模块2.7.2 · 数学与统计能力 · 随机分析
鞅与风险中性定价
stochastic-calculus · martingale · filtration · conditional-expectation · discrete-time · stopping-time · optional-stopping · brownian-motion
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中文题目题目2863 · 概率
A Batch-Size Compound Poisson Desk Flow
Trades arrive according to $N\sim\mathrm{Poisson}(\lambda)$. Each trade contributes a batch size $B$ taking values $0,1,2$ with probabilities $1/2,1/3,1/6$, independently across trades and from $N$. Let \[ S=\sum_{k=1}^N B_k. \] Find the MGF of $S$, and compute $E[S]$ and $\mathr
打开 →题目2844 · 概率
A Geometric Number of Exponential Stages
A task takes a geometric number of stages: $N$ has support $\{1,2,\dots\}$ with $P(N=n)=p(1-p)^{n-1}$. Each stage duration is i.i.d. $\mathrm{Exponential}(\beta)$, independent of $N$. Let $T=\sum_{i=1}^N X_i$. Use MGFs to identify the law of $T$.
打开 →题目2676 · 机器学习
A Tiny Tail Probability Can Dominate Average PnL
A strategy makes +0.04% on 98% of days and loses -2.5% on the remaining 2% of days. What is the unconditional average daily return?
打开 →题目2857 · 概率
A Two-Volatility Mixture Is Not Gaussian
A return $R$ is conditionally Gaussian: \[ R\mid V=\sigma \sim N(0,\sigma^2), \] where $V$ equals $1$ or $2$ with probability $1/2$ each. Compute the characteristic function of $R$ and explain why $R$ is not itself Gaussian.
打开 →题目3794 · 随机过程
Absorbing Boundary as Path Termination
How should an absorbing boundary condition be interpreted inside a Feynman-Kac representation?
打开 →题目3192 · 概率
Aggregate Slippage Over a Poisson Number of Orders
Let $X_1,X_2,\dots$ be i.i.d. increments with $E[X_i]=2$ and $\mathrm{Var}(X_i)=3$. Let $N$ be independent of the increments and distributed as Poisson(4). For the stopped sum $S_N=\sum_{i=1}^N X_i$, compute $E[S_N]$ and $\mathrm{Var}(S_N)$.
打开 →题目3176 · 概率
Aggressive vs Defensive Quote After a Signal
There are two possible actions. `Aggressive` pays 10 in the good state and -8 in the bad state. `Defensive` pays 4 in the good state and -1 in the bad state. The good state has prior probability $\frac{2}{5}$. Before acting, you may see a binary signal that is correct with probab
打开 →题目3177 · 概率
Allocate Between Fast and Safe Books
There are two possible actions. `Aggressive` pays 9 in the good state and -6 in the bad state. `Defensive` pays 5 in the good state and 1 in the bad state. The good state has prior probability $\frac{1}{2}$. Before acting, you may see a binary signal that is correct with probabil
打开 →题目2940 · 概率
Almost Sure Extinction in Every Subcritical Process
A Galton-Watson process has offspring mean $m<1$. What is its extinction probability?
打开 →题目1697 · 统计
Any False Sector Winner After Within-Sector Mining
A desk has 12 sectors, each containing 5 genuinely null variants. In each sector it keeps only the smallest p-value, and it flags the sector if that winning p-value is below 1%. Assuming independence, what is the probability at least one sector is falsely flagged?
打开 →题目3020 · 概率
API Error Before Database Error
Api errors and database errors arrive as independent Poisson processes with rates $\lambda_A=5$ and $\lambda_B=5$ per hour. What is the probability that the next arrival in the merged stream comes from stream A?
打开 →题目1823 · 统计
AR(1) Multi-Step Forecast 3
A signal follows X_t = -1 + 0.8 X_(t-1) + e_t with Var(e_t) = 1 and current value X_t = 3. What is the h = 4 step forecast E[X_(t+4) | X_t]?
打开 →题目6026 · 统计
ARCH(1) as the Beta-Zero Special Case
A GARCH(1,1) reduces to ARCH(1) when $\beta=0$: $h_t=\omega+\alpha r_{t-1}^2$. With $\omega=0.7$ and $\alpha=0.3$, compute the unconditional variance $\bar h$ as a decimal.
打开 →题目5654 · 数理金融
Asian Path Payoff 4
A simulated path for an arithmetic-average Asian call is [95, 92, 90, 97] with strike 94. What payoff does this single path contribute to the Monte Carlo estimator?
