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I have the following two SDEs

$$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$

$$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$

where $W$ is the standard Brownian motion/Wiener process. This isn't homework, I'm just curious. I can solve the first one but the second one is in terms of $N_1$ and $N_2$ so I don't know how to go about it. I'm new to SDEs so any help is appreciated!

$$N_1(t)=N_1(0)exp\left\{((2a-1)p-\frac{1}{2}\alpha_1^2)t+\alpha_1 W_1\right\}$$

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  • $\begingroup$ @ Carol : I suggest you use Girsanov theorem to find a probability measure under wich the second equation take the same for mas the first one and then using the Radon Nikodym process express the solution into the original probability space (of course you have to check under which conditions you can use Girsanov with respct to $N_1$). Best regards $\endgroup$
    – TheBridge
    Dec 4, 2014 at 8:19
  • $\begingroup$ It's a 2d Geometric Brownian Motion, with triangular coefficient matrices: have you tried looking for that on the web? $\endgroup$
    – SBF
    Dec 4, 2014 at 10:23
  • $\begingroup$ Thanks for the replies @TheBridge and Ilya. I am a second-year undergraduate management student so my math background is weak. I will look into your suggestions. $\endgroup$
    – Carol
    Dec 4, 2014 at 10:56
  • $\begingroup$ Solve the first equation (it's linear, just gbm), plug into the second and solve it with variation of constants method. $\endgroup$
    – zhoraster
    Jul 18, 2018 at 15:23
  • $\begingroup$ @zhoraster Would you provide more hints on how to treat the stochastic process of $N_1$ in the second equation? Thanks a lot! $\endgroup$
    – huighlh
    Aug 3, 2018 at 1:59

1 Answer 1

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As mentioned in the comments the SDE

$$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$

is decoupled and so one can solve it

$$N_{1,t}=N_{1,0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{1,t}\right).$$

for $\mu:=(2a-1)p$ and $\sigma:=\alpha_1$. Then since $W_{1}$ is independent of $W_{2}$ we can condition by it and so $N_{1,t}$ is simply a continuous process.

Then for the second equation,

$$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$

we simply use the Solution to General Linear SDE

\begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t \big) dB_t. \end{align}

\begin{align*} N_{2,t} =& \exp\left( \int_0^t\left( a(s) - \frac{1}{2}g^2(s) \right) \mathrm{d}s + \int_0^t g(s)\mathrm{d}B_s\right) \\ &\cdot \left(N_{2,0}+ \int_0^{t} b(s)\exp\left( \int_0^s\left( \frac{1}{2}g^2(r) - a(r)\right) \mathrm{d}r - \int_0^s g(r)\mathrm{d}B_r\right)\mathrm{d}s\right). \end{align*}

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