Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 202 (Stochastic Calculus)

Use a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t),\mathbb{P})$, and put ${\mathcal{F}}_{0-}={\mathcal{F}}_0$. The predictable sigma-algebra on $\Omega \times [0,\infty )$ isEquivalently it is generated by left-continuous adapted processes. A real-valued previsible process is a function $(\omega ,t)\mapsto H_t(\omega )$ that is measurable for $\mathcal{P}$ and the Borel sigma-algebra of $\mathbb{R}$.

A simple predictable process has the formwhere $h_{-1}$ is bounded and ${\mathcal{F}}_0$-measurable, and each $h_i$ is bounded and ${\mathcal{F}}_{t_i}$-measurable. Set $H=0$ after $t_n$; allowing a final infinite interval gives an equivalent dense class. The values may be restricted to finitely many values if desired, by approximating bounded measurable functions by simple functions. The initial-time term matters for arbitrary finite measures on $\mathcal{P}$: a measure can charge time zero, even though the stochastic integral against a continuous martingale does not.

For fixed $u>0$, define the left-limit sigma-algebra by ${\mathcal{F}}_{u-}=\sigma ({\bigcup }_{s<u}{\mathcal{F}}_s)$. The section at $u$ of a generating rectangle $A\times (s,t]$ of the predictable sigma-algebra is $A$ when $s<u\le t$, and empty otherwise. In the first case $A\in {\mathcal{F}}_s\subseteq {\mathcal{F}}_{u-}$. Sections of the time-zero generators are empty for $u>0$.

The collection of subsets of sample-time space whose section at $u$ belongs to ${\mathcal{F}}_{u-}$ is a sigma-algebra: sections commute with complements and countable unions. It therefore contains all of $\mathcal{P}$. Apply this to $\{(\omega ,t):H_t(\omega )\in E\}$ for every Borel set $E$. Thus predictable sections at deterministic times satisfyAt $u=0$ the same argument gives ${\mathcal{F}}_0$-measurability, using the convention ${\mathcal{F}}_{0-}={\mathcal{F}}_0$. The PDF asks for information strictly before $u$; the TeX aid drops the minus sign.

Let $V$ be the closure of the simple predictable processes in the Hilbert space $L^2(\mathcal{P},\mu )$. Common refinement of deterministic partitions makes these processes a vector space. To prove density of simple predictable processes for finite measures, suppose $g\in V^{\perp }$. Since $\mu$ is a finite measure, the Cauchy-Schwarz inequality gives $g\in L^1(\mu )$, sodefines a finite signed measure. The indicator of each predictable generating rectangle, including each time-zero rectangle, is a simple predictable process; hence $\nu$ vanishes on these rectangles.

Finite intersections of the rectangles are again rectangles or empty. Add the whole space to this pi-system. Its $\nu$-mass is zero because $1_{\Omega \times [0,n]}$ is simple and converges to $1$ in $L^2(\mu )$ by the dominated convergence theorem. The zero sets of $\nu$ form a lambda-system now that its total mass is zero. The Pi-lambda theorem therefore gives $\nu (E)=0$ for every $E\in \mathcal{P}$. Taking the sets where $g>0$ and $g<0$ gives $g=0$ $\mu$-almost everywhere.

Thus $V^{\perp }=\{0\}$ and the orthogonal complement criterion for a closed subspace of a Hilbert space yieldsOmitting the time-zero term would make this conclusion false when $\mu$ is concentrated at time zero.

For the simple predictable process from part (a), define its stochastic integral byThe time-zero value contributes nothing. Write $D_i=M_{t\wedge t_{i+1}}-M_{t\wedge t_i}$. If $i<j$ and $t>t_j$, then $h_i{D}_i{h}_j$ is ${\mathcal{F}}_{t_j}$-measurable and $\mathbb{E}[D_j\mid {\mathcal{F}}_{t_j}]=0$. If $t\le t_j$, then $D_j=0$. Boundedness ensures integrability, so all off-diagonal terms vanish by conditional expectation. ConsequentlyThe quadratic variation theorem says $M^2-[M]$ is a local martingale. A bounded continuous martingale is square-integrable, and localization gives the conditional second-moment identityMultiplication by $h_i^2$ and summation therefore give the Itô isometryFor Brownian motion, $[M{]}_s=s$. More generally the Itô isometry holds for every predictable process with finite right-hand side, against a continuous square-integrable martingale. Indeed, on a fixed horizon the quadratic-variation measure ${\mu }_M(E)=\mathbb{E}{\int }_0^t{1}_{E}d[M]$ is finite. Part (c) approximates the integrand by simple predictable processes; the displayed equality makes the elementary integrals Cauchy in $L^2$ and defines their unique limit. The finite sum is exactly the elementary case of the Itô isometry.

