Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

Abstract

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.

Appendices

Appendix A: Proofs of the Maximum Principles for FBSDE Games

We first recall some basic concepts and results from Banach space theory. Let V be an open subset of a Banach space $\mathcal{X}$ with norm ∥⋅∥, and let F:V→ℝ.

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

  $$D_y F(x) := \lim_{\varepsilon\rightarrow0} \frac{1}{\varepsilon} \bigl(F(x + \varepsilon y) - F(x)\bigr) $$

  exists.

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

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

  such that

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

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

  $$L = \nabla_x F. $$

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

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

Proof of Theorem 2.1

(Sufficient maximum principle) We first prove that

$$J_1(u_1,\hat{u}_2) \leq J_1( \hat{u}_1, \hat{u}_2)\quad \text{ for all } u_1 \in \mathcal{A}_1. $$

To this end, fix $u_{1} \in\mathcal{A}_{1}$ and consider

$$ \Delta:= J_1(u_1, \hat{u}_2) - J_1 (\hat{u}_1, \hat{u}_2) = I_1 + I_2 + I_3, $$

(A.1)

where

(A.2)

(A.3)

(A.4)

By (8) we have

(A.5)

By the concavity of φ ₁, (10), and the Itô formula,

(A.6)

By the concavity of ψ ₁, (5), (9), and the concavity of φ _,

(A.7)

Adding (A.5), (A.6), and (A.7), we get

(A.8)

Since $\hat{\mathcal{H}}_{1}(x,y,z,k)$ is concave, it follows by a standard separating hyperplane argument (see, e.g., [20], Chap. 5, Sect. 23) that there exists a supergradient $a=(a_{0}, a_{1}, a_{2}, a_{3}(\cdot)) \in \mathbb{R}^{3} \times\mathcal{R}$ for $\hat{\mathcal{H}}_{1}(x,y,z,k)$ at $x = \hat {X}(t),\ y = \hat{Y}_{1}(t),\ z = \hat{Z}_{1}(t^{-})$, and $k = \hat{K}_{1}(t^{-}, \cdot)$ such that if we define

then

$$\varphi_1(x,y,z,k) \leq0\quad \text{ for all } x,y,z,k. $$

On the other hand, we clearly have

$$\varphi_1\bigl(\hat{X}(t), \hat{Y}(t), \hat{Z}(t), \hat{K}_1(t,\cdot)\bigr) = 0. $$

It follows that

Combining this with (A.8), we get

Hence,

$$J_1(u_1, \hat{u}_2) \leq J_1( \hat{u}_1,\hat{u}_2) \quad\text{ for all } u_1 \in\mathcal{A}_1. $$

The inequality

$$J_2(\hat{u}_1,u_2) \leq J_2( \hat{u}_1, \hat{u}_2)\quad \text{ for all } u_2 \in \mathcal{A}_2 $$

is proved similarly. This completes the proof of Theorem 2.1. □

Proof of Theorem 2.2

(Necessary maximum principle) Consider

(A.9)

By (10), (13), and the Itô formula,

(A.10)

By (9), (13), and the Itô formula,

(A.11)

Adding (A.10) and (A.11), we get, by (A.9),

(A.12)

If D ₁=0 for all bounded $\beta_{1} \in\mathcal{A}_{1}$, then this holds in particular for β ₁ of the form in (a1), i.e.,

$$\beta_1(t) = \chi_{(t_0,T]}(t) \alpha_1(\omega), $$

where α ₁(ω) is bounded and $\mathcal{E}^{(1)}_{t_{0}}$-measurable. Hence,

$$E \biggl[ \int_{t_0}^T E \biggl[ \frac{\partial H_1}{\partial u_1}(t) \mid\mathcal{E}^{(1)}_t \biggr] \alpha_1 \,dt \biggr] = 0. $$

Differentiating with respect to t ₀, we get

$$E \biggl[ \frac{\partial H_1}{\partial u_1} (t_0) \alpha_1 \biggr] = 0 \quad\text{ for a.a. } t_0. $$

Since this holds for all bounded $\mathcal{E}^{(1)}_{t_{0}}$-measurable random variables α ₁, we conclude that

$$E \biggl[ \frac{\partial H_1}{\partial u_1}(t) \mid\mathcal{E}^{(1)}_t \biggr] = 0 \quad\text{ for a.a. } t \in[0,T]. $$

A similar argument gives that

$$E \biggl[ \frac{\partial H_2}{\partial u_2}(t) \mid\mathcal{E}^{(2)}_t \biggr] = 0, $$

provided that

$$D_2 := \frac{d}{ds} J_2(u_1,u_2 + s \beta_2) \mid_{s=0} = 0\quad \text{ for all bounded } \beta_2 \in\mathcal{A}_2. $$

This shows that (i) ⇒ (ii). The argument above can be reversed, to give that (ii) ⇒ (i). We omit the details. □

Appendix B: Linear BSDEs with Jumps

Lemma B.1

(Linear BSDEs with jumps)

Let Λ be an $\mathcal{F}_{T}$-measurable and square-integrable random variable. Let β and ξ ₀ be bounded predictable processes, and ξ ₁ a predictable process such that ξ ₁(t,ζ)≥C ₁ with C ₁>−1 and |ξ ₁(t,ζ)|≤C ₂(1∧|ζ|) for a constant C ₂≥0. Let φ be a predictable process such that $E[\int_{0}^{T} \varphi^{2}(t) \,dt] < \infty$. Then the linear BSDE

(B.1)

has the unique solution

$$ Y(t) = E\biggl[\varLambda\varUpsilon(t, T) + \int _t^T \varUpsilon(t,s) \varphi(s) \, ds \mid \mathcal{ F}_t\biggr], \quad0 \leq t \leq T, $$

(B.2)

where ϒ(t,s), 0≤t≤s≤T, is defined by

(B.3)

i.e.,

(B.4)

Hence,

$$\varUpsilon(t,s) = \frac{\varUpsilon(0,s)}{\varUpsilon(0,t)}. $$

Proof

For completeness, we give the proof, also given in [18]. The existence and uniqueness follow by general theorems for BSDEs with Lipschitz coefficients. See, e.g., [17]. Hence, it only remains to prove that if we define Y(t) to be the solution of (B.1), then (B.2) holds. To this end, define

$$\varUpsilon(s) = \varUpsilon(0,s). $$

Then by the Itô formula (see, e.g., [1], Chap. 1),

Hence, $\varUpsilon(t) Y(t) + \int_{0}^{t} \varUpsilon(s)\varphi(s) \, ds$ is a martingale, and therefore

$$\varUpsilon(t) Y(t) + \int_0^t \varUpsilon(s) \varphi(s) \, ds = E\biggl[ \varLambda\varUpsilon(T) + \int_0^T \varUpsilon(s)\varphi(s) \, ds \mid \mathcal{F}_t\biggr] $$

or

$$Y(t) = E\biggl[ \varLambda\frac{\varUpsilon(T)}{\varUpsilon(t)} + \int_t^T \frac {\varUpsilon (s)}{\varUpsilon(t)} \varphi(s) \, ds \mid\mathcal{F}_t\biggr], $$

as claimed. □