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The positive state-price density alternative characterizes one-period arbitrage. Interpret $A_0$ as the vector of current asset prices and $A_1$ as their terminal payments. A fixed vector of holdings produces initial cost $x^{\top }{A}_0$ and terminal payment $x^{\top }{A}_1$. The first alternative is a nonpositive-cost investment with a nonnegative payment and strict improvement either initially or on an event of positive probability. A strictly positive state-price density prices every such payment and excludes both forms of arbitrage, as proved below.

If there is a traded bank account costing one and paying a deterministic $R>0$, the density satisfies $1=R\mathbb{E}\nu$. Hence $dQ/dP=R\nu$ defines an equivalent martingale measure andConversely this pricing relation under an equivalent martingale measure gives $\nu =R^{-1}dQ/dP$. Without such an account, $\nu$ is a pricing density, not necessarily a probability density or even an integrable random variable by itself; the stated requirement is integrability of $\nu A_1$. This distinction keeps the abstract alternative valid without an unstated asset paying one.

Specify the trading conventions first. Let the bank account have value $R_r>0$, with $R_0=1$ and its next gross return known at time $r$. Write $B_r=1/R_r$ for the discount factor. Let $S_r$ be the ex-dividend asset-price vector, and $d_r$ the vector of cash dividends paid per unit at time $r$; set $d_r=0$ for non-dividend-paying assets. Under an equivalent martingale measure $Q$, each discounted cumulative asset gainis a martingale. Equivalently, ${\mathbb{E}}_Q[B_{r+1}(S_{r+1}+d_{r+1})\mid {\mathcal{F}}_r]=B_r{S}_r$. This is the gain-process form of risk-neutral pricing, rather than a martingale assumption on raw ex-dividend prices.

Let $X_r$ be the ${\mathcal{F}}_r$-measurable holdings after trading and distributions at time $r$. The previous holdings $X_{r-1}$ earn the next price change and dividends. Define the strategy's net cash payment and its residual portfolio wealth byA negative payment is an external infusion; a self-financing portfolio with no consumption has $D_r^X=0$. The discounted budget equation isAssume these gains and payments are integrable under $Q$, for example by bounded predictable holdings and integrable discounted asset gains. Thenis a true martingale, by conditioning the displayed one-step gains. Merely assuming a local martingale without appropriate integrability would not justify the conditional equality.

Conditioning $M_n$ on ${\mathcal{F}}_r$ proves the general multi-period dividend pricing identityFor a strategy liquidated at time $n$, $X_n=0$ and all liquidation proceeds are included in $D_n^X$. The terminal value disappears, giving exactly the requested dividend-only relation. That liquidation convention is necessary: holding one unit of the bank account with no payments would otherwise have a positive left side and zero right side.

An attainable claim $H$ paid at $n$ can be replicated by a strategy with no earlier payments and final liquidation payment $D_n^X=H$. Its time-$r$ value is thereforeTwo admissible replicating strategies have the same value by this identity, and the value is the same under every equivalent martingale measure for which the replication gains are true martingales. For unattainable claims this expectation is a possible model price, but the replication argument does not establish a unique price. With density process $Z_r={\mathbb{E}}_P[dQ/dP\mid {\mathcal{F}}_r]$, the equivalent state-price density form is $V_r=(B_r{Z}_r{)}^{-1}{\mathbb{E}}_P[B_n{Z}_{n}H\mid {\mathcal{F}}_r]$.

Use the physical dynamics of the Black-Scholes model, $d{S}_t/S_t=\mu dt+\sigma d{W}_t$, with bank rate $\rho$ and $\sigma >0$. Write $\lambda =(\mu -\rho )/\sigma$ for the market price of risk. If $w_t$ is portfolio wealth, ${\pi }_t$ the dollar amount in stock, and $c_t\ge 0$ the consumption rate, the budget dynamics areAdmissible controls are progressively measurable, locally integrable in the required drift and diffusion norms, and keep $w_t\ge 0$. Such a solvency restriction excludes doubling-type strategies. In a complete Brownian market, the investment-consumption problem is to maximizewhere the utility functions are increasing and concave. A terminal-only problem omits the integral. Strict concavity gives uniqueness of optimal consumption and terminal wealth when an optimum exists.

