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I need to find solutions for SDE of types: $$ dX_t=a\,dt+b^1\,dW_t^1+b^2\,dW_t^2\\ dX_t=a\,dt+b^1\,dW_t^1+b^2\,X_t\,dW_t^2 $$

As far I've found solutions for SDE with 2-dimensional noise in an amazing book of P. Kloeden - Numerical Solution For Stochastic Differential Equations.

It has a solution for this type of equation: $$ dX_t=aX_t\,dt+b^1X_t\,dW_t^1+b^2X_t\,dW_t^2\\ X_t=X_0\exp\Biggl(\Biggl(a-\frac12((b^1)^2+(b^2)^2)\Biggr)t+b^1W_t^1+b^2W_t^2\Biggr) $$

Is there some generic solution for SDE with 2-dimensional noise like for the General Linear SDE?
Maybe it is possible to use the upper solution here?

UPDATE: As lutzl proposed, the first equation can be reduced to this type: $$dX_t=adt+\sqrt{b_1^2+b_2^2}dW_t$$ Where: $$X_t=at+\sqrt{b_1^2+b_2^2}W_t$$ Is it correct?

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mirik
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  • You have $b^1W^1_t+b^2W^2_t=\sqrt{(b^1)^2+(b^2)^2}\tilde W$ where $\tilde W$ is itself a Brownian motion. Thus the first and third problem are essentially with one-dimensional noise, the third reduces to the standard geometric Brownian motion. – Lutz Lehmann Oct 12 '18 at 17:48
  • You can also integrate the first directly, $X_t=X_0+at+b^1,W^1_t+b^2,W^2_t$. Note the integration constant. – Lutz Lehmann Oct 13 '18 at 07:21

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