On the same measurable space, absolute continuity of measures meansThe Radon-Nikodym theorem then supplies a nonnegative measurable density $Z$ with ${\mathbb{E}}_{\mathbb{P}}Z=1$ and $\tilde{\mathbb{P}}(A)={\mathbb{E}}_{\mathbb{P}}[Z{1}_A]$. The converse follows directly from this formula. Strict positivity of the Radon-Nikodym derivative gives equivalence of the two probability measures; absolute continuity of measures alone need not give equivalence.

For a continuous local martingale $M$, its stochastic exponential isThe Itô formula gives $d{Z}_t=Z_{t}d{M}_t$ and $Z_0=1$, so $Z$ is a strictly positive local martingale. The finite-horizon Girsanov theorem says: if $Z$ is a true martingale on $[0,T]$ and $d\tilde{\mathbb{P}}=Z_{T}d\mathbb{P}$ on ${\mathcal{F}}_T$, then for every continuous local martingale $N$ under $\mathbb{P}$,For an infinite-horizon change of measure, assume $Z$ is a uniformly integrable martingale and use its terminal density $Z_{\infty }$. In particular, if $M=\int \theta dB$, the process $\tilde{B}=B-\int \theta ds$ is Brownian motion under the new probability measure, by the Lévy characterization of Brownian motion.

The quadratic variation cannot change. Adding the continuous finite-variation process $-[N,M]$ leaves $[\tilde{N}]=[N]$. Also the same pathwise partition sums defining $[N]$ have the same limit under an absolutely continuous change of measure; convergence in probability transfers by integrability of the density. Thus the bracket of the unchanged process $N$, now a semimartingale, is also unchanged. The local martingale property itself need not survive. This is quadratic covariation under an absolutely continuous measure change.

The Novikov condition holds on every horizon, sinceTherefore $Z=\mathcal{E}(M)$ is a true martingale and the Girsanov theorem applies through each finite $T$.

The deterministic bound also gives a genuine infinite-horizon density. For any $p>1$, the stochastic exponential identity isA nonnegative local martingale is a supermartingale, soThis $L^p$ bound gives uniform integrability. Also $M$ is a L2-bounded continuous martingale, since localization and the Itô isometry give $\mathbb{E}{M}_t^2=\mathbb{E}[M{]}_t\le C$. The L2 martingale convergence theorem gives a finite limit $M_{\infty }$; hence $Z_{\infty }=\exp (M_{\infty }-[M{]}_{\infty }/2)>0$ and $\mathbb{E}{Z}_{\infty }=1$. Weighting by $Z_{\infty }$ defines an equivalent probability measure on ${\mathcal{F}}_{\infty }$. This is the bounded-bracket criterion for a stochastic exponential.

The reference Brownian motion must start at $x$, and both laws must use the same stopped-path rule. Under the reference probability measure let $R_s=x+W_s$, let $\tau$ be its exit time from $(a,b)$, and put $T=t\wedge \tau$ and $c=(\delta -1)/2$. The Girsanov density for a stopped Bessel process, as a functional of the stopped reference path, isUse the predictable process ${\theta }_s=c{R}_s^{-1}{1}_{\{s\le \tau \}}$, with any harmless choice at the endpoint. On $[0,t]$, $\int {\theta }_s^{2}ds\le c^{2}t/a^2$. The Novikov condition holds; under the density $D$, the Girsanov theorem makes ${\tilde{W}}_s=W_s-{\int }_0^s{\theta }_{u}du$ a Brownian motion. Before exit, $d{R}_s=c{R}_s^{-1}ds+d{\tilde{W}}_s$, which is the required stochastic differential equation. The drift is a locally Lipschitz function on $(0,\infty )$, so pathwise uniqueness and uniqueness in law hold up to this exit time.

For a version involving just the endpoint and an ordinary integral, the Itô formula for $\log R$ givesThusThe stopped density is measurable from the stopped path, so it is also the Radon-Nikodym derivative of the stopped-path laws. The stopping times in this formula are evaluated on the reference path, rather than taken from a different realization of $X$. A zero-start reference Brownian motion would have a different deterministic initial value and could not dominate this law.

