for optimality are presented.