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Michael Hardy
Michael Hardy
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A solution to stochastic PDE du$du(t)= a(t)u(t)\,dt +s(t)dz\,dz$

Is there a general (integral) solution to $du(t)= -a(t)u(t)dt +\sigma(t)dz$$du(t)= -a(t)u(t)\,dt +\sigma(t)\,dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dz(v)$$u(t)=e^{-\int_{t_0}^{t} \alpha(s) \, ds}u(t_0)+\int_{t_0}^t \sigma(v) e^{-\int_v^t a(s) \, ds} \, dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard WiennerWiener process. Is there a good reference for it?

A solution to stochastic PDE du(t)= a(t)u(t)dt +s(t)dz

Is there a general (integral) solution to $du(t)= -a(t)u(t)dt +\sigma(t)dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wienner process. Is there a good reference for it?

A solution to stochastic PDE $du(t)= a(t)u(t)\,dt +s(t)\,dz$

Is there a general (integral) solution to $du(t)= -a(t)u(t)\,dt +\sigma(t)\,dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s) \, ds}u(t_0)+\int_{t_0}^t \sigma(v) e^{-\int_v^t a(s) \, ds} \, dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wiener process. Is there a good reference for it?

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Adam
Adam
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Is there a general (integral) solution to $du(t)= a(t)u(t)dt +\sigma(t)dz$$du(t)= -a(t)u(t)dt +\sigma(t)dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wienner process. Is there a good reference for it?

Is there a general (integral) solution to $du(t)= a(t)u(t)dt +\sigma(t)dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wienner process. Is there a good reference for it?

Is there a general (integral) solution to $du(t)= -a(t)u(t)dt +\sigma(t)dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wienner process. Is there a good reference for it?

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Adam
Adam
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Is there a general (integral) solution to $du(t)= a(t)u(t)dt +\sigma(t)dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dW(v)$$u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wienner process. Is there a good reference for it?

Is there a general (integral) solution to $du(t)= a(t)u(t)dt +\sigma(t)dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dW(v)$ correct (which I have seen claimed without a justification)? Is there a good reference for it?

Is there a general (integral) solution to $du(t)= a(t)u(t)dt +\sigma(t)dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s)ds}u(t_0)+\int_{t_0}^{t} \sigma(v)e^{-\int_v^{t}a(s)ds}dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wienner process. Is there a good reference for it?

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Adam
Adam
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  • 12
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