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$$\textbf{Preface}$$ I am assuming log normal asset but this is not clear from the question? Or rather I have misinterpreted the question! Well as I see it from a a purely mathematical exercise $$d\left(\dfrac{S_t}{M_t}\right) =\frac{1}{M_t}dS_t - \frac{S_t}{M_t^2}dM_t +O(dt^2)$$ using Ito's lemma. Then we can sub in the original processes yields ...
$$\dfrac{dP_1}{dP}=\dfrac{f(B_1)}{f(B)}=\dfrac{\frac{1}{\sqrt{2\pi}\sigma_1} e^{ -\frac{(x-\mu_1)^2}{2\sigma_1^2} })}{\frac{1}{\sqrt{2\pi}\sigma} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }}=\dfrac{\sigma_1}{\sigma}e^{\dfrac{(x-\mu)^2}{2\sigma^2} -\dfrac{(x-\mu_1)^2}{2\sigma_1^2}}=\dfrac{t}{t}e^{\dfrac{(x-0)^2}{2t^2} -\dfrac{(x-\mu t)^2}{2t^2}}=e^\dfrac{x^2+x^2-2x\mu ... 3 Consider an (arithmetic) Ornstein-Uhlenbeck process as a model of the asset price X_t:$$dX_t = \kappa(\mu-S_t)dt + \sigma dW_t$$where \mu is the mean-reversion level, \sigma is a volatility parameter, W_t is Brownian motion, and \kappa is the reversion speed. An Ornstein-Uhlenbeck process will revert to the mean infinitely often if \kappa ... 0 Two cointegrated series contain a single unit root. Each series can be formulated as the sum of a common unit root plus a stationary component. Most textbooks covering cointegration will cover such formulations - see Hamilton's (1994) discussion of Phillips' "triangular representation" of a cointegrated vector, for example. Simulating is likely to be easy ... 0 Non-Stationary process can be analyzed and there are various models available that can be used . For example, Autoregressive Integrated Moving Average model (ARIMA) models are used to explain homogeneous non-stationary models as well as random walk with drift can be used for explaining several such series. have a look at this link : ... 2 You are asking an interesting question. Firstly, a Submartingale has increasing or equal expectation (not decreasing). Secondly, the process dX_t=X_tdW_t is a true martingale (not strictly local), since its solution (by Ito):$$X_t=X_0e^{W_t-\frac{t}{2}}$$has E(X_t)=X_0 constant expectation (e^{-\frac{t}{2}}E(e^{W_t})=1, W_t\sim N(0,t)). The ... 3 Note that$$P(X_i >s)= \exp\Big(-\int_0^s \lambda_i(u) du \Big),$$for i=1, 2. Then,$$P(\min(X_1, X_2) >s) = P((X_1>s)\cap (X_2>s)) = P(X_1>s)P(X_2>s) = \exp\Big(-\int_0^s (\lambda_1(u)+\lambda_2(u)) du \Big).$$That is, the hazard function for \min(X_1, X_2) is \lambda_1(s)+\lambda_2(s). Alternatively, note that$$\lambda_i(s) = ...