Advanced Probability Theory

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Questions and Answers

Let $T$ be a random variable. Which of the following conditions is both necessary and sufficient for $T$ to be a stopping time with respect to a filtration $(\mathcal{F}n){n \geq 0}$?

$\forall n \in \mathbb{N}, \{T = n\} \in \mathcal{F}_n$ (correct)
$\forall n \in \mathbb{N}, \{T \leq n\} \in \mathcal{F}_{n-1}$
$\forall n \in \mathbb{N}, \{T = n\} \in \mathcal{F}_{n+1}$
$\forall n \in \mathbb{N}, \{T \geq n\} \in \mathcal{F}_n$

Consider a sequence of random variables $(X_n)_{n \geq 0}$ and a measurable set $B \in \mathcal{B}(\mathbb{R})$. Let $T = \inf {n : X_n \in B}$ and $S = \sup {n : X_n \in B}$. Which of the following statements is generally true?

Neither $T$ nor $S$ can be stopping times.
$S$ is a stopping time, but $T$ is not necessarily a stopping time.
Both $T$ and $S$ are always stopping times.
$T$ is a stopping time, but $S$ is not necessarily a stopping time. (correct)

Let $T$ be a stopping time with respect to a filtration $(\mathcal{F}n){n \geq 0}$, and $X_n$ a sequence of random variables. If $X_T(\omega) = X_{T(\omega)}(\omega)$ on ${T < \infty}$ and 0 otherwise, how is the $\sigma$-algebra $\mathcal{F}_T$ defined, encapsulating information up to the random time $T$?

$\mathcal{F}_T = \{A \in \mathcal{F}_\infty : A \cap \{T > n\} \in \mathcal{F}_n \}$
$\mathcal{F}_T = \{A \in \mathcal{F}_\infty : A \cap \{T < n\} \in \mathcal{F}_n \}$
$\mathcal{F}_T = \{A \in \mathcal{F}_\infty : A \cap \{T \leq n\} \in \mathcal{F}_n \}$ (correct)
$\mathcal{F}_T = \{A \in \mathcal{F}_\infty : A \cap \{T = n\} \in \mathcal{F}_n \}$

Consider stopping times $S$ and $T$. Which of the following statements regarding the relationship between $\mathcal{F}_S$ and $\mathcal{F}_T$ is correct if $S \leq T$ almost surely?

$\mathcal{F}_S \subseteq \mathcal{F}_T$ (A) Signup and view all the answers

Let $(X_n){n \geq 0}$ be a sequence of random variables and $T$ a stopping time. Which of the following expressions defines the stopped process $(X{T \wedge n})_{n \geq 0}$ correctly, emphasizing the path-wise behavior of the process up to the stopping time?

$(X_{T(\omega) \wedge n}(\omega))_{n \geq 0}$ (D) Signup and view all the answers

Given stopping times $T_n$ for $n \geq 0$, which of the following operations on the sequence $(T_n)_{n \geq 0}$ does not necessarily result in another stopping time?

$\frac{1}{N} \sum_{n=1}^{N} T_n$, where $N$ is a fixed positive integer. (A) Signup and view all the answers

Consider a random variable $X_n^* = \max_{k \leq n} |X_k|$ representing the running maximum of a sequence of random variables $(X_n)_{n \geq 0}$. For $p > 1$ and $k > 0$, and defining $X_n^* \wedge k = \min(X_n^, k)$, which inequality relating the $L^p$ norm of $X_n^ \wedge k$ and $X_n$ is most accurate, reflecting a maximal inequality concept?

$||X_n^* \wedge k||_p^p \leq \frac{p}{p-1} E[|X_n|^p]$ (D) Signup and view all the answers

Let $X$ be a stochastic process and $T$ a stopping time with respect to the filtration $(\mathcal{F}t)$. Suppose that for every bounded martingale $M$, the process $M_t^{T} = M{t \land T}$ is also a martingale. Which of the following provides the most precise description of what can be concluded about the process $X$ and the stopping time $T$?

