<gh_stars>1-10
import numpy as np

import sampler as spl

class Equation(object):
    """Base class for defining PDE related function."""
    
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        self.dim = dimension
        self.dim_system = eqn_config.dim_system
        self.total_time = eqn_config.total_time
        self.start_time = eqn_config.start_time
        self.num_iteration = num_iteration
        
    def mu(self, t, x, v):
        """Drift coefficient in the SDE."""
        raise NotImplementedError
        
    def sigma(self, t, x, v):
        """Diffusion coefficient in the SDE."""
        raise NotImplementedError
        
    def dx_sigma(self, t, x, v):
        """Derivative of diffusion coefficient in the SDE."""
        raise NotImplementedError
        
    def dXsample(self, t_start, t_end, x_0, dW):
        """Sample forward SDE."""
        raise NotImplementedError
        
    def dIsample(self, t_start, t_end, x, dW, Xsample):
        """Sample derivative process of forward SDE."""
        raise NotImplementedError

    def f(self, t, x, v):
        """Generator function in the PDE."""
        raise NotImplementedError

    def g(self, t, x):
        """Terminal condition of the PDE."""
        raise NotImplementedError
        
        
class SemilinearBlackScholes(Equation):
    """
    Semilinear Black-Scholes PDE from paper arxiv.org/abs/2005.10206v2
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(SemilinearBlackScholes, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.ones(self.dim) * 50.0
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"): 
            if(self.num_gridpoint < 1):
                raise ValueError("SDE Approximation schemes with <=1 gridpoint are not valid.")
            else:
                self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
                self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
                
        self.dx_sigmaMatrix = np.zeros((self.dim, self.dim, self.dim))
        for i in range(self.dim):
            self.dx_sigmaMatrix[i, i, i] = 1.0
                
                
    def mu(self, x, t):
        return x
    
    def sigma(self, x, t):
        return np.diag(x)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return x
        
    def dx_sigma(self, x, t):
        return self.dx_sigmaMatrix
        
    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW = {}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            return (x_0 * np.exp(0.5*(t_end - t_start) +  np.array([np.sum(np.vstack(dW[(1,)]), axis=0)])), dW)
            

    def f(self, t, x, v):
        return np.array([v / (1 + v**2)])

    def g(self, x):
        return np.array([np.log(0.5 * (1 + np.linalg.norm(x)**2))])
        
        
class SystemSemilinearHeatEquation(Equation):
    """
    System of semilinear heat PDEs from paper arxiv.org/abs/2005.10206v2
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(SystemSemilinearHeatEquation, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.zeros(self.dim)
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"):
            self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
            self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
    def mu(self, x, t):
        return np.zeros(self.dim)
    
    def sigma(self, x, t):
        return np.sqrt(2)*np.eye(self.dim)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return np.ones(self.dim)
        
    def dx_sigma(self, x, t):
        return np.zeros((self.dim, self.dim, self.dim))
    
    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW={}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 + np.sqrt(2) * dW[(1,)][0], dW)
            return (x_0 + np.sqrt(2) * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)]), dW)
            
    def f(self, t, x, v):
        return np.array([v[1] / (1 + v[1]**2), (2*v[0]) / 3])

    def g(self, x):
        return np.array([1 / (2 + 0.4 * np.linalg.norm(x)**2), np.log(0.5 * (1 + np.linalg.norm(x)**2))])


class AllenCahn(Equation):
    """
    Allen-Cahn PDE from paper arxiv.org/abs/2005.10206v2
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(AllenCahn, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.zeros(self.dim)
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"):
            self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
            self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
    def mu(self, x, t):
        return np.zeros(self.dim)
    
    def sigma(self, x, t):
        return np.sqrt(2)*np.eye(self.dim)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return np.ones(self.dim)
        
    def dx_sigma(self, x, t):
        return np.zeros((self.dim, self.dim, self.dim))
    
    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW={}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 + np.sqrt(2) * dW[(1,)][0], dW)
            return (x_0 + np.sqrt(2) * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)]), dW)
            
    
    def f(self, t, x, v):
        return np.array([v - v**3])

    def g(self, x):
        return np.array([1 / (2 + 0.4 * np.linalg.norm(x)**2)])
    
    
class SineGordon(Equation):
    """
    Sine-Gordon type PDE from paper arxiv.org/abs/2005.10206v2
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(SineGordon, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.zeros(self.dim)
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"):
            self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
            self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
    def mu(self, x, t):
        return np.zeros(self.dim)
    
    def sigma(self, x, t):
        return np.sqrt(2)*np.eye(self.dim)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return np.ones(self.dim)
        
    def dx_sigma(self, x, t):
        return np.zeros((self.dim, self.dim, self.dim))
    
