91Ó°ÊÓ

For an individual starting at position \(x_0\) and taking \(n\) steps on the real line, we are given that while being at position \(x\), he moves to a location with mean 0 and variance \(\beta x^{2}\). Let \(X_n\) denote the position of the individual after n steps.

To find the expected position, we need to compute the expected value of the function \(X_{n}\). Using the linearity of expected values, we can find the expectation for each step and then sum them up over \(n\) steps. So for the first step: \[E[X_1] = E[x_0 + M_1] = E[x_0] + E[M_1] = x_0,\] where \(M_1\) is the movement in the first step with mean 0. For the subsequent steps, \[E[X_{n}] = E[X_{n-1} + M_n],\] where \(M_n\) represents the movement in the \(n\)th step. Since \(E[M_n] = 0\), we have: \[E[X_{n}] = E[X_{n-1}],\] Which results in a constant expected position throughout the process: \[E[X_n] = x_0 \quad \text{for} \quad n \geq 1.\]

To find the variance, we can use the property of variances that states \(\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)\), when \(X\) and \(Y\) are independent. In this case, \(X_n = X_{n-1} + M_n\). Let's compute the variance for the first step: \[\operatorname{Var}(X_1) = \operatorname{Var}(x_0 + M_1) = \operatorname{Var}(M_1) = \beta x_0^{2}.\] Now, let's compute the variance for the next steps: \[\operatorname{Var}(X_n) = \operatorname{Var}(X_{n-1} + M_n) = \operatorname{Var}(X_{n-1}) + \operatorname{Var}(M_n) = \operatorname{Var}(X_{n-1}) + \beta(X_{n-1})^2.\] To solve this recursively, we already know that \(\operatorname{Var}(X_1) = \beta x_0^{2}\). Therefore, we can calculate the subsequent variances as follows: \[\operatorname{Var}(X_n) = \sum_{i=0}^{n-1} \beta (X_i)^2 \quad \text{for} \quad n \geq 1.\] To get a sense of the recursive formula, you can compute the first few variances explicitly as: \[\operatorname{Var}(X_2) = \operatorname{Var}(X_1) + \beta (X_1)^2 = \beta x_0^2 + \beta(E[X_1])^2\] \[\operatorname{Var}(X_3) = \operatorname{Var}(X_2) + \beta (X_2)^2 = \beta x_0^2 + \beta (E[X_1])^2 + \beta (E[X_2])^2\] And so on. Overall, we derived both the expected position \(E[X_n]\) and variance \(\operatorname{Var}(X_n)\) after \(n\) steps using the provided information about the relationship between step size and variance.