Stochastic Processes and Financial Mathematics
(part two)
13.5 Exercises on Chapter 13
In all the following questions, \(B_t\) denotes a Brownian motion and \(\mc {F}_t\) denotes its generated filtration.
On Ito’s formula
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13.1 Write the following equations in integral form.
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(a) \(dX_t=2t\,dt+B_t\,dB_t\) over the time interval \([0,t]\),
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(b) \(dY_t= t\,dt\) over the time interval \([t,T]\).
Write down a differential equation satisfied by \(Y_t\). Is \(X_t\) differentiable?
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13.2 Apply Ito’s formula to find an expression for the stochastic differential of \(Z_t=t^3X_t\), where \(dX_t=\alpha \,dt+\beta \,dB_t\) and \(\alpha ,\beta \in \R \) are deterministic constants.
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13.3 In each case, find the stochastic differential \(dZ_t\), with coefficients in terms of \(t\), \(B_t\) and \(Z_t\).
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(a) \(Z_t=tB_t^2\)
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(b) \(Z_t=e^{\alpha t}\), where \(\alpha >0\) is a deterministic constant.
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(c) \(Z_t=(X_t)^{-1}\), where \(dX_t=t^2\,dt+B_t\,dB_t\).
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(d) \(Z_t=\sin (X_t)\), where \(dX_t=\cos (X_t)\,dt+\cos (X_t)\,dB_t\).
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13.4 Find \(dF_t\) where \(F_t=B_t^n\), where \(n\geq 2\). Hence, show that
\[\E [B_t^n]=\frac {n(n-1)}{2}\int _0^t\E [B_u^{n-2}]\,du.\]
Check that this is consistent with the formula obtained in part (c) of exercise 11.4.
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13.5 Show that the following processes are martingales:
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(a) \(X_t=e^{t/2}\cos (B_t)\),
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(b) \(Y_t=(B_t+t)e^{-B_t-t/2}\).
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13.6 Use Ito’s formula to show that
\[tB_t=\int _0^t u\,dB_u+\int _0^t B_u\,du.\]
On stochastic differential equations
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13.7 Suppose that \(X_t\) satisfies \(X_0=1\) and \(dX_t= (2+2t)\,dt+ B_t\,dB_t\).
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(a) Find \(\E [X_t]\) as a function of \(t\).
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(b) Let \(Y_t=X_t^2\). Calculate \(dY_t\) and hence find \(\var (X_t)\) as a function of \(t\).
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(c) Suppose that \(X'_t\) satisfies \(X'_t=1\) and \(dX'_t=(2+2t)\,dt+G_t\,dB_t\), where \(G_t\) is some unknown function of \(t\) and \(B_t\). Based on your solutions to (a) and (b), comment on whether \(X_t\) and \(X'_t\) are likely to have the same mean and/or variance.
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13.8 Suppose that \(X_t\) satisfies \(X_0=1\) and \(dX_t=\alpha X_t\,dt+\sigma _t\,dB_t\), where \(\alpha \) is a deterministic constant and \(\sigma _t\) is a stochastic process. Find \(\E [X_t]\) as a function of \(t\).
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13.9 Suppose that \(X_t\) satisfies \(X_t=1\) and \(dX_t=X_t\,dB_t\). Show that \(\var (X_t)=e^t-1\).
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13.10 \(\msconly \) Let \((X_t)\) be the Ornstein-Uhlenbeck process introduced in Section 13.3. Calculate the autocovariance function \(C(s,t)=\cov (X_t,X_s)\), where \(s\leq t\), in terms of the parameters \(\mu ,\theta \) and \(\sigma \). Hence show that \(\var (X_t)\to \frac {\sigma ^2}{2\theta }\) as \(t\to \infty \).
Hint: For \(s\leq t\) and a deterministic \(f:\R \to \R \), the random variables \(\int _0^{s} f(u) dB_u\) and \(\int _s^t f(u) dB_u\) are independent. Can you explain why?
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13.11 Consider the stochastic differential equation
\[dX_t=3X_t^{1/3}\,dt+3X_t^{2/3}\,dB_t\]
with the initial condition \(X_0=0\).
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(a) Show that \(Z_t=B_t^3\) is a solution of this equation.
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(b) Can you think of another solution?
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13.13 Check that (13.14) is a solution of (13.13).
Challenge Questions
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(a) Let \(Y\) be an \(\mc {F}_T\) measurable random variable such that \(Y\in L^2\). Show that \(M_t=\E [Y\|\mc {F}_t]\) is a martingale for \(t\in [0,T]\).
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(b) You may assume that the stochastic process \(M_t\) in part (a) is continuous. Hence, for any given \(Y\in \mc {F}_T\), the martingale representation theorem, from Section 13.4, tells us that there exists a stochastic process \(h_t\) such that
\[M_t=M_0+\int _0^t h_u\,dB_u.\]
for all \(t\in [0,T]\). Find an explicit formula for \(h_t\) in each of the following cases.
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(i) \(Y=B_T^2\)
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(ii) \(Y=B_T^3\)
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(iii) \(Y=e^{\sigma B_T}\), where \(\sigma >0\) is a deterministic constant.
Hint: Use the various connections that we’ve already found (in lemmas, exercises, examples, etc) between Brownian motion and martingales. For example, for (i) you might look at the formula for \(dZ_t\) where \(Z_t=B_t^2\).
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