打开 →题目432 · 概率
Asymmetric Penalties in an Exponential Race
Two independent alarms go off at $\operatorname{Exp}(4)$ and $\operatorname{Exp}(6)$ times respectively. If alarm 1 fires first you pay $\$3$; if alarm 2 fires first you pay $\$5$. After the first alarm fires, the remaining alarm is reset (memoryless restart) and you pay an addit
打开 →题目3063 · 概率
At Least One Arrival in the Final Quarter Given Four Total
A Poisson process is observed on $[0,1]$ hour. Conditional on exactly 4 arrivals in the hour, what is the probability that at least one arrival falls in the final quarter-hour?
打开 →题目6043 · 随机过程
Autocorrelation of a Stationary OU Process at a Lag
A stationary OU process has mean-reversion speed kappa = 0.7. What is the autocorrelation between X_t and X_{t+2}?
打开 →题目3700 · 随机过程
Auxiliary Process Drift After Calibrating a Target Process
Under P, two processes share the same Brownian driver: dX_t = 0.3dt + 0.2dW_t and dY_t = 0.45dt + 0.1dW_t. If Q is chosen so that X has drift -0.05 under Q, what drift does Y have under Q?
打开 →题目2673 · 机器学习
Average Edge Across Opposing Regimes
A signal earns +6 bps on 70% of days in calm markets and -10 bps on 30% of days in stressed markets. What is its unconditional average daily edge in bps?
打开 →题目1756 · 统计
Backing Out Omitted Covariance From a Slope Drop
A desk regresses slippage Y on inventory pressure X. Without an urgency control, the OLS slope on X is 0.90. After adding a perfect measure of urgency U, the slope falls to 0.60. Suppose the structural model is Y = beta X + 0.5 U + noise and Var(X)=1. What is Cov(X,U)?
打开 →题目5970 · 概率
Ballot Problem via Martingale
In an election candidate A receives 7 votes and candidate B receives 3 votes; the 10 votes are counted in a uniformly random order. Using a martingale / optional stopping argument, find the probability that A is strictly ahead of B throughout the entire count.
打开 →题目3807 · 随机过程
Barrier Knock-Out as Boundary Data
For a knock-out style problem, why does the boundary condition typically encode zero continuation value after a hit?
打开 →题目2421 · 机器学习
Bayes Act Under Weighted Squared Loss 1
Suppose the loss is L(a,Y)=W(Y-a)^2 where W>0 is observed at prediction time. What predictor minimizes E[L(a,Y)|X,W]?
打开 →题目355 · 概率
Beta-Binomial Moments via Adam's and Eve's Laws
Let $P \sim \operatorname{Beta}(2, 3)$ and, given $P = p$, let $X \sim \operatorname{Binomial}(10, p)$. Using Adam's law ($E[X] = E[E[X \mid P]]$) and Eve's law ($\operatorname{Var}(X) = E[\operatorname{Var}(X \mid P)] + \operatorname{Var}(E[X \mid P])$), derive $E[X]$ and $\oper
打开 →题目371 · 概率
Beta-Uniform Prior on Binomial Success Probability
Let $P \sim \operatorname{Uniform}(0,1)$ and, given $P = p$, let $X \mid P{=}p \sim \operatorname{Binomial}(10, p)$. Using the tower property, find $E[X]$.
打开 →题目2398 · 机器学习
Bias Budget Implied by a Variance Reduction
A regularization change reduces a model's variance term from 0.30 to 0.11 while leaving irreducible noise unchanged. How much extra bias squared could you add before the total MSE stops improving?
打开 →题目667 · 概率
Biased Corridor Hit Probability 1
A random walk starts at 2, moves +1 with probability 3/5 and -1 with probability 2/5, and stops when it first hits 0 or 7. What is the probability that it reaches 7 before 0?
打开 →题目1667 · 统计
Bootstrap Bias of the Sample Maximum on a Tiny Sample
A sample is {1,4}. Under the nonparametric bootstrap with resample size 2, compute the bootstrap expectation of the sample maximum and the resulting bootstrap bias estimate for the original maximum statistic.
打开 →题目1668 · 统计
Bootstrap Variance of a Bernoulli Mean From the Plug-In Law
A sample of size n has empirical success rate p_hat. Under the nonparametric bootstrap, what is the variance of the resampled sample mean conditional on the observed data?
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