    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW={}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 + np.sqrt(2) * dW[(1,)][0], dW)
            return (x_0 + np.sqrt(2) * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)]), dW)
    
    def f(self, t, x, v):
        return np.array([np.sin(v)])

    def g(self, x):
        return np.array([1 / (2 + 0.4 * np.linalg.norm(x)**2)])
    
    
class RecursivePricingDefaultRisk(Equation):
    """
    Recursive pricing with default risk from paper https://arxiv.org/abs/1607.03295v4
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(RecursivePricingDefaultRisk, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.ones(self.dim) * 100
        self.sigmaC = eqn_config.sigma
        self.muC = eqn_config.mu
        self.delta = eqn_config.delta
        self.R = eqn_config.R
        self.vh = eqn_config.vh[eqn_config.dim.index(dimension)]
        self.vl = eqn_config.vl[eqn_config.dim.index(dimension)]
        self.gammah = eqn_config.gammah
        self.gammal = eqn_config.gammal
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"): 
            if(self.num_gridpoint < 1):
                raise ValueError("SDE Approximation schemes with <=1 gridpoint are not valid.")
            else:
                self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
                self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
        self.dx_sigmaMatrix = np.zeros((self.dim, self.dim, self.dim))
        for i in range(self.dim):
            self.dx_sigmaMatrix[i, i, i] = self.sigmaC

    def mu(self, x, t):
        return self.muC * x
    
    def sigma(self, x, t):
        return self.sigmaC*np.diag(x)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return self.sigmaC*x
        
    def dx_sigma(self, x, t):
        return self.dx_sigmaMatrix

    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW = {}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * dW[(1,)][0]), dW)
            return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)])), dW)

    def f(self, t, x, v):
        return (np.array([-(1-self.delta) * v * ((v < self.vh) * self.gammah 
                            + (v >= self.vl) * self.gammal
                            + ((v >= self.vh) and (v < self.vl)) * (((self.gammah - self.gammal)/(self.vh - self.vl)) * (v - self.vh) + self.gammah))
                            - self.R * v]))

    def g(self, x):
        return np.array([min(x)])


class PricingCreditRisk(Equation):
    """
    Semilinear PDE for Valuing derivative contracts with counterparty credit risk from paper https://arxiv.org/abs/1607.03295v4
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(PricingCreditRisk, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.ones(self.dim) * 100
        self.sigma = eqn_config.sigma
        self.beta = eqn_config.beta
        self.K1 = eqn_config.K1[eqn_config.dim.index(dimension)]
        self.K2 = eqn_config.K2[eqn_config.dim.index(dimension)]
        self.L = eqn_config.L[eqn_config.dim.index(dimension)]
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"): 
            if(self.num_gridpoint < 1):
                raise ValueError("SDE Approximation schemes with <=1 gridpoint are not valid.")
            else:
                self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
                self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
        self.dx_sigmaMatrix = np.zeros((self.dim, self.dim, self.dim))
        for i in range(self.dim):
            self.dx_sigmaMatrix[i, i, i] = self.sigmaC

    def mu(self, x, t):
        return np.zeros(self.dim)
    
    def sigma(self, x, t):
        return self.sigmaC*np.diag(x)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return self.sigmaC*x
        
    def dx_sigma(self, x, t):
        return self.dx_sigmaMatrix

    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW = {}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 * np.exp(((-0.5)*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * dW[(1,)][0]), dW)
            return (x_0 * np.exp(((-0.5)*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)])), dW)

    def f(self, t, x, v):
        return np.array([self.beta*(max(v,0) - v)])

    def g(self, x):
        return np.array([max(min(x)-self.K1,0.0) - max(min(x)-self.K2,0.0) - self.L])


class SemilinearBlackScholesAmericanOption(Equation):
    """
    Pricing a American Put Option with an Semilinear Black-Scholes PDE from paper https://link.springer.com/article/10.1007%2Fs007800200091
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(SemilinearBlackScholesAmericanOption, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.ones(self.dim) * 40.0
        self.sigma = eqn_config.sigma
        self.mu = eqn_config.mu
        self.R = eqn_config.R
        self.K = eqn_config.K
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"): 
            if(self.num_gridpoint < 1):
                raise ValueError("SDE Approximation schemes with <=1 gridpoint are not valid.")
            else:
                self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
                self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
        self.dx_sigmaMatrix = np.zeros((self.dim, self.dim, self.dim))
        for i in range(self.dim):
            self.dx_sigmaMatrix[i, i, i] = self.sigmaC

    def mu(self, x, t):
        return self.muC * x
    
    def sigma(self, x, t):
        return self.sigmaC*np.diag(x)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return self.sigma*x
        
    def dx_sigma(self, x, t):
        return self.dx_sigmaMatrix

    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW = {}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * dW[(1,)][0]), dW)
            return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)])), dW)

    def f(self, t, x, v):
        return np.array([self.R*self.K*(self.g(x) >= v)])

    def g(self, x):
        return np.array([max(self.K - max(x), 0.0)])
                            
