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Stochastic Calculus
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Generator of a Diffusion
L2
Generator of a Diffusion
Senior Quant · Stochastic Calculus
Question
Define the generator
A
\mathcal{A}
A
of a diffusion
d
X
=
μ
d
t
+
σ
d
W
dX = \mu\,dt + \sigma\,dW
d
X
=
μ
d
t
+
σ
d
W
. Compute
A
f
\mathcal{A}f
A
f
for
f
∈
C
2
(
R
)
f \in C^2(\mathbb{R})
f
∈
C
2
(
R
)
. Show that
f
(
X
t
)
−
∫
0
t
A
f
(
X
s
)
d
s
f(X_t) - \int_0^t \mathcal{A}f(X_s)\,ds
f
(
X
t
)
−
∫
0
t
A
f
(
X
s
)
d
s
is a local martingale (Dynkin's formula).
Show Answer
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