Show that B d (t) as defined in Problem 21.1 is almost surely a continuous process. Problem
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Show that Bd(t) as defined in Problem 21.1 is almost surely a continuous process.
Problem 21.1
Suppose that the tenor structure T−1, T0,..., TN is equispaced, withΔt = Tn − Tn−1 and T−1 = 0. For 0 ≤ t ≤ TN, the discretely rebalanced bank account Bd(t) has present value.
with
where θ(x) is again the Heaviside-θ-function. Use Theorem 17.2 on page 390 to prove that Xt has the dynamics
Theorem 17.2:
Let Xt be a jump-diffusion process of the kind
with deterministic or stochastic jumps Yt. If z(x, t) is a sufficiently smooth function, then the stochastic process Zt = z(Xt , t) is also a jump-diffusion and has the dynamics
where the partial derivatives are to be evaluated at x = Xt.
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