The stopped process $X^T$ is adapted to the filtration $(\mathcal{F}_t)$, and $E[X_T | \mathcal{F}_0] = X_0$. (B) Signup and view all the answers

Consider a stochastic process $(X_t)_{t \geq 0}$. Which of the following statements, concerning the relationship between finite-dimensional distributions and the law of $X$, is most accurate under general conditions?

Knowledge of all finite-dimensional distributions is sufficient to uniquely determine the law of $X$ because the cylinder sets form a $\pi$-system generating the $\sigma$-algebra. (A) Signup and view all the answers

Suppose $(X_t)_{t \geq 0}$ is a stochastic process. Which of the following statements most accurately describes the relationship between the process being cadlag and its sample paths?

The stochastic process $(X_t)_{t \geq 0}$ is cadlag if and only if almost surely, the map $t \mapsto X_t(\omega)$ is continuous from the right and has limits from the left. (D) Signup and view all the answers

Let $X: [0, \infty) \rightarrow \mathbb{R}$ be a function. Which of the following is the most accurate interpretation of the statement that $X$ is a cadlag function?

For all $t \in [0, \infty)$, $\lim_{s \downarrow t} X_s = X_t$ and $\lim_{s \uparrow t} X_s$ exists, regardless of whether they are equal. (A) Signup and view all the answers

In the context of Kolmogorov's criterion, given random variables $(\rho_t)_{t \in I}$ where $I \subseteq [0, 1]$ is dense, and assuming $k\rho_t - \rho_s k_p \leq C|t - s|^{\beta}$ for some $p > 1$ and $\beta > \frac{1}{p}$, what is the most precise interpretation of the role of the condition $\beta > \frac{1}{p}$?

It provides a sufficient condition for the existence of a continuous modification $(X_t)_{t \in [0, 1]}$ and ensures that the Hölder exponent $\alpha$ can be chosen such that the sample paths are almost surely $\alpha$-Hölder continuous for $\alpha \in [0, \beta - \frac{1}{p})$. (A) Signup and view all the answers

Consider the space $D([0, \infty), \mathbb{R})$ of cadlag functions. Which of the following statements is most accurate concerning the $\sigma$-algebra typically used on this space?

The $\sigma$-algebra on $D([0, \infty), \mathbb{R})$ is typically generated by the coordinate functions $X_t \mapsto X_s$ for all $s \geq 0$, and it coincides with the Borel $\sigma$-algebra induced by the Skorokhod metric. (B) Signup and view all the answers

Suppose we have a collection of random variables $(\rho_t){t \in I}$, where $I \subseteq [0, 1]$ is a dense set. Under the conditions of Kolmogorov's criterion, specifically assuming that for some $p > 1$ and $\beta > \frac{1}{p}$, we have $E[|\rho_t - \rho_s|^p] \leq C|t - s|^{\beta}$ for all $t, s \in I$, which of the following additional conditions is absolutely necessary to ensure the uniqueness (in the almost sure sense) of the continuous modification $(X_t){t \in [0,1]}$ such that $X_t = \rho_t$ almost surely for all $t \in I$?

No additional conditions are needed; Kolmogorov's criterion inherently provides uniqueness. (C) Signup and view all the answers

Consider a measurable space $(\Omega, \mathcal{F})$ and two probability measures $P$ and $Q$ on it. If $Q$ is absolutely continuous with respect to $P$, which of the following statements must hold regarding the Radon-Nikodym derivative $\frac{dQ}{dP}$?

$\frac{dQ}{dP}$ is measurable with respect to the completion of $\mathcal{F}$ under $P$, accommodating scenarios where $\mathcal{F}$ is not complete but Q remains absolutely continuous with respect to P. (C) Signup and view all the answers

Let $P$ and $Q$ be two probability measures on a measurable space $(\Omega, \mathcal{F})$. Suppose that $Q$ is absolutely continuous with respect to $P$. Which of the following conditions is sufficient to ensure the existence of a bounded Radon-Nikodym derivative $\frac{dQ}{dP}$?

