Math Problems Vault

Problem 18.1

Let $\left(B_{t}\right)_{t \geqslant 0}$ be $B M^{1}$ . Use Itô's formula to obtain representations of

X_{t} = \int _{0}^{t} \exp \left(B_{s}\right) d B_{s} \quad \text{ and } \quad Y_{t} = \int _{0}^{t} B_{s} \exp \left(B_{s}^{2}\right) d B_{s}

which do not contain Itô integrals.

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Problem 18.2

Let $\Pi = \left\{ 0 = t_{0} < t_{1} < \cdots < t_{N} = T\right\}$ be a generic partition of the interval $[0, T]$ and denote by $\left|\Pi \right| = \max_{j}\left|t_{j} - t_{j - 1}\right|$ the mesh of the partition. Show that the limit

\lim _{\left|\Pi \right| \rightarrow 0} \sum _{j = 1}^{N}\left(t_{j} - t_{j - 1}\right)^{y}

is $\infty , T$ or 0 according to $\gamma < 1, \gamma = 1$ or $\gamma > 1$ .

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Problem 18.3

Show that the limits (18.5) and (18.6) actually hold in $L^{2}(\mathbb {P})$ and uniformly for $T$ from compact sets.

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Problem 18.4

Give a direct proof for the identity

\begin{aligned} \mathbb {E} & {\left[\left(\sum _{l = 1}^{N} g\left(B_{t_{l - 1}}\right)\left[\left(B_{t_{l}} - B_{t_{l - 1}}\right)^{2} - \left(t_{l} - t_{l - 1}\right)\right]\right)^{2}\right] } \\ & = \mathbb {E}\left[\sum _{l = 1}^{N}\left|g\left(B_{t_{l - 1}}\right)\right|^{2}\left[\left(B_{t_{l}} - B_{t_{l - 1}}\right)^{2} - \left(t_{l} - t_{l - 1}\right)\right]^{2}\right] . \end{aligned}

appearing in the proof of Lemma 18.4.

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Problem 18.5

The following exercise contains an alternative proof of Itô's formula (18.1) for a one - dimensional Brownian motion $\left(B_{t}\right)_{t \geqslant 0}$ .

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(a)

Let $f \in \mathcal{C}^{1}(\mathbb {R})$ . Show that for $\Pi = \left\{ t_{0} = 0 < t_{1} < \cdots < t_{n} = t\right\}$ and suitable intermediate values $\xi_{l}$

\begin{gathered} B_{t} f\left(B_{t}\right) = \sum _{l} B_{t_{l}}\left(f\left(B_{t_{l}}\right) - f\left(B_{t_{l - 1}}\right)\right)+\sum _{l} f\left(B_{t_{l - 1}}\right)\left(B_{t_{l}} - B_{t_{l - 1}}\right) \\ +\sum _{l} f^{\prime }\left(\xi _{l}\right)\left(B_{t_{l}} - B_{t_{l - 1}}\right)^{2} . \end{gathered}

Conclude from this that the differential expression $B_{t} d f\left(B_{t}\right)$ is well - defined.

(b)

Show that $d\left(B_{t} f\left(B_{t}\right)\right) = B_{t} d f\left(B_{t}\right)+f\left(B_{t}\right) d B_{t}+f^{\prime }\left(B_{t}\right) d t$ .

(c)

Use (b) to find a formula for $d B_{t}^{n}, n \geqslant 2$ , and show that for every polynomial $p(x)$ Itô's formula holds:

d p\left(B_{t}\right) = p^{\prime }\left(B_{t}\right) d B_{t}+\frac{1}{2} p^{\prime \prime }\left(B_{t}\right) d t .

(d)

Show the following stability result for Itô integrals:

L^{2}(\mathbb {P}) - \lim _{n \rightarrow \infty } \int _{0}^{T} g_{n}(s) d B_{s} = \int _{0}^{T} g(s) d B_{s}

for any sequence $\left(g_{n}\right)_{n \geqslant 1}$ which converges in $\mathcal{L}_{T}^{2}$ .