The state-price density isThe Itô formula givesThus $H_t{w}_t+{\int }_0^t{H}_s{c}_{s}ds$ is a nonnegative local martingale, hence a supermartingale, yielding the state-price budget constraintConversely, a nonnegative consumption plan and terminal payoff $\xi$ satisfying budget equality and the necessary integrability can be financed in the Brownian complete market. DefineThe Brownian martingale representation theorem for the total future budget supplies a stochastic-integral coefficient; comparing it with $H_t(\sigma {\pi }_t-\lambda w_t)$ defines ${\pi }_t$. This reconstructs the wealth equation with $w_t\ge 0$. Localization gives the representation for integrable positive budgets when an $L^2$ assumption is not available.

For smooth strictly concave utilities with the usual marginal-utility range, let $I_u=(u^{\prime }{)}^{-1}$ and $I_U=(U^{\prime }{)}^{-1}$ be the inverse marginal utility functions. The Lagrange-multiplier solution iswhere $y>0$ makes the budget an equality. This is a global argument, not merely formal differentiation: the concave supporting-tangent inequality giveswith the analogous terminal inequality. Taking expectations, integrating, and applying the budget bound makes the difference in objectives nonpositive. Replication of the candidate budget proves attainability and optimality whenever these expectations are well-defined and finite. If the marginal-utility ranges or integrability conditions fail, existence must be established separately rather than inferred from the first-order equations.

A detailed example is constant relative risk aversion utility $u(c)=c^{1-\gamma }/(1-\gamma )$ and $U(w)=\eta w^{1-\gamma }/(1-\gamma )$, with $\gamma >0$, $\gamma \ne 1$, $\eta >0$, and $\delta \ge 0$. PutThe optimal plans from the marginal-utility equations are $c_t^{\ast }=(y{e}^{\delta t}{H}_t{)}^{-1/\gamma }$ and ${\xi }^{\ast }={\eta }^{1/\gamma }(y{e}^{\delta T}{H}_T{)}^{-1/\gamma }$. The normal distribution of $W_t$ gives, for $a=1-1/\gamma$,The budget therefore becomes $y^{-1/\gamma }b(T)=w_0$, determining the multiplier explicitly. Independent Brownian increments in the conditional budget formula give $w_t^{\ast }=c_t^{\ast }b(T-t)$. The diffusion coefficient of $c_t^{\ast }$ is $(\lambda /\gamma )c_t^{\ast }$, so matching the wealth diffusion determines the dollar stock holding. Consequently the finite-horizon power-utility investment and consumption optimum isAll candidate moments are finite on a finite horizon, since powers of a lognormal state-price density have finite moments. The budget and tangent argument above therefore verify optimality in this example, including admissibility and positive wealth.

For another verification of the formulas, the Hamilton-Jacobi-Bellman equation for a value function with discount measured from the current time isSubstitute $V(w,t)=b(T-t{)}^{\gamma }{w}^{1-\gamma }/(1-\gamma )$. The two concave maximizations give the displayed controls, and the remaining equation is $b^{\prime }(\tau )=1-kb(\tau )$ with $b(0)={\eta }^{1/\gamma }$, exactly solved by the formula above. ThusFor $k=0$, interpret the integral as $\tau$, so $b(\tau )=\tau +{\eta }^{1/\gamma }$. Unlike the infinite-horizon Merton consumption-investment problem, finite horizons do not require $k>0$. Consumption vanishes in the terminal-only formulation; then the integral term in $b$ is absent and the optimal stock fraction remains $(\mu -\rho )/(\gamma {\sigma }^2)$. Logarithmic utility is handled directly with inverse marginal utility $1/z$ and the same wealth-to-consumption formula at $\gamma =1$, with $k=\delta$.

Write $T=t_0$, and specify the physical Black-Scholes model as $d{S}_t/S_t=\mu dt+\sigma d{W}_t$, with constant $\sigma >0$ and bank rate $\rho$. The physical drift $\mu$ is needed for an expected utility calculation; a pricing model specified only under a risk-neutral measure does not determine the investor's preferences over risky positions. PutHere $\vartheta$ is the market price of risk, $Z_T$ is the density of the risk-neutral measure $Q$, and $\xi$ is the state-price density. The Girsanov theorem makes $W_t^Q=W_t+\vartheta t$ a Brownian motion under $Q$.

