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1

One way to start thinking about this is to work out a couple of Discrete versions of Ito's lemma Øksendal (6th edition) Example 3.1.9: almost surely, $$B_t^2 - t = \int_0^t 2B_s dB_s$$ This has a discrete version which holds everywhere: let $X_n=\pm 1$ and $S_n=\sum_{i=1}^n X_i$, then $$S^2_n-n = 2\sum_{i=0}^{n-1} S_i X_{i+1}$$ To verify ...

1

We have $$V_t = a_t S_t + b_t \beta_t.$$ By Ito's product rule, \begin{align*} dV_t & = d(a_t S_t) + d(b_t \beta_t) \\ & = a_t dS_t + S_t da_t + da_t dS_t + b_t d\beta_t + \beta_t db_t + db_td\beta_t. \end{align*} Since $da_t$ and $db_t$ have no $dW_t$ term, the cross terms are both zero and we have \begin{align*} dV_t & = a_t dS_t + S_t da_t ...

1

In the Black-Scholes model, you would have $d S_t = \mu\, d t + \sigma\, d W_t$ where $W$ is a Brownian motion. So if $V_t = a_t S_t + b_t \beta_t$, then $$dV_t = a_t\, d S_t + S_t\, d a_t + da_t\,dS_t + b_t\,d\beta_t + \beta_t\,d b_t + db_t\, d\beta_t$$ by the product rule. In your case, when $a_t = 1-t$ you will have  dV_t = (1-t) \, dS_t - S_t\, dt + ...

3

For the first question, since by definition, \begin{align*} \varepsilon_t^{if} = e^{i \int_0^{t}f\big(\frac{1}{\xi}\langle M\rangle_s\big)\frac{dM_s}{\sqrt{\xi}} + \frac{1}{2}\int_0^t f\big(\frac{1}{\xi}\langle M\rangle_s\big)\frac{d\langle M\rangle_s}{\xi}}, \end{align*} then, \begin{align*} d\varepsilon_t^{if} = i \varepsilon_t^{if} ...

0

So I have a "+" sign for the second term (not negative) dr =r[(θ(t)+ d(lnσ)/dt * lnr + 1/2*σ^2)dt + σdW] I left out the subscript t's..... You can let V = log r then Apply Ito and solve for A and B... where B = r*σ

2

The portfolio is self-financing. You simply forgot a term in $b$ and a $-t$ term in $V$: \begin{eqnarray} V_t &=& a_t S_t + b_t \beta_t = (2B_t ) (10+ B_t) + (- t - B_t^2 - 20B_t)1 \\ &=& 20B_t + 2B_t^2 - t - B_t^2 - 20B_t \\ &=& B_t^2 - t \end{eqnarray} Applying Ito's lemma \begin{eqnarray} dV_t &=& (2B_t dB_t + ...

0

Your choice of $a_t$ and $b_t$ is feasible. For a self-financing portfolio, the units invested should be static within an infinitesimal time interval, that is, no extra investing or withdrawing during this period. In other words, the portfolio value changes only through its underlyings. For a further discussion, See the article ...

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