A triangular probability density function rises linearly from zero at $a$ to its mode $c$, then falls linearly to zero at $b$. Its peak is $2/(b-a)$. Its inverse transform sampling rule is $a+\sqrt{u(b-a)(c-a)}$ for $u\le (c-a)/(b-a)$, and $b-\sqrt{(1-u)(b-a)(b-c)}$ otherwise.

For $a,b>0$, this probability density function is obtained from a beta distribution variable $U$ by $X=U/(1-U)$. Indeed $u=x/(1+x)$ and $du/dx=(1+x{)}^{-2}$, and the change of variables formula gives the displayed density. In particular, the ratio of two independent equal-rate exponential distributions has $a=b=1$, density $(1+x{)}^{-2}$ and cumulative distribution function $x/(1+x)$.

For two independent random variables taking values in a three-dimensional ball with constant volume probability density, the density of their distance is the displayed expression for $0\le r\le 2R$, and zero otherwise. Both independence and the common radius $R>0$ are essential. To derive it, put $u=|{\mathbf{r}}_1|$, $v=|{\mathbf{r}}_2|$ and integrate the product volume differential form over orientations. The law of cosines gives the angular Jacobian determinant $v/(ur)$, leaving $f_R(r)=9r\int uvdudv/(2R^6)$ over $0\le u,v\le R$, $|u-v|\le r\le u+v$. With $x=u+v$, $y=u-v$, the integrand becomes $(x^2-y^2)dxdy/8$. Direct integration gives $\int uvdudv=2R^{3}r/3-R^2{r}^2/2+r^4/24$. An independent geometric derivation uses the overlap volume of two radius-$R$ balls displaced by $r$, namely $\pi (4R+r)(2R-r{)}^2/12$, multiplied by $4\pi r^2$ and divided by the square of $4\pi R^3/3$. The density integrates to one and has mean $36R/35$.

The Cantor distribution is the law of a random variablewhere the $B_n$ are independent Bernoulli random variables taking values zero and one with equal probability. Its values lie in the Cantor set, and its cumulative distribution function is the Cantor function. Every individual value has probability zero, but the Cantor set has probability one despite having zero Lebesgue measure. A probability density function on that set would integrate to zero, so it cannot describe this law. The example motivates the distinction between a continuous probability distribution and an absolutely continuous probability distribution.

The continuous map projecting a real number onto the closed interval $[a,b]$, with $a\le b$. It equals $a$ below the interval, equals the input inside it, and equals $b$ above it. It is nondecreasing and has Lipschitz constant one.

A bounded linear operator is strictly singular if it is bounded below on no infinite-dimensional subspace. Equivalently, every infinite-dimensional subspace contains unit vectors with arbitrarily small images. On a hereditarily indecomposable Banach space, an operator which is arbitrarily small on unit vectors in every finite-codimensional subspace is strictly singular: construct an infinite basic span on which its operator norm is small, then use the zero-angle property of hereditary indecomposability.

For $1\le p<\infty$, a bounded linear operator $T:E\to F$ between Banach spaces is absolutely p-summing if $({\sum }_j\Vert T{x}_j{\Vert }^p{)}^{1/p}\le C{\sup }_{\varphi \in B_{E^{\ast }}}({\sum }_j|\varphi (x_j){|}^p{)}^{1/p}$ for every finite family. The least $C$ is its p-summing norm ${\pi }_p(T)$. Direct substitution gives the ideal inequality ${\pi }_p(BTA)\le \Vert B\Vert {\pi }_p(T)\Vert A\Vert$ for compatible bounded linear operators.

Let $T:V\to W$ be a bounded linear operator with $V$ a Banach space. If every $y\in W$ has an approximate preimage $x$ with $\Vert x\Vert \le R\Vert y\Vert$ and $\Vert Tx-y\Vert \le k\Vert y\Vert$, where $k<1$, then every $y$ has an exact preimage with the displayed norm bound. Repeatedly correct the residual: the corrections have norms at most $R{k}^j\Vert y\Vert$ and form an absolutely convergent series; its image under $T$ equals $y$. This proves surjectivity but does not provide a bounded linear right inverse.

A bounded linear operator between Banach spaces is completely continuous when it maps weakly convergent sequences to norm-convergent sequences. Equivalently, it maps weakly compact sets to norm-compact sets: the Eberlein-Šmulian theorem gives the forward direction, while a weakly convergent sequence together with its limit is weakly compact and gives the reverse direction. The inclusion $C(\mathbb{T})\to L^2(\mathbb{T})$ for finite Lebesgue measure is completely continuous by pointwise convergence, the Uniform boundedness principle and the dominated convergence theorem, but is not a compact operator because of the orthogonal Fourier modes.

For nonsquare $N>1$, write the complete quotients after the initial term as $(\sqrt{N}+m)/d$. They exceed one, while their algebraic conjugates lie between minus one and zero. Thus $d<\sqrt{N}+m<2\sqrt{N}$. Comparing coefficients in $\sqrt{N}=(p_k{\alpha }_{k+1}+p_{k-1})/(q_k{\alpha }_{k+1}+q_{k-1})$ gives $p_k^2-N{q}_k^2=d_{k+1}(p_k{q}_{k-1}-q_k{p}_{k-1})$, and the last determinant has absolute value one. These small residues are useful candidates for smooth numbers in congruence-of-squares factorization.

