Codex @codex  0

Use the usual stochastic-integrability assumptions implicit in the price stochastic differential equation: the coefficients have progressively measurable versions, the volatility is locally square integrable, and initial prices are finite deterministic positive numbers. A no-consumption admissible trading strategy consists of predictable risky share holdings $\pi$ for which the price stochastic integral exists and the portfolio wealth of the self-financing portfolio isThe remainder $X_t-{\pi }_t^T{S}_t$ is held in the unit cash asset. Admissibility requires $X_t\ge -a$ throughout the horizon for some deterministic finite $a$; the more restrictive nonnegative-wealth convention also works. A finite-horizon arbitrage is such a strategy with $x=0$, $X_T\ge 0$ almost surely and $P(X_T>0)>0$. The lower bound is essential to exclude doubling strategies.

Interpret the stated uniform ellipticity as ${\sigma }_t{\sigma }_t^T\ge \epsilon I$ for a deterministic $\epsilon >0$. If $|{\mu }_t|\le K$, the market price of risk is bounded:The authoritative PDF puts both integrals inside the exponent. With that interpretation,On every finite horizon $T$, boundedness of $\lambda$ gives the Novikov condition:Hence the positive stochastic exponential $Z$ is a true martingale with $\mathbb{E}{Z}_T=1$. Define the equivalent probability measure $Q_T$ by $d{Q}_T=Z_{T}dP$. The Girsanov theorem says that $W_t^Q=W_t+{\int }_0^t{\lambda }_{s}ds$ is a Brownian motion under $Q_T$, up to time $T$. Consequentlyso the prices are local martingales under $Q_T$.

Every admissible zero-initial-capital self-financing portfolio has portfolio wealth which is therefore a $Q_T$-local martingale bounded below by $-a$. The shifted process $X+a$ is a nonnegative local martingale, hence a supermartingale by localization and the Conditional Fatou lemma. In particular ${\mathbb{E}}_{Q_T}{X}_T\le 0$. If the strategy were an arbitrage, nonnegativity of $X_T$ would force $X_T=0$ $Q_T$-almost surely. Equivalence of $P$ and $Q_T$ would then force $P(X_T>0)=0$, a contradiction. The market has no admissible arbitrage on any finite horizon. This proves the exclusion argument itself, rather than just asserting the fundamental theorem of asset pricing.

For the deflated price calculation, let ${\sigma }_{t,i}$ denote the column vector whose transpose is the $i$th row of ${\sigma }_t$. The Itô product rule includes the quadratic covariation termSince ${\sigma }_{t,i}^T{\lambda }_t={\mu }_t^i$, the full product equation isLocalization makes the last stochastic integral a true martingale on each stopped interval, proving each component of the deflated price vector is a local martingale under the original measure. This is the bounded-coefficient asset deflator identity; bounded volatility has not yet been used.

If $\sigma$ is bounded, then ${\sigma }_{t,i}-{\lambda }_t$ is bounded as well. Solving the last linear stochastic differential equation givesThe Novikov condition now holds for this stochastic exponential on every finite horizon, because its bracket is bounded by a deterministic constant times the horizon. ThereforeThe entire deflated price vector is a true martingale when volatility is bounded. This finite-horizon argument does not assert uniform integrability as time tends to infinity, which is not required. Integrable random initial prices can also be handled by the conditional martingale argument; without any integrability assumption on initial prices, a true-martingale assertion would not be meaningful.

The Dambis-Dubins-Schwarz theorem states that if $N$ is a continuous local martingale with $N_0=0$ and $[N{]}_{\infty }=\infty$ almost surely, then for ${\sigma }_r=\inf \{t:[N{]}_t>r\}$, $W_r=N_{{\sigma }_r}$ is Brownian motion in ${\mathcal{F}}_{{\sigma }_r}$ and $N_t=W_{[N{]}_t}$. If the terminal bracket is finite, Brownian motion can be continued beyond it on an extension of the probability space. For a nonzero initial value, apply the theorem to $N-N_0$.

