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Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems

This paper develops a new methodology for studying continuous-time Nash equilibrium in a financial market with asymmetrically informed agents. This approach allows us to lift the restriction of risk neutrality imposed on market makers by the current literature. It turns out that, when the market makers are risk averse, the optimal strategies of the agents are solutions of a forward-backward system of partial and stochastic differential equations. In particular, the price set by the market makers solves a nonstandard"quadratic"backward stochastic differential equation. The main result of the paper is the existence of a Markovian solution to this forward-backward system on an arbitrary time interval, which is obtained via a fixed-point argument on the space of absolutely continuous distribution functions. Moreover, the equilibrium obtained in this paper is able to explain several stylized facts which are not captured by the current asymmetric information models.


Introduction
In this paper we address a long-standing open problem of existence of an equilibrium in a financial market with asymmetrically informed traders and risk averse market makers in continuous-time with finite horizon. In this market the price is an equilibrium outcome of a game between the market makers and an informed trader who possesses superior information. Both market makers and informed trader choose their controls adapted to their filtrations. As we assume that the market makers obtain their information via interaction with the traders and have the obligation to absorb the total demand for the asset, their filtration is the one generated by the total demand process, Y . Informed trader, on the other hand, has the filtration jointly generated by the market prices and her private information. In this game the market makers' control is the price, S, while the control of the informed trader is her trading strategy, X. Thus, the equilibrium price satisfies the following conditions: i) the informed trader's problem has a solution, and ii) given this solution, S fulfils the market makers' objectives.
The study of this game goes back to [30], which is the canonical model in Market Microstructure Theory for the analysis of strategic trading in the presence of private information (see [3], [11], and [37] for a review of Kyle's model as well as a discussion of its relationship with other market microstructure models). Various extensions of this model exist in the literature: see, among others, [2], [4], [5], [7], [14], [16], [17], [18], and [22].
The original model and all these extensions assume that the market makers are risk-neutral and compete in a Bertrand fashion for the total demand. This means in equilibrium the utility of any market maker is a martingale. Since the utility is linear this in turn implies that the optimal strategy for the market makers is to set the price to be the conditional expectation of the fundamental value of the asset given their filtration. In particular, in these models there is always a unique price satisfying the objective of the market makers for any control of the informed trader. Furthermore, the martingale property of the price results in optimal strategy of the informed trader being inconspicuous in the equilibrium; that is, the law of Y in its own filtration is the same as that of Y − X in its own filtration.
Whereas the risk-neutrality of the market makers makes the model tractable, it is not consistent with the empirical evidence which shows that the market makers are risk averse and exercise their control so that the total demand mean reverts around a target level at a speed determined by their risk aversion (see [25] and [35] for New York Stock Exchange, [23] for London Stock Exchange, [12] for Foreign Exchange; for a survey of related literature and results, see Sections 1.2 and 1.3 in [9]).
Although relaxing the assumption of market makers' risk neutrality is natural and prompted by the empirical evidence, there has been only one attempt in the literature to investigate the effect of such an extension. Subrahmanyam in [40] considered a one-period model where the market makers with identical exponential utilities set the price that makes their utilities martingales. This assumption is the direct analogue of the original Kyle model discussed earlier in the context of risk averse market makers. The tractability of the model considered in [40] lies in the fact that in a one-period setting there exists an optimal response for the market makers for any strategy of the insider. However, in a multi-period setting the existence of such responses is uncertain which led Subrahmanyam to the conclusion that an extension of his model to a multi-period setting is not possible due to the strategic behaviour of the agents.
The same problem persists in the continuous time model that we consider here. In particular, given a trading strategy of the informed trader, the optimal response of the market makers is found via solving the following backward stochastic differential equation (BSDE): where c > 0 is a constant, V is a bounded random variable representing the fundamental value of the asset, Y is a given total demand process, and β is a Brownian motion with respect to F Y -the filtration of the market makers generated by Y . The BSDE above is said to admit a solution if there exits a pair (Z, S) of F Y -adapted processes satisfying (1.1) and (1.2). When this BSDE admits a solution, P would be the price that makes the utilities of the market makers martingales.
