  \begin{algorithmic}[1] 
    \State \textbf{Input:} Initial parameters $\left(\theta^P_0, \theta^Q_0,\ldots, \theta^Q_{N-1}\right)$, learning rate $\eta$; batch size $M$; number of iteration $K$.
    \State \textbf{Data:} Simulated Brownian increments $\left\{ \Delta W_{t_i, k} \right\}_{0\leq i\leq N-1, 1\leq k\leq K}$
    \State \textbf{Output:} The triple $(X_{t_i}, P_{t_i}, Q_{t_i})$
    \For{$k = 1$ to $K$}
        \State $X_{t_0, k}^{\pi} = x_0$, $P_{t_0, k}^{\pi} = \mu_0^\pi(x_0; \theta^P_{0} ) $ 
        \For{$i = 0$ to $N-1$}
        \State $ Q_{t_i, k}^{\pi} = \phi_i^\pi\left(X_{t_i}^\pi; \theta^Q_{i}\right) $ 
        \State $ u_{t_i, k}^{\pi} = \mathcal{M}\left(t_i, X_{t_i, k}^{\pi}, P_{t_i,k}^{\pi}, Q_{t_i,k}^{\pi} \right)$
        \State $ X_{t_{i+1},k}^{\pi} =  X_{t_i,k}^{\pi} + \bar{b} \left(t_i, X_{t_i,k}^{\pi},u_{t_i, k}^{\pi} \right) \Delta t_i  + \bar{\sigma} \left(t_i, X_{t_i,k}^{\pi}, u_{t_i, k}^{\pi} \right) \Delta W_{t_i, k} $
        \State $ P_{t_{i+1},k}^{\pi} = P_{t_i,k}^{\pi} -  \bar{F} \left(t_i, X_{t_i,k}^{\pi}, u_{t_i, k}^{\pi} \right) \Delta t_i + Q_{t_i,k}^{\pi} \Delta W_{t_i, k} $ 
        \EndFor
    \State $\text{Loss} = \frac{1}{M} \sum_{j=1}^M  \left\| -\nabla_x g\left( X_{t_N,k}^{\pi} \right) - P_{t_N,k}^{\pi}\right\|^2 $
    \State $  \left(\theta^P_0, \theta^Q_0,\ldots, \theta^Q_{N-1} \right)  \longleftarrow  \left(\theta^P_0, \theta^Q_0,\ldots, \theta^Q_{N-1} \right) - \eta \nabla \text{Loss} $
    \EndFor
  \end{algorithmic}

\begin{algorithmic}[1]
\State \textbf{Input:} Initial parameters $\left(\theta^P_0, \theta^Q_0,\ldots, \theta^Q_{N-1}\right)$, learning rate $\eta$; batch size $M$; number of iteration $K$.
\State \textbf{Data:} Simulated Brownian increments $\left\{ \Delta W_{t_i, k} \right\}_{0\leq i\leq N-1, 1\leq k\leq K}$
\State \textbf{Output:} The triple $(X_{t_i}, P_{t_i}, Q_{t_i})$
\For{$k = 1$ to $K$}
\State $X_{t_0, k}^{\pi} = x_0$, $P_{t_0, k}^{\pi} = \mu_0^\pi(x_0; \theta^P_{0} ) $
\For{$i = 0$ to $N-1$}
\State $ Q_{t_i, k}^{\pi} = \phi_i^\pi\left(X_{t_i}^\pi; \theta^Q_{i}\right) $
\State $ u_{t_i, k}^{\pi} = \mathcal{M}\left(t_i, X_{t_i, k}^{\pi}, P_{t_i,k}^{\pi}, Q_{t_i,k}^{\pi} \right)$
\State $ X_{t_{i+1},k}^{\pi} = X_{t_i,k}^{\pi} + \bar{b} \left(t_i, X_{t_i,k}^{\pi},u_{t_i, k}^{\pi} \right) \Delta t_i + \bar{\sigma} \left(t_i, X_{t_i,k}^{\pi}, u_{t_i, k}^{\pi} \right) \Delta W_{t_i, k} $
\State $ P_{t_{i+1},k}^{\pi} = P_{t_i,k}^{\pi} - \bar{F} \left(t_i, X_{t_i,k}^{\pi}, u_{t_i, k}^{\pi} \right) \Delta t_i + Q_{t_i,k}^{\pi} \Delta W_{t_i, k} $
\EndFor
\State $\text{Loss} = \frac{1}{M} \sum_{j=1}^M \left\| -\nabla_x g\left( X_{t_N,k}^{\pi} \right) - P_{t_N,k}^{\pi}\right\|^2 $
\State $ \left(\theta^P_0, \theta^Q_0,\ldots, \theta^Q_{N-1} \right) \longleftarrow \left(\theta^P_0, \theta^Q_0,\ldots, \theta^Q_{N-1} \right) - \eta \nabla \text{Loss} $
\EndFor
\end{algorithmic}
{
    "title": "convergence of the deep bsde method for stochastic control problems   formulated through the stochastic maximum principle",
    "id": "2401.17472",
    "abstract": "it is well-known that decision-making problems from stochastic control can be formulated by means of forward-backward stochastic differential equation (fbsde). recently, the authors of ji et al. 2022 proposed an efficient deep learning-based algorithm which was based on the stochastic maximum principle (smp). in this paper, we provide a convergence result for this deep smp-bsde algorithm and compare its performance with other existing methods. in particular, by adopting a similar strategy as in han and long 2020, we derive a posteriori error estimate, and show that the total approximation error can be bounded by the value of the loss functional and the discretization error. we present numerical examples for high-dimensional stochastic control problems, both in case of drift- and diffusion control, which showcase superior performance compared to existing algorithms.",
    "categories": "math.oc cs.na math.na q-fin.cp",
    "doi": "",
    "created": "2024-01-30",
    "updated": "",
    "authors": [
        "zhipeng huang",
        "balint negyesi",
        "cornelis w. oosterlee"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.17472"
}
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