Introduction to Stochastic Calculus tulo 1 Introduction to Stochastic Calculus Exercise 1.1 Show...
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Cap´ ıtulo 1 Introduction to Stochastic Calculus Exercise 1.1 Show that a sum of martingales is still an martingale. Exercise 1.2 Let M = {M t ,t ∈ IN 0 } be a martingale, with E[M 2 t ] < ∞, ∀t, defined on the filtred probability space (Ω, F , F , P ), where F = {F t , t∈IN 0 } is the filtration. Let η = {η t ,t ∈ IN 0 } be a stochastic process defined on the same probability space with the following property: η t is F t-∞ -measurable, such that E[η 2 t ] < ∞, ∀t. Show that the process N = {N t ,t ∈ IN 0 } defined by N t = N 0 + t X k=1 η k (M k - M k-1 ) is a martingale as long as N 0 is F 0 -measurable. Exercise 1.3 Let X be a random variable such that E P [|X |] < ∞, where (Ω, F , F , P ) is its filtrered probability space. Show that the stochastic process M = {M t ,t ∈ IN 0 }, where M t = E P [X |F t ], t≥0 is a martingale. Exercise 1.4 Suppose that S follows a geometric Brownian motion, with drift μ and volatility σ: dS = μSdt + σSdW (where W is the Brownian motion). Show that S n also follows a geometric Brownian motion. Exercise 1.5 Let X and Y be two processes adapted to the same Brownian motion W , such that: dX = μ X dt + σ X dW ; dY = μ Y dt + σ Y dW. Apply Ito’s formula to 1 2 ((X + Y ) 2 - X 2 - Y 2 )= XY to derive d(XY ). Now suppose that X and Y are adpated to two independent Brownian motions. Then prove that in this case the chain rule for the Ito’s calculus is the same as in the deterministic chain rule. Exercise 1.6 Show that if B is a zero-volatility process and X is any stochastic process, then d(BX )= BdX + XdB. 1
Transcript of Introduction to Stochastic Calculus tulo 1 Introduction to Stochastic Calculus Exercise 1.1 Show...
Capıtulo 1
Introduction to Stochastic Calculus
Exercise 1.1 Show that a sum of martingales is still an martingale.
Exercise 1.2 Let M = Mt, t ∈ IN0 be a martingale, with E[M2t ] < ∞,∀t, defined on the
filtred probability space (Ω,F ,F ,P), where F = Ft,t ∈ IN′ is the filtration. Let η = ηt, t ∈IN0 be a stochastic process defined on the same probability space with the following property:
ηt is Ft−∞-measurable, such that E[η2t ] < ∞,∀t. Show that the process N = Nt, t ∈ IN0defined by
Nt = N0 +t∑
k=1
ηk(Mk −Mk−1)
is a martingale as long as N0 is F′-measurable.
Exercise 1.3 Let X be a random variable such that EP [|X|] < ∞, where (Ω,F ,F ,P) is its
filtrered probability space. Show that the stochastic process M = Mt, t ∈ IN0, where
Mt = EP [X|Ft], t ≥ ′
is a martingale.
Exercise 1.4 Suppose that S follows a geometric Brownian motion, with drift µ and volatility
σ:
dS = µSdt+ σSdW
(where W is the Brownian motion). Show that Sn also follows a geometric Brownian motion.
Exercise 1.5 Let X and Y be two processes adapted to the same Brownian motion W , such
that:
dX = µXdt+ σXdW ; dY = µY dt+ σY dW.
Apply Ito’s formula to 12((X + Y )2 −X2 − Y 2) = XY to derive d(XY ).
Now suppose that X and Y are adpated to two independent Brownian motions. Then prove
that in this case the chain rule for the Ito’s calculus is the same as in the deterministic chain
rule.
Exercise 1.6 Show that if B is a zero-volatility process and X is any stochastic process, then
d(BX) = BdX +XdB.
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Exercise 1.7 Use Ito’s formula to compute E[W 4(t)], where W is a Brownian motion.
Exercise 1.8 Compute the stochastic differential of X when:
a) X(t) =∫ t0 g(s)dW (s) (g is a stochastic process adapted to W )
b) X(t) = eαW (t)
c) X(t) = eαY (t) where Y follows the following SDE:
dY = µdt+ σdW
d) X(t) = Y 2(t), where Y follows the following SDE:
dY = αY dt+ σY dW
Exercise 1.9 Let g be a deterministic function and W a Brownian motion; define X as follows:
X(t) =
∫ t
0g(s)dW (s).
Prove that for a ∈ IR:
E[eiyX(t) = e−y2
2
∫ t0 g
2(s)ds
and therefore X(t) ∼ N(0,∫ t0 g
2(s)ds).
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