Although the terminal condition is unconventional, as Y and V are given, the right hand side of (1.2) is a fixed F Y 1 -measurable random variable. Thus, we can rewrite the terminal condition as P 1 = ξ and, due to the boundedness of V , ξ would be bounded as well. On the other hand, the form of the driver poses a real difficulty since the process Y multiplying Z 2 is in general unbounded. This renders the system (1.1)-(1.2) outside the realm of quadratic BSDEs.
Price response of the market makers is only one side of the equilibrium. To characterise an equilibrium we also need to find the equilibrium level of Y which results from the informed agent's optimal trading strategy. Consistent with the literature the total demand is driven by a Brownian motion and has a drift which is determined by the informed trader. Hence, an equilibrium consists of (α, S) where α is the optimal drift given S and S satisfies the following forward-backward stochastic differential equation (FBSDE): whereα is the F Y -optional projection of α. It is well-known that the existence of a solution for FBSDEs is quite delicate even if the driver is globally Lipschitz and satisfies a linear growth condition. [1] showed the existence and uniqueness of a solution over a small time interval via a fixed-point algorithm on a Banach space of processes. This result has been extended by [21] and [19] to arbitrary time intervals by pasting solutions obtained for small time intervals. An alternative technique for solving FBSDEs is the so-called four-step scheme introduced by [33] which requires strong smoothness on the coefficients of the system and is based on the link between quasi-linear partial differential equations. When the driver is quadratic, the problem becomes more complicated and only few results are available. Moreover, since available results originate from the solvability of quadratic BSDEs, the standard assumption in the current literature is that the driver is bounded by k(1 + z 2 ) for some constant k (see, e.g., [26]). However, as (1.4) does not fit into the current paradigm of quadratic BSDEs, these results are not applicable to our setting. Despite these difficulties one can obtain a solution to this system when the market makers set the price as a smooth function of total demand, i.e. S t = H(t, Y t ), and α is the optimal drift of the informed trader given S. This solution provides a Markovian equilibrium for the model that we consider. As we show later, in this case the system (1.3)-(1.5) transforms into provided Y has a smooth transition density, where the last equality is an equality in distribution. This is still a forward-backward system of a forward SDE and a backward PDE such that the terminal condition of the PDE depends on the solution of the SDE, which in turn depends on H. This coupling between the SDE and the PDE makes it natural to employ a fixed-point algorithm.
Indeed, if we are given a continuous distribution for Y 1 , (1.8) yields a function H(1, y), which is increasing in y. This allows us to obtain H(t, y) via (1.6), and Y via (1.7). Hence, this procedure defines a mapping from the space of distributions into itself. We show in Theorem 4.1, via Schauder's fixed point theorem, that this mapping has a fixed point under the assumption that V = f (η) for some increasing and bounded f satisfying some mild regularity conditions, and a standard normal random variable η.
The validity of Schauder's fixed point theorem in our setting relies heavily on the properties of solutions of (1.7) for any given function H satisfying (1.6) with a bounded and increasing terminal condition. These properties are explored in Lemmata 4.1 -4.3. In particular, we obtain a remarkable connection between the laws of Y 1 and that of Brownian motion as follows: for all x > 0, where C is a constant that depends only on the bound on H. We also show that Y has a smooth transition density.
The existence of solution to the system (1.6)-(1.8) ensures the existence of a Markovian solution for the price process which makes the utilities of market makers martingales once the drift of total demand, Y has the form given in equation (1.7). However, in order for such a drift to appear in equilibrium, it should be optimal for the insider to choose a drift whose F Y -optional projection has this form.
To this end, we establish in Proposition 3.1 that the sole criteria of optimality for the insider is that the strategy fulfils the bridge condition H(1, Y 1 ) = V . Thus, if Markovian equilibrium exists, the equilibrium pair (H, Y ) solves the system (1.6)-(1.8) and satisfies H(1, Y 1 ) = V . The existence of such a pair is precisely the result of the Theorem 5.1 which allows us to establish the existence of the equilibrium in Theorem 5.2.
