Stochastic Differential Games in Insider Markets via Malliavin Calculus

Abstract

In this paper, we use techniques of Malliavin calculus and forward integration to present a general stochastic maximum principle for anticipating stochastic differential equations driven by a Lévy type of noise. We apply our result to study a general stochastic differential game problem of an insider.

Appendix: Proof of Theorem 4.1

Proof

The proof relies on a combination of arguments of Refs. [1] and [2].

• (i) Suppose $(\widehat{\pi},\widehat{\theta})\in\mathcal{A}_{\varPi}\times \mathcal{A}_{\varTheta}$ is a Nash-equilibrium. Since 1 and 2 hold for all π and $\theta, (\widehat{\pi}, \widehat{\theta})$ is a directional critical point for J _i (π,θ) for i=1,2 in the sense that, for all bounded $\beta\in\mathcal{A}_{\varPi}$ and $\eta\in \mathcal{A}_{\varTheta}$, there exists δ>0 such that $\widehat{\pi }+y\beta\in\mathcal{A}_{\varPi}, \widehat{\theta}+v\eta\in\mathcal {A}_{\varTheta}$ for all y,v∈]−δ,δ[. Then we have

  

  (A.1)

  where

  

  (A.2)

  We study the three summands separately. Using the short notation

  $$\frac{\partial}{\partial x }f_1\bigl(t,\widehat{X}(t),\widehat{\pi },\widehat{\theta},z\bigr)=\frac{\partial}{\partial x }f_1(t,z),\quad \nabla_\pi f_1 \bigl(t,X^{(\pi,\widehat{\theta})}(t),\pi,\widehat{\theta },z\bigr)\big \vert _{\widehat{\pi}=\pi}$$

  and similarly for $\frac{\partial}{\partial x }b, \nabla_{\pi}b, \frac{\partial}{\partial x }\sigma, \nabla_{\pi}\sigma, \frac {\partial}{\partial x }\gamma$ and ∇ _π γ. By the duality formulas (20) and (23) and the Fubini theorem, we get

  

  Changing notation z ₁→z, this becomes

  

  (A.3)

  Here, we used the multidimensional product rule for Malliavin derivatives. Similarly, by using both Fubini and duality formulas (20) and (23), we get

  

  Changing notation t ₁→t and z ₁→z, this becomes

  

  (A.4)

  Recall that

  

  so

  

  (A.5)

  By combining (A.3)–(A.4), we get

  

  (A.6)

  Now, apply this to $\beta=\beta_{\alpha}\in\mathcal{A}_{\varPi}$ given as β _α (s):=αχ _[t,t+h](s), for some t,h∈]0,T[, t+h≤T, where, α=α(ω) is bounded and $\mathcal{G}^{2}_{t}$-measurable. Then $Y^{(\beta_{\alpha})}(s)=0$ for 0≤s≤t and (A.6) becomes

  

  (A.7)

  where

  

  Note that, by the definition of Y, with $Y(s)=Y^{(\beta_{\alpha })}(s)$ and s≥t+h, the process Y(s) follows the dynamics

  

  (A.8)

  for s≥t+h with initial condition Y(t+h) in time t+h. By the Itô formula for forward integrals, this equation can be solved explicitly, and we get

  

  (A.9)

  where, in general, for s≥t,

  

  Note that G(t,s) does not depend on h, but Y(s) does. Defining $H^{1}_{0}$ as in (30), it follows that

  

  where $\widehat{H}^{1}_{0}(s)=H^{1}_{0}(s,\widehat{X}(s),\widehat{\pi},\widehat {\theta} )$. Differentiating with respect to h at h=0, we get

  

  Since Y(t)=0, we see that

  $$\frac{ \,\mathrm {d}}{ \,\mathrm {d}h }E^x \biggl[\int_t^{t+h} \frac{\partial H_0}{\partial x }(s)Y(s)\,\mathrm {d}s \biggr]_{h=0}=0. $$

