Let $W$ be a $R^n-$valued Wiener process under the real$-$world measure $P.$ Assume that $$ is a

$mn$ real $($constant$)$ matrix, $in{R}^m$ is a constant column vector and $r$ is a real constant.

Let the risk$-$free asset be given by

$d{S}_0(t)=r{S}_0(t)dt$

and let the $m$ risky assets be each given by

$d{S}_i(t)=S_i(t){}_{i}dt+S_i(t){\displaystyle \underset{j=1}{\overset{n}{}}}{}_{ij}d{W}_j(t),$

for $i=1,2,$dots,$m.$ Suppose also that $r=0\mathrm{.}2.$ For the following situations, determine if there

are uncountably many, one unique or no risk$-$neutral measure$($s$).$ Justify your answer.

$($a$)$

$n=3,m=2,$

${}_1=0\mathrm{.}5,{}_2=0\mathrm{.}4,$ and

${}_{11}={}_{12}={}_{21}={}_{22}=2,{}_{13}={}_{23}=-1$

$($b$)$

$n=2,m=3,$

${}_1=0\mathrm{.}6,{}_2=0\mathrm{.}8,{}_3=0\mathrm{.}9,$ and

${}_{11}={}_{12}=2,{}_{21}={}_{22}={}_{31}=3,{}_{32}=4$

$($c$)$

$n=m,$

${}_i=0\mathrm{.}4$ for all $i=1,2,$dots,$m$

${}_{ij}=2$ for any $i=j,i=1,$dots,$m,j=1,$dots,$n,$ and

${}_{ij}=-1$ for any $ij,i=1,$dots,$m,j=1,$dots,$n.$

Hint: You may use that $||=3^{m-1}(3-m).$