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Let $(B_t)$ be Brownian motion in ${\mathbb{R}}^3$ started at a nonzero point. The function $h(x)=|x{|}^{-1}$ is a positive harmonic function on ${\mathbb{R}}^3\setminus \{0\}$ because its Laplacian vanishes there. Stopping on annuli and applying Itô formula shows that $h(B_t)$ is a local martingale. Letting the inner boundary shrink to zero also shows that three-dimensional Brownian motion does not hit the origin.

A positive local martingale is a supermartingale, so $h(B_t)$ is $L^1$-bounded by $h(B_0)$. The upcrossing proof from part (c), which applies verbatim to a positive supermartingale, therefore gives a finite almost-sure limit

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For $u\in C^2(\overline{D})$, Itô formula shows thatis a martingale. Take expectations and let $t\to \infty$. The function $u$ is bounded on the compact set $\overline{D}$, while $\Delta u$ is bounded and ${\mathbb{E}}_{x}T<\infty$ by part (a). The dominated convergence theorem therefore gives Dynkin formula for Brownian motion

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Apply Itô formula to $Z_t^{(A)}=\exp (A{X}_t-A^{2}t/2)$. Its semimartingale decomposition isThe second term is a continuous finite-variation process. Since $Z^{(A)}$ is assumed to be a local martingale, uniqueness of the semimartingale decomposition makes this term identically zero. Both $A$ and $Z^{(A)}$ are nonzero, so the quadratic variation of $X$ is $[X{]}_t=t$.

The Lévy characterization of Brownian motion now says that $X_t-X_0$ is a Brownian motion. Consequentlyis a constant multiple of an exponential Brownian martingale. It is therefore a true martingale for every $a\in \mathbb{R}$.

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Use first the test function $f(x)=x$. The assumed martingale problem says thatis a continuous local martingale. Next use $f(x)=x^2$ to see thatis a local martingale. On the other hand, Itô formula applied to $X=X_0+\int b(X_s)ds+Y$ shows thatis a local martingale. Their difference is both a continuous local martingale and a finite-variation process, so

Part (b), with $H_s=\sigma (X_s{)}^2>0$, supplies a Brownian motion $W$ such thatThereforeso $X$ is a weak solution of a stochastic differential equation to the stated equation.

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PutThen $f_n\to |\cdot |$ uniformly, $f_n^{\prime }(x)=\tanh (nx)$, and $f_n^{\prime \prime }(x)=n{\mathrm{sech}}^2(nx)$. Itô formula givesThe bounded predictable integrands $\tanh (nX)$ converge pointwise to $\mathrm{sgn}(X)$, with $\mathrm{sgn}(0)=0$. Part (c) therefore makes the stochastic integrals converge u.c.p. The left-hand side converges u.c.p. to $|X|$, so the increasing continuous processesalso converge u.c.p. Their limit $A$ has a continuous increasing version: extract almost-sure locally uniform convergence from each compact interval and use a diagonal argument. We obtainthe Tanaka formula with $A=L_t^0(X)$. It expresses $|X|$ as a continuous local martingale plus a continuous finite-variation process, so $|X|$ is a continuous semimartingale.

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In the Black-Scholes model,Conditioning on ${\mathcal{F}}_t$ and using the moment-generating function of the independent Gaussian increment $W_T-W_t$ givesThe function $c$ satisfies the zero-rate Black-Scholes equation, so Itô formula leaves only its stochastic term:Consequently the required delta hedge is

Solved by gpt-5.6-sol high.