$Wiener process probability$

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$Wiener process probability$

Calculate with what probability it will happen that the random trajectory of the Wiener process will acquire a value greater than 2 at time 2 and a value less than 2 at time 3.

Probability Stochastic Analysis Statistics

Kingroma99

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Let $X = W_2 - W_0$ and $Y = W_3 - W_2$. Then $X,Y$ are independent and normal (with mean $0$ and variances $2-0=2$ and $3-2=1$) and

$$p = \mathbb{P}[\{X > 2\} \cap \{X+Y < 2\}] = \int_{\{s>2\}} \mathbb{P}[Y < 2 - s | X = s] \, \mathrm{d}s$$
By independence, one has
$$p = \int_{2}^{\infty} \int_{-\infty}^{2-s} \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^2}{2}\right) \cdot \frac{1}{2 \sqrt{2\pi}}\exp\left(-\frac{(s/2)^2}{2}\right) \mathrm{d}t \, \mathrm{d}s$$

This evaluates to

$$p = 0.0345114\ldots$$

Matchmaticians Alumnus

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Kingroma99

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Hello, thank you for your answer! Could you please tell me how did you calculate the double integral? Thank you for your response.
Matchmaticians

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@Green: Please respond to Kingroma99 question above.
Matchmaticians Alumnus

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@Kingroma99, the double integral was evaluated numerically. Otherwise one can compute it by first treating t as a constant, and integrating from negative infinity to 2-s. This involves the error function, which is the antiderivative of the Gaussian. Then, the result can be integrated with respect to t from 2 to infinity. To get a closed form in terms of error functions, one follows the above procedure while doing a few integral substitutions.

$Wiener process probability$

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