                        
class PricingDifferentInterestRates(Equation):
    """
    Pricing Problem of an European option in a financial market with different interest rates for borrowing and lending. from paper https://arxiv.org/abs/1607.03295v4
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(PricingDifferentInterestRates, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.ones(self.dim) * 100.0
        self.sigmaC = eqn_config.sigma
        self.muC = eqn_config.mu
        self.Rl = eqn_config.Rl
        self.Rb = eqn_config.Rb
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"): 
            if(self.num_gridpoint < 1):
                raise ValueError("SDE Approximation schemes with <=1 gridpoint are not valid.")
            else:
                self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
                self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
        self.dx_sigmaMatrix = np.zeros((self.dim, self.dim, self.dim))
        for i in range(self.dim):
            self.dx_sigmaMatrix[i, i, i] = self.sigmaC

    def mu(self, x, t):
        return self.muC * x
    
    def sigma(self, x, t):
        return self.sigmaC*np.diag(x)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return self.sigmaC*x
        
    def dx_sigma(self, x, t):
        return self.dx_sigmaMatrix

    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW = {}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * dW[(1,)][0]), dW)
            return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)])), dW)
    
    def dIsample(self, t_start, t_end, x, dW, Xsample):
        return np.concatenate([np.ones(1), np.sum(np.vstack(dW[(1,)]), axis=0) / (t_end - t_start)])

    def f(self, t, x, v):
        return np.array([[-self.Rl * v[0][0] - ((self.muC - self.Rl) / self.sigmaC) * sum(v[0][1:]) + (self.Rb - self.Rl) * max((sum(v[0][1:]) / self.sigmaC) - v[0][0],0.0)]])

    def g(self, x):
        if(self.dim == 1):
            return np.array([[max(x[0]-100.0, 0.0)]])
        else:
            return np.array([[max(max(x)-120.0,0.0) - 2*max(max(x)-150.0,0.0)]])
        
        
class SystemSemilinearHeatEquationTEST(Equation):
    """
    System of semilinear heat PDEs from paper arxiv.org/abs/2005.10206v2
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(SystemSemilinearHeatEquationTEST, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.zeros(self.dim)
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"):
            self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
            self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
    def mu(self, x, t):
        return np.zeros(self.dim)
    
    def sigma(self, x, t):
        return np.sqrt(2)*np.eye(self.dim)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return np.ones(self.dim)
        
    def dx_sigma(self, x, t):
        return np.zeros((self.dim, self.dim, self.dim))
    
    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW={}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 + np.sqrt(2) * dW[(1,)][0], dW)
            return (x_0 + np.sqrt(2) * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)]), dW)
    
    def dIsample(self, t_start, t_end, x, dW, Xsample):
        dI = np.zeros(self.dim + 1)
        dI[0] = 1.0
        return dI

    def f(self, t, x, v):
        return np.array([[v[1][0] / (1 + v[1][0]**2)], [(2*v[0][0]) / 3]])

    def g(self, x):
        return np.array([[1 / (2 + 0.4 * np.linalg.norm(x)**2)], [np.log(0.5 * (1 + np.linalg.norm(x)**2))]])


class ExampleExplicitSolution(Equation):
    """
    System of semilinear heat PDEs from paper arxiv.org/abs/2005.10206v2
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(ExampleExplicitSolution, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.zeros(self.dim)
        self.sigmaC = eqn_config.sigma
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"): 
            if(self.num_gridpoint < 1):
                raise ValueError("SDE Approximation schemes with <=1 gridpoint are not valid.")
            else:
                self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
                self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])

    def mu(self, x, t):
        return np.zeros(self.dim)
    
    def sigma(self, x, t):
        return self.sigmaC*np.eye(self.dim)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return self.sigmaC*np.ones(self.dim)
        
    def dx_sigma(self, x, t):
        return np.zeros((self.dim, self.dim, self.dim))

