Let us consider a quantum system with the dynamics described by the master equation \[ \frac{\partial}{\partial t}\rho(t)=\frac{1}{\mathrm{i}\hbar}(H_{\mathrm{eff}}\rho -\rho H_{\mathrm{eff}}^{\dagger})+C\rho C^{\dagger}\,. \] The last term of this equation is interpreted as describing quantum jumps, the remaining terms describe the jump-free evolution. The operators \(H_{\mathrm{eff}}\) and \(C\) are non-Hermitian in general.
We express the the density matrix \(\rho\) in the following form: \[ \rho(t)=\overline{|\Psi(t)\rangle\langle\Psi(t)|}\,, \] where \(\overline{\cdot}\) denotes averaging over different realizations of trajectories \(|\Psi(t)\rangle\). Let us consider \(|\Psi(t)\rangle\) obeying the Itô stochastic differential equation \[ \mathrm{d}|\Psi(t)\rangle=\frac{1}{\mathrm{i}\hbar}H_{\mathrm{eff}}|\Psi(t)\rangle\mathrm{d}t +C|\Psi(t)\rangle\mathrm{d}W_t\,,\label{schr-sde}\tag{1} \] where \(W_t\) is a real Wiener process having the properties \[ \overline{\mathrm{d}W_t}=0\,,\qquad\overline{\mathrm{d}W_t^2}=\mathrm{d}t\,. \] Then \[ \begin{aligned} |\Psi(t+\mathrm{d}t)\rangle\langle\Psi(t+\mathrm{d}t)|=|\Psi\rangle\langle\Psi| &+ \frac{1}{\mathrm{i}\hbar}\bigl(H_{\mathrm{eff}}|\Psi\rangle\langle\Psi| -|\Psi\rangle\langle\Psi|H_{\mathrm{eff}}^{\dagger}\bigr)\mathrm{d}t \\ & +\bigl(C|\Psi\rangle\langle\Psi|+|\Psi\rangle\langle\Psi|C^{\dagger}\bigr)\mathrm{d}W_t +C|\Psi\rangle\langle\Psi|C^{\dagger}\mathrm{d}W_t^2\,. \end{aligned} \] Taking average and using Itô calculus one has \[ \rho(t+\mathrm{d}t)-\rho(t)=\frac{1}{\mathrm{i}\hbar}(H_{\mathrm{eff}}\rho -\rho H_{\mathrm{eff}}^{\dagger})\mathrm{d}t+C\rho C^{\dagger}\mathrm{d}t\,. \] This is the required equation for the density matrix. Therefore, the stochastic Schrödinger equation (1) for the wave function \(|\Psi(t)\rangle\) leads to the master equation for the density matrix \(\rho(t)\). Instead of equation (1) we can take the stochastic differential equation \[ \mathrm{d}|\Psi(t)\rangle=\frac{1}{\mathrm{i}\hbar}H_{\mathrm{eff}}|\Psi(t)\rangle\mathrm{d}t +\frac{1}{2}a^{*}(2C-a)|\Psi(t)\rangle\mathrm{d}t+(C-a)|\Psi(t)\rangle\mathrm{d}W_t\,, \] where \(a\) is a complex number.
Equation (1) does not conserve the norm of the wave function \(|\Psi(t)\rangle\). Indeed, using Itô calculus we get \[ \mathrm{d}\Vert\Psi\Vert^{2}= \frac{1}{\mathrm{i}\hbar}\langle\Psi|H_{\mathrm{eff}}-H_{\mathrm{eff}}^{\dagger}|\Psi\rangle\mathrm{d}t +\langle\Psi|C^{\dagger}C|\Psi\rangle\mathrm{d}t+\langle\Psi|C +C^{\dagger}|\Psi\rangle\mathrm{d}W_t\,. \] The sum of the first two terms in this equation is equal to zero because the master equation for the density matrix should be trace preserving. Therefore \[ \mathrm{d}\Vert\Psi\Vert^{2}=\langle\Psi|C+C^{\dagger}|\Psi\rangle\mathrm{d}W_{t}\,. \] Using the second form of the stochastic Schrödinger equation we obtain instead \[ \mathrm{d}\Vert\Psi\Vert^{2}=\langle\Psi|C+C^{\dagger}-a-a^{*}|\Psi\rangle\mathrm{d}W_{t}\,. \] If \(a=\langle\Psi(t)|C|\Psi(t)\rangle\equiv\langle C\rangle_t\) then \(\mathrm{d}\Vert\Psi\Vert^{2}=0\). Thus, the stochastic Schrödinger equation \[ \mathrm{d}|\Psi(t)\rangle=\frac{1}{\mathrm{i}\hbar}H_{\mathrm{eff}}|\Psi(t)\rangle\mathrm{d}t +\frac{1}{2}\langle C\rangle_t^{*}\bigl(2C-\langle C\rangle_t\bigr)|\Psi(t)\rangle\mathrm{d}t +\bigl(C-\langle C\rangle_t\bigr)|\Psi(t)\rangle\mathrm{d}W_t \] conserves the norm. However, this equation is explicitly nonlinear.