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Suppose we have an HJB equation of the form $$ \frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\delta^{a},q-1)-v(t,s,x,q)\right]\right\}+max_{\delta^{b}}\left\{ \lambda^{b}(\delta^{b})\left[v(t,s,x-s+\delta^{b},q+1)-v(t,s,x,q)\right]\right\} $$ with terminal condition $$ v(T,s,x,q)=-e^{-\gamma(x+qs)} $$ We will search a solution of the form $$ v(t,s,x,q)=-e^{-\gamma\left(x+\theta(t,s,q)\right)}=f(x,\theta(t,s,q)) $$ by direct substitution into HJB equation and application of the chain rule we get $$ \frac{\partial f(x,\theta(t,s,q))}{\partial\theta(t,s,q)}\frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\frac{\partial f(x,\theta(t,s,q))}{\partial\theta(t,s,q)}\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}+\frac{\partial^{2}f(x,\theta(t,s,q))}{\partial\theta(t,s,q)^{2}}\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[f(x+s+\delta^{a},\theta(t,s,q-1))-f(x,\theta(t,s,q))\right]\right\} +max_{\delta^{b}}\left\{ \lambda^{b}(\delta^{b})\left[f(x-s+\delta^{b},\theta(t,s,q+1))-f(x,\theta(t,s,q))\right]\right\} $$ taking derivatives of $f$ $$ \gamma f(x,\theta(t,s,q))\frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\gamma f(x,\theta(t,s,q))\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}-\gamma^{2}f(x,\theta(t,s,q))\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[f(x+s+\delta^{a},\theta(t,s,q-1))-f(x,\theta(t,s,q))\right]\right\}+max_{\delta^{b}}\left\{ \lambda^{b}(\delta^{b})\left[f(x-s+\delta^{b},\theta(t,s,q+1))-f(x,\theta(t,s,q))\right]\right\} $$ dividing by $\gamma f(x,\theta(t,s,q))$ $$ \frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}-\gamma\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \frac{\lambda^{a}(\delta^{a})}{\gamma}\left[e^{-\gamma\left(s+\delta^{a}+\theta(t,s,q-1)-\theta(t,s,q)\right)}-1\right]\right\}+max_{\delta^{b}}\left\{ \frac{\lambda^{b}(\delta^{b})}{\gamma}\left[e^{\gamma\left(s-\delta^{b}-\theta(t,s,q+1)+\theta(t,s,q)\right)}-1\right]\right\} $$ Is this correct? It is claimed that with this ansatz we should instead have $$ \frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}-\gamma\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \frac{\lambda^{a}(\delta^{a})}{\gamma}\left[1-e^{-\gamma\left(s+\delta^{a}+\theta(t,s,q-1)-\theta(t,s,q)\right)}\right]\right\} +max_{\delta^{b}}\left\{ \frac{\lambda^{b}(\delta^{b})}{\gamma}\left[1-e^{\gamma\left(s-\delta^{b}-\theta(t,s,q+1)+\theta(t,s,q)\right)}\right]\right\} $$ Not really sure why signs are different, but think I am missing something really trivial.

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  • $\begingroup$ derivatives had a wrong sign (had to be γ(-f) instead of γf for example) $\endgroup$
    – sle
    Commented Jan 2, 2022 at 12:14

1 Answer 1

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derivatives had a wrong sign (had to be γ(-f) instead of γf for example)

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