The rest of the paper is structured as follows: Section 2 describes the model we consider, Section 3 is devoted to the (formal) derivation of the system (1.6)-(1.8) and characterisation of the optimal strategy of the informed trader, Section 4 establishes the existence of solution to the system (1.6)-(1.8), Section 5 proves the existence of the equilibrium, and Section 6 concludes.

Market structure
Let (Ω, F , (F t ) t∈[0,1] , P) be a filtered probability space satisfying the usual conditions of right continuity and P-completeness. We suppose that F 0 is not trivial and there exists an F 0 -measurable standard normal random variable, η. Moreover, the filtered probability space also supports a standard Brownian motion, B, with B 0 = 0 and, thus, B is independent of η. Let's define V := f (η) for some bounded and strictly increasing function f with a continuous derivative. As all the randomness in our model will depend only on V and B, we shall take F = σ(Ñ ,F), whereF is the minimal σ-field with respect to which V and (B t ) t∈[0,1] are measurable andÑ = {E : E ⊂ F for some F ∈F with P(F ) = 0}. Moreover, in view of the independence of V and B, we may assume the existence of a family of probability measures, (P v ) such that the disintegration formula P(E) = f (R) P v (E)P(V ∈ dv) holds for all E ∈ F , and for all v ∈ f (R) the measure P v satisfies P v (E) = P(E|V = v). The existence of such a family is easily justified when we consider We consider a market in which the risk free interest rate is set to 0 and a single risky asset is traded. The fundamental value of this asset equals V , which will be announced at time t = 1.
There are three types of agents that interact in this market: i) Liquidity traders who trade for reasons exogenous to the model and whose cumulative demand at time t is given by σB t for some constant σ > 0.
ii) A single informed trader, who knows V from time t = 0 onwards, and is risk neutral. We will call the informed trader insider in what follows and denote her cumulative demand at time t by X t . The filtration of the insider, F I , is generated by observing the price of the risky asset and V . Thus, an insider who has the information that V = v possesses the minimal right continuous filtration generated by V and the price process, and completed with the null sets of P v .
iii) Market makers observe only the net demand of the risky asset, Y = X + σB, thus, their filtration, F M , is the minimal right-continuous filtration generated by Y and completed with P-null sets. The number of market makers is assumed to be N ≥ 2.
We also assume that the market makers have identical preferences described by the common utility function, U(x) = −e −ρx , and compete in a Bertrand fashion for the net demand of the risky asset. In case of several market makers quoting the same winning price, we adopt the convention that the total order is equally split among them.
Similar to [2] we assume that the market makers set the price of risky security, S, as S t = H(t, Y t ) for some function H.
To understand the subtlety of the equilibrium derived later, it is important to observe that an insider who is given the information that V = v has the probability measure P v on (Ω, F ) while the probability measure of the market makers is given by P, and these measures are singular with respect to each other as P v (V = v) = 1, whereas P(V = v) = 0 in our settings. As we described the agents in this market, we are now in the position to define admissibility of functions H for the market makers (which will be called pricing rule in what follows) and admissibility of the trading strategy of the insider. The conditions we impose are standard in the literature and were first introduced in [2]. The integrability conditions (2.9) and (2.10) prevent the insider from following doubling strategies (see [2] for the discussion). The absolute continuity of insider's strategies is without loss of generality since strategies with a martingale component and/or jumps are strictly suboptimal as shown in [2]. Definition 2.1 A function H : R + × R → R is a pricing rule if H ∈ C 1,2 , strictly increasing in y, and satisfies The class of such functions is denoted with H.
Note that since any pricing rule is strictly monotone, it is easy to see that B is adapted to F I . The admissible strategies for the insider is defined in the following.
Definition 2.2 An insider strategy, X, is admissible for a given pricing rule, H, if X t = t 0 α s ds for some F I -adapted α, and satisfies The class of admissible strategies for a given pricing rule H will be denoted by A(H).