There exists a constant $M > 0$ such that for all $A \in \mathcal{F}$, $Q(A) \leq M \cdot P(A)$. (C) Signup and view all the answers

Given a stochastic process $(X_t)_{t \geq 0}$, which of the following statements best differentiates between the concepts of continuity and cadlag properties in the context of stochastic processes?

A stochastic process is continuous if its sample paths are continuous, while it is cadlag if its sample paths are right-continuous with left limits, allowing for jumps but maintaining a certain regularity in the discontinuities. (D) Signup and view all the answers

Consider a probability space $(\Omega, \mathcal{F}, P)$ and a sub-sigma algebra $\mathcal{G} \subseteq \mathcal{F}$. Let $X$ be a random variable such that $E[|X|] < \infty$. If the conditional expectation $E[X | \mathcal{G}]$ is pathwise unique (i.e., any version of the conditional expectation is equal $P$-almost surely), what deeper property does this imply about the structure of $\mathcal{G}$ and its relationship to $X$?

The sigma-algebra $\mathcal{G}$ is atomic up to $P$-null sets, where each atom $A \in \mathcal{G}$ either almost surely determines the value of $X$ or is irrelevant. (C) Signup and view all the answers

Let $X_t$ be a stochastic process indexed by $t \in [0, \infty)$. Suppose $X_t$ satisfies Kolmogorov's criterion on a dense subset $I \subseteq [0, 1]$. Which of the following statements regarding the Hölder continuity of the resulting continuous modification is the most precise?

For any $\alpha < \beta - \frac{1}{p}$, there exists a random variable $K_\alpha \in L^p$ such that $|X_s - X_t| \leq K_\alpha |s - t|^\alpha$ for all $s, t \in [0, 1]$, but there's no guarantee that such a $K_\alpha$ exists for $\alpha = \beta - \frac{1}{p}$. (B) Signup and view all the answers

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Suppose $(X_n)_{n \geq 0}$ is a martingale with respect to a filtration $(\mathcal{F}n){n \geq 0}$. Under what conditions does the martingale converge both almost surely and in $L^1$?

The martingale $(X_n)_{n \geq 0}$ is bounded in $L^p$ for some $p > 1$, ensuring control over the tail behavior of the martingale. (B) Signup and view all the answers

Consider a measurable space $(\Omega, \mathcal{F})$ and a $\sigma$-finite measure $\mu$ on it. If a positive function $f$ satisfies $\int_A f d\mu = 0$ for all $A \in \mathcal{F}$, what can be definitively concluded about the function $f$?

The function $f$ is zero $\mu$-almost everywhere on $\Omega$, allowing for a set of $\mu$-measure zero where $f$ is non-zero. (C) Signup and view all the answers

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $(X_n){n \geq 1}$ be a sequence of i.i.d. random variables with $E[X_1] = 0$ and $E[X_1^2] = 1$. Define $S_n = \sum{i=1}^n X_i$. Which of the following statements accurately describes the asymptotic behavior of $\frac{S_n}{\sqrt{n}}$?

$\frac{S_n}{\sqrt{n}}$ converges in distribution to a standard normal distribution $N(0, 1)$ as $n \to \infty$, as predicted by the central limit theorem. (B) Signup and view all the answers

Suppose $(\Omega, \mathcal{F}, P)$ is a probability space and $X$ is an integrable random variable. Let $(\mathcal{F}n){n \geq 0}$ be a filtration. If $X_n = E[X | \mathcal{F}n]$, under what condition is $(X_n){n \geq 0}$ guaranteed to be a uniformly integrable martingale?

The random variable $X$ is integrable, belonging to $L^1(P)$, and the conditional expectations are well-defined. (D) Signup and view all the answers

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Consider a sequence of random variables $(X_n)_{n \geq 1}$ converging in probability to a random variable $X$. Under what additional condition does this convergence imply convergence in $L^1$ (i.e., $E[|X_n - X|] \to 0$)?

