n the two-dividend American call option model, we assume discrete dividends of amount D₁ and D₂ are paid out by the underlying asset at times t₁ and t₂, respectively. Let S̃_t denote the asset price at time t, net of the present value of escrowed dividends and S̃^∗_t1 (S̃^∗_t2) denote the optimal exercise price at time t₁ (t₂) above which the American call should be exercised prematurely. Let r,σ,X and T denote the riskless interest rate, volatility of S̃, strike price and expiration time, respectively. Let C (S̃_t,t) denote the value of the American call at time t. Show that S̃^∗_t1 and S̃^∗_t2 are given by the solution of the following nonlinear algebraic equations

The American call price is given by (Welch and Chen, 1988)