For a dollar holding ${\pi }_t$ in the stock and consumption rate $c_t$, the portfolio wealth equation isSince $d{\xi }_t=-\rho {\xi }_{t}dt-\vartheta {\xi }_{t}d{W}_t$, integration by parts gives the useful deflated wealth equation with consumptionFor strategies with a true deflated gain martingale, terminal wealth $X=V_T$ and consumption obey the state-price budget constraintFor nonnegative wealth and consumption, the admissibility condition can instead give a budget inequality; increasing utilities use the full budget at an optimum. Conversely, suitable integrable payoff and consumption plans with this budget are financeable in the Brownian complete market. To construct the strategy, setBy the Brownian martingale representation theorem, write $d{M}_t=H_{t}d{W}_t$. The required dollar stock holding is ${\pi }_t=(H_t/{\xi }_t+\vartheta V_t)/\sigma$; substitution in the deflated gain equation verifies financing. Remaining wealth goes into the bank account. For square-integrable data the representation holds directly, and localization handles the customary integrable admissible data.

First take no consumption. For an increasing strictly concave differentiable utility function whose marginal derivative can be inverted, let $I_u=(u^{\prime }{)}^{-1}$. Choose $\lambda >0$ so thatThis candidate is not merely a first-order guess. The concave supporting-tangent inequality gives, pointwise,Taking expectations and using the budget proves $\mathbb{E}u(X)\le \mathbb{E}u(X^{\ast })$ for every feasible competitor. Finance the optimizer by its risk-neutral pricing value $V_t=e^{-\rho (T-t)}{\mathbb{E}}_Q[X^{\ast }\mid {\mathcal{F}}_t]$, using the preceding representation to find the holdings.

For a merely concave differentiable utility function, the general formulation is to select $X^{\ast }(\omega )$ from the maximizers of $u(x)-\lambda {\xi }_T(\omega )x$ over its wealth domain, and choose the multiplier to satisfy the budget. If increasing utility is not assumed, the equality-budget multiplier may have either sign; the positive-multiplier inverse formula above is for increasing utility. Flat portions allow multiple maximizers; boundaries require one-sided inequalities rather than equality of marginal derivatives. Existence of a multiplier, integrability and an admissible maximizer must be checked. Concavity alone does not guarantee a well-posed problem: with unrestricted dollar positions, linear utility $u(x)=x$ and $\mu >\rho$ permits unbounded expected terminal wealth. The explicit utilities below satisfy the needed conditions in their stated domains.

For exponential utility $u(x)=(1-e^{-ax})/a$ on $\mathbb{R}$, $u^{\prime }(x)=e^{-ax}$ and $I_u(y)=-\log y/a$. ThereforeThe budget is equivalently ${\mathbb{E}}_Q{X}^{\ast }=w_0{e}^{\rho T}$, since ${\xi }_T=e^{-\rho T}dQ/dP$. Under $Q$, ${\mathbb{E}}_Q{W}_T=-\vartheta T$, giving $\log \lambda =\rho T-{\vartheta }^{2}T/2-a{w}_0{e}^{\rho T}$. HenceMatching the Brownian coefficient of $dV$ with $\sigma {\pi }_t$ gives the finite-horizon exponential-utility portfolioHold ${\pi }_t^{\ast }/S_t$ shares and invest $V_t-{\pi }_t^{\ast }$ in the bank account. This is a dollar amount, not a fixed wealth fraction. The Gaussian exponential formula also gives the optimal expected utilityThis calculation uses unrestricted wealth and an admissibility class permitting the displayed Gaussian terminal wealth, with integrable deflated gains and the relevant exponential moment. A nonnegative-wealth constraint would change the optimizer; no such constraint is imposed on this exponential-utility example.

For both terminal and running utility function, choose $X$ and a nonnegative adapted consumption rate $c$ to maximize ${\mathbb{E}}_P[u(X)+{\int }_0^{T}v(c_t)dt]$ subject to the combined budget. Pointwise maximization of $u(x)-\lambda {\xi }_{T}x$ and $v(c)-\lambda {\xi }_{t}c$ gives, for interior strictly concave cases,Apply the concave supporting-tangent inequality to $u$ and to $v$ at every time, then integrate and take expectations. The difference in objectives is at most $\lambda$ times the difference in budget expenditures, which is nonpositive. This proves optimality, not just the marginal conditions. The financing value isThe question's consumption rate $R_t$ is denoted $c_t$ here to keep it distinct from interest-rate notation.

