A voltage with boundary values $u(a)$ and $u(b)$ is a function $u:V\to \mathbb{R}$ that is harmonic at every other vertex:A current flow is an antisymmetric function $i(x,y)=-i(y,x)$ satisfying Kirchhoff's node law ${\sum }_{y}i(x,y)=0$ away from its source and sink. Voltage and current are related by Ohm's law,The effective resistance is the voltage drop divided by the total current from $a$ to $b$. Equivalently, it is the voltage drop generated by a unit current flow:

By Tonelli theorem,If $p<p_c$, part (a) bounds the summand by $e^{-c\Vert x{\Vert }_{\infty }}$. There are only polynomially many vertices at each radius, so the series converges and $\chi (p)<\infty$.

Conversely, suppose $\chi (p)<\infty$. Then $p<1$. Choose $q>p$ so close to $p$ thatUse the standard sprinkling coupling for Bernoulli percolation: first expose the $p$-open clusters, then independently open each remaining edge with probability $\alpha =(q-p)/(1-p)$. Explore the $q$-cluster of the origin cluster by following sprinkled edges. Each discovered $p$-cluster has at most $2d$ times its number of vertices as many incident edges, so the exploration is dominated by a Galton-Watson process of mean at most $2d\alpha \chi (p)<1$. This process dies out almost surely, and hence there is no infinite $q$-open cluster. Thus $q\le p_c$, and $p<q$ implies $p<p_c$. Therefore

Because $p<p_c={\tilde{p}}_c$, choose a finite set $S\ni 0$ with ${\varphi }_p(S)=\rho <1$. Let $L$ exceed the ${\ell }^{\infty }$-distance from $0$ to every endpoint of an edge in $\partial S$, and putOn the one-arm event to distance $n>L$, take the first oriented boundary edge $(x,y)\in \partial S$ used by an open self-avoiding path. The connection $0\stackrel{S}{\leftarrow }x$, the open edge $(x,y)$, and the remaining connection from $y$ to $\partial B(n)$ occur disjointly. The van den Berg-Kesten inequality and translation invariance therefore giveIteration yields $q_n\le {\rho }^{\lfloor n/L\rfloor }$ up to an inessential finite-scale adjustment. Since $p<1$, each of the finitely many remaining $q_n$ is strictly below one, so reducing the exponent if necessary produces a constant $c>0$ valid for every $n\ge 1$:This is the finite-size proof of exponential decay of subcritical percolation.

PutFix $k\ge 1$ and write $n=qk+r$ with $0\le r<k$. Repeated subadditivity givessoThe finitely many values $x_0,\dots ,x_{k-1}$ are bounded, henceTaking the infimum over $k$ gives $\mathrm{limsup}{x}_n/n\le \alpha$, while the definition of $\alpha$ gives $x_n/n\ge \alpha$ for every $n$. ThereforeThis is Fekete lemma.

The terms independent of $v$ in the substituted PDE satisfysoIntegrating backward from $T$ yieldsThus the requested constant is

Forone hasSubstitution into the PDE and collection of the coefficient of $v$ give the Riccati differential equationThe terminal condition also requires $B(T)=0$.

Apply the multidimensional Itô formula to ${\xi }_t=F(t,S_t,v_t)$. The stated PDE cancels its drift to $rFdt$, leavingConsequently $e^{-rt}{S}_t$ and $e^{-rt}{\xi }_t$ are local martingales under the physical measure $P$. Thus $P$ itself is an equivalent local martingale measure for the augmented market relative to the bank account. The continuous-time fundamental theorem of asset pricing says that existence of such a measure for locally bounded prices implies no free lunch with vanishing risk, and hence no arbitrage. The terminal condition also gives ${\xi }_T=\sqrt{S_T}$ as required.

Risk-neutral valuation givesFor $s\ge t$,ThereforeThe future ${\zeta }_j$ are independent and identically distributed under the stated model, sowith an empty product equal to one.

If $T\mapsto P_t^T$ is nonincreasing, thenso $1+r_t=(P_t^{t+1}{)}^{-1}\ge 1$ and $r_t\ge 0$.

Conversely, if every spot rate is nonnegative, then $B_{T+1}=B_T(1+r_T)\ge B_T$. Under a risk-neutral measure,Hence

The one-period spot interest rate is defined byand the bank account byA probability measure $Q$ equivalent to the physical measure is a risk-neutral measure when every discounted zero-coupon bond priceis a $Q$-martingale. Equivalently,

At maturity, $P_T^T=1$. If $P_t^T\le 0$ on an event $A\in {\mathcal{F}}_t$ of positive probability, buying one bond on $A$ has nonpositive cost and certain payoff $1$ on $A$ at $T$; any negative purchase cost can also be consumed or retained. This is an arbitrage. Therefore absence of arbitrage implies

Suppose a numéraire strategy $\eta$ existed and write $N_t={\eta }_{t+1}\cdot P_t={\eta }_t\cdot P_t>0$, using zero consumption. Since ${\eta }_{t+1}$ is ${\mathcal{F}}_t$-measurable and $M_t=(-1{)}^t{Z}_t{P}_t$ is a martingale,ThusorThe left side is nonnegative and the right side nonpositive, so $Z_t{N}_t=0$ almost surely. Strict positivity of $N_t$ implies $Z_t=0$ almost surely for every $t$, contradictingTherefore the market has no numéraire strategy.

A numéraire strategy $\eta$ has zero consumption and strictly positive wealthat every date. Given an investment-consumption arbitrage $H$, retain its holdings and invest each nonnegative consumption $C_s^{0,H}$ in the numéraire. Withthe self-financing identity for $\eta$ gives zero intermediate consumption for $K$. At a deterministic $T$ after a date at which positive consumption occurs with positive probability, liquidating givesIt is strictly positive with positive probability. Hence $K$ is a terminal-consumption arbitrage.

For initial capital zero and a predictable strategy $H$, let $C_t^{0,H}$ denote consumption after the time-$t$ portfolio payoff and before choosing the next holdings. An investment-consumption arbitrage haswith strictly positive consumption at some date with positive probability. A terminal-consumption arbitrage is a finite-horizon such strategy whose consumption is zero before its terminal date $T$, while $C_T^{0,H}\ge 0$ almost surely and $\mathbb{P}(C_T^{0,H}>0)>0$.

Apply Itô formula to $V(t,S_t)$. Its Brownian coefficient isThe self-financing portfolio's Brownian coefficient is ${\pi }_t\sigma S_t$. Since $S_t\sigma >0$, equality of the two value processes forcesThus the stock holding is the claim's option delta.

For constant coefficients, the density process $Y_t{B}_t/B_0$ is a true exponential martingale and defines the risk-neutral measure $Q$. Under $Q$,The minimal value process is thereforeThe Markov property and the lognormal transition law make this a deterministic function of $(t,s)$, and

SetThe boundedness of ${\xi }_T$ and positivity of the deflator make $M$ a nonnegative true martingale. By the Brownian martingale representation theorem, $d{M}_t={\eta }_{t}d{W}_t$. A self-financing wealth process with stock holding ${\pi }_t$ satisfiesThe product $YV$ then has diffusion coefficientChooseThen $Y_t{V}_t=M_t$, so $V_T={\xi }_T$ and the strategy replicates the claim. It is admissible because $V=M/Y$ is nonnegative.

For any other admissible replicating wealth $\tilde{V}$, the nonnegative local martingale $Y\tilde{V}$ is a supermartingale. HenceThe constructed strategy hasso this is the minimal replication cost.