STOCHASTIC DIFFERENTIAL EQUATIONS.
PROBLEM SET 3
DUE BY THE END OF THE SPRING SESSION
Problem 1. Suppose that Xtx is a one dimensional diffusion that satisfies
dXtx = b(Xtx )dt + σ(Xtx )dBt ,
X0x = x,
(Bt ) is a standard one dimensional Brownian motion and b(x), σ(x) are regular enough so
there exists a weak solution of the above equation. We can assume that they are both
Lipschitz. In addition, let σ(x) ≥ σ0 > 0 for all x ∈ R.
x
Suppose that τ[a,b]
is the exit time of the diffusion from the interval [a, b], where a < x < b.
x .
i) Compute M (x) := Eτ[a,b]
x
x
= a] and P[Xτ[a,b]
= b].
ii) Compute P[Xτ[a,b]
Problem 2. Suppose that the price of a stock (St ) satisfies the following stochastoc
differential equation
dSt = µSt dt + σSt dBt , S0 := S,
where (Bt ) is a one dimensional Brownian motion. A derivative financial instrument (contingency claim) is an instrument whose value Vt depends, at a given time t, on the stock value
St and equals Vt := V (t, St ). Usually at certain final time T > 0 (expiration of the contract)
the value function is given and then VT = f (ST ) The main problem is to find a deterministic
function V (t, S) such that Vt = V (t, St ).
i) Note that the portfolio containing a contingency claim and −∂S V (t, S) units of the
stock is riskless, i.e. its value given by
Z t
(0.1)
Πt := Π0 +
[dVs − ∂S V (s, Ss )dSs ]
0
is a deterministic function.
ii) Suppose that there exists a bank acount that is riskless and returns yield r > 0, i.e.
any capital A deposited, returns AerT , after time T > 0. Show, using an arbitrage
argument, that
Πt = Π0 ert
As a corollary derive the equation on V (t, S).
iii) Prove that
Z
√
σ2
2
−rt
V (t, S) = e
f S exp
r−
t + σ tx
e−x /2 dx.
2
R
iv) The European call option has at the expiration time T > 0 value f (S) := (S − K)+ ,
where K is the striking price of the option. Compute V (t, S).
1
2
DUE BY THE END OF THE SPRING SESSION
Suppose that the price of a stock (St ) satisfies the following stochastoc differential equation
dSt = µSt dt + σSt dBt ,
S0 := S,
where (Bt ) is a one dimensional Brownian motion. A derivative financial instrument (contingency claim) is an instrument whose value Vt depends, at a given time t, on the stock value
St and equals Vt := V (t, St ). Usually at certain final time T > 0 (expiration of the contract)
the value function is given and then VT = f (ST ) The main problem is to find a deterministic
function V (t, S) such that Vt = V (t, St ).
i) Note that the portfolio containing a contingency claim and −∂S V (t, S) units of the
stock is riskless, i.e. its value given by
Z t
(0.2)
Πt := Π0 +
[dVs − ∂S V (s, Ss )dSs ]
0
is a deterministic function.
ii) Suppose that there exists a bank acount that is riskless and returns yield r > 0, i.e.
any capital A deposited, returns AerT , after time T > 0. Show, using an arbitrage
argument, that
Πt = Π0 ert
As a corollary derive the equation on V (t, S).
iii) Prove that
Z
√
σ2
2
−rt
f S exp
r−
t + σ tx
e−x /2 dx.
V (t, S) = e
2
R
iv) The European call option has at the expiration time T > 0 value f (S) := (S − K)+ ,
where K is the striking price of the option. Compute V (t, S).
Problem 3) Recall that τx is the hitting time of level x of a one dimensional Brownian
motion.
i) Show that the process (τx )x≥0 satisfies
τx1 , τx2 − τx1 , . . . , τxn − τxn−1
ii)
iii)
vi)
v)
are independent for any 0 < x1 < . . . < xn . In addition, for a fixed h ≥ 0, the law of
τx+h − τx does not depend on x. We say that the process has independent and time
homogeneous increments. The processes of such a class are called Levy processes.
Show that any Levy process is Markovian. Determine its transition semigroup.
Show that for a given a > 0 the law of (a2 τax ) is identical with that (τx ). The process
self-similar with the Hurst exponent H = 2.
Find the density of τx for a given x and its generating function φ(λ) := Ee−λτx .
Show that the process (τx ) is increasing and right continuous (i.e. cadlag). An
increasing Levy process is called a subordinator.
Problem 4) Suppose that (Bt ) is a standard Brownian motion and (Yt ) independent of it
subordinator consider in the previous exercise. Show that Xt := BYt is a Levy process. Find
the density of Xt for a given t and its generating function φ(λ) := Ee−λXt . This process is
called a Cauchy process.
SDE PROBLEMS
3
Problem 5) Suppose that µ is Borel measure on a topological space (X, T ). The set
supp µ consisting of such points x that there exists a neighborhood U 3 x such that µ(U) > 0
is called a (topological) support of µ.
i) Show that supp µ is closed.
ii) Show that the function c(t) ≡ 0 belongs to the support of the Wiener measure W on
C[0, 1].
iii) Show that a given C 1 function f (t) belongs to the support of the Wiener measure.
vi) Prove that supp W = C[0, 1].
(1)
(2)
Exercise 6) Suppose that Bt = (Bt , Bt ) is a standard 2 dimensional Brownian motion.
Let τ be the exist time of e2 + Bt from the upper half-plane H := [(x, y) : y > 0], where
(1)
e2 := (0, 1). Let X := Bτ . Compute Ee−λτ for any λ > 0. Find the distribution
of X.
a-ij-3
x
Exercise 7) Assume that the generator L of a diffusion (Xt ), given by (??), satisfies
x is an exit time of the diffusion from a bounded
assumptions A1 − A3 . Suppose that τD
2+α
domain D that is C
-regular for some α > 0.
Show that there exist +∞ > λ∗ ≥ λ∗ > 0 such that
x
E exp {λτD
} < +∞,
for λ < λ∗ and
∀ x ∈ D.
x
} = +∞,
E exp {λτD
∀ x ∈ D.
for all λ > λ∗ .
Exercise 8) Prove that in the previous problem λ∗ = λ∗ . In addition there exists u(x) > 0,
x ∈ D such that
−Lu(x) = λ∗ u(x),
u(x) = 0, x ∈ ∂D.