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split equation onto multiple lines
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user5810
user5810

Hello,

Can anyone help me see how one can get from the following integral

$$\lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,y)d\alpha$$ equation\begin{eqnarray*} \lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,y)d\alpha & =\\ \lambda\int_{\beta=0}^{y}\int_{\alpha=s}^{Q-\beta}e^{-\mu(\alpha-s)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,0)d\alpha\end{eqnarray*}$$ = $$ $$\lambda\int_{\beta=0}^{y}\int_{\alpha=s}^{Q-\beta}e^{-\mu(\alpha-s)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,0)d\alpha$$ to this second order partial differntial equation $\frac{\partial^{2}}{\partial x\partial y}f(x,y)-\mu\frac{\partial}{\partial y}f(x,y)-\frac{\lambda}{k}\frac{\partial}{\partial x}f(x,y)=0$, where $\mu,k,\lambda,Q,s$ are all constants.

Hello,

Can anyone help me see how one can get from the following integral equation\begin{eqnarray*} \lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,y)d\alpha & =\\ \lambda\int_{\beta=0}^{y}\int_{\alpha=s}^{Q-\beta}e^{-\mu(\alpha-s)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,0)d\alpha\end{eqnarray*} to this second order partial differntial equation $\frac{\partial^{2}}{\partial x\partial y}f(x,y)-\mu\frac{\partial}{\partial y}f(x,y)-\frac{\lambda}{k}\frac{\partial}{\partial x}f(x,y)=0$, where $\mu,k,\lambda,Q,s$ are all constants.

Hello,

Can anyone help me see how one can get from the following integral

$$\lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,y)d\alpha$$ $$ = $$ $$\lambda\int_{\beta=0}^{y}\int_{\alpha=s}^{Q-\beta}e^{-\mu(\alpha-s)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,0)d\alpha$$ to this second order partial differntial equation $\frac{\partial^{2}}{\partial x\partial y}f(x,y)-\mu\frac{\partial}{\partial y}f(x,y)-\frac{\lambda}{k}\frac{\partial}{\partial x}f(x,y)=0$, where $\mu,k,\lambda,Q,s$ are all constants.

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Submartingale
Submartingale
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Writing an integral equation as a partial differential equation

Hello,

Can anyone help me see how one can get from the following integral equation\begin{eqnarray*} \lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,y)d\alpha & =\\ \lambda\int_{\beta=0}^{y}\int_{\alpha=s}^{Q-\beta}e^{-\mu(\alpha-s)}f(\alpha,\beta)d\alpha d\beta+k\int_{\alpha=s}^{x}f(\alpha,0)d\alpha\end{eqnarray*} to this second order partial differntial equation $\frac{\partial^{2}}{\partial x\partial y}f(x,y)-\mu\frac{\partial}{\partial y}f(x,y)-\frac{\lambda}{k}\frac{\partial}{\partial x}f(x,y)=0$, where $\mu,k,\lambda,Q,s$ are all constants.