Lecture10.pdf - Lecture 10 Verification Theorem and...

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Lecture 10: Verification Theorem and Black–Scholes Market 1 Vericiation Theorem Recall the HJB equation is v t ( t, x ) + sup u U 1 2 tr ( σ ( t, x, u ) > v xx ( t, x ) σ ( t, x, u ) ) + b ( t, x, u ) > v x ( t, x ) + f ( t, x, u ) = 0 , ( t, x ) [0 , T ) × R n , v ( T, x ) = h ( x ) , x R n . (1) The following theorem tells how to derive optimal feedback control via HJB equation. Theorem 1.1 (Verification theorem) . Let v C 1 , 2 ([0 , T ] × R n ) solve the HJB equation (1). Assume that for every ( t, x ) [0 , T ] × R n , u ( t, x ) = argmax u U 1 2 tr ( σ ( t, x, u ) > v xx ( t, x ) σ ( t, x, u ) ) + b ( t, x, u ) > v x ( t, x ) + f ( t, x, u ) exists, and that u is a feedback control. Then u = { u ( t, x t ) } t [0 ,T ] is an optimal control. This gives rise to the following general algorithm to solve a stochastic control problem: 1. Solve the HJB equation (1) (either analytically if you are lucky, or numerically in most cases) to get v ; 2. Solve an optimization problem, for each ( t, x ): Maximize G ( t, x, u ) := 1 2 tr ( σ ( t, x, u ) > v xx ( t, x ) σ ( t, x, u ) ) + b ( t, x, u ) > v x ( t, x ) + f ( t, x, u ) , subject to u U to get u ( t, x ).

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