A weak solution of a stochastic differential equation consists of a filtered probability space, an adapted continuous process $X$, and a Brownian motion $B$ relative to that filtration, with specified initial distribution, such that for every finite $t$ the coefficient integrals are finite andIn one dimension the integrability conditions are ${\int }_0^t\sigma (X_s{)}^{2}ds<\infty$ and ${\int }_0^t|b(X_s)|ds<\infty$. The noise integral is the Itô integral. A solution on a prescribed stochastic basis with prescribed $B$ is distinguished from this weak construction.

A strong solution of a stochastic differential equation uses the prescribed $B$ and initial variable, and is adapted to their augmented natural filtration; thus no additional randomness is needed. For deterministic $X_0=x$, each $X_t$ is determined by the driving Brownian motion through time $t$. Merely being adapted to a larger filtration is insufficient for this definition.

Uniqueness in law means that any two weak stochastic solutions with the same initial distribution have the same law of $X$ on continuous path space. Pathwise uniqueness means that two solutions on one stochastic basis, with the same $B$ and almost surely equal initial values, are indistinguishable stochastic processes. The latter compares paths under the same noise, rather than just comparing their probability distributions.

Fix $X_0=x$ and a finite horizon $T$. Begin under Wiener measure $\mathbb{P}$ with $X_t=x+W_t$, and let $K$ bound $|b|$. The bounded progressively measurable process $b(X)$ is an admissible Itô integral integrand. PutThe Novikov condition follows from ${\int }_0^{T}b(X_s{)}^{2}ds\le K^{2}T$. With $d\mathbb{Q}=Z_{T}d\mathbb{P}$, the Girsanov theorem says that $B=W-{\int }_0^{\cdot }b(X_s)ds$ is Brownian motion. Therefore a weak stochastic solution exists, sinceConsistent finite-horizon densities on canonical continuous path space give a solution for all finite times. No uniform integrability at infinite time is needed for this extension.

For uniqueness in law, take any weak stochastic solution under a measure $\mathbb{Q}$ and reverse the change of measure:Again the Novikov condition holds. Under $\mathbb{P}$, $W=B+\int b(X_s)ds=X-x$ is Brownian motion, so the law of $X$ is the fixed shifted Wiener measure. Substitution of $dB=dX-b(X)dt$ givesUnder shifted Wiener measure, the stochastic integral on the right has a fixed measurable version as a functional of the canonical path. It can be defined by simple predictable processes converging in the integrand $L^2$ norm and an almost surely convergent subsequence of their integrals. Equivalence of the two finite-horizon measures preserves that version. Thus for every bounded measurable path functional $F$,The right-hand side depends only on $b,x,T$, not on the chosen solution. This proves uniqueness in law on every finite horizon, and hence on the whole continuous path space. This is Weak existence and uniqueness in law for an additive-noise SDE with bounded drift. For merely measurable $b$, one should not assume that naive deterministic left-endpoint samples of $b(X)$ converge; the predictable $L^2$ approximation is the appropriate definition.

The Girsanov theorem construction proves a weak stochastic solution, but does not by itself establish a strong stochastic solution. It constructs $X=x+W$ first, and then constructs a new driving Brownian motion $B=W-\int b(X_s)ds$. Adaptation of $X$ to the original filtration of $W$ does not by itself prove adaptation to the smaller natural filtration of $B$. Inverting this relation requires an additional result; uniqueness in law alone is insufficient in general.

For the particular coefficients in part (b), however, the literal claim that a strong solution may fail is incorrect. The strong existence theorem for additive-noise SDEs with bounded measurable drift gives strong existence and pathwise uniqueness for bounded Borel state-dependent $b$ and unit diffusion. Consequently every weak realization here is determined by its driving noise and initial value. This theorem is deeper than the Girsanov theorem argument and no proof is needed here. Examples with path-dependent drift or with a nonconstant diffusion coefficient do not contradict it.

The valid distinction is therefore: the argument in part (b) has established weak existence and uniqueness in law; strong existence requires an additional theorem, and in this class that theorem is available. A primary reference for the correction is On strong solutions and explicit formulas for solutions of stochastic integral equations.