(e)

Use a suitable version of Weierstraß' approximation theorem and the stopping technique from the proof of Theorem 18.1, Step $4^{\circ }$ , to deduce from c) Itô's formula for $f \in \mathcal{C}^{2}$ .

Problem 18.6

(a)(b)

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(a)

Use the (two - dimensional, deterministic) chain rule $d(F \circ G) = F^{\prime } \circ G d G$ to deduce the formula for integration by parts for Stieltjes integrals:

\int _{0}^{t} f(s) d g(s) = f(t) g(t) - f(0) g(0) - \int _{0}^{t} g(s) d f(s) \quad \text{ for all } f, g \in \mathcal{C}^{1}([0, \infty ), \mathbb {R})

(b)

Use the Itô formula for a Brownian motion $\left(b_{t}, \beta_{t}\right)_{t \geqslant 0}$ in $\mathbb {R}^{2}$ to show that

\int _{0}^{t} b_{s} d \beta _{s} = b_{t} \beta _{t} - \int _{0}^{t} \beta _{s} d b_{s}, \quad t \geqslant 0

What happens if $b$ and $\beta$ are not independent?

Problem 18.7

Prove the following time - dependent version of Itô's formula: let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $f:[0, \infty ) \times \mathbb {R} \rightarrow \mathbb {R}$ be a function of class $\mathcal{C}^{1,2}$ . Then

f\left(t, B_{t}\right) - f(0,0) = \int _{0}^{t} \frac{\partial f}{\partial x}\left(s, B_{s}\right) d B_{s}+\int _{0}^{t}\left(\frac{\partial f}{\partial t}\left(s, B_{s}\right)+\frac{1}{2} \frac{\partial ^{2} f}{\partial x^{2}}\left(s, B_{s}\right)\right) d s .

Prove and state the $d$ - dimensional counterpart.

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Problem 18.8

Prove Theorem 5.6 using Itô's formula.

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Problem 18.9

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ . Use Itô's formula to verify that the following processes are martingales:

X_{t} = e^{t / 2} \cos B_{t} \quad \text{ and } \quad Y_{t} = \left(B_{t}+t\right) \exp \left( - B_{t} - t / 2\right) \text{. }

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Problem 18.10

(a)(b)

Let $B_{t} = \left(b_{t}, \beta_{t}\right), t \geqslant 0$ be a $\mathrm{BM}^{2}$ and set $r_{t}: = \left|B_{t}\right| = \sqrt{b_{t}^{2}+\beta_{t}^{2}}$ .

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(a)

Show that the stochastic integrals $\int_{0}^{t} b_{s} / r_{s} d b_{s}$ and $\int_{0}^{t} \beta_{s} / r_{s} d \beta_{s}$ exist.

(b)

Show that $W_{t}: = \int_{0}^{t} b_{s} / r_{s} d b_{s}+\int_{0}^{t} \beta_{s} / r_{s} d \beta_{s}$ is a $\mathrm{BM}^{1}$ .

Problem 18.11

Let $B_{t} = \left(b_{t}, \beta_{t}\right), t \geqslant 0$ be a $\mathrm{BM}^{2}$ and $f(x+i y) = u(x, y)+i v(x, y), x, y \in \mathbb {R}$ , an analytic function. If $u_{x}^{2}+u_{y}^{2} = 1$ , then $\left(u\left(b_{t}, \beta_{t}\right), v\left(b_{t}, \beta_{t}\right)\right), t \geqslant 0$ , is a $\mathrm{BM}^{2}$ .

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Problem 18.12

Show that the $d$ - dimensional Itô formula remains valid if we replace the real - valued function $f: \mathbb {R}^{d} \rightarrow \mathbb {R}$ by a complex function $f = u+i v: \mathbb {R}^{d} \rightarrow \mathbb {C}$ .