The sequence $(X_n)_{n \geq 1}$ is stochastically dominated by an integrable random variable $Y$, i.e., $P(|X_n| > t) \leq P(Y > t)$ for all $t > 0$ and all $n$, and $E[Y] < \infty$. (B) Signup and view all the answers

Given a standard Brownian motion $(B_t)_{t \geq 0}$ and a stopping time $T$, and defining a reflected process $\tilde{B}t$ as $\tilde{B}t = B_t \cdot 1{t < T} + (2B_T - B_t) \cdot 1{t \geq T}$, what is the most precise characterization of the relationship between the increments of $B^*$ and $\tilde{B}$ as $n \to \infty$, considering $T_n$ as an approximation of $T$?

The increments of $B^*$ converge almost surely to the increments of $\tilde{B}$, given the continuity of $B$ and the convergence of $T_n$ to $T$ almost surely. (C) Signup and view all the answers

Considering a bounded domain $D \subset \mathbb{R}^d$ and a function $u \in C^2(D)$ satisfying Laplace's equation, which of the following statements regarding the mean value property is most accurate in the context of characterizing harmonic functions?

The mean value property is both a necessary and sufficient condition for $u$ to be harmonic in D, provided $u$ is continuous and the integrals exist. (C) Signup and view all the answers

Let $(B_t){t \geq 0}$ be a standard Brownian motion in $\mathbb{R}^d$, and let $u: \mathbb{R}^d \rightarrow \mathbb{R}$ be a harmonic function such that $E[|u(x + B_t)|] < \infty$ for any $x \in \mathbb{R}^d$ and $t \geq 0$. Which of the following modifications to Itô's lemma would be most pertinent in demonstrating that $(u(B_t)){t \geq 0}$ is a martingale with respect to $(F_t)_{t \geq 0}$?

Itô's lemma must be adapted to account for the stochastic integral term vanishing due to the harmonicity of $u$, resulting in a simplification of the stochastic process. (D) Signup and view all the answers

Consider a stochastic process $X_t = f(B_t)$, where $B_t$ is a standard Brownian motion and $f$ is a continuously differentiable function. Under what condition does $X_t$ qualify as a local martingale but not necessarily a true martingale?

The stochastic integral $\int_0^t f'(B_s) dB_s$ must satisfy the integrability condition for local martingales but $E[|X_t|] = \infty$ for some $t$, thus preventing it from being a true martingale. (B) Signup and view all the answers

Let $B_t$ represent standard Brownian motion. If a functional $F[B]$ is invariant under time reversal, what profound implication does this have for characterizing the statistical properties of $B_t$ in relation to its time-reversed counterpart?

The invariance dictates that $B_t$ and $B_{1-t}$ (for $0 \leq t \leq 1$) have the same finite-dimensional distributions, directly linking forward and backward statistical behaviors. (A) Signup and view all the answers

Suppose a stochastic process $X_t$ satisfies the stochastic differential equation $dX_t = \mu(X_t)dt + \sigma(X_t)dB_t$, where $B_t$ is a standard Brownian motion. Under what condition does the scale function $s(x)$ guarantee that $X_t$ is inaccessible from the interval $(a, b)$?

When $s(x)$ is unbounded as $x$ approaches either $a$ or $b$, ensuring that the boundaries are impenetrable and act as reflecting barriers. (C) Signup and view all the answers

Given a martingale $(M_t)_{t \geq 0}$ with continuous paths and $M_0 = 0$, and defining its quadratic variation process as $\langle M \rangle_t$, what is the precise interpretation of the statement $\langle M \rangle_t = 0$ for all $t \geq 0$ almost surely?

This implies that $M_t$ is constant almost surely i.e. $M_t = 0$ for all $t$, indicating the absence of any stochastic fluctuations whatsoever. (A) Signup and view all the answers

Let $X$ be a random variable representing the payoff of a derivative security at maturity $T$, and consider a risk-neutral measure $\mathbb{Q}$. What is the economic implication of the statement that $E^{\mathbb{Q}}[X] = 0$?