Observe that if X ∈ A(H), the terminal wealth of the insider is given by The first term in the terminal wealth corresponds to continuous trading in the risky asset, while the second term exists due to a potential discontinuity in the asset price when the value becomes public knowledge at time t = 1. The second expression for the wealth follows from integration by parts. Given the definition of a pricing rule and admissible trading strategies, we can now define an equilibrium as follows: is an equilibrium if H * ∈ H, X * ∈ A(H * ), and i) given H * , the insider's strategy X * solves her optimization problem: ii) Given X * , the pricing rule H * is such that the market makers' wealth satisfies zero-utility gain condition , i.e. U (G) is a (F M , P)-martingale, where The above is the formulation of a Markovian Nash equilibrium in our model. The condition for the optimality of insider's strategy is a straightforward description of the best response of the insider for a given pricing rule. The market makers' optimality condition follows the tradition of Kyle models where the market makers' utility remain a martingale due to the Bertrand competition among them. Indeed, suppose that one of the market maker's, say MM i , decides to deviate at some time t from this pricing rule by, e.g., selling at a higher price than H would suggest in order to achieve a positive utility gain. However, the other market makers could then offer to sell at a slightly lower price which would still allow them to make a positive utility gain. Moreover, as this lower price is more favourable to the traders, no one will trade with MM i eliminating any opportunities for a utility gain. Deviation from the zero-utility gain condition by buying at a lower price is also suboptimal for a similar reason. Clearly, buying (resp. selling) at a higher (resp. lower) price is suboptimal since it leads to a loss in the utility. Thus, a pricing rule satisfying the zero-utility gain condition is the best response of the market makers. The zero-utility gain condition is also a direct continuous-time analogue of the concept of autarky utility defining the equilibrium in the one-period Kyle model of [40] studying the effects of the risk aversion of market makers on equilibrium. Recall that the market makers are identical by assumption and, therefore, they offer the same price quotes in equilibrium and the order is split equally among them due to our order splitting convention when there are more than one winning quote.

Characterisation of equilibrium
In this section we show that a Markovian equilibrium of this game is described by a forward-backward system of stochastic and partial differential equations given by (1.6)-(1.8) by first studying the optimal response of the market makers for a given strategy of the insider, and then characterising the profit maximising strategies for the insider.
Suppose that X is an admissible trading strategy of the insider so that Y in its own filtration satisfies The best response of the market makers is to choose a price, S, that will satisfy the zero-utility gain condition. Let price S follow for some predictable process Z and an optional process µ that are to be determined by the market makers. As there is a potential discrepancy between S 1 and V , there is a possibility of a jump in the market makers' wealth at time 1. More precisely, However, The zero-utility gain condition implies On the other hand, if we compute the dynamics of U(G) for t < 1 by Ito's formula, we obtain dU Reiterating the zero-utility gain condition for t < 1 shows that we must have Therefore, the zero-utility gain condition stipulates that the price S follows 14) and the market makers' problem is to find (Z, S) to solve (3.14) with the terminal condition (3.13) given the total demand process Y . The BSDE in (3.14) is reminiscent of the quadratic BSDEs, which have been studied extensively, and the connection of which to problems arising in mathematical finance is well-established (see, e.g., [29], [13], [6], and the references therein). The essential deviation of (3.14) from the BSDEs considered in these papers is that the coefficient of Z 2 t in (3.14) is ρ 2N Y t , which is in general unbounded. This makes the direct application of the results contained in the current literature for quadratic BSDEs to (3.14) impossible.
However, if we turn to a Markovian equilibrium, i.e. consider S t = H(t, Y t ), it is natural to expect that in equilibriumα t =α t (t, Y t , S t , Z t ) for some deterministic functionα so that Thus, if a Markovian equilibrium can be attained it will provide a Markovian solution to the FBSDE defined by (3.13)-(3.15), whereα is the optimal drift chosen by the insider.