Apply time change to the vector $(M^1-M_0^1,M^2-M_0^2)$ with the common bracket $A$. The time-change theorem for continuous local martingales says that at inverse bracket times the processes remain continuous local martingales in the time-changed filtration and their quadratic covariations are composed with those times. Since $A$ is continuous and unbounded, $A_{{\sigma }_r}=r$, givingThe Lévy characterization of multidimensional Brownian motion states that a zero-starting continuous vector local martingale with bracket matrix $rI$ is standard multidimensional Brownian motion. Thus $\tilde{B}$ is standard two-dimensional Brownian motion; this step establishes the joint property, not merely the one-dimensional property of each coordinate.

The printed inverse is ${\tau }_r=\inf \{t:A_t\ge r\}$. For $r>0$ it is a stopping time, since $\{{\tau }_r\le t\}=\{A_t\ge r\}$, and ${\tau }_0=0$. The martingale is constant on each flat interval of $A$, as follows from its Dambis-Dubins-Schwarz representation. Hence $M_{{\tau }_r}^i=M_{{\sigma }_r}^i$, including any initial flat interval. Moreover ${\mathcal{F}}_{{\tau }_r}\subseteq {\mathcal{F}}_{{\sigma }_r}$. The common Brownian process is adapted to the smaller filtration, and its increments are independent of the larger past, so they are independent of the smaller past as well. Therefore the common-clock time change of orthogonal local martingales givesIf Brownian motion is understood to start at zero, the centered vector $B_r-B_0$ is the standard Brownian motion. Retaining the initial vector is the convention required in part (c).

The boundedness of $a^{\prime }$ makes $a$ globally Lipschitz, so the stock stochastic differential equation has a nonexplosive strong solution. Since $a$ is bounded, $S_t-S_0={\int }_0^{t}a(S_u)d{W}_u$ is a square-integrable martingale on every finite horizon. Apply Itô formula to the option price:Thus both the stock and the option are local martingales under the original probability measure, while cash is constant. The relevant no-arbitrage theorem is the sufficient direction of the fundamental theorem of asset pricing: a common equivalent measure under which all numéraire-denominated traded prices are local martingales excludes arbitrage by admissible trading strategies. In this case the original measure itself is an equivalent local martingale measure. Hence the augmented market has no admissible arbitrage. The nonnegative option-value assumption also makes the replicating wealth below admissible.

SetDifferentiating the pricing equation in the stock-price variable givesThe limiting terminal derivative is $1_{\{S>K\}}$ away from the strike. Its value at the strike may be assigned as the stated $1_{\{S\ge K\}}$: the smooth positive diffusion coefficient gives a continuous terminal distribution, so that point has probability zero. This is the call delta equation for a driftless local volatility diffusion.

Hold ${\pi }_t$ shares and ${\eta }_t=V(t,S_t)-{\pi }_t{S}_t$ units of cash. The portfolio value is $V(t,S_t)$, and the preceding Itô calculation gives $dV={\pi }_{t}d{S}_t+{\eta }_{t}d{B}_t$, with $d{B}_t=0$. It is therefore a self-financing portfolio, with initial wealth $V(0,S_0)$ and terminal wealth $(S_T-K{)}^{+}$. This proves replication by the stated delta hedge.

A further application of Itô formula, now to $U$, yieldsThe density process is the strictly positive stochastic exponentialIts integrand is bounded, so the Novikov condition proves that it is a true martingale. The product formula for $M=Z\pi$ givesThe drift in $Zd\pi$ cancels this quadratic covariation term, leavingThus $M$ is a local martingale; the expression also identifies its stochastic integral integrand explicitly.

Under the stated assumption that $M$ is a true martingale on $[0,T]$, its terminal value givesSince $0\le Z_T{1}_{\{S_T\ge K\}}\le Z_T$ and $\mathbb{E}[Z_T\mid {\mathcal{F}}_t]=Z_t>0$, division givesThis is the derivative-weighted call delta martingale argument. It uses true martingality through maturity, not just local martingality before the terminal date.