The derivative security is fairly priced, reflecting a market equilibrium where the expected payoff under the risk-neutral measure equals its initial cost, normalized to zero. (C) Signup and view all the answers

Given a sequence of independent and identically distributed random variables $X_1, X_2, ..., X_n$ with sample mean $\bar{x}$ and $S_n = X_1 + ... + X_n$, and considering Cramér's theorem concerning large deviations, what is the significance of the Legendre transform $\psi^*(a)$ in determining the rate of decay for $P(S_n \geq an)$ where $a > \bar{x}$?

$\psi^(a)$ represents the exact exponential decay rate, such that $\lim_{n \to \infty} \frac{1}{n} \log P(S_n \geq an) = -\psi^(a)$, accurately quantifying the speed of convergence. (C) Signup and view all the answers

Suppose $b_n = -\log P(S_n \geq an)$ forms a sub-additive sequence, where $S_n$ is the sum of $n$ independent and identically distributed random variables. According to Fekete's lemma, what can be rigorously inferred about the asymptotic behavior of $\frac{b_n}{n}$?

$\lim_{n \to \infty} \frac{b_n}{n}$ exists and is equal to the infimum of the sequence ${\frac{b_k}{k}}_{k=1}^{\infty}$. (B) Signup and view all the answers

Consider a scenario where $P(X_1 \leq 0) = 1$ for a random variable $X_1$. Based on the provided context relating to large deviations theory, what is the precise expression for $\frac{1}{n} \log P(S_n \geq 0)$, where $S_n$ represents the sum of $n$ independent and identically distributed random variables each distributed as $X_1$?

$\frac{1}{n} \log P(S_n \geq 0) = \log P(X_1 = 0)$ (A) Signup and view all the answers

In the context of Cramér's theorem, if the moment generating function $M(\lambda) = E[e^{\lambda X_1}]$ exists for a random variable $X_1$, and $\psi(\lambda) = \log M(\lambda)$, how does the Legendre transform $\psi^*(a)$ relate to the Chernoff bound for $P(S_n \geq an)$, where $S_n = \sum_{i=1}^{n} X_i$?

$\psi^*(a)$ is equal to the exponent in the tightest possible Chernoff bound, providing the precise exponential rate of decay for $P(S_n \geq an)$. (D) Signup and view all the answers

Consider a series of independent, identically distributed random variables and let $\psi(\lambda)$ represent the cumulant generating function. What specific optimization problem must be solved to compute the rate function $\psi^*(a)$ in Cramér's theorem, and what constraints, if any, apply to the optimization variable $\lambda$?

Maximize $a\lambda - \psi(\lambda)$ with respect to $\lambda$ subject to the constraint that $\lambda$ is real and non-negative. (C) Signup and view all the answers

Given Cramér's theorem, under what precise condition concerning the relationship between $a$ and $\bar{x}$ (the mean of the i.i.d. random variables) does the large deviation principle become relevant for analyzing the tail behavior of $P(S_n \geq an)$, and why is this condition necessary?

The large deviation principle is relevant if and only if $a > \bar{x}$, because this ensures that the event $S_n \geq an$ is a rare event whose probability decays exponentially. (C) Signup and view all the answers

Suppose one aims to establish a lower bound on the probability $P(S_n \geq 0)$ using large deviation theory. What specific assumptions or transformations are invoked to simplify the analysis, and what critical challenge arises when $a = 0$ in applying Cramér's theorem?

Translating $X_i$ by a constant to enforce a zero mean simplifies the analysis; however when $a = 0$, the infimum of $\psi(\lambda)$ over non-negative $\lambda$ may be zero, thereby complicating the lower bound estimation. (A) Signup and view all the answers

In the context of large deviations and Cramér's theorem, if we define $\psi(\lambda) = \log E[e^{\lambda X_1}]$ and $\psi^(a) = \sup_{\lambda \geq 0} (a\lambda - \psi(\lambda))$, how does the non-negativity of $\psi^(a)$ relate to the fundamental properties of moment generating functions and their implications for bounding probabilities?

