1. **Problem Statement:** We want to verify the expression for the fourth moment of a Wiener process $W_t$, specifically the expectation $\mathbb{E}[W_{t_1} W_{t_2} W_{t_3} W_{t_4}]$. 2. **Background:** A Wiener process is a Gaussian process with mean zero and covariance $\mathbb{E}[W_s W_t] = \min(s,t)$. For Gaussian processes, the fourth moment can be expressed in terms of second moments using Isserlis' (or Wick's) theorem: $$ \mathbb{E}[X_1 X_2 X_3 X_4] = \mathbb{E}[X_1 X_2] \mathbb{E}[X_3 X_4] + \mathbb{E}[X_1 X_3] \mathbb{E}[X_2 X_4] + \mathbb{E}[X_1 X_4] \mathbb{E}[X_2 X_3] $$ 3. **Apply to Wiener Process:** Using $X_i = W_{t_i}$, we have $$ \mathbb{E}[W_{t_1} W_{t_2} W_{t_3} W_{t_4}] = \mathbb{E}[W_{t_1} W_{t_2}] \mathbb{E}[W_{t_3} W_{t_4}] + \mathbb{E}[W_{t_1} W_{t_3}] \mathbb{E}[W_{t_2} W_{t_4}] + \mathbb{E}[W_{t_1} W_{t_4}] \mathbb{E}[W_{t_2} W_{t_3}] $$ 4. **Covariance Values:** Since $\mathbb{E}[W_s W_t] = \min(s,t)$, assuming $t_1 \leq t_2 \leq t_3 \leq t_4$, we get $$ \mathbb{E}[W_{t_1} W_{t_2}] = t_1, \quad \mathbb{E}[W_{t_3} W_{t_4}] = t_3, \quad \mathbb{E}[W_{t_1} W_{t_3}] = t_1, \quad \mathbb{E}[W_{t_2} W_{t_4}] = t_2, \quad \mathbb{E}[W_{t_1} W_{t_4}] = t_1, \quad \mathbb{E}[W_{t_2} W_{t_3}] = t_2 $$ 5. **Substitute and Simplify:** $$ \begin{aligned} \mathbb{E}[W_{t_1} W_{t_2} W_{t_3} W_{t_4}] &= t_1 t_3 + t_1 t_2 + t_1 t_2 \\ &= t_1 t_3 + 2 t_1 t_2 \end{aligned} $$ 6. **Check Your Expression:** The expression you wrote, $$ \mathbb{E}[W_{t_1} W_{t_2} W_{t_3} W_{t_4}] = t_2 t_4 + 2 t_3 t_4 $$ does not match the correct form based on the covariance structure. The indices and minimums must be carefully considered. **Summary:** Your approach using the sum of products of covariances is correct in principle, but the final substitution and simplification of terms need to respect the covariance $\min(t_i, t_j)$. The final formula should be expressed in terms of these minimums, not just products of $t_i t_j$ without ordering. Hence, the reasoning makes sense, but the final numeric expression needs correction based on the ordering of $t_i$ and the covariance definition.