Show that B_d(t) as defined in Problem 21.1 is almost surely a continuous process.

Problem 21.1

Suppose that the tenor structure T₋₁, T₀,..., T_N is equispaced, withΔt = T_n − T_n−1 and T₋₁ = 0. For 0 ≤ t ≤ T_N, the discretely rebalanced bank account B_d(t) has present value.

with

where θ(x) is again the Heaviside-θ-function. Use Theorem 17.2 on page 390 to prove that X_t has the dynamics

Theorem 17.2:

Let X_t be a jump-diffusion process of the kind

with deterministic or stochastic jumps Y_t. If z(x, t) is a sufficiently smooth function, then the stochastic process Z_t = z(X_t , t) is also a jump-diffusion and has the dynamics

where the partial derivatives are to be evaluated at x = X_t.

Transcribed Image Text:

Ba(1) = X,PN,(1), %3D N-1 N, X, = [] and N, = 0(t - T„), n=0 P„(Tn-1) n=0