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3799Why Boundary Conditions Matter Even When the Terminal Payoff Is KnownIf the terminal payoff is already specified, why do boundary conditions still matter in many Feynman-Kac problems?随机过程中等essay未尝试面试订阅3800Discounting as State-Dependent KillingWhy is the term -r(x,t)u often interpreted as a state-dependent killing rate in the Feynman-Kac picture?随机过程中等essay未尝试面试订阅3801Mapping a PDE with No Source and No DiscountIf the PDE has u t + L u = 0 with only a terminal condition, what is the corresponding expectation structure?随机过程中等essay未尝试面试订阅3802Mapping a PDE with Discount and Source TogetherIf a PDE contains both -r u and +f, what two separate pathwise ingredients should appear in the expectation representation?随机过程中等essay未尝试面试订阅3803Why the Diffusion Coefficient Touches the Second DerivativeWhy does the diffusion strength of the state process appear in front of the second spatial derivative in the PDE?随机过程中等essay未尝试面试订阅3804Why Drift Touches the First DerivativeWhy does the drift coefficient show up next to the first spatial derivative in the PDE?随机过程中等essay未尝试面试订阅3805Reading a Running Penalty from the PDEA backward PDE includes a negative source term. What economic story should you suspect in the underlying expectation?随机过程中等essay未尝试面试订阅3806Why Feynman-Kac Is a Bridge, Not Two Separate FactsWhy is it better to think of Feynman-Kac as a bridge between probabilistic expectations and PDEs rather than as two separate formulas to memorize?随机过程中等essay未尝试面试订阅3807Barrier Knock-Out as Boundary DataFor a knock-out style problem, why does the boundary condition typically encode zero continuation value after a hit?随机过程中等essay未尝试面试订阅3808Why a Source-Free PDE Still Carries Rich EconomicsIf a PDE has no source term at all, why is it still not 'empty' from an economic point of view?随机过程中等essay未尝试面试订阅3809Expectation with a Stopping Time Instead of Fixed TIf an expectation stops at a hitting time rather than at a fixed maturity, what change should you expect on the PDE side?随机过程中等essay未尝试面试订阅3810Why State Dependence in r or f Matters PathwiseWhat changes in the expectation picture when discount or running reward depends on the current state X s rather than on time alone?随机过程中等essay未尝试面试订阅3811Why Conditional Expectation Solves the PDE Rather Than Merely Satisfies It AccidentallyWhy is the Feynman-Kac expectation not just a lucky candidate but the natural solution to the PDE problem?随机过程中等essay未尝试面试订阅3812How to Tell Whether a PDE Corresponds to a Reward or a Cost ProblemWithout memorizing signs, how can you reason whether a PDE corresponds to a running reward problem or a running cost problem?随机过程中等essay未尝试面试订阅3813Why the Same Diffusion Can Generate Many Different PDEsHow can the same underlying diffusion process lead to many different Feynman-Kac PDEs?随机过程中等essay未尝试面试订阅3815Why the Martingale Argument Sits Underneath Feynman-KacWhat is the martingale intuition that secretly sits underneath the Feynman-Kac theorem?随机过程中等essay未尝试面试订阅4966Probability the First Observed Jump by Time 0.5 Is to BFrom state A, the jump rates to B and C are 0.8 and 1.2. What is the probability that by time 0.5 a jump has occurred and that first jump was to B?随机过程简单数值题未尝试面试订阅4967Probability the First Observed Jump by Time 2 Is to CFrom state A, the jump rates to B, C, and D are 0.3, 0.5, and 0.2. What is the probability that by time 2 the first jump has occurred and it was to C?随机过程简单数值题未尝试面试订阅4968Probability of No Jump by Time 0.4From state A, the jump rates to B and C are 1.5 and 0.5. What is the probability that no jump has occurred by time 0.4?随机过程简单数值题未尝试面试订阅4969Probability the First Observed Jump by Time 1 Is to DFrom state A, the jump rates to B, C, and D are 0.6, 0.9, and 0.5. What is the probability that by time 1 the first jump has occurred and it was to D?随机过程简单数值题未尝试面试订阅