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Problem 18.13

(a)(b)(c)(d)

Let $\chi \in \mathcal{C}_{c}^{\infty }( - 1,1)$ with $0 \leqslant \chi \leqslant 1, \chi (0) = 1$ and $\int \chi (x) d x = 1$ . Set $\chi_{n}(x): = n \chi (n x)$ .

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(a)

Show that supp $\chi_{n} \subset [ - 1 / n, 1 / n]$ and $\int \chi_{n}(x) d x = 1$ .

(b)

Let $f \in \mathcal{C}(\mathbb {R})$ and $f_{n}: = f \star \chi_{n}$ . Show that $\left|\partial^{k} f_{n}(x)\right| \leqslant n^{k} \sup_{y \in \mathbb {B}(x, 1 / n)}\left|f(y)\right|\left\| \partial^{k} \chi \right\|_{L^{1}}$ .

(c)

Let $f \in \mathcal{C}(\mathbb {R})$ uniformly continuous. Show that $\lim_{n \rightarrow \infty }\left\| f \star \chi_{n} - f\right\|_{\infty } = 0$ . What can be said if $f$ is continuous but not uniformly continuous?

(d)

Let $f$ be piecewise continuous. Show that $\lim_{n \rightarrow \infty } f \star \chi_{n}(x) = f(x)$ at all $x$ where $f$ is continuous.

Problem 18.14

Show that $\beta_{t} = \int_{0}^{t} \operatorname {sgn}\left(B_{s}\right) d B_{s}$ is a $\mathrm{BM}^{1}$ .

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Problem 18.15

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}, \Phi \in \mathcal{C}_{b}^{2}(\mathbb {R}), \Phi (0) = 0$ and $f, g \in \mathcal{L}_{T}^{2}$ for some $T > 0$ . Show that for all $t \in [0, T]$

\begin{aligned} & \mathbb {E}\left[\int _{0}^{t} f(r) d B_{r} \cdot \Phi \left(\int _{0}^{t} g(r) d B_{r}\right)\right] \\ & = \mathbb {E}\left[\int _{0}^{t} f(s) g(s) \Phi ^{\prime }\left(\int _{0}^{s} g(r) d B_{r}\right) d s\right] \\ & +\frac{1}{2} \mathbb {E}\left[\int _{0}^{t} \int _{0}^{s} f(r) d B_{r} \Phi ^{\prime \prime }\left(\int _{0}^{s} g(r) d B_{r}\right) g^{2}(s) d s\right] . \end{aligned}

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Problem 18.16

Let $\sigma_{\Pi }, b_{\Pi } \in \mathcal{S}_{T}$ which approximate $\sigma , b \in \mathcal{L}_{T}^{2}$ as $\left|\Pi \right| \rightarrow 0$ , and consider the Itô processes $X_{t}^{\Pi } = X_{0}+\int_{0}^{t} \sigma_{\Pi }(s) d B_{s}+\int_{0}^{t} b_{\Pi }(s) d s$ and $X_{t} = X_{0}+\int_{0}^{t} \sigma (s) d B_{s}+\int_{0}^{t} b(s) d s$ . Adapt the proof of Lemma 18.5 to show that

\int _{0}^{T} g\left(X_{s}^{\Pi }\right) d X_{s}^{\Pi } \xrightarrow [\left|\Pi \right| \rightarrow 0]{\mathbb {P}} \int _{0}^{T} g\left(X_{s}\right) d X_{s} \quad \text{ for all } g \in \mathcal{C}_{b}(\mathbb {R})

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Problem 18.17

Let $X_{t} = M_{t}+A_{t}$ and $X_{t} = M_{t}^{\prime }+A_{t}^{\prime }$ where $M, M^{\prime }$ are continuous local martingales and $A, A^{\prime }$ are continuous processes which have locally a.s. bounded variation. Show that $M - M^{\prime }$ and $A - A^{\prime }$ are a.s. constant.

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