Module merton_jump 

Merton’s log-utility portfolio problem with a jump in the risky asset.

A validating instance for the neural jump BSDE: a diffusion-plus-jump wealth process whose optimal control and value still have closed forms, but where the jump term genuinely changes the optimal policy (unlike the no-jump Merton, where the policy is constant in the jump parameters).

Dynamics

The risky asset follows a geometric jump-diffusion and the wealth x is invested with fraction u in the risky asset and the rest at the risk-free rate r:

dS/S  = mu dt + sigma dW + (y - 1) dN
dx    = x [ r + u (mu - r) ] dt + x u sigma dW + x u (y - 1) dN

where N is a Poisson process with intensity lambda and y > 0 is a deterministic multiplicative jump, so a jump scales wealth by 1 + u (y - 1).

Objective

Maximize expected log utility of terminal wealth, J(u) = E[ ln x_T ].

HJB equation

0 = d_t V + sup_u { x (r + u (mu - r)) V_x + 0.5 (x u sigma)^2 V_xx
     + lambda [ V(x (1 + u (y - 1))) - V(x) ] }

Exact solution

The value function is V(t, x) = ln x + B (T - t) with

B = r + u* (mu - r) - 0.5 (u* sigma)^2 + lambda ln(1 + u* (y - 1)),

and the optimal fraction u* solves the quadratic

0 = (mu - r) - sigma^2 u* + lambda (y - 1) / (1 + u* (y - 1)).

Setting lambda = 0 or y = 1 reduces this to the no-jump Merton policy u* = (mu - r) / sigma^2.

Structs

MertonJump

Merton log-utility portfolio with a deterministic multiplicative jump.