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The question is

about derivatives in finance. Big tips!! Thanks

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Now consider an Ornstein-Uhlenbeck process X = (Xt)t20, defined by the stochastic differential
dXt = -XXtdt + odZt,
where Z = (Zt)t>o is a standard Brownian motion under the probability measure P and > > 0.
(i) Given a starting value, Xo = 0, solve the stochastic differential equation (2).
(ii) By simulating N paths of the Ornstein-Uhlenbeck process described above, approximate the
transition density of the process and plot it for particular values of A and o. In your code, you
will need to make NV large. How large will be up to you - you will face a trade-off between
running time and accuracy. Compare the transition density of the Ornstein-Uhlenbeck
process with (x, t), the transition density of the standard Brownian motion.
(iii) The exact transition density for the Ornstein-Uhlenbeck process starting at zero is given by
p(x, t) =
e 2 v (t ) ,
u (t ) =
0 2
27 ( 1 - e-2xt).
Show that (3) satisfies the following partial differential equation
1 2 22
2 2x2 P(x, t) +. (xp(x, t)) = P(x, t).
(iv) Find lime->op(x, t) and limt->cop(x, t). How would you describe the long-run behavior of
an Ornstein-Uhlenbeck process?

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