For logarithmic utility with $u(x)=a\log x$, $v(x)=b\log x$, assume $a,b,w_0>0$ and positive wealth and consumption. The marginal conditions and budget giveEach state-price-weighted payment is deterministic, so conditional valuation immediately gives $V_t=[a+b(T-t)]/(\lambda {\xi }_t)$. Put $A_t=a+b(T-t)$. Since the Itô formula gives $d(1/{\xi }_t)=(1/{\xi }_t)[(\rho +{\vartheta }^2)dt+\vartheta d{W}_t]$, this value satisfiesComparison with the wealth equation verifies the complete finite-horizon logarithmic investment and consumption strategy:Thus the optimal share holding is ${\pi }_t^{\ast }/S_t$, and the remaining value $V_t-{\pi }_t^{\ast }$ is in the bank account; borrowing or shorting is allowed if that value or the stock holding is negative. Terminal wealth is positive and consumption is positive throughout. Finally $\mathbb{E}\log {\xi }_t=-(\rho +{\vartheta }^2/2)t$, so the optimal objective, if desired explicitly, isFinite-horizon lognormal moments ensure the budget and expected utilities are finite, completing the feasibility and optimality checks.

For a competitive equilibrium with one productive asset, the primitives are the probability law and information filtration of total output $d$, each agent's utility function and discount factor, the initial ownership shares ${\theta }_0^j$ summing to one, and the permitted consumption and trading strategies. Prices are not primitives of the equilibrium problem. The derived objects are the ex-dividend price $S$, each agent's consumption $C^j$ and holdings ${\theta }^j$, and their marginal-utility pricing processes. At a common price, every agent solves the individual problem, and resource and share clearing requireThe individual discrete dividend budget equations, optimality Euler equations and terminal conditions from part (i), together with these clearing equations, define equilibrium even when the market is incomplete. Summing the individual budget constraints is consistent with total fruit consumption.

For the given constant absolute risk aversion utility, $U_j^{\prime }(c)=e^{-{\gamma }_{j}c}$. Hence each interior optimum satisfiesThe same $S$ must satisfy all these equations, but their conditional expectation identities do not by themselves identify the different agents' pricing densities pointwise.

To illustrate explicit exponential-utility risk sharing, first consider a complete or otherwise implementable interior allocation. With positive constant welfare weights $a_j$, a pointwise resource multiplier ${\nu }_t$ satisfiesWriteSolving the marginal equations and summing the consumptions givesThe output risk is shared in proportion to risk tolerance ${\gamma }_j^{-1}$; differences in patience produce the time-dependent transfer term. The common normalized state-price density is ${\zeta }_t={\nu }_t/{\nu }_0$. The weights, up to a common scale, are determined by the initial wealth budgetsand, for a fundamental price,If consumption is required to be nonnegative and the interior formula produces a negative allocation, replace it byFor $d_t>0$ this determines the resource multiplier uniquely. At zero total output all constrained consumptions are zero and the corresponding marginal inequalities apply.

A single traded tree need not span the contingent transfers in the general risk-sharing formula. A genuine equilibrium example within the literal one-tree market is obtained by taking all ${\beta }_j=\beta$ and initial shares ${\theta }_0^j=\Gamma /{\gamma }_j$. ThenEvery agent simply keeps the initial shares, all budget constraints and clearing conditions hold, and their discounted marginal utilities coincide. The fundamental price above satisfies every Euler equation. For example, nonnegative bounded output ensures the required price and utility integrability and the terminal terms vanish; the supporting-line argument from part (i) then verifies individual optimality. For heterogeneous discount factors or arbitrary initial ownership, the displayed coupled Euler-budget-clearing system remains the correct one-tree formulation; using the unconstrained Pareto allocation as a traded equilibrium requires an additional financing check.

Take $S_t$ to be the ex-dividend stock price: the date-$t$ output has already been paid when trading occurs. Receiving ${\theta }_t{d}_t$ units of fruit and selling ${\theta }_t-{\theta }_{t+1}$ shares gives the discrete dividend budget equationThe entering holding ${\theta }_t$ is chosen with date-$(t-1)$ information, while $C_t$ and the next holding ${\theta }_{t+1}$ may use date-$t$ information. The output is the consumption good and price numéraire; its stochastic law is exogenous to the individual's price-taking choice.

