{"id":901,"date":"2015-01-17T15:00:38","date_gmt":"2015-01-17T21:00:38","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=901"},"modified":"2015-02-20T18:17:42","modified_gmt":"2015-02-21T00:17:42","slug":"thieles-differential-equation","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/policy-values\/6-policy-values-with-continuous-cash-flows\/thieles-differential-equation\/","title":{"rendered":"Thiele’s differential equation"},"content":{"rendered":"

To get an idea as to how policy values evolve over time, we write the policy value at time \\(t\\) in a single equation as
\n\\begin{eqnarray*}
\n~_t V &=& \\int_0^{\\infty}
\n\\left\\{ (b_{t+s}+E_{t+s})\\mu_{[x]+t+s} – (P_{t+s}-e_{t+s})
\n\\right\\}
\n\\frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds \\\\
\n&=& \\int_t^{\\infty}
\n\\left\\{ (b_r+E_r)\\mu_{[x]+r} – (P_r-e_r) \\right\\}
\n\\frac{v(r)}{v(t)} \\frac{~_r p_{[x]}}{~_t p_{[x]}} dr
\n\\end{eqnarray*}
\nusing \\(r=t+s\\) and the relation \\(~_{r-t} p_{[x]+r}=\\frac{~_r p_{[x]}}{~_t p_{[x]}}\\). This yields
\n\\begin{eqnarray*}
\nv(t) ~_t p_{[x]} ~_t V
\n&=& \\int_t^{\\infty} \\left\\{ (b_r+E_r)\\mu_{[x]+r} – (P_r-e_r) \\right\\} v(r) ~_r p_{[x]} dr
\n\\end{eqnarray*}
\nNow take a (partial) derivative with respect to \\(t\\). On the right-hand side, by the fundamental theorem of calculus, we have
\n\\begin{eqnarray*}
\n\\frac{\\partial}{\\partial t}RHS &=& – \\left\\{ (b_t+E_t)\\mu_{[x]+t} – (P_t-e_t) \\right\\} v(t) ~_t p_{[x]}.
\n\\end{eqnarray*}
\nOn the left-hand side, by the chain-rule of differentiation (twice), we have
\n\\begin{eqnarray*}
\n\\frac{\\partial}{\\partial t}LHS &=&
\nv(t) ~_t p_{[x]} \\left\\{ \\frac{\\partial}{\\partial t}~_t V – (\\delta_t + \\mu_{[x]+t}) ~_t V \\right\\}.
\n\\end{eqnarray*}
\nEquating both sides yields Thiele’s differential equation<\/em> <\/p>\n\n\n\n\n
\\(\\frac{\\partial}{\\partial t} ~_t V\\)<\/td>\n\\(=P_t-e_t\\)<\/td>\n\\(+(\\delta_t + \\mu_{[x]+t}) ~_t V\\)<\/td>\n\\(-
\n (b_t+E_t)\\mu_{[x]+t}\\)<\/td>\n<\/tr>\n
change in reserve<\/td>\n= net premium income<\/td>\n+ increase in reserve due to interest and mortality<\/td>\n– benefit outgo<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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To get an idea as to how policy values evolve over time, we write the policy value at time \\(t\\) in a single equation as \\begin{eqnarray*} ~_t V &=& \\int_0^{\\infty} \\left\\{ (b_{t+s}+E_{t+s})\\mu_{[x]+t+s} – (P_{t+s}-e_{t+s}) \\right\\} …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":897,"menu_order":2,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-ex","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/901"}],"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=901"}],"version-history":[{"count":4,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/901\/revisions"}],"predecessor-version":[{"id":1579,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/901\/revisions\/1579"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/897"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=901"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}