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Reflected paths ending below the level

HardProbabilityStochastic Processes~7m

Standard Brownian motion runs on [0,1][0,1]. What is the probability that its running maximum reaches at least 11 yet the path finishes at or below 0.50.5 - that is, P(max⁡0≤s≤1Ws≥1,  W1≤0.5)P\big(\max_{0\le s\le 1} W_s \ge 1,\; W_1 \le 0.5\big)? Use Φ(1.5)=0.9332\Phi(1.5) = 0.9332. (to 4 decimals)

Show hints (2)+
  1. Reflect the path about the barrier level a=1a=1 after its first hitting time; the endpoint yy maps to 2a−y2a-y.
  2. The joint event equals P(W1≥2a−0.5)=P(W1≥1.5)P(W_1 \ge 2a - 0.5) = P(W_1 \ge 1.5).

Answer

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0.0668 (± 0.003)

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