For the interior differentiable case, attach an adapted Lagrange multiplier ${\Lambda }_t$ to each budget constraint. On a finite horizon, with the terminal holding fixed or its continuation value included, the optimization Lagrangian isAn adapted variation in $C_t$ gives ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. An arbitrary date-$t$ measurable variation in ${\theta }_{t+1}$ affects two adjacent terms. Its coefficient must have zero conditional mean, soThese are the Euler equations of marginal utility pricing in a dividend economy. If the utility function is merely concave, a supporting slope replaces $U^{\prime }$; binding consumption or trading restrictions give the corresponding inequalities rather than unconstrained equalities.

With known initial information and ${\Lambda }_0>0$, normalize ${\zeta }_t={\Lambda }_t/{\Lambda }_0$. This is a state-price density for the traded asset. In particular,The ratio is the stochastic discount factor: payment in a state with greater marginal utility has a higher marginal value. For a priced contingent payoff $H$ at $t+1$, its price is ${\mathbb{E}}_t[({\zeta }_{t+1}/{\zeta }_t)H]$; pricing arbitrary claims uniquely also requires market completeness. The deflated cumulative gains ${\zeta }_t{S}_t+{\sum }_{s=1}^t{\zeta }_s{d}_s$ form a martingale whenever the displayed terms are integrable.

Iterating the Euler equation givesUnder the dividend-price transversality condition and the needed integrability, this becomes the fundamental priceThus an infinite-horizon price is not automatically just a dividend sum: its terminal term must vanish. Telescoping the individual budget constraints similarly givesA no-Ponzi lower restriction on this final term gives the consumption budget inequality; its vanishing gives equality. Together with the supporting-line inequality for a concave utility function, the Euler equations and these terminal conditions supply the usual sufficiency argument for an admissible candidate. In an incomplete market, each optimal agent's marginal utility can supply a different state-price density pricing the same traded tree.

Use ${\theta }_t$ for the dollar amount invested in the stock; the number of shares is ${\theta }_t/S_t$. The bank account contains $w_t-{\theta }_t$ dollars. With no intermediate consumption, the self-financing portfolio equation isIf instead $\theta$ denotes share holdings, replace each dollar ${\theta }_t$ here by $S_t{\theta }_t$. The equation says that trading only reallocates existing portfolio wealth; gains come from cash interest and stock returns.

For the standard unconstrained complete-diffusion formulation, assume the completed natural Brownian filtration, initial wealth $w_0>0$, and nonnegative admissible wealth. Define the market price of risk ${\kappa }_t=({\mu }_t-r_t)/{\sigma }_t$. The boundedness assumptions make $\kappa$ bounded. The state-price density normalized at zero isIts money-market-adjusted process $Z_t=e^{{\int }_0^t{r}_{s}ds}{\zeta }_t$ is a true martingale by the Novikov condition. Under $d\mathbb{Q}=Z_{T}d\mathbb{P}$, the Girsanov theorem gives $W_t^{\mathbb{Q}}=W_t+{\int }_0^t{\kappa }_{s}ds$ as a Brownian motion, and the discounted stock has zero drift.

The Itô product rule applied to the wealth equation givesFor nonnegative admissible wealth this is a nonnegative local martingale, hence a supermartingale. Therefore every affordable terminal payoff $X=w_T$ satisfies the state-price budget constraint $\mathbb{E}[{\zeta }_{T}X]\le w_0$. Nonzero volatility and the Brownian martingale representation theorem allow replication of nonnegative claims with finite state-price cost.

For a differentiable strictly concave utility function with the usual conditions ensuring an interior integrable optimizer, put $I=(U^{\prime }{)}^{-1}$, the inverse marginal utility. Pointwise maximization of $U(x)-y{\zeta }_{T}x$ givesThe second equation determines the positive constant $y$. To prove optimality, use the concave supporting-tangent inequality:Taking expectations and using the budget bound proves that the candidate dominates every admissible affordable payoff, whenever the utility expectations are well defined. This is the complete-market terminal utility optimizer.