A continuous martingale is an adapted process with almost surely continuous paths, $\mathbb{E}|M_t|<\infty$ for every $t$, andA continuous local martingale is continuous and adapted and has a localizing sequence of stopping times $T_n\uparrow \infty$ almost surely such that every $M_t^{T_n}=M_{t\wedge T_n}$ is a true martingale. The initial value is normally integrable; the frequently used class ${\mathcal{M}}_{c,\mathrm{loc}}$ is normalized to start at zero.

The usual conditions for a filtration are completeness and right continuity. Completeness means that ${\mathcal{F}}_0$ contains every subset of each $\mathbb{P}$-null set in the ambient $\mathcal{F}$; right continuity means ${\mathcal{F}}_t={\bigcap }_{u>t}{\mathcal{F}}_u$. These conditions concern the filtered probability space, whereas continuity of a martingale concerns its sample paths.

For a continuous vector semimartingale $X$ and $f\in C^{1,2}$, the Itô formula isThe Itô product rule, or stochastic integration by parts, isThe additional second-order term is the quadratic covariation.

For general semimartingales with càdlàg paths, use left limits in the integrands. The Itô formula for semimartingales with jumps, for time-independent $f\in C^2$, isHere $X^c$ denotes the continuous local martingale part; the continuous bracket is that of these parts. For time-dependent $f\in C^{1,2}$ add $\int {\partial }_{s}f(s,X_{s-})ds$ and insert $s$ in the other evaluations of $f$. The integration by parts formula becomesFor continuous paths the jump sum vanishes, recovering the first two formulas.

Use two standard results: for every continuous local martingale $U$, there is a unique continuous adapted increasing quadratic variation $[U]$, starting at zero, such that $U^2-U_0^2-[U]$ is a local martingale; and a continuous finite-variation local martingale is constant.

Define the quadratic covariation by the polarization identityThis is continuous, adapted, starts at zero, and is of finite variation on compact time intervals. Since $(M+N{)}^2-(M-N{)}^2=4MN$, subtraction of the two quadratic-variation identities shows that $MN-M_0{N}_0-[M,N]$ is a continuous local martingale. In the zero-start convention this is exactly $MN-[M,N]\in {\mathcal{M}}_{c,\mathrm{loc}}$.

If $A$ is another continuous adapted finite-variation process with $A_0=0$ and the same local martingale property, then $A-[M,N]$ is both a continuous local martingale and of finite variation. It is constant, and its initial value is zero, so it vanishes. Uniqueness is up to indistinguishability of stochastic processes.

The normalization $[M,N{]}_0=0$ is essential. If arbitrary nonzero initial constants are allowed in the martingale class and this normalization is omitted, adding a constant to the bracket preserves the claimed property. The usual definition of quadratic covariation always includes the zero initial value.

The Lévy characterization of Brownian motion states that an adapted continuous real process $M$ with $M_0=0$ is standard Brownian motion relative to $({\mathcal{F}}_t)$ if and only if it is a continuous local martingale and $[M{]}_t=t$. The multidimensional version replaces the bracket identity by $[M^i,M^j{]}_t={\delta }_{ij}t$.

For the converse direction, a standard Brownian motion is a continuous martingale, and its independent centered increments give quadratic variation $t$. Distinct independent coordinates have zero quadratic covariation.

For the substantive direction, fix a vector $\theta \in {\mathbb{R}}^d$. The Itô formula applied toshows that its drift is zero: the second-order term is $-\frac{1}{2}|\theta {|}^2{E}_{t}dt$ and cancels the time derivative. Thus $E(\theta )$ is a complex local martingale. On $[0,T]$ its modulus is the deterministic value $e^{|\theta {|}^{2}t/2}\le e^{|\theta {|}^{2}T/2}$; localization and the dominated convergence theorem make it a true martingale. HenceThis conditional characteristic function is the deterministic characteristic function of $N(0,(t-s)I)$. For $A\in {\mathcal{F}}_s$, multiplying by $1_A$ and taking expectations, then using uniqueness of characteristic functions, proves that $M_t-M_s$ has this multivariate normal distribution and is independent of ${\mathcal{F}}_s$. Iterated conditioning gives independent increments with Gaussian distributions. Continuity, the zero initial value and these increment laws are the definition of standard Brownian motion. The case $d=1$ proves the requested statement; the vector form will also be used below.

