{"id":501,"date":"2015-01-06T17:07:26","date_gmt":"2015-01-06T17:07:26","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=501"},"modified":"2015-02-20T18:41:33","modified_gmt":"2015-02-21T00:41:33","slug":"constant-transition-forces-in-the-multiple-decrement-table","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/4-fractional-age-assumptions\/constant-transition-forces-in-the-multiple-decrement-table\/","title":{"rendered":"Constant Transition Forces in the Multiple Decrement Table"},"content":{"rendered":"

Under this assumption, transition forces are constant within the year<\/em>,
\n\\begin{eqnarray*}
\n\\mu_{x+t}^{0j} = \\mu_{x}^{0j} , \\text{ for } 0 \\leq t \\lt 1 .
\n\\end{eqnarray*}
\nRecall that
\n\\begin{eqnarray*}
\n~ _t p_x^{00} &=& \\exp \\left\\{- \\int_0^t sum_{j=1}^n \\mu_{x+s}^{0j} ~ds
\n\\right\\}
\n= \\exp \\left\\{- \\int_0^t \\mu_{x+s}^{0\\bullet} ~ds \\right\\},
\n\\end{eqnarray*}
\nwhere \\(\\mu_{x}^{0\\bullet}= \\sum_{j=1}^n \\mu_{x}^{0j}\\) is the total force of transition out of state 0 at age \\(x\\). Thus, under the constant force assumption, we have
\n\\begin{eqnarray*}
\n~ _t p_x^{00} &=& \\exp \\left\\{-t \\times \\mu_{x}^{0\\bullet}\\right\\},
\n\\end{eqnarray*}
\nand, taking the limit as \\(t \\rightarrow 1\\), we have \\(p_x^{00} = e^{-
\n\\mu_{x}^{0\\bullet}}\\). Thus, we can write
\n\\begin{eqnarray*}
\n~ _t p_x^{00} &=& \\left\\{p_x^{00}\\right\\}^t .
\n\\end{eqnarray*}
\nFurther,
\n\\begin{eqnarray*}
\n_t p_x^{0j} &=& \\int_0^t ~ _s p_x^{00} \\mu_{x+s}^{0j} ~ ds =
\n\\mu_{x}^{0j} \\int_0^t ~ e^{- s \\mu_{x}^{0\\bullet}} ~ ds \\\\
\n&=& \\mu_{x}^{0j} \\frac{1-e^{- t\\mu_{x}^{0\\bullet}}}{\\mu_{x}^{0\\bullet}}~=~ \\frac{\\mu_{x}^{0j}}{\\mu_{x}^{0\\bullet}} (1- (p_x^{00})^t)
\n\\end{eqnarray*}
\nAgain, taking the limit as \\(t \\rightarrow 1\\), we have
\n\\begin{eqnarray*}
\n\\frac{ p_x^{0j}}{1- p_x^{00}} = \\frac{ p_x^{0j}}{p_x^{0\\bullet}}=
\n\\frac{\\mu_{x}^{0j}}{\\mu_{x}^{0\\bullet}} .
\n\\end{eqnarray*}
\nThus, we may write
\n\\begin{eqnarray*}
\n_t p_x^{0j} = \\frac{ p_x^{0j}}{p_x^{0\\bullet}} \\left(1- (p_x^{00})^t\\right) .
\n\\end{eqnarray*}<\/p>\n

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Under this assumption, transition forces are constant within the year, \\begin{eqnarray*} \\mu_{x+t}^{0j} = \\mu_{x}^{0j} , \\text{ for } 0 \\leq t \\lt 1 . \\end{eqnarray*} Recall that \\begin{eqnarray*} ~ _t p_x^{00} &=& \\exp \\left\\{- \\int_0^t …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":495,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-85","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/501"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=501"}],"version-history":[{"count":4,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/501\/revisions"}],"predecessor-version":[{"id":1617,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/501\/revisions\/1617"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/495"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=501"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}