Nonlinear Option Pricing

130 Early Exercise Problems

Finally, by taking the expectation, and assuming that

R

t

0

Du(s, X

s

) dX

s

is a

true Q-martingale, we get

u(t, X

t

) = E

Q

[g(X

T

)|F

t

] −

Z

T

t

E

Q

[(∂

s

u + Lu)(s, X

s

)1

{u(s,X

s

)=g(X

s

)}

|F

t

] ds

As ∂

t

u + Lu ≤ 0,

−(∂

t

u + Lu)1

{u(t,x)=g(x)}

= (∂

t

u + Lu)

−

1

{u(t,x)=g(x)}

= (Lg(x))

−

1

{u(t,x)=g(x)}

Here we have assumed that the exercise domain D = {(t, x) | u(t, x) = g(x)}

is an open set so that at the points where u = g we also have Lu = Lg, and

we have disregarded the discontinuity of ∂

2

x

u at the boundary ∂D. Leaving

apart these technical difficulties, we obtain

u(t, X

t

) = E

Q

[g(X

T

)|F

t

] + E

Q

"

Z

T

t

(Lg(X

s

))

−

1

{u(s,X

s

)=g(X

s

)}

ds











F

t

#

We complete the proof using