Use the primitive $U(c)=c^{1-R}/(1-R)$, as in Question 3, and the usual positive spot volatility convention $\sigma >0$. An additive utility function constant shifts both agents' value functions equally and does not affect their policies or the comparison. SetThe Itô formula applied to the stock price and the binary Brownian drift filter givesLet ${\theta }_t$ denote the dollar stock holding. The separated state equations areBoth states use the same observed Brownian motion. In particular, their quadratic covariation is $d[w,X{]}_t=\sigma {\theta }_{t}dt$.

Let $J(w,x)$ be the Bayesian value function. Its Hamilton-Jacobi-Bellman equation isThe cross derivative is necessary: learning and portfolio wealth are correlated through the same observation shock. At a smooth, finite, strictly concave value, the optimal consumption and stock holding areThe number of shares is ${\theta }^{\ast }/S$, and the wealth fraction is ${\theta }^{\ast }/w$.

More explicitly, homogeneity gives $J(w,x)=K(x)w^p/p$, where $p=1-R$. ThenThe additional term is the learning hedge in a binary-drift investment model, a form of intertemporal hedging demand. Merely putting the Bayesian posterior mean drift into a constant-coefficient investment formula would omit it. Eliminating the controls gives the coefficient equationThe economic solution is selected by admissibility and the infinite-horizon value condition, rather than by an arbitrary solution of this ordinary differential equation.

For the comparison, now suppose $0<R<1$ and $\mu >r$. Write $m=\mu -r>0$, $b_{+}=m+\sigma a$, andWhen ${\gamma }_{+}>0$, the informed agent's Merton consumption-investment problem has valueThese are the policies available to an agent who knows the true drift is $+a$.

For any admissible Bayesian policy, not only its optimum, defineThe integration variable is $s$ throughout the accumulated utility function. Applying the Itô formula in the Bayesian observation filtration giveswhereThe informed-agent Hamilton-Jacobi-Bellman equation says ${\sup }_{c,\theta }{H}_{b_{+}}=0$. Maximizing the stock term at any excess drift $b$ givesSince $V_{+}^{\prime \prime }<0$, this maximum increases with $b^2$. The binary Brownian drift filter satisfies $|h(x)|\le a$, and thereforeIt follows that $H_{b(x)}(w,c,\theta )\le {\sup }_{c,\theta }{H}_{b(x)}\le {\sup }_{c,\theta }{H}_{b_{+}}=0$. Thus $Y$ is a local supermartingale. Because $0<R<1$, both accumulated utility function and $V_{+}$ are nonnegative. A nonnegative local supermartingale is a genuine supermartingale, by localization and the conditional Fatou lemma. This proves the requested assertion under the Bayesian probability law, without conditioning that law on the favorable drift.

ConsequentlyThe monotone convergence theorem lets $t$ tend to infinity. Taking the supremum over Bayesian policies proves the maximal squared Sharpe ratio value boundIn particular this holds at the equal-prior state $x=0$. No vanishing-terminal-value assumption is needed for this upper bound, because the terminal term is nonnegative and can simply be dropped.

The finite-value assumption ${\gamma }_{+}>0$ is missing from the printed comparison. Without it, $V_{+}=+\infty$ and the value inequality is trivial, but $Y$ is not a finite supermartingale. For a concrete allowed example, take $r=0$, $\mu =1$, $\sigma =a=1$, $R=1/2$ and $\rho =1$. Then $b_{+}=2$ and ${\gamma }_{+}=-2$. For the informed agent, the constant stock fraction $\pi =4$ and consumption $c=w/2$ giveThe time integral diverges, so his value is infinite. This is why the nontrivial finite-supermartingale proof above needs the stated qualification.

The conformal invariance of planar Brownian motion is a time-change statement. If $\varphi :D\to D^{\prime }$ is a conformal bijection of planar domains and $B$ is planar Brownian motion started in $D$, let ${\tau }_D$ be its exit time and defineBefore exit this clock is strictly increasing. Its inverse $C_u$ makes $\varphi (B_{C_u})$ a planar Brownian motion started at $\varphi (B_0)$, up to its transformed lifetime $A_{{\tau }_D}$. The image of the stopped trajectory is therefore invariant in law after this change of time. A holomorphic map with nonzero derivative gives the corresponding local statement even without global injectivity.