We now turn to the optimisation problem for the insider when S t = H(t, Y t ) for an admissible pricing rule H. Observe that from the point of view of the insider the total demand process follows for a given insider's strategy X t = t 0 α s ds. And the value function, Ψ, can be defined as Then, a formal application of the dynamic programming principle leads to the HJB equation Since the term to be maximised is linear in α, the only way to ensure the finiteness of solution is to set which yields Ψ t + σ 2 2 Ψ yy = 0. Then, by straightforward calculations we see that H must satisfy a backward heat equation and therefore Ito's formula will yield that S should satisfy Combining this with (3.14) and (3.15) implies as soon as we note that z = σH y (t, y) by the choice of S. The above form ofα is necessary in order for the market makers to quote a Markovian pricing rule. However, in order for suchα to appear in equilibrium, it should be optimal for the insider to choose a drift whose F M -optional projection has this form. In Proposition 3.1 we will show that the sole criteria of optimality for the insider is that the strategy fulfils the bridge condition H(1, Y 1 ) = V . Thus, if a Markovian equilibrium exists, and H solves the backward heat equation above and satisfies H(1, Y 1 ) = V . As we show in Section 4 and 3 a pair (H, Y ) satisfying the above conditions exists for some admissible insider trading strategy and that it indeed constitutes an equilibrium. In order to see that this equilibrium is indeed feasible, suppose that we have a pair (H, Y ) which solves the following system of equations: with Y 0 = 0 where β is a Brownian motion on some given probability space and Y is understood to be a strong solution of the forward SDE. Further assume that the transition probability of Y possesses a smooth density, p. Then, the theory of filtration enlargements give us that Y solves the SDE whereβ is a Brownian motion with respect to the natural filtration of Y initially enlarged with the random variable Y 1 and, in particular, independent of Y 1 . Thus, ifṼ is a random variable with the same distribution as V and independent ofβ, we can replace Y 1 with H −1 (1,Ṽ ) in (3.21) and obtain the SDE Now, suppose that the solutions of this SDE are unique in law. Then,Ỹ will have the same law as Y , which yields in particular thatỸ 1 = H −1 (1,Ṽ ) and in its own filtrationỸ follows for some Brownian motion BỸ . The above discussion makes it clear what the optimal strategy of the insider should be given H. Since V is independent of B, the optimal number of shares of the risky asset held by the insider at time t equals This ensures that Y follows (3.17) in its own filtration and H(1, Y 1 ) = V achieving the optimality conditions for the insider as well as those for the market makers. These considerations imply that the question of existence of the equilibrium can be reduced to the problem of existence of a solution to the system (3.18)-(3.20) with process Y admitting a smooth transition density. Despite the apparent simplicity, the existence of a solution to this system is far from being a trivial matter. Indeed, in order to determine H via the basic PDE in (3.18), we first need to know its boundary condition. However, the boundary condition for H, (3.20), requires the knowledge of the distribution of Y 1 which can only be determined if we know H. Thus, this problem is appropriate for the employment of a fixed point theorem which indeed yields the existence of the solution as demonstrated in the next section.
We end this section by proving the optimality criteria for the insider that we used in order to establish the above system.

Proposition 3.1 Suppose H is a pricing rule satisfying
X is an optimal strategy for the insider.
Proof. We adapt the arguments in [2] and [41] to our case. Consider the function Therefore from (3.25) and Ito's formula it follows that where the last equality is due to (2.10). The conclusion follows from the fact that Ψ(1, H(1, y) is strictly increasing. Therefore, an insider trading strategy which gives H(1, Y 1 ) = V is optimal.

The main result and its proof
In this section we state and prove the main result of this paper that establishes the existence of a solution to the system given by   is a pair (H, Y ), which solves the system of equations (3.18)- (3.20). Moreover, 0 < H y (t, y) ≤ C 1 √ 1−t for all (t, y) ∈ [0, 1) × R and for some constant C. Furthermore, Y is the unique strong solution of (3.19) and for any bounded and continuous g, E[g(Y t )|Y s = y] = R g(y)p(s, y; t, z)dz for all 0 ≤ s ≤ t ≤ 1 for some p such that, for any fixed (t, z), p(s, y; t, z) > 0 on (0, t) × R and is C 1,2 (0, t) × R).
The theorem will be proved by an application of Schauder's fixed point theorem.
Observe that if we start with an absolutely continuous probability measure on R with a full support, (3.20) yields an increasing function H(1, ·), which defines an H solving (3.18). If we then plug this function into the SDE of (3.19), we arrive at a new probability measure on R associated with the distribution of Y 1 . This procedure defines a transformation from the space of probability measures on R into itself. Application of Schauder's fixed point theorem requires a suitable choice of a closed and convex subset, D, of probability measures on R such that the above transformation maps D into itself and satisfies the conditions of Schauder's fixed point theorem.
