Exercise 24.1 Consider the Vasiek model, where we always assume that a > 0. (a) Solve...

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Exercise 24.1 Consider the Vasiček model, where we always assume that a > 0. (a) Solve the Vasiček SDE explicitly, and determine the distribution of r(t). Hint: The distribution is Gaussian (why?), so it is enough to compute the expected value and the variance. (b) As too, the distribution of r(t) tends to a limiting distribution. Show that this is the Gaussian distribution N[b/a, o/√2a]. Thus we see that, in the limit, r will indeed oscillate around its mean reversion level b/a. (c) Now assume that r(0) is a stochastic variable, independent of the Wiener process W, and by definition having the Gaussian distribution obtained in (b). Show that this implies that r(t) has the limit distribution in (b), for all values of t. Thus we have found the stationary distribution for the Vasiček model. (d) Check that the density function of the limit distribution solves the time invariant Fokker-Planck equation, i.e. the Fokker-Planck equation with the-term equal to zero. Exercise 24.1 Consider the Vasiček model, where we always assume that a > 0. (a) Solve the Vasiček SDE explicitly, and determine the distribution of r(t). Hint: The distribution is Gaussian (why?), so it is enough to compute the expected value and the variance. (b) As too, the distribution of r(t) tends to a limiting distribution. Show that this is the Gaussian distribution N[b/a, o/√2a]. Thus we see that, in the limit, r will indeed oscillate around its mean reversion level b/a. (c) Now assume that r(0) is a stochastic variable, independent of the Wiener process W, and by definition having the Gaussian distribution obtained in (b). Show that this implies that r(t) has the limit distribution in (b), for all values of t. Thus we have found the stationary distribution for the Vasiček model. (d) Check that the density function of the limit distribution solves the time invariant Fokker-Planck equation, i.e. the Fokker-Planck equation with the-term equal to zero.

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ANSWER a The Vasicek model is given by the following stochastic differential equation SDE drt ab rtdt dWt where rt represents the shortterm interest rate at time t a is the speed of mean reversion b i... View the full answer