Optimal Control for Stochastic Delay Systems Under Model Uncertainty: A Stochastic Differential Game Approach

Abstract

In this paper, we study a robust recursive utility maximization problem for time-delayed stochastic differential equation with jumps. This problem can be written as a stochastic delayed differential game. We suggest a maximum principle of this problem and obtain necessary and sufficient condition of optimality. We apply the result to study a problem of consumption choice optimization under model uncertainty.

Appendix A

Lemma A.1

Suppose that δ>0 is a given constant, $\beta, \theta_{0} \in L^{2}_{\mathcal{F}}(-\delta,T+\delta), \ell\in L^{2}_{\mathcal{F}}(0,T), \theta_{1}(t,z)>-1+\varepsilon$ and θ ₁∈H ²(−δ,T+δ). Moreover, suppose that β,θ ₀,θ ₁ are uniformly bounded, and F is such that

$$F\in S^2_{\mathcal{F}}(T,T+\delta) \quad\textit{and}\quad E \Bigl[ \underset{0\leq t\leq T}{\sup}\big|F(t)\big|^2 \Bigr]< \infty. $$

Then the linear anticipated BSDE

$$ \left \{ \begin{array}{rcl} \mathrm {d}Y(t) & = &- \displaystyle \biggl( \ell(t) +\beta(t) Y(t)+\theta_0(t) Z(t) +\int_{\mathbb{R}_0} \theta_1(t,z)K(t,z) \nu(\mathrm {d}z) \biggr)\,\mathrm {d}t\\ & & +Z(t)\,\mathrm {d}B(t) +\displaystyle\int_{\mathbb{R}_0}K(t,z) \widetilde{N}(\mathrm {d}z,\mathrm {d}t);\quad t \in[ 0,T] , \\ Y(t) &=& F(t);\quad t \in[T,T+\delta] ,\\ Z(t)&=&0,\quad t \in[T,T+\delta] ,\\ K(t,z)&=&0,\quad t \in[T,T+\delta] \end{array} \right . $$

(95)

has the unique solution

$$\begin{aligned} Y(t)=E \biggl[ F(T)G(t,T)+\int_t^TG(t,s)l(s) \,\mathrm {d}s \Big| \mathcal {F}_t \biggr], \end{aligned}$$

(96)

where G(t,s) is defined by

$$ \left \{ \begin{array}{rcl} \mathrm {d}G^{\theta} (t,s) & = & G^{\theta} (t,s^-) \biggl(\beta(s)\,\mathrm {d}s + \theta_0(s)\,\mathrm {d}B(s) + \displaystyle\int_{\mathbb{R}_0}\theta _1(s,z)\widetilde{N}(\mathrm {d}z,\mathrm {d}s) \biggr);\\ && \quad s \in[ t,T+\delta ], \\ G^{\theta} (t,t) &=& 1, \\ G^{\theta} (t,s)&=& 0 ,\quad s \in[t-\delta, t[. \end{array} \right . $$

(97)

Proof

The existence and uniqueness results follow by a general theorem for time-advanced BSDEs; see [30].

Equation (97) has a unique solution. In fact, for s∈[t,t+δ], (97) becomes

$$ \left \{ \begin{array}{rcl} \mathrm {d}G^{\theta} (t,s) & = & G^{\theta} (t,s^-) \biggl(\beta(s)\,\mathrm {d}s + \theta_0(s)\,\mathrm {d}B(s) +\displaystyle\int_{\mathbb{R}_0}\theta_1(s,z) \widetilde{N}(\mathrm {d}z,\mathrm {d}s) \biggr);\\ && \quad s \in[ t,t+\delta] , \\ G^{\theta} (t,t) &=& 1. \end{array} \right . $$

(98)