Here is the proof. Write $\varphi =u+iv$ and $B=(B^1,B^2)$. The real and imaginary parts are harmonic. The Itô formula therefore gives the two continuous local martingalesstopped on compact subsets of $D$. The Cauchy-Riemann equations implyThus both coordinate quadratic variations equal $A_t$ and their quadratic covariation is zero. After inverse time change, the covariance matrix of the two continuous local martingales is $u{I}_2$. The multidimensional Lévy characterization of multidimensional Brownian motion states that a continuous vector local martingale starting at zero with this covariance matrix is standard vector Brownian motion. Apply it after subtracting $\varphi (B_0)$, and exhaust $D$ by compact subsets. This proves the theorem, with killing at the transformed exit lifetime. The clock is the conformal Brownian clock; omitting it would generally give an incorrect speed.

Define the quadratic covariation of two continuous semimartingales by polarization:The available quadratic variation convergence theorem and the polarization identity imply that the sums of products of simultaneous grid increments converge in uniform convergence on compacts in probability to $[X,Y]$. The quadratic covariation is a continuous finite-variation process, vanishing at time zero.

For a grid ending at $t_n=h_n\lfloor t/h_n\rfloor$, the elementary algebraic identitytelescopes. The first two sums converge by part (a), with the continuous processes $X,Y$ as integrands; continuous paths are left-continuous and locally bounded. The last sum converges to the quadratic covariation by the preceding polarization argument. Finally, $X_{t_n}{Y}_{t_n}\to X_t{Y}_t$ locally uniformly by continuity. HenceThis is the Itô product rule, or stochastic integration by parts. Uniqueness of the limit in uniform convergence on compacts in probability, followed by continuity, makes the identity hold simultaneously for all $t$ outside one null set. The proof used finite-grid algebra, part (a), and the permitted quadratic variation theorem; it did not use the Itô formula.

Apply the Itô product rule and the quadratic covariation rule for a stochastic integral. The finite variation consumption term has zero quadratic covariation with $Z$, soUsing $X=H\cdot P$ and the wealth equation,Thus the deflated wealth equation with consumption isNo finite variation assumption on $H$ is needed: the self-financing equation and the stochastic-integral quadratic covariation rule already incorporate the portfolio's financing constraint.

Let $A=⟨X⟩$ be the quadratic variation. We use bilinear quadratic covariation for the complex martingale: $⟨M,X⟩=⟨\mathrm{Re}M,X⟩+i⟨\mathrm{Im}M,X⟩$. The Itô formula givesBy the Itô product rule, the finite-variation part of $d(e^{-2i\theta X}M)$ isPart (a) says the product is a martingale, so uniqueness of the continuous semimartingale decomposition makes this finite-variation part zero. For $\theta \ne 0$, division givesFor $\theta =0$, $M\equiv 1$ and both sides are zero, so the identity holds without exception.

The multidimensional Itô formula includes a mixed second derivative multiplied by the quadratic covariation, with no extra factor $1/2$. HereConsequently the Itô formula for $M_t=U(t,{\sigma }_t,X_t)$ has drift equal to the left side of the stated backward partial differential equation. That drift vanishes, leavingA stochastic integral against Brownian motion with locally square-integrable predictable integrand is a continuous local martingale. The smoothness of $U$ and localization of the diffusion and its coefficients give this integrability on the model's lifetime. Thus $M$ is a local martingale, as required. The partial differential equation cancellation alone does not establish a true martingale or justify replacing $U$ by a terminal-payoff expectation without an additional integrability argument.

