# Solved – show \$M(t)=e^{frac{t}{2}}sin(B(t))\$ is a martingale by using Ito s formula

Show that \$M(t)=e^{frac{t}{2}}sin(B(t))\$ is a martingale by using Ito s formula.

\$B(t)\$ is brownian-motion.

i must show \$s le t \$ then \$E(M(t)|mathcal F_{s})=M(s)\$ but i dont know how use ito formula
thanks for help

Contents

Your exact question is answered on Quant Stackexchange here. Essentially, if you can express your stochastic process \$M(t)\$ as \$M(t) = int A(t) dB(t)\$ where \$B(t)\$ is a martingale, then \$M(t)\$ is also a martingale. Since Brownian motion is a martingale, it suffices to express \$M(t)\$ as \$int A(t) dB(t)\$ where \$A(t)\$ is something and \$B(t)\$ is Brownian motion. In the notation of stochastic calculus, you have to show \$\$dM(t) = A(t) dB(t)\$\$ for some \$A(t)\$.

Ito's formula allows you to calculate \$dM(t)\$ because \$M(t) = f(t, B(t))\$ where \$f(x,y) = e^{x/2}sin(y)\$ and \$B(t)\$ is Brownian motion. Ito's formula gives you

\$\$dM(t) = left(frac{partial f}{partial x} + frac{1}{2} frac{partial^2 f}{partial y^2}right) dt + frac{partial f}{partial y}(t, B(t)) dB(t)\$\$

It doesn't matter what \$frac{partial f}{partial y}\$ is. You just need to show that

\$\$frac{partial f}{partial x} + frac{1}{2} frac{partial^2 f}{partial y^2} = 0\$\$

and it follows that \$M(t) = (something) times dB(t)\$ is a martingale.

Edit: @Lost1 points out in the comments that this only proves that \$M(t)\$ is a local martingale. It also needs to satisfy some grwoth conditions in order to be a martingale. According to the Wikipedia article on "local martingale" it is sufficient that for every \$varepsilon > 0\$ and every \$t\$ there exits a constant \$C(varepsilon, t)\$ such that \$|f(s,x)| le Ce^{varepsilon x^2}\$ for all \$x in mathbb{R}\$ and all \$0 le s le t\$. In this case, \$|f(s,x)| le e^{s/2} le e^{t/2}e^0\$ seems to be sufficient. Feel free to edit if you don't think this is a good approach.

Rate this post
{"ft_lang_label":"__label__en","ft_lang_prob":0.8409323,"math_prob":0.9996358,"size":2133,"snap":"2023-40-2023-50","text_gpt3_token_len":663,"char_repetition_ratio":0.14466886,"word_repetition_ratio":0.058479533,"special_character_ratio":0.30145335,"punctuation_ratio":0.06363636,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999943,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-01T15:36:12Z\",\"WARC-Record-ID\":\"<urn:uuid:79fc1ed3-df75-4925-93a0-28ef69ba13ff>\",\"Content-Length\":\"72544\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3264e006-de45-4103-8f41-bd87e4d1f9a3>\",\"WARC-Concurrent-To\":\"<urn:uuid:cb99d753-81d3-4a99-8295-a45d5abda637>\",\"WARC-IP-Address\":\"172.67.156.74\",\"WARC-Target-URI\":\"https://answerofmath.com/solved-show-mtefract2sinbt-is-a-martingale-by-using-ito-s-formula/\",\"WARC-Payload-Digest\":\"sha1:WBIBDAC7JOHJ24SVBUE57MZURQROVAIF\",\"WARC-Block-Digest\":\"sha1:OMTGX7XWW4TB4FKRJOKXTAPB33LLA4N4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510903.85_warc_CC-MAIN-20231001141548-20231001171548-00421.warc.gz\"}"}