ECE478 Financial Signal Processing

Problem Set IV: Stochastic Calculus

Notation here follows Shreve, Stochastic Calculus for Finance, vols. I & II, Springer, 2004.

Theoretical Problems

1. Let Ft be the Öltration generated by a Wiener process W (t). Let R (t) be the interest

rate process used to deÖne the discount process D (t). Assume there exists a unique

risk-neutral measure, leading to the Wiener process W~ (t) with respect to P~. If V (T)

is a random variable that is FT -measurable, and V (t) is deÖned via:

V (t) = 1

D (t)

E~ (D (T) V (T)jFt)

then D (t) V (t) is a martingale.

(a) Suppose V (T) > 0 a.s. Show that V (t) > 0 a.s. (from its deÖnition above).

(b) Show that there exists an adapted process ( ~ t) such that:

dV (t) = R (t) V (t) dt +

( ~ t)

D (t)

dW~ (t)

Hint: Start with a formula for d (D (t) V (t)) as per the martingale representation

theorem, then as you expand this out recognize that dV dt = 0.

(c) Show that there exists an adapted process (t) such that we can write:

dV (t) = R (t) V (t) dt + (t) V (t) dW~ (t)

By the way, (t) can be random and in particular it is Öne if the formula for

(t) you derived involves V (t). The point is there is SOME process you can

put there that works! How did we use strict positivity? (Think of D (t), V (t)

as continuous processes; D (t) is intrinsically positive, but what happens if V (t)

can take on negative as well as positive values?) This shows that V (t) is a

generalized geometric Brownian motion process. The point of this problem is

that every strictly positive asset is a generalized geometric Brownian motion.

2. Let X (t); Y (t) be ItÙ processes given by:

dX (t) = a (t) dt + b (t) dW (t)

dY (t) = c (t) dt + d (t) dW (t)

where a; b; c; d are adapted processes. Assume Y (t) > 0 a.s., and V (t) = X (t) =Y (t).

Obtain an SDE satisÖed by V (t), simpliÖed so it has the above form (i.e., in the form

of an ItÙ process). Note that X (t); Y (t), but not dX (t) or dY (t), can appear in your

Önal expression for dV (t).