The actual wealth and dollar portfolio are as explicit as general adapted coefficients permit. LetThe Brownian martingale representation theorem, with localization if only $L^1$ integrability is initially known, gives $h$. The dollar portfolio from a deflated wealth martingale isThe bank holds $w_t^{\ast }-{\theta }_t^{\ast }$ dollars. Equivalently, with $B_t=e^{{\int }_0^t{r}_{s}ds}$, $w_t^{\ast }=B_t{\mathbb{E}}^{\mathbb{Q}}[w_T^{\ast }/B_T\mid {\mathcal{F}}_t]$. These formulas need no unjustified differentiability of a value function with random coefficients.

The printed assumption that $U$ is only increasing and concave does not guarantee that the interior formula exists, or even that the optimal value is finite. In general replace the inverse formula by an affordable selectionwhere $\mathcal{D}$ is the wealth domain; boundary choices or supporting slopes handle nondifferentiability. Existence, finite cost and replication still have to hold. A concrete obstruction is the unbounded linear terminal-wealth utility: take $U(x)=x$, $r=0$, $\mu =\sigma =1$, and the Brownian filtration. Then ${\zeta }_T=e^{-W_T-T/2}$. For $A_n=\{{\zeta }_T<1/n\}$, the bounded nonnegative claimis replicable, costs $w_0$, and has $\mathbb{E}{X}_n>n{w}_0$. Thus there is no finite optimal expected utility under the literal utility hypotheses alone. Also, bounded coefficients do not ensure market completeness in an arbitrarily enlarged filtration containing untraded randomness; the full replication formula uses the Brownian market information assumption stated above.

Assume, as in the Bayesian model, that the unknown drift $\alpha$ is independent of the driving Brownian motion, with known positive prior precision ${\tau }_0$. Recover the observed signal from the positive stock price:For a fixed time $t$, the centered observed path $Y_s-(s/t)Y_t$ is a Brownian bridge independent of $(\alpha ,Y_t)$: it is independent of the prior and has zero covariance with $W_t$ in their joint Gaussian law. Thus the whole observed history has the same posterior information about $\alpha$ as its endpoint. This proves Brownian endpoint sufficiency for a constant drift rather than assuming it.

The likelihood of that endpoint, as a function of $\alpha$, is proportional to $\exp (\alpha Y_t-{\alpha }^{2}t/2)$. Multiplying by the normal distribution prior and completing the square gives the Gaussian Brownian drift filterDefine the observed innovation process ${\hat{W}}_t=Y_t-{\int }_0^t{\hat{\alpha }}_{s}ds$. It is a continuous observed-filtration martingale: for $s<u$, independence of future noise and the tower property of conditional expectation give $\mathbb{E}[\alpha -{\hat{\alpha }}_u\mid {\mathcal{F}}_s^S]=0$, so the conditional mean of its increment vanishes. Its quadratic variation is $t$, since subtracting the finite-variation drift from $Y$ does not change quadratic variation. The Lévy characterization of Brownian motion makes $\hat{W}$ an observed-filtration Brownian motion. Differentiating the posterior mean givesConversely, the last equation determines $\hat{\alpha }$ from $\hat{W}$ with deterministic coefficients and known initial value, and $Y=\hat{W}+\int \hat{\alpha }dt$. Thus the observed and innovation filtrations coincide, supplying the natural Brownian information needed for replication. Consequently the observed stock dynamics are $d{S}_t/S_t=\sigma d{\hat{W}}_t+\sigma {\hat{\alpha }}_{t}dt$. Its observed market price of risk is ${\kappa }_t={\hat{\alpha }}_t-r/\sigma$, and its normalized state-price density solvesThe posterior mean is unbounded, so bounded-coefficient Novikov condition reasoning from Question 2 is not available automatically. A direct finite-horizon argument supplies true pricing. Relative to driftless Wiener measure for $Y$, the Bayesian mixture has observation densityA law with observation drift $k=r/\sigma$ instead has density $e^{k{Y}_t-k^{2}t/2}$. ThereforeThis is the finite-horizon pricing density for Gaussian drift learning. The ratio changes the physical observed law into a genuine probability law, hence has mean one. Indeed, differentiation of the explicit likelihood, using $d{Y}_t=d{\hat{W}}_t+{\hat{\alpha }}_{t}dt$, givesThe Itô formula therefore gives $d{D}_t=-D_t{\kappa }_{t}d{\hat{W}}_t$, agreeing with the stochastic exponential above. Under this pricing law the stock drift is $\sigma k=r$.