Interpret $f$ as a conformal bijection onto $D^{\prime }$. If it is merely an injective map into a larger target, the correct exit domain is $f(D)$ instead. Let $\zeta$ be the exit time from $D$, write $f=u+iv$, and identify planar Brownian motion with $B^1+i{B}^2$. The Cauchy-Riemann equations give $u_x=v_y$, $u_y=-v_x$, and both $u,v$ are harmonic functions. The Itô formula, localized on compact subsets of $D$, yieldsThus both coordinates are continuous local martingales, withSince $f^{\prime }$ never vanishes, $A$ is strictly increasing before $\zeta$. Let ${\eta }_s=\inf \{t:A_t>s\}$, defined for $s<A_{\zeta -}$. The optional time-change theorem gives continuous local martingale coordinates for ${\beta }_s=f(B_{{\eta }_s})$, with bracket matrix $sI$. The Lévy characterization of multidimensional Brownian motion makes $\beta$ a standard planar Brownian motion started at $f(z)$, up to its lifetime. Therefore conformal invariance of planar Brownian motion takes the formTo identify the lifetime, exhaust $D$ by relatively compact open sets $D_n$ with nested closures. The images $f(D_n)$ exhaust $D^{\prime }$ because $f$ is a homeomorphism onto $D^{\prime }$. Each stopped transformed path exits $f(D_n)$ at precisely the clock value of its exit from $D_n$. Increasing these exit times gives the exit lifetime of Brownian motion from $D^{\prime }$, including a possibly infinite lifetime. This identifies $A_{\zeta -}$ with that lifetime and needs no global extension of $f$ to the boundary.

Use the open strip $S=\{x+iy:0<y<\pi \}$; the printed closed strip denotes the region together with its exit boundary. Starting at $z=x+iy$, the conformal bijection $e^z$ sends $S$ onto the complex upper half-plane and the bottom edge $a\in \mathbb{R}$ onto $v=e^a>0$. The upper edge maps to the negative half-axis. The transformed initial point is $e^x\cos y+i{e}^x\sin y$.

By conformal invariance of planar Brownian motion, transform the given Poisson kernel for the upper half-plane and include the Jacobian determinant $dv/da=e^a$. The bottom-boundary harmonic measure of a strip isThis is the unconditional exit density on the bottom edge, not the density conditional on using that edge. Its total mass isIndeed substitution $v=e^{a-x}$ gives the positive-half-axis mass of the Poisson kernel, equal to $\frac{1}{2}+{\pi }^{-1}\arctan (\cot y)=1-y/\pi$ for $0<y<\pi$. To obtain the conditional bottom-edge density, divide by $1-y/\pi$. The top-edge density is $\sin y/[2\pi (\cosh (a-x)+\cos y)]$ and has mass $y/\pi$, a useful normalization check.

For a harmonic function $u$ on $\mathbb{C}\cong {\mathbb{R}}^2$, the Itô formula givesStop on leaving increasing discs and at increasing deterministic times. On each such interval the gradient is bounded, so the stochastic integral is a martingale. This proves that $u(B)$ is a continuous local martingale.

If $|u|\le K$, this is a bounded local martingale and hence a true martingale. Start Brownian motion at an arbitrary $z$. Its Gaussian heat kernel givesFor a unit vector $e$, ${\partial }_e{p}_t(v)=-(e\cdot v)p_t(v)/t$, so $\int |{\partial }_e{p}_t(v)|dv=\sqrt{2/\pi }/\sqrt{t}$. Integrating the directional derivative along the segment from $z$ to $w$ yieldsLetting $t\to \infty$ gives the harmonic Liouville theorem:This Gaussian heat-kernel proof of the harmonic Liouville theorem uses the bounded martingale identity established above.

Let the infinitesimal generator be the second-order operatorwhere $a(x)$ is symmetric positive semidefinite and $b(x)$ is a vector. In the standard bounded-coefficient setting take $a,b$ bounded and measurable. An L-diffusion is a continuous adapted process $X$ for which, for every $g\in C_b^2({\mathbb{R}}^d)$,Here $C_b^2$ includes boundedness of the function and its derivatives through order two. This is the true-martingale version of the diffusion martingale problem. The matrix $a$ is the diffusivity, and $b$ is the drift. An Itô diffusion with coefficient $\sigma$ has $a=\sigma {\sigma }^{\mathsf{T}}$.