$\psi^*(a)$ is always non-negative because it stems from maximizing $a\lambda - \psi(\lambda)$ and is fundamentally linked to the properties of moment generating functions, ensuring that the probability bounds remain meaningful. (D) Signup and view all the answers

Consider a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}t){t \geq 0}, P)$ and a stochastic process $(X_t){t \geq 0}$. Which of the following conditions is sufficient to ensure that $(X_t){t \geq 0}$ is a martingale with respect to the filtration $(\mathcal{F}t){t \geq 0}$?

$E[X_t | \mathcal{F}_s] = X_s$ almost surely for all $s \leq t$, $E[|X_t|] < \infty$ for all $t \geq 0$, and $X_t$ is $\mathcal{F}_t$-measurable. (B) Signup and view all the answers

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion. Which of the following stochastic integrals is pathwise the least regular (i.e., possesses the worst sample path properties)?

$\int_{0}^{t} B_{s}^{2} dB_s$ (B) Signup and view all the answers

Suppose $(X_n){n \geq 1}$ is a sequence of independent and identically distributed random variables with characteristic function $\phi(t)$. According to Cramér’s large deviation theorem, under suitable conditions, the probability $P(\sum{i=1}^{n} X_i > na)$ decays exponentially in $n$. Which of the following large deviation rate functions, $I(a)$, correctly characterizes this exponential decay?

$I(a) = \sup_{t} [at - \log(E[e^{tX_1} ])]$ (B) Signup and view all the answers

Consider a Lévy process $(X_t)_{t \geq 0}$ with Lévy exponent $\psi(u)$. Which of the following statements is not generally true regarding the properties of $\psi(u)$?

$\psi(u)$ is infinitely differentiable. (B) Signup and view all the answers

Let $X$ and $Y$ be random variables on $(\Omega, \mathcal{F}, P)$. Which of the following statements regarding conditional expectation is always true without additional assumptions?

$E[aX + bY | \mathcal{G}] = aE[X | \mathcal{G}] + bE[Y | \mathcal{G}]$ for any sub-$\sigma$-algebra $\mathcal{G} \subseteq \mathcal{F}$ and constants $a, b \in \mathbb{R}$. (B) Signup and view all the answers

Let $(M_t)_{t \geq 0}$ be a continuous martingale with $M_0 = 0$. Define its quadratic variation as $[M]_t$. Which of the following statements is not a direct consequence of the definition or properties of quadratic variation?

If $M_t$ is a Brownian motion, then $[M]_t$ is absolutely continuous with respect to Lebesgue measure. (B) Signup and view all the answers

Consider a Poisson random measure $N(dt, dx)$ on a space $E$ with intensity measure $\lambda(dx)$. For a measurable function $f: E \to \mathbb{R}$, the integral $\int_E f(x) N(dt, dx)$ is well-defined. Under which condition is the integral $\int_E f(x) N(dt, dx)$ a martingale?

$\int_E f(x) \lambda(dx) = 0$ (D) Signup and view all the answers

Suppose $(B_t)_{t \geq 0}$ is a standard Brownian motion. Let $\tau = \inf{t \geq 0 : B_t = a}$ be the hitting time of level $a > 0$. Which of the following statements regarding the strong Markov property applied to $\tau$ is generally correct?

$(B_{\tau + t} - a){t \geq 0}$ is a Brownian motion independent of $\mathcal{F}\tau$. (C) Signup and view all the answers

Flashcards

Sigma-algebra

A family of subsets of a set, closed under complement, countable unions, and containing the empty set.

Measure

A function that assigns a non-negative real number or +∞ to subsets of a set. Satisfies measure axioms (non-negativity, null empty set, countable additivity).

Filtration

An increasing sequence of sigma-algebras, representing the information available at different points in time.