Before we present the proof of the fixed point result, we collect some useful facts on the behaviour of the solutions of (3.19) in the following lemmata. The first lemma observes a striking relationship between the time 1 laws of the solutions of (3.19) and that of B σ 2 . An immediate consequence of this lemma is that the law of Y 1 , where Y is the solution of (3.19) for a given H, has a full support on R. This property allows us to compute the law of Y 1 via a Girsanov transform using the law of B 1 , which is achieved in the second lemma.
and some constant C. Let c ≥ 0 be a constant, then the stochastic differential equation has a unique strong solution on [0, 1]. Moreover, for any x > 0 and, in particular, P(Y 1 ≤ y) ∈ (0, 1) for all y ∈ R.
Proof. Since yH y (t, y) is locally Lipschitz on [0, T ] × R for any T < 1, the above equation has a unique strong solution on [0, T ] upto an explosion time τ . Since T is arbitrary this implies the existence of a unique continuous strong solution on [0, 1∧τ ). Moreover, Thus, by Ito's isometry and the elementary inequality Therefore, Gronwall inequality yields E[Y 2 t ] ≤ (1 + σ 2 )e 4σ 2 for all t ∈ [0, 1]. In particular, Y never explodes and the SDE has a non-exploding strong solution.
To obtain the estimates (4.30) and (4.31) let Thus,Ỹ t = W Tt for some Brownian motion W and the time change T t satisfying Thus, by the Optional Sampling Theorem, for any K ∈ R we have which implies (4.30) and (4.31).

Lemma 4.2
Let h be a bounded, nondecreasing, and absolutely continuous function, which is not constant. Consider the solution, H, of (3.18) with the terminal condition h. Then, |H(t, ·)| ≤ h ∞ for t ≤ 1, and 0 < H y (t, ·) ≤ C 1 √ 1−t for t < 1, where C = 2 σ 2 π h ∞ . Consequently, there exists a unique, strong solution, Y , of (4.29) and, for any bounded and continuous function g and T ≤ 1, we have where W is a Brownian motion on a filtered probability space (Ω,F, (F t ) t∈[0,1] , Q) and (M t ) t∈[0,1] is a strictly positive ((F t ), Q)-martingale given by where τ is any stopping time with respect to the natural filtration of W such that τ ≤ 1, Q-a.s., W * t = sup s≤t |W s |, and K is some constant that depends only on σ and h ∞ .

Proof. Observe that
where q(t, x) is the probability density of a normal random variable with mean 0 and variance t. Then, clearly, |H(t, y)| ≤ R |h(z)|q(σ 2 (1 − t), z − y)dz ≤ h ∞ . Moreover, H y (t, y) is strictly positive whenever t < 1. Indeed, differentiating above, we have On the other hand, where C = h(z) ∞ 2 σ 2 π . Hence, Lemma 4.1 implies the existence and the uniqueness of a strong solution to (4.29).
Next, we will characterise the distribution of Y on [0, T ] for T < 1 by constructing a weak solution to (4.29) via a Girsanov transform. To this end let W be a Brownian motion on some filtered probability space (Ω,F, (F t ) t∈[0,1] , Q). Then, M is a martingale on [0, T ] by Corollary 3.5.16 in [28]. Thus, if we defineP on (Ω,F) bỹ P/dQ = M T , σW solves (4.29) underP on [0, T ]. Due to the uniqueness in law of the solutions of (4.29), for any continuous and bounded function g we therefore have We next to aim to extend the above equality to T = 1, which would follow from the Dominated Convergence Theorem once we demonstrate that M is a bounded martingale. Direct calculations lead to where B(t, y) = − c σ 2 implying where M 1 := lim T →1 M T . Our next goal is to prove the estimate in (4.33) which will, in turn, imply that M 1 is strictly positive. Let τ be a stopping time with respect to the natural filtration of W and bounded by 1. Then, where B(t, y) is given by (4.35). A simple application of integration by parts on B(t, y) yields that |B(t, y)| ≤ 2c for some K that depends on σ and h ∞ only. The above estimate also shows that c t 0 W s H y (s, σW s )dW s is a square integrable martingale on [0, 1] with As {ω : M 1 (ω) = 0} ⊆ {ω : 1 0 W 2 s (ω) (H y (s, σW s (ω))) 2 ds = ∞}, this yields that M is strictly positive on [0, 1], Q-a.s. and Next lemma is not needed for the fixed point algorithm that we will consider in order to show the existence of a solution to the system (3.18)- (3.20). On the other hand, it shows that the solutions to (3.19) has a smooth transition density which is necessary to construct the equilibrium in our model.