  Therefore, by (A.9),

  

  where Y(t+h) is given by

  

  Therefore, by the two preceding equalities,

  

  where

  

  and

  

  Applying once more the duality formula, we have

  

  where we have

  $$F_1(t,s)=\frac{\partial\widehat{H}^1_0}{\partial x }(s)G(t,s). $$

  Since Y(t)=0, we see that A _1,2=0. We conclude that

  

  (A.10)

  Moreover,

  

  (A.11)
  

  (A.12)
  

  (A.13)

  On the other hand, differentiating A ₃ with respect to h at h=0, we get

  

  Since Y(t)=0, we get

  

  Using the definition of $\widehat{p}$ and $\widehat{H}_{1}$ given respectively by (39) and (38) in the theorem, it follows from (A.7) that

  $$ E \bigl[ \nabla_\pi\widehat{H}_1\bigl(t, \widehat{X }(t),\widehat {u}(t)\bigr)\vert \mathcal{G}^2_{t} \bigr] + E[A]=0\quad\text{a.e. in\ }(t,\omega), $$

  (A.14)

  where

  

  (A.15)

  Similarly, we have

  

  (A.16)

  where

  

  (A.17)

  Define

  $$D(s)=D(t+h)G(t+h,s);\quad s \geq t+h, $$

  where G(t,s) is defined as in (42). Using similar arguments, we get

  $$E \bigl[ \nabla_\pi\widehat{H}_2\bigl(t, \widehat{X }(t),\widehat {u}(t)\bigr)\vert \mathcal{G}^1_{t} \bigr] + E[B]=0\quad\text{a.e. in \ }(t,\omega), $$

  where B is given in the same way as A. This completes the proof of (i).

• (ii) Conversely, suppose that there exists $\widehat{\pi}\in\mathcal{A}_{\varPi}$ such that (36) holds. Then, by reversing the previous arguments, we obtain that (A.7) holds for all $\beta_{\alpha}(s):=\alpha\chi_{[ t,t+h]}(s) \in\mathcal {A}_{\varPi}$, where

  

  for some t,h∈]0,T[,t+h≤T, where α=α(ω) is bounded and $\mathcal{G}^{2}_{t}$-measurable. Hence, these equalities hold for all linear combinations of β _α . Since all bounded $\beta\in\mathcal{A}_{\varPi}$ can be approximated pointwise boundedly in (t,ω) by such linear combinations, it follows that (A.7) holds for all bounded $\beta\in\mathcal{A}_{\varPi}$. Hence, by reversing the remaining part of the previous proof, we conclude that

  $$\frac{\partial J_1}{\partial y}(\widehat{\pi}+y\beta, \widehat {\theta})\vert _{y=0}=0,\quad\text{for all } \beta. $$

  Similarly, suppose that there exists $\widehat{\theta}\in\mathcal {A}_{\varTheta}$ such that 37 holds. Then, the above argument leads us to conclude that

  $$\frac{\partial J_2}{\partial v}(\widehat{\pi}, \widehat{\theta }+v\eta)\vert _{v=0}=0,\quad\text{for all } \eta. $$

  On the other hand, assume moreover that $\pi\rightarrow J_{1}(\pi, \widehat{\theta})$, then

  

  Taking the limit for y→0 and using the fact that ${\lim}_{y\rightarrow0}\frac{1}{y} (J_{1}(\frac{\pi}{1-y},\widehat {\theta})-J_{1}(\pi, \widehat{\theta}) )=0$, we obtain that $0\geq J_{1}(\beta, \widehat{\theta})-J_{1}(\widehat{\pi},\widehat{\theta})$. Since β can be chosen within the set $\mathcal{A}_{\pi}$, we obtain by formally setting β=π that

  

  (A.18)

  Analogously, we obtain

  

  (A.19)

  This means that $(\widehat{\pi},\widehat{\theta})$ is a Nash-equilibrium for the market. is concave in each π or θ.

  This completes the proof.

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