We can then get a unique solution ξ(t,⋅) for (98). When s∈[t+δ,T+δ], (97) can be written has

$$ \left \{ \begin{array}{rcl} \mathrm {d}G^{\theta} (t,s) & = & G^{\theta} (t,s^-) \biggl(\beta(s)\,\mathrm {d}s + \theta_0(s)\,\mathrm {d}B(s) +\displaystyle\int_{\mathbb{R}_0}\theta_1(s,z) \widetilde{N}(\mathrm {d}z,\mathrm {d}s) \biggr);\\ && \quad s \in [t+\delta,T+\delta ], \\ G^{\theta} (t,s) &=& \xi(t,s),\quad s\in[t,t+\delta]. \end{array} \right . $$

(99)

Equation (99) is a classical stochastic delay differential equation (SDDE), and therefore has a unique solution. It only remains to prove that if Y(t) is defined to be the solution of (95), then (96) holds. By Itô formula, we have

$$\begin{aligned} \mathrm {d}\bigl(G(t,s)Y(s)\bigr)&=G\bigl(t,s^-\bigr)\,\mathrm {d}Y(s)+Y(s)\,\mathrm {d}G(t,s)+ \mathrm {d}(GY) (s) \\ &=G\bigl(t,s^-\bigr) \biggl\{ - \biggl(\ell(t) +\beta(t) Y(t)+ \theta_0(t) Z(t) \\&\quad{} +\int_{\mathbb{R}_0} \theta_1(t,z)K(t,z) \nu(\mathrm {d}z) \biggr)\,\mathrm {d}t \\ &\quad{} +Z(t)\,\mathrm {d}B(t) +\int_{\mathbb{R}_0}K(t,z) \widetilde {N}(\mathrm {d}z, \mathrm {d}t) \biggr\} \\&\quad{} + Y(s)G\bigl(t,s^-\bigr) \biggl[\beta(s)\,\mathrm {d}s + \theta_0(s)\,\mathrm {d}B(s) \\ &\quad{} +\int_{\mathbb{R}_0}\theta_1(s,z) \widetilde{N}(\mathrm {d}z,\mathrm {d}s) \biggr] \\&\quad{} +G\bigl(t,s^-\bigr) \biggl[ \theta_0(s)Z(s) \,\mathrm {d}s+\int_{\mathbb{R}_0}\theta _1(s,z)K(t,z) \nu(\mathrm {d}z)\,\mathrm {d}s \biggr]. \end{aligned}$$

Integrating from 0 to T and taking the conditional expectation under $\mathcal{F}_{t}$, we have

$$E \bigl[G(t,T)Y(T) \big| \mathcal{F}_t \bigr] - G(t,t)Y(t)=-E \biggl[\int_t^T G\bigl(t,s^-\bigr)\ell(s)\,\mathrm {d}s \Big| \mathcal{F}_t \biggr] . $$

Since G(t,t)=1, we obtain

$$Y(t)=E \biggl[G(t,T)Y(T) + \int_t^T G \bigl(t,s^-\bigr)\ell(s) \,\mathrm{d}s \Big| \mathcal{F}_t \biggr] . $$

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Remark A.1

Let V be an open subset of a Banach space $\mathcal{X}$ and let $F: V \rightarrow\mathbb{R}$.

• We say that F has a directional derivative (or Gateaux derivative) at x∈V in the direction $y\in\mathcal{X}$ if

  $$D_yF(x):=\underset{\varepsilon\rightarrow0}{\lim} \frac{1}{\varepsilon }\bigl(F(x + \varepsilon y)-F(x)\bigr)\quad \text{exists.} $$

• We say that F is Fréchet differentiable at x∈V if there exists a linear map

  $$L:\mathcal{X} \rightarrow\mathbb{R} $$

  such that

  $$\underset{\underset{h \in\mathcal{X}}{h \rightarrow0}}{\lim} \frac {1}{\|h\|}\big|F(x+h)-F(x)-L(h)\big|=0. $$

  In this case, we call L the Fréchet derivative of F at x, and we write

  $$L=\nabla_x F. $$

• If F is Fréchet differentiable, then F has a directional derivative in all directions $y \in\mathcal{X}$ and

  $$D_yF(x)= \nabla_x F(y). $$