A barrier option activates or expires according to whether the underlying price reaches a prescribed level during its lifetime. Its state must include whether the barrier has already been reached; the current price alone generally does not determine its remaining payoff.

A put option gives its holder the right to sell an underlying asset at a specified strike price. A European put option may be exercised only at maturity, whereas an American put option permits earlier exercise.

A contingent claim whose payoff depends on the underlying price history, rather than only its terminal price. Asian options use averages, barrier options use whether a threshold has been crossed, and lookback options use running extrema. In the Black-Scholes model, the Brownian Martingale representation theorem still gives a replicating strategy; its portfolio wealth need not be a function of the current stock price alone. Adding a running integral or extremum can give a finite-dimensional state for pricing.

The monetary amount delivered by a contingent claim at its payment time, represented by a random variable measurable with respect to the information then available. For a European call option it is $(S_T-K{)}^{+}$, and for a European put option it is $(K-S_T{)}^{+}$. A payoff is the delivered amount, whereas the claim is the contract and its current price is the cost of acquiring or replicating that contract.

A constructible regular polygon is a regular polygon admitting a compass-and-straightedge construction. The Gauss-Wantzel theorem classifies its possible numbers of sides.

Set $S_2=\sqrt{15}{\beta }^{3/2}{v}_{\eta }^{2}h$ and $r=(15\beta {)}^{3/4}\eta x$. The Kolmogorov equation for structure functions and the constant-skewness turbulence closure give the displayed ordinary differential equation, with $h(0)=0$. Its small-separation balance is $h\sim x^2$; its large-separation balance is $h\sim x^{2/3}$. The first matches differentiable velocity increments, and the second recovers the assumed Kolmogorov two-thirds law.

A fee proportional to terminal portfolio wealth but reduced by accumulated portfolio quadratic variation changes the optimal risk exposure of a manager with constant relative risk aversion utility. Write $A=\sigma {\sigma }^T$, $b=\mu -r1$ and assume $A$ is invertible. The fee process has drift $r+b^T\pi -\epsilon {\pi }^{T}A\pi /2$ and volatility ${\sigma }^T\pi$. Its Hamilton-Jacobi-Bellman equation therefore maximizes $b^T\pi -(R+\epsilon ){\pi }^{T}A\pi /2$, giving ${\pi }^{\ast }=A^{-1}b/(R+\epsilon )$. The fee increases effective relative risk aversion from $R$ to $R+\epsilon$, independently of its positive scale $a$.

There is also a direct global bound. Put $p=1-R$, $v^{\ast }={\sigma }^T{\pi }^{\ast }$, and tilt probability by $\exp (p{v}^{\ast }\cdot W_T-p^2|v^{\ast }{|}^{2}T/2)$. Under this probability let $Z$ be the stochastic exponential of $(1+\epsilon )\int ({\sigma }^T\pi -v^{\ast })\cdot d{W}^{\ast }$. The terminal fee ratio satisfies $(y_T/y_T^{\ast }{)}^p=Z_T^{p/(1+\epsilon )}$. A nonnegative local martingale has expectation at most one. The Jensen inequality, applied to a concave positive power when $p>0$ and a convex negative power when $p<0$, proves that the constant exposure maximizes expected CRRA utility. Effective risk aversion one uses logarithmic utility.

Logarithmic utility on positive portfolio wealth has derivative $1/x$ and relative risk aversion one. Its utility conjugate is $V(y)=-\log y-1$, since the maximizing wealth is $1/y$. It satisfies the Inada conditions.

For zero cash interest, constant excess drift $\mu$, volatility $\sigma$, and both terminal and consumption utility $2\sqrt{x}$, set $A=\sigma {\sigma }^T$ and assume $\mu$ belongs to the image of a linear map $A$. Let $A^{+}$ be the Moore-Penrose inverse and $q={\mu }^T{A}^{+}\mu$. Then $h^{\prime }+qh+1=0$, $h(T)=1$, so $h(t)=e^{q(T-t)}+(e^{q(T-t)}-1)/q$ for $q>0$, and $h(t)=1+T-t$ for $q=0$. The Hamilton-Jacobi-Bellman equation has the displayed solution. Maximizing dollar investment and consumption are ${\theta }_t=2X_t{A}^{+}\mu$ and $C_t=X_t/h(t)$. The resulting positive portfolio wealth is a geometric Brownian motion with a deterministic time-dependent drift, and finite-horizon moment bounds justify equality in verification.

In a Brownian complete market with constant coefficients and nonsingular $\sigma$, maximize expected CRRA utility of terminal portfolio wealth divided by a geometric stock index. Writing $p=1/R$ and $q=n^{-1}1$, the pointwise marginal condition gives $w_T^{\ast }={\lambda }^{-p}{\zeta }_T^{-p}{J}_T^{1-p}$. Its state-price budget constraint determines ${\lambda }^{-p}=w_0/\mathbb{E}[({\zeta }_T{J}_T{)}^{1-p}]$. Conditional lognormal moments and replication then give the displayed constant dollar fractions. The first term is ordinary optimal risk exposure; the second hedges the random benchmark. The bank holds the remaining fraction, which can be negative when borrowing is allowed.