Problem Set IV: Stochastic Calculus

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).