Dynamic Hedging with Stochastic Differential Utility
Dynamic Hedging with Stochastic Differential Utility
Dynamic Hedging with Stochastic Differential Utility
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Proposition 10 Assume u ∈ L and δ > 0. Then the HJB equation is:<br />
u(c t ) − δJ + A d J =0,<br />
as:<br />
Proof. (You may apply the theorem as well) Define the value function<br />
Consequently:<br />
J(z t )=E t<br />
·Z<br />
J(z t+ε )=E t+ε<br />
·Z<br />
s≥t<br />
s≥t+ε<br />
Take the conditional expectations at t:<br />
T ε J(z t )=E t<br />
·Z<br />
¸<br />
e −δ(s−t) u (c s ) ds ,t≥ 0.<br />
¸<br />
e −δ(s−t−ε) u (c s ) ds ,t+ ε ≥ 0.<br />
s≥t+ε<br />
Subtract the first equation from the last one:<br />
T ε J (z t ) − J (z t )=E t<br />
·Z<br />
s≥t+ε<br />
¸<br />
e −δ(s−t−ε) u (c s ) ds .<br />
Z<br />
e −δ(s−t−ε) u (c s ) ds −<br />
s≥t<br />
¸<br />
e −δ(s−t) u (c s ) ds .<br />
Divide by ε and take the limit when ε ↓ 0:<br />
LHS<br />
lim<br />
ε↓0 ε<br />
T ε J (z t ) − J (z t )<br />
= lim<br />
= A d J.<br />
ε↓0 ε<br />
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