A more general local martingale problem may permit unbounded coefficients and require only a local martingale. That convention is weaker and must not silently replace the true martingale property used here and in part (b). The bounded-coefficient convention guarantees all finite-horizon integrability needed below; a sufficient more general replacement is $\mathbb{E}{\int }_0^T({\sum }_i|b_i(X_s)|+{\sum }_i{a}_{ii}(X_s))ds<\infty$ for every $T$.

We prove the extension from spatial test functions to time-dependent test functions for a diffusion martingale problem directly, without assuming a driving Brownian motion. Fix $0\le s<t$ and a deterministic partition $s=t_0<\cdots <t_n=t$. For each $j$, $g_j(x)=f(t_j,x)$ lies in $C_b^2$. The defining martingale problem givesCall the sum $S_n$. Telescoping separates the spatial and temporal changes:The sample path $X$ is uniformly continuous on $[s,t]$ and has compact range. Joint continuity of ${\partial }_{t}f$, the spatial first derivatives, and the spatial second derivatives therefore gives, as the mesh tends to zero,almost surely. For bounded $a,b$ and bounded derivatives of $f$, both integrals are uniformly bounded on this finite horizon, as are the two endpoint terms. The dominated convergence theorem permits passage to the limit in the expectation against $1_A$. Thus $M_t^f$ is integrable and $\mathbb{E}[1_A(M_t^f-M_s^f)]=0$ for every $A\in {\mathcal{F}}_s$, provingAdaptation and continuity follow from the formula, so $M^f$ is a continuous martingale. Under the more general integrability condition in part (a), the same proof uses an integrable bound proportional to $1+{\int }_s^t({\sum }_i|b_i(X_u)|+{\sum }_i{a}_{ii}(X_u))du$; positive semidefiniteness bounds every off-diagonal entry by the diagonal ones. With only local assumptions this reasoning proves a local martingale, and an integrability hypothesis is needed to upgrade that conclusion.

The Itô formula for $f(t,x)=e^{x+at}$ gives the stochastic differential equationWrite $N_t={\int }_0^t{Z}_{u}d{B}_u$ for its continuous local martingale part. Its clock is the quadratic variationBecause $Z_t>0$, $A$ is strictly increasing. For $s<A_{\infty }$, its inverse ${\tau }_s$ satisfies $A_{{\tau }_s}=s$ and $d{\tau }_s/ds=Z_{{\tau }_s}^{-2}$. The Dambis-Dubins-Schwarz theorem, or the Lévy characterization of Brownian motion after the optional time-change theorem, makes ${\beta }_s=N_{{\tau }_s}$ a Brownian motion for the time-changed filtration. With $Y_s=Z_{{\tau }_s}$,This is the Bessel process equation with dimension $\boxed{2+2a}$, on the positive stochastic interval. This establishes the Exponential Brownian-to-Bessel time change.

The lifetime convention is important: the inverse is defined only for $s<A_{\infty }$. If $a<0$, the strong law for Brownian motion gives $B_t/t\to 0$, so $Z_t\to 0$ exponentially and $A_{\infty }<\infty$. Then $Y_s\to 0$ as $s\uparrow A_{\infty }$, and this is its first hit of zero after continuous extension. If $a>0$ the clock diverges by the same strong law. If $a=0$ it diverges by infinite occupation time of one-dimensional Brownian motion on a bounded interval, since $e^{2B_t}$ is bounded below there. Thus the lifetime is infinite when $a\ge 0$. Negative dimensions are interpreted by the same positive-state equation killed at zero; no post-zero Bessel process behavior is asserted.

Work before the first zero of $Z$, with $Z_0=z>0$. For the Bessel process equation, the Itô formula applied to $x^r$ givesIts quadratic variation isThis clock is strictly increasing on the positive-state interval. For $s$ before its terminal clock value, let $Y_s=W_{{\sigma }_s}$ and let ${\beta }_s=({\int }_0^{\cdot }r{Z}_u^{r-1}d{B}_u{)}_{{\sigma }_s}$. The Dambis-Dubins-Schwarz theorem makes $\beta$ a Brownian motion there. Since $d{\sigma }_s=ds/(r^2{Z}_{{\sigma }_s}^{2r-2})$, the drift changes toConsequently the Power time change of a Bessel process givesThe initial value is $z^r$, and this equation identifies a Bessel process of dimension ${\delta }^{\prime }$ until its first zero. Localize both equations on compact subintervals of $(0,\infty )$ to justify the inverse and all stochastic integrals for arbitrary $r>0$.