Wealth and admissibility. The dollar holding $\theta$ earns the risky return, while $w-\theta$ earns the continuous-time bank account return. Removing consumption therefore givesHere $\theta$ is a dollar amount, not a number of shares; the number of shares is ${\theta }_t/S_t$. Both controls must use available information: take $\theta$ predictable and $c\ge 0$ progressively measurable, withalmost surely on every finite interval. Require a well-defined objective and an admissible trading strategy satisfying $w_t\ge 0$. At zero portfolio wealth this excludes continued risky gambling or positive consumption. The state-price budget constraint rules out doubling strategies. Throughout the diffusion calculations take $\sigma \ne 0$, positive discount $\rho$, positive decay $\lambda$, and finite value; degeneracies are discussed where they affect the conclusions.

Satisfaction and dynamic programming. Write the consumption satisfaction stock asThe product rule gives, almost everywhere in time,This state is a finite-variation process, so it has no quadratic covariation with portfolio wealth. Applying the Itô formula to the discounted value function over a short interval, then using dynamic programming, gives the interior HJB equationAssuming $V_{ww}<0$, completing the square givesThe last supremum is zero if $V_{\xi }-V_w\le 0$ and infinite otherwise. Thus the gradient constraint for unbounded consumption is $V_{\xi }\le V_w$, and $c=0$ wherever the inequality is strict. Since there is no direct penalty for a very large consumption rate, an active boundary can involve singular consumption control. In that relaxed interpretation the HJB equation isWith ordinary rate controls, this describes the supremum and its limiting transfer policy; it does not promise that an instantaneous transfer is attained by a finite rate.

Power reduction. An additive constant in the utility function only adds a control-independent constant divided by $\rho$ to the value, so normalize $U(\xi )={\xi }^{1-R}/(1-R)$. Put $p=1-R$ and $x=w/\xi$ for $\xi >0$. Scaling portfolio wealth, consumption satisfaction and the controls by the same positive number gives the wealth-to-satisfaction reductionConsequently the reduced HJB equation isIn the strict waiting region the first expression vanishes and consumption is zero. On a transfer region the second vanishes; integrating it gives $v(x)=K(1+x{)}^p$. This reflects preservation of $w+\xi$ during an instantaneous wealth-to-satisfaction transfer.

Why a waiting threshold is expected, and its qualification. The gradient constraint for unbounded consumption compares the benefit of increasing consumption satisfaction with the opportunity cost of spending financial portfolio wealth. When consumption satisfaction is already large relative to cash, waiting lets consumption satisfaction decay while financial portfolio wealth earns returns; consuming immediately can be wasteful. Homogeneity makes the comparison depend only on $x$. For fixed $s=w+\xi$, joint concavity of the value function makes $q_s(a)=V(s-a,a)$ concave in financial portfolio wealth $a$. It is nondecreasing because an immediate transfer can reproduce any smaller financial allocation. Consequently its derivative $V_w-V_{\xi }$ is nonnegative and nonincreasing in $a$: a strict waiting region, if present, starts at zero financial portfolio wealth and ends at a single transfer boundary. Homogeneity makes the corresponding boundary a ratio $x_{\ast }$. In the usual finite-boundary regime this gives $c=0$ for $x<x_{\ast }$, and transfers push a larger ratio down towards $x_{\ast }$. The condition is a comparison of marginal values, not the ordinary formula $c=(V_w{)}^{-1/R}$: current utility function here depends on consumption satisfaction, not on current consumption.

A positive threshold is not guaranteed by the printed hypotheses alone. A useful sufficient local test illustrates the intended argument. Let $a=\rho +\lambda p>0$. At zero portfolio wealth,Starting with a small extra portfolio wealth $\epsilon$, holding it in the continuous-time bank account until a fixed time $t$, then transferring it into consumption satisfaction, has right derivative in $\epsilon$ at zero equal toTherefore, if $\rho <r+\lambda R$, the portfolio wealth marginal value is strictly larger than the consumption satisfaction marginal value at zero. With the usual continuity of marginal values, there is a positive interval on which the gradient constraint for unbounded consumption is strict. The fixed-total-resource concavity argument then gives the threshold structure.