    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW = {}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 +  self.sigmaC * dW[(1,)][0], dW)
            return (x_0 +  self.sigmaC * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)]), dW)
    
    def dIsample(self, t_start, t_end, x, dW, Xsample):
        return np.concatenate([np.ones(1), np.sum(np.vstack(dW[(1,)]), axis=0) / (t_end - t_start)])

    def f(self, t, x, v):
        return np.array([[self.sigmaC * (v[0][0] - ((2+(self.sigmaC**2)*self.dim) / (2*(self.sigmaC**2)*self.dim))) * sum(v[0][1:])]])

    def g(self, x):
        return np.array([[np.exp(self.total_time + sum(x)) / (1 + np.exp(self.total_time + sum(x)))]])  


class HJBEquation(Equation):
    """
    Hamilton-Jacobi-Bellmann (HJB) Equation wich admits a explicit solution from paper https://doi.org/10.1007/s40304-017-0117-6
    """
    
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(HJBEquation, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.zeros(self.dim)
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"):
            self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
            self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
    def mu(self, x, t):
        return np.zeros(self.dim)
    
    def sigma(self, x, t):
        return np.sqrt(2)*np.eye(self.dim)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return np.ones(self.dim)
        
    def dx_sigma(self, x, t):
        return np.zeros((self.dim, self.dim, self.dim))
    
    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW={}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 + np.sqrt(2) * dW[(1,)][0], dW)
            return (x_0 + np.sqrt(2) * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)]), dW)
    
    def dIsample(self, t_start, t_end, x, dW, Xsample):
        return np.concatenate([np.ones(1), np.sum(np.vstack(dW[(1,)]), axis=0) / (t_end - t_start)])

    def f(self, t, x, v):
        return np.array([[np.linalg.norm(v[0][1:])**2]])

    def g(self, x):
        return np.array([[np.log(0.5 * (1 + np.linalg.norm(x)**2))]])



class PricingDifferentInterestRates(Equation):
    """
    Pricing Problem of an European option in a financial market with different interest rates for borrowing and lending. from paper https://arxiv.org/abs/1607.03295v4
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(PricingDifferentInterestRates, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.ones(self.dim) * 100.0
        self.sigmaC = eqn_config.sigma
        self.muC = eqn_config.mu
        self.Rl = eqn_config.Rl
        self.Rb = eqn_config.Rb
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        
        if(self.samplingMethod != "Explicit"): 
            if(self.num_gridpoint < 1):
                raise ValueError("SDE Approximation schemes with <=1 gridpoint are not valid.")
            else:
                self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
                self.sampleNeeded = self.sampleMethod.sampleNeeded
        else:
            self.sampleNeeded = np.array([True,True,False,False,False,False,False,False])
            
        self.dx_sigmaMatrix = np.zeros((self.dim, self.dim, self.dim))
        for i in range(self.dim):
            self.dx_sigmaMatrix[i, i, i] = self.sigmaC

    def mu(self, x, t):
        return self.muC * x
    
    def sigma(self, x, t):
        return self.sigmaC*np.diag(x)
    
    # only needed for itoSRID2 sampling method
    #def sigma(self, x, t):
    #   return self.sigmaC*x
        
    def dx_sigma(self, x, t):
        return self.dx_sigmaMatrix

    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
        else:
            if dW == None:
                dW = {}
                dW[(1,)] = np.array([np.random.normal(0.0,np.sqrt(t_end-t_start),(1,self.dim))])
            #return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * dW[(1,)][0]), dW)
            return (x_0 * np.exp((self.muC - 0.5*(self.sigmaC**2))*(t_end - t_start) +  self.sigmaC * np.array([np.sum(np.vstack(dW[(1,)]), axis=0)])), dW)
    
    def dIsample(self, t_start, t_end, x, dW, Xsample):
        return np.concatenate([np.ones(1), np.sum(np.vstack(dW[(1,)]), axis=0) / (t_end - t_start)])

    def f(self, t, x, v):
        return np.array([[-self.Rl * v[0][0] - ((self.muC - self.Rl) / self.sigmaC) * sum(v[0][1:]) + (self.Rb - self.Rl) * max((sum(v[0][1:]) / self.sigmaC) - v[0][0],0.0)]])

    def g(self, x):
        if(self.dim == 1):
            return np.array([[max(x[0]-100.0, 0.0)]])
        else:
            return np.array([[max(max(x)-120.0,0.0) - 2*max(max(x)-150.0,0.0)]])
        
