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Actuary /
BrownianActuary.Brownian HistoryHide minor edits - Show changes to output Changed line 58 from:
!!Useful to:
!!Useful Computation Changed lines 56-62 from:
Use multiplication rules to simplify the result. to:
Use multiplication rules to simplify the result. !!Useful computation *Involving expectation *Involving variance Changed lines 24-25 from:
*{$ X(t+s)-X(t) $} is normally distributed with mean {$\alpha s$} and variance {$ \sigma^2 s $}; in particular, {$ X(s) $} is normally distributed with mean {$\alpha s$} and variance {$ \sigma^2 s $}. to:
*{$ X(t+s)-X(t) $} is normally distributed with mean {$\alpha s$} and variance {$ \sigma^2 s $}; in particular, {$ X(s) $} is normally distributed with mean {$\alpha s$} and variance {$ \sigma^2 s $}. Changed line 26 from:
The term {$\alpha dt$} is called the ''drift'' in the process; it's the deterministic part of the equation and deals with the long term behavior of the process. The term {$\sigma dZ(t)$} is the random part, which deals with the short term behavior. to:
->The term {$\alpha dt$} is called the ''drift'' in the process; it's the deterministic part of the equation and deals with the long term behavior of the process. The term {$\sigma dZ(t)$} is the random ''noise'' part, which deals with the short term behavior. Changed lines 19-21 from:
->{$$\lim_{n\rightarrow \infty} \sum_{i=1}^n |Z(iT/n)-Z((i-1)T/n) = \infty. $$} to:
->{$$\lim_{n\rightarrow \infty} \sum_{i=1}^n |Z(iT/n)-Z((i-1)T/n)| = \infty. $$} Changed line 40 from:
{$$E(X(t)) = X(0)e^{\alpha t},$$ to:
{$$E(X(t)) = X(0)e^{\alpha t},$$} Changed line 17 from:
The quadratic variation of a Brownian motion is deterministic. This implies higher-ordered variations are zero. to:
The quadratic variation of a Brownian motion is deterministic. This implies higher-ordered variations are zero. Added line 30:
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*We can verify that {$$X(t)= X(0)e^{(\alpha - 0.5 \sigma^2)t+\sigma\sqrt{t}Z(t)}, $$} using Ito's lemma below. *A calculation shows {$$E(X(t)) = X(0)e^{\alpha t},$$ which means {$ \alpha $} is the ''expected continuously compounded return'' on {$X$}. Changed lines 43-44 from:
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!!Ito Processes These are processes following {$$ dX(t) = \alpha(X) dt + \sigma(X) dZ(t),$$} where {$ \alpha $} and {$ \sigma $} are functions of {$X$}. All processes above are examples. !!Multiplication Rules Basically approximations using 1st power of {$dt$}. {$$ dt dZ = 0; $$} {$$ (dt)^2 =0; $$} {$$ (dZ)^2 = dt $$} !!Ito's Lemma Let {$C(a, b)$} be a {$C^{2,1}$} function, and let {$S(t)$} be an Ito process. Then {$$d C(S, t) = C_a(S,t) dS + \frac{1}{2} C_{aa}(S, t) (dS)^2 + C_b(S,t) dt. $$} Use multiplication rules to simplify the result. Changed line 25 from:
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{$ X(s) $} is normally distributed with mean {$\alpha s$} and variance {$ \sigma^2 s $}. Changed lines 28-29 from:
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*We also have {$$ X(t)= \alpha t + \sigma Z(t).$$} Changed line 31 from:
Here the process follows to:
*Here the process follows Changed lines 34-36 from:
{$$ dX(t)/X(t) = to:
{$$ dX(t)/X(t) = d \ln X(t) = \alpha dt + \sigma dZ(t).$$} *{$$X(t)= X(0)e^{(\alpha - 0.5 \sigma^2)t+\sigma\sqrt{t}Z(t)}. $$} Changed lines 19-21 from:
->{$$\lim_{n\rightarrow \infty} \sum_{i=1}^n |Z(iT/n)-Z((i-1)T/n) to:
->{$$\lim_{n\rightarrow \infty} \sum_{i=1}^n |Z(iT/n)-Z((i-1)T/n) = \infty. $$} Changed lines 23-36 from:
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{$X(t)$} is modified from {$Z(t)$} by introducing two parameters, {$ \alpha $} and {$ \sigma $}. *{$ X(t+s)-X(t) $} is normally distributed with mean {$\alpha s$} and variance {$ \sigma^2 s $}; in particular, {$ Z(s) $} is normally distributed with mean {$\alpha s$} and variance {$ \sigma^2 s $}. *For an incremental time {$dt$}, {$$ dX(t) = X(t+dt)-X(t) = \alpha dt + \sigma dZ(t).$$} The term {$\alpha dt$} is called the ''drift'' in the process; it's the deterministic part of the equation and deals with the long term behavior of the process. The term {$\sigma dZ(t)$} is the random part, which deals with the short term behavior. !!Geometric Brownian Motion Here the process follows {$$ dX(t) = X(t+dt)-X(t) = \alpha X(t) dt + \sigma X(t) dZ(t),$$} or {$$ dX(t)/X(t) = = \alpha dt + \sigma dZ(t).$$} Changed lines 5-7 from:
*{$ Z(0)=0 $} to:
*{$ Z(0)=0 $}. *{$ Z(t+s)-Z(t) $} is normally distributed with mean 0 and variance {$ s $}; in particular, {$ Z(s) $} is normally distributed with mean 0 and variance {$s$}. *Nonoverlapping intervals are independently distributed. That is, {$ Z(t+a)-Z(t) $} is independent of {$Z(t)- Z(t-b)$}, for all {$a, b >0$}. Changed lines 13-16 from:
*Binomial approximation: *quadratic variation *total to:
*Binomial approximation: ->Let{$ dt $} be a short change in time. Then {$$ dZ(t) = Z(t+dt)-Z(t) = Y(t)\sqrt{dt},$$} where {$ Y(t) $} is the binomial distribution taking values {$\pm 1$} with probability 0.5. *quadratic variation: ->{$$\lim_{n\rightarrow \infty} \sum_{i=1}^n (Z(iT/n)-Z((i-1)T/n))^2 = T. $$} The quadratic variation of a Brownian motion is deterministic. This implies higher-ordered variations are zero. This makes the multiplication rules below possible. *total variation: ->{$$\lim_{n\rightarrow \infty} \sum_{i=1}^n |Z(iT/n)-Z((i-1)T/n)}^2 = \infty. $$} !!Arithmetic Brownian Motion Added lines 1-15:
!!Brownian Motion A ''stochastic process'' is collection of random variables indexed by time {$t$}, or a function from the reals to a set of random variables. A ''Brownian motion'' is a stochastic process that is a random walk occurring in continuous time, with continuous movement. More precise definition: Let {$ Z(t) $} be the value of a Brownian motion at time {$ t $}. It has the following properties: *{$ Z(0)=0 $}; *{$ Z(t+s)-Z(t) $} is normally distributed with mean 0 and variance {$ s $}; *Nonoverlapping intervals are independently distributed. That is, {$ Z(t+a)-Z(t) $} is dndependent of {$Z(t)- Z(t-b)$}, for all {$a, b >0$}. *{$ Z(t) $} is continuous. !!!Properties: *{$ Z(t) $} is a ''martingale'', that is, {$ E[Z(t+s) | Z(t)] = Z(t) $}. This is also called a ''diffusion process''. *Binomial approximation: Assume that the change in {$ Z(t) $} is a binomial distribution, with a scale factor *quadratic variation *total variation |