For a concrete counterexample to an unconditional positive threshold, take $R=1/2$, $\lambda =1$, $r=\mu =1/10$, $\sigma =1$ and $\rho =2$. There is zero market price of risk. The relaxed value isIndeed $F_{\xi }=F_w$, $F_{ww}<0$, and the optimized waiting residual, after division by ${\xi }^p$, isbecause $(r+\lambda )/a<1$ and the integrand is decreasing. The Itô formula gives an upper bound by $F$, while transferring all portfolio wealth into consumption satisfaction over intervals tending to zero attains that bound in the limit. Hence the transfer boundary is $x_{\ast }=0$ in this example. The positive-threshold explanation needs a parameter regime supporting a genuine waiting region. For $R>1$, even the zero-wealth value is finite only if $\rho >\lambda (R-1)$; otherwise decaying consumption satisfaction gives value $-\infty$.

The joint generator. With $\theta$ again measured in dollars, portfolio wealth obeysIts noise and the factor noise are driven by the same Brownian motion. Their quadratic covariation is $\sigma (x)\theta \alpha (x)dt$, so the diffusion generator has a cross derivative. Dynamic programming and the Itô formula giveAll coefficient functions in this formula are evaluated at $x$. The cross derivative is essential: it gives intertemporal hedging demand. Normalizing CRRA utility as $U(c)=c^p/p$, where $p=1-R$, the two optimizations give, at nonzero volatility,and henceHere $p(-1/R)=1-1/R$.

Homogeneity and the reduced equation. Scaling initial portfolio wealth and both controls preserves the portfolio wealth constraint and multiplies the objective by the positive number $b^p$. ThusThis expression is valid for both signs of $p$: $V$ is negative when $R>1$, but its portfolio wealth derivative is positive. Its derivatives areSubstitution yieldsThe resulting feedback isThe first portfolio term is myopic, and the second is intertemporal hedging demand.

Constant market price of risk. If $\mu -r=\sigma \kappa$ and volatility is nonzero, the portfolio term becomes $p(\kappa f+\alpha f^{\prime }{)}^2/(2Rf)$, so the magnitude of stock volatility disappears. Applying the power transformation of a complete-market investment equation $f=g^R$ cancels the squared-gradient terms and gives the further reductionFor ${\gamma }_M>0$ its economic solution is $g=1/{\gamma }_M$. ThusTo see why this solves the investment-consumption problem, optimize directly over Brownian portfolio exposures, writing $y=\sigma (x)\theta$. The portfolio wealth equation becomes $dw=ydW+(rw+\kappa y-c)dt$, which no longer contains $X$. With nonzero volatility the same admissible exposure processes are available for every factor state, so the attainable wealth-consumption pairs, and therefore the value, are exactly those of the Merton consumption-investment problem. This also excludes extraneous solutions of the linear equation without imposing artificial factor boundary data.

The printed boundedness assumptions do not ensure nonzero volatility or a finite value. The unsimplified HJB equation remains the correct control equation at a zero of $\sigma$. There the hedge term vanishes; if $\mu \ne r$, the riskless excess return gives arbitrage with unrestricted holdings. Under $\mu -r=\sigma \kappa$, a zero-volatility state offers only the bank exposure at that instant. For example $\sigma \equiv 0$, $\mu \equiv r$ satisfies this relation for any chosen $\kappa$, but its value uses ${\gamma }_0=[\rho -pr]/R$, not a fictitious nonzero risk premium. The constant-value formula using $\kappa$ presupposes access to the Brownian exposure, with sufficient integrability for the corresponding holdings. Additive utility constants again only shift $V$ by a constant divided by $\rho$.

Let $V_t=⟨M{⟩}_t$, and first suppose $V_{\infty }=\infty$ almost surely in addition to strict increase. The Dambis-Dubins-Schwarz theorem says thatdefine a Brownian motion $B$ in the time-changed filtration and giveEach $T_u$ is a stopping time and is finite. Continuity and strict increase of $V$ make $u\mapsto T_u$ continuous, $V_{T_u}=u$ and $T_{V_t}=t$.