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?
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中文题目题目1519 · 数学
Adjusted Gross Multiplier 4
Use a second-order Taylor approximation around 0 to estimate (1+3x)^(2) * (1+1x)^(-1) at x=1/40.
打开 →题目5850 · 数学
Alternating Geometric Sum
Evaluate the infinite series sum_(n=0)^inf 5 * (-2/3)^n.
打开 →题目1451 · 数学
Alternating-Input Recurrence 1
A sequence satisfies x_(n+1) = 1/2 x_n + c_n, where c_n = 1 on odd n and c_n = 3 on even n. Assuming any starting value x_1, what are the limits of the even and odd subsequences?
打开 →题目1443 · 数学
Arcsine Cubic Remainder
Compute lim_{x->0} [arcsin x - x] / x^3.
打开 →题目3262 · 数学
Area of a Parallelogram from a Unit Square
The map $x=u+v,\ y=u-v$ sends the unit square $0\le u,v\le 1$ to a parallelogram. What is its area?
打开 →题目3265 · 数学
Area of a Polar Wedge
Use the polar Jacobian to compute the area of the region $0\le r\le 2,\ 0\le \theta\le \pi/3$.
打开 →题目3261 · 数学
Area of an Ellipse from the Unit Disk
The map $x=2u,\ y=3v$ sends the unit disk $u^2+v^2\le 1$ to an ellipse in the $(x,y)$-plane. What is the area of that ellipse?
打开 →题目3264 · 数学
Area Scaling Under a Triangular Map
The map $x=u,\ y=u+2v$ sends the unit square to a parallelogram. Compute the image area.
打开 →题目5857 · 数学
Arithmetico-Geometric Series
Evaluate the infinite series sum_(n=1)^inf n * (1/2)^n.
打开 →题目1523 · 数学
Asymmetric Buffer Calibration 3
Choose c so that exp(-1x + c x^2) - 1 - -1x has quadratic coefficient 4.
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3554 · 随机过程
Average Exit Time With Uniform Start on [-3, 3]
If X is uniform on [-3, 3], what is the expected exit time from [-3, 3]?
打开 →题目3552 · 随机过程
Average Exit Time With Uniform Start on [0, 6]
If the starting point X is uniform on [0, 6], what is the expected exit time from [0, 6]?
打开 →题目3675 · 随机过程
Average Exit Time with Uniform Start on [0,6]
The starting point X is uniform on [0,6]. Brownian motion starts at X and stops on first exit from [0,6]. What is E[tau] on average?
打开 →题目3670 · 随机过程
Average Upper-Exit Probability with Uniform Start on [0,8]
The starting point X is uniform on [0,8]. Brownian motion starts at X and stops on first exit from [0,8]. What is the average probability of exiting through 8?
打开 →题目3551 · 随机过程
Average Upper-Hit Probability With Uniform Start on [0, 6]
If the starting point X is uniform on [0, 6], what is the average probability of hitting 6 before 0?
打开 →题目2031 · 数学
Backing Out the Stress State From a Reciprocal Plug-In Score 11
A funding-buffer score uses phi(L)=1/(1+L). Suppose leverage L equals 0 with probability 1/2 and H with probability 1/2. If phi(E[L]) = 1/3, what is H and what is E[phi(L)]?
打开 →题目1962 · 数学
Balanced Exponential Penalties 17
A desk minimizes J(x)=6 e^x + 3 e^{-2x}. What x is optimal?
打开 →题目1966 · 数学
Balanced Softplus Tradeoff 21
The linear reward and saturation penalty balance exactly at a central point. The desk maximizes K(x) = 2 x - 4 ln(1+e^x). What x is optimal?
打开 →题目2008 · 数学
Barrier Cost With Strong Quadratic Curvature 13
Both the quadratic inventory term and the capacity wall are active sources of curvature. Show that f(q) = 4 q^2 + 3/(1-q) is strictly convex on q<1.
打开 →题目3519 · 随机过程
Barrier Giving Touch Probability 0.02 Over Horizon 0.25
For standard Brownian motion over horizon t = 0.25, what barrier a gives touch probability 0.02?
打开 →题目3518 · 随机过程
Barrier Giving Touch Probability 0.1 Over Horizon 9
For standard Brownian motion over horizon t = 9, what barrier a gives touch probability 0.1?
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3522 · 随机过程
Barrier Matching a Unit-Horizon Risk Across Horizon 4
A barrier of 1.2 over horizon 1 gives some touch or survival risk. What barrier over horizon 4 preserves that same standardized ratio?
打开 →题目3524 · 随机过程
Barrier Matching a Unit-Horizon Risk Across Horizon 9
A barrier of 1 over horizon 1 gives some touch or survival risk. What barrier over horizon 9 preserves that same standardized ratio?
打开 →题目3533 · 随机过程
Barrier Needed for 90% Survival Over One Time Unit
For standard Brownian motion over t = 1, what barrier a gives P(max_{0<=s<=1} W_s < a) = 0.90?
打开 →题目3528 · 随机过程
Barrier Needed for a 5% Joint Tail With Endpoint Cap 0
For standard Brownian motion over t = 1, what barrier a gives P(max_{0<=s<=1} W_s >= a, W_1 <= 0) = 0.05?
打开 →题目3684 · 随机过程
Barrier Needed for a 90% Drifted Hit Probability with Sigma One
A diffusion satisfies dX_t = 0.5dt + 1dW_t, starts at 2, and stops on first exit from [0,b]. What upper barrier b makes the upper-exit probability equal 0.9?
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