# Numerically integrate the pricing differential equation

### Description

Use an implicit integration scheme to numerically integrate
the pricing differential equation for each of the given instruments,
backwardating from time `Tmax`

to time 0.

### Usage

1 2 3 | ```
integrate_pde(z, min_num_time_steps, S0, Tmax, instruments, stock_level_fcn,
discount_factor_fcn, default_intensity_fcn, variance_cumulation_fcn,
dividends = NULL)
``` |

### Arguments

`z` |
Space grid value morphable to stock prices using |

`min_num_time_steps` |
The minimum number of timesteps used. Calls, puts and coupons may result in extra timesteps taken. |

`S0` |
Time zero price of the base equity |

`Tmax` |
The maximum time on the grid, from which all backwardation steps will take place. |

`instruments` |
A list of instruments to be priced. Each
one must have a |

`stock_level_fcn` |
A function for changing space grid value to stock
prices, with arguments |

`discount_factor_fcn` |
A function for computing present values to
time |

`default_intensity_fcn` |
A function for computing default intensity
occurring during this timestep, dependent on time and stock price, with
arguments |

`variance_cumulation_fcn` |
A function for computing total stock variance
occurring during this timestep, with arguments |

`dividends` |
A |

### Value

A grid of present values of derivative prices, adapted to `z`

at
each timestep. Time zero value will appear in the first index.

### See Also

Other Implicit Grid Solver: `construct_implicit_grid_structure`

,
`find_present_value`

,
`form_present_value_grid`

,
`infer_conforming_time_grid`

,
`iterate_grid_from_timestep`

,
`take_implicit_timestep`

,
`timestep_instruments`