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4

Consider a financial market with a filtered probability space $\left(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb P\right)$ satisfying usual conditions equipped with a stock price process $S_t$. Suppose there exists a risk-free asset who is governed by $\mathrm{d}B_t=r_tB_t\mathrm{d}t$. Suppose the market is free of arbitrage, i.e. there exists a probability ...

4

For the first one, we have: $$dS_t = \mu_t S_t dt + \sigma_tS_t dW_t$$ and note that $$(dS_t)^2 = \sigma_t ^2 S_t^{2} dt.$$ We apply Ito formula to $$f(S_t) = \ln S_t.$$ As $f'(x) = x^{-1}$ and $f^{''}(x)= -x^{-2}$, we get: $$d \ln S_t = S_t^{-1} dS_t - 0.5 S_t^{-2} (dS_t)^2$$ which is equivalent to  d \ln S_t = S_t^{-1} dS_t - 0.5 \...

3

KeSchn and I pointed out in the comments that this it is not possible to represent all stock dynamics using the Generalized Black Scholes model. For example, there can be jumps at random moments and not just at random moments but also jumps of random size. These jumps can affect either $\mu_t$ or $\sigma_t$. Models with too many sources of randomness are not ...

6

The way I understand it is: In equation 2 $x_{j, t + 1}$ is defined as the change in of $p_j$ over the period $t$ to $t + 1$. The formula says that the expectation of the change is zero which is the same as saying that the expectation of the original variable at $t+1$ is equal to its current value.

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