Lemma 4.3
Let h be a non constant, bounded, nondecreasing, absolutely continuous function, the derivative of which is bounded on compacts. Consider the solution, H, of (3.18) with the terminal condition h. Then the unique strong solution, Y , of (4.29) admits a transition density p(s, y; t, z) in the sense that for any continuous and bounded g, E[g(Y t )|Y s = y] = R g(y)p(s, y; t, z)dz for all 0 ≤ s ≤ t ≤ 1. Moreover, for any fixed (t, z), p(s, y; t, z) > 0 on (0, t) × R and is C 1,2 ((0, t) × R).
Proof. Due to the Lemma 4.2 we have 0 < H y (t, y) ≤ C 1 √ 1−t for t < 1 and y ∈ R, where H is the solution of (3.18) with the terminal condition h, and C = 2 σ 2 π h ∞ . Furthermore, there exists a unique the solution, Y , to (3.19) and for any bounded function g and 0 ≤ t < u ≤ 1, Thus, we can define r(t, y; u, z) for all 0 ≤ t < u ≤ 1 by and the transition density of Y is given by p(t, y; u, z) = q(σ 2 (u − t), z − y)r(t, y; u, z). Using Ito formula and the PDE (3.18) satisfied by H we can obtain (recall that B(t, y) is given by (4.35)) r(t, y; u, z) where the first inequality is due to bounds on H y (t, y) and the last one due to the bounds on H(t, y) obtained in Lemma 4.2 .
In order to show that r(t, y; u, z) > 0 for all u ≤ 1 it suffices to show that Indeed, due the uniform bounds on H y , the nonnegative random variable inside the conditional expectation in (4.37) is zero only if u t Y 2 s H 2 y (s, Y s )ds = ∞. To this end fix an ω and observe that K t,u := sup t≤s≤u Y 2 s satisfies Q(0 < K t,u < ∞|Y t = y, Y u = z) = 1. Therefore, Observe that, for the fixed ω, Y is a continuous function of time and, therefore, takes values in a compact set, which implies that h ′ (z + Y s ) is bounded for z ∈ [−1, 1] and all s ∈ [t, u]. This implies that the first integral is finite since To see the finiteness of the second integral apply integration by parts to get Note thath is positive, therefore, the above integral is bounded from above by The third integral can be shown to be finite in the same way.
In order to show that p(t, y; u, z) ∈ C 1,2 ([0, u) × R) for fixed (u, z), where u < 1, we will show that it is the fundamental solution of a parabolic differential equation (see p. 3 of [20] for the definition of fundamental solutions). In view of the relationship between the fundamental solutions of PDEs and transition densities of diffusion processes (see the discussion following Definition 5.7.9 in [28]), let's consider the PDE i.e. H yy is uniformly bounded on [0, T ] × R. Thus, we have shown that condition ii) was satisfied. Since the constant functions solve the (4.39), condition iii) is satisfied as well; thus, a fundamental solution, Γ(t, y; s, z) to (4.39) exists. In particular, if one consider this PDE with the boundary condition u(T, y) = g(y) for some bounded g, the solution is given by On the other hand, since the SDE (4.29) satisfies the hypotheses of Theorem 5.7.6 in [28] on the time interval [0, T ], u has the following stochastic representation by this theorem: Thus, R g(z)p(t, y; T, z) dz = R g(z)Γ(t, y; T, z) dz, and since g is arbitrary, we deduce p(t, y; T, z) = Γ(t, y; T, z) for all 0 < t < T < 1 and, thus, it is C 1,2 on (0, T ) × R for T < 1.
