Consider a geometric Brownian motion martingale with stochastic volatility. The logarithm of the latent volatility follows a fractional Brownian motion, with an unknown “volatility of volatility” and Hurst exponent H. (a) Write the stochastic differential equations for the system as described. (b) Assuming the model represents a stock price and given a historical price sampled every second, describe your approach for fitting the model. How would you estimate the latent volatility? Be specific. (c) The current volatility model is non-stationary. Suggest and justify an adjustment to make it stationary. What are the revised equations?

We play a game where each of us secretly draws a random number uniformly between 0 and 1. Each player can choose to keep this value or discard it and draw a new number, also uniformly random between 0 and 1, without observing the other player’s number or decision. The player with the higher final number wins $100. What strategy would you use?

Consider the following code: 1 double f ( double x ) { 2 if ( x == 0) 3 return 1.0; 4 return f (0.5 * x ) + f (0.3 * x ) ; 5 } Analyze the time and memory complexities of this function. Provide asymptotically tight bounds.