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1. Brownian Motion - a limit

If \$f:,infty) to mathbbR\$ is a continuous function and \$D subset ,infty)\$ a dense set, then \$\$forall s in ,t]: f(s) geq -a iff forall s in ,t] cap D: f(s) geq -a.\$\$ The implication "\$Rightarrow\$" is obvious and the other implication follows from the continuity of \$f\$ (proof by contradiction: if we had \$f(s) -a

ight). tag1\$\$Finally, note that\$\$mathbbP(forall s in ,t]: B_s geq -a) = mathbbP(forall s in ,t]: B_s > -a) tag2\$\$which follows, for instance, from the reflection principle or from the strong Markov property of Brownian motion (see this answer). Combining \$(1)\$ and \$(2)\$ proves the desired identity.

2. Function of brownian motion is a martingale

You can apply the following theorem to prove \$u(t,B_t)\$ a martingale. The main idea is -as Ilya wrote- to use the fact that the density \$p(t,x)\$ of a Gaussian distribution with mean 0, variance \$t\$ solves \$\$fracpartialpartial t p(t,x) = frac12 fracpartial^2partial x^2 p(t,x) qquad (x in mathbbR,t>0) tag1\$\$.Theorem Let \$(B_t)_t geq 0\$ a Brownian motion and \$u in C^1,2 cap C(,infty) times mathbbR)\$ such that \$u\$ and its partial derivatives are exponentially bounded. Then \$\$M_t^u := u(t,B_t) - u(0,B_0) - int_0^t underbraceleft( fracpartial partial t frac12 fracpartial^2partial x^2

ight) u(r,B_r)_=:Lu(r,B_r) , dr\$\$ is a martingale.Sketch of the proof: By the Markov property of the Brownian motion, one can show that\$\$beginalign* mathbbE(M_t^u-M_s^u mid mathcalF_s) &= mathbbE left( u(t,B_t)-u(s,B_s) - int_s^t Lu(r,B_r) , dr mid mathcalF_s

ight) &= mathbbE left( u(t,B_t-sz)-u(s,B_0z) - int_0^t-s Lu(rs,B_rz) , dr

ight) bigg|_z=B_s &= mathbbE bigg( underbracevarphi(t-s,B_t-s) - varphi(0,B_s) - int_0^t-s Lvarphi(r,B_r) , dr_M_t-s^varphi-M_0^varphi bigg) bigg|_z=B_s endalign*\$\$where \$varphi(t,x) := u(ts,xz)\$. To prove that this expression equals \$0\$, use Fubini's Theorem, partial integration and \$(1)\$ to show that\$\$mathbbE(M_t^varphi-M_varepsilon^varphi) = int (p(t,x) cdot varphi(t,x)-p(varepsilon,x) cdot varphi(varepsilon,x)) , dx - int_varepsilon^t p(r,x) cdot left( fracpartialpartial t frac12 fracpartial^2partial x^2

ight) varphi(r,x) , dx , dr = 0\$\$for \$t > varepsilon>u\$. Finally, by Doob's maximal inequality and the exponential boundedness, one can apply dominated converence to conclude that \$\$mathbbE(M_t^varphi-M_0^varphi) = lim_varepsilon to 0 mathbbE(M_t^varphi-M_varepsilon^varphi) = 0\$\$This finishs the proof.(For more details, see e.g. Ren L. Schilling/Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Theorem 5.6.)Actually, the mentioned theorem is a special case of the following theorem:Theorem Let \$(X_t)_t geq 0\$ a Feller process with generator \$A\$. Then \$\$f(X_t)-f(X_0) - int_0^t Af(X_r) , dr\$\$ is a martingale for any \$f\$ in the domain of \$A\$.So, in your case the process is given by \$X_t := (t,B_t)\$. With some knowledge about generators, it's not that difficult to show that the generator \$A\$ is given by \$\$A = fracpartialpartial t frac12 fracpartial^2partial x^2.\$\$ (This leads to the topic of transition semigroup and its generator. It's also contained in the book, chapter 7.)

3. Why does the Arctic ocean not freeze when it has been -25 to -35 degrees at Alert, Nunavut for the last two months?

Nobody pointed out yet that air temperature does not translate into water temperature. Then take into account that water is always in motion, it has kinetic energy, it is not still like the water you put in an ice cube tray in your freezer

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