        
class SemilinearStochasticLotkaVolterra(Equation):
    """
    Semilinear Stochastic Lotka Volterra
    """
    def __init__(self, eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint):
        super(SemilinearStochasticLotkaVolterra, self).__init__(eqn_config, dimension, num_iteration, samplingMethod, num_gridpoint)
        self.x_init = np.ones(self.dim)*0.1
        self.samplingMethod = samplingMethod
        self.num_gridpoint = num_gridpoint
        self.alpha = eqn_config.alpha
        self.beta1 = eqn_config.beta1 /self.dim
        self.beta2 = eqn_config.beta2 / self.dim
        
        if(self.samplingMethod != "Explicit"):
            self.sampleMethod = getattr(spl, self.samplingMethod)(eqn_config, dimension, num_gridpoint)
            self.sampleNeeded = self.sampleMethod.sampleNeeded
            
    def mu(self, x, t):
        #x1 = np.array([x,]*self.dim) - np.diag(x + (1.0/self.beta2)*self.beta1*x)
        #return self.alpha*x+self.beta2*np.dot(x1.T,x)
        return self.alpha*x
    
    def sigma(self, x, t):
        return self.alpha*(np.diag(x[:-1], -1) + np.diag(x, 0) + np.diag(x[1:], 1))
    
    def Dmu(self, x, t, states):
        D = x.reshape((self.dim,self.dim))
        #X = states[t] # shape(d,)
        tmp = np.zeros((self.dim,self.dim))
        for i in range(0, self.dim):
            for j in range(0, self.dim):
                #tmp[i,j] = (i==j)*(self.alpha + np.sum(x)*self.beta2 - x[i]*(self.beta2 - 2*self.beta1)) + (i!=j)*self.beta2*x[i]
                tmp[i,j] = (i==j)*self.alpha
        return np.dot(tmp,D).reshape(-1)
    
    def Dsigma(self, x, t):
        D = x.reshape((self.dim,self.dim))
        tmp = np.ones(self.dim)
        deriv = np.diag(tmp[:-1], -1) + np.diag(tmp, 0) + np.diag(tmp[1:], 1)
        res = np.empty((0,self.dim**2))
        res2 = np.zeros((self.dim**2,self.dim**2))
        for j in range(0,self.dim):
            tmp2 = np.zeros((self.dim,self.dim))
            tmp2[:,j] = deriv[:,j]
            col = np.dot(tmp2,D).reshape(-1)
            res = np.vstack([res,col])
        res2[0:(self.dim**2), 0:self.dim] = res.T
        return res2
        
    def dXsample(self, t_start, t_end, x_0, dW=None):
        if(self.samplingMethod != "Explicit"):
            return self.sampleMethod.dXsampling(self.mu, self.sigma, self.dx_sigma, t_start, t_end, x_0, dW)
    
    def dIsample(self, t_start, t_end, x, dW, Xsample):
        grid = self.sampleMethod.find_grid(float(t_start), float(t_end))
        states = dict(zip(grid, Xsample))
        dW_reshape = self.sampleMethod.reshapeD(grid, self.sampleMethod.sampleNeeded, dW)
        Dstates = self.sampleMethod.dXsampling(lambda x,t: self.Dmu(x,t,states), self.Dsigma, self.dx_sigma, t_start, t_end, np.diag(np.ones(self.dim)).reshape(-1), dW_reshape)[0]
        dI = np.zeros(self.dim)
        sigma_inv = np.linalg.inv(self.sigma(Xsample[0],t_start))
        integrand = np.dot(sigma_inv, Dstates[0].reshape((self.dim,self.dim))).T
        dI = dI + np.dot(integrand, dW[(1,)][0].flatten())
        if(len(grid) != 2):
            for k in range(1,len(Dstates)-2):
                sigma_inv = np.linalg.inv(self.sigma(Xsample[k],grid[k]))
                integrand = np.dot(sigma_inv, Dstates[k].reshape((self.dim,self.dim))).T
                dI = dI + np.dot(integrand, dW[(1,)][1][k-1])
            sigma_inv = np.linalg.inv(self.sigma(Xsample[-2],grid[-2]))
            integrand = np.dot(sigma_inv, Dstates[-2].reshape((self.dim,self.dim))).T
            dI = dI + np.dot(integrand, dW[(1,)][2].flatten())
        return np.concatenate([np.ones(1), np.dot(self.sigma(Xsample[0], t_start).T, dI) / (t_end - t_start)])

    def f(self, t, x, v):
        return np.array([[np.sqrt(np.sum(v[0][1:]**2) + 1)]])

    def g(self, x):
        return np.array([[1 / (2 + 0.4 * np.linalg.norm(x)**2)]])