Having collected all the prerequisites we can now prove our main theorem. Proof of Theorem 4.1. In the setting of Lemma 4.2 M defines an equivalent change of measure between the laws of Y and σW . Thus, if we define r(y) by and, therefore the probability density of Y 1 under P is given by  [20]) states that if D is a closed convex subset of a Banach space and T : D → D is a continuous operator, then it has a fixed point if the space T D is precompact, i.e. every sequence in T D has a subsequence which converges to some element of the Banach space. In order to apply this theorem, we first need to find a suitable Banach space which contains a class of probability distribution functions on R that is large enough to contain the distribution of Y 1 where Y is one of the components of the solution to the system of equations (3.18)- (3.20). In view of the above discussion, the distribution of Y 1 will be continuous, in fact it will admit a density. Thus, we may take C b (R), the space of bounded continuous functions on R, equipped with the sup norm as our underlying Banach space and set P as the space of absolutely continuous distribution functions on R, i.e. P ∈ P if P is increasing, P (−∞) = 0, P (∞) = 1, and there exists a measurable function P ′ such that P (y) = y −∞ P ′ (z) dz. Then, we can define the set The reason for this judicious choice of C * will become apparent when we define the operator T . We will prove the existence of a fixed point in four steps.
Step 1: D is a closed convex set. It is clear that D is convex. To see it is also closed, suppose that P n is a sequence of elements in D converging to some element, P , of the Banach space in the sup norm. Clearly, P is increasing with P (−∞) = 0 and P (∞) = 1. Moreover, for any x ≤ y in R, it follows from Fatou's lemma that since each P ′ n is bounded from above by the same integrable function, which in turn is an upper bound to the positive function lim sup n→∞ P ′ n . However, this implies that P is absolutely continuous and, in particular, there exists a function P ′ with 0 ≤ P ′ (z) ≤ lim sup n→∞ P ′ n (z) ≤ C * q(σ 2 , z) for all z ∈ R. To finish the proof that D is closed, we need to show Since P n converges P weakly, there exists a probability space supporting random variables (Y n ) n≥0 and Y such that Y n → Y , a.s., Y n has distribution P n , and Y has distribution P . Note that one can directly verify that which shows the uniform integrability of the sequence (Y n − x) + . Therefore, Similar arguments show the other inequality. Thus, D is closed.
Step 2: Defining the operator T . For any P ∈ D, let H : [0, 1] × R → R be the unique function which solves the following boundary value problem.
where Φ is the cumulative distribution function of a standard normal random variable.
Thus, by Lemma 4.2, for all t < 1, 0 < H y (t, y) ≤ C 1 To this H one can associate a unique process Y which solves (4.29) for c = σ 2 ρ 2N and Y 1 is a continuous random variable with the probability density q(σ 2 , y)r(y), where r is defined in (4.41). Thus, we can define Note that T P belongs to D due to (4.30), (4.31), and (4.42).
Step 3: T is precompact. Since T D is an equicontinuous family of functions, by a version of Ascoli-Arzela Theorem (see Corollary III.3.3 in [31]), if P n is a sequence in T D then it admits a subsequence which converges pointwise to P ∈ C b (R). Moreover, this convergence is uniform on every compact subset of R. This would mean that T D is precompact once we show that the convergence is uniform over all R.
To do so let's assume without loss of generality that P n itself is the convergent subsequence and consider any ε > 0. Due to the definition of D, there exist x * and x * such that Since P n converges to P pointwise, we also have with the same x * and x * that Since the convergence is uniform on the compact [x * , x * ], there exist a K such that for all n ≥ K sup Thus, for any n ≥ K we have Thus, we have shown that the convergence of P n to P is uniform on R, i.e. T D is precompact in C b (R) equipped with the sup norm. Hence, Schauder's fixed point theorem yields T has a fixed point provided T is a continuous operator, which we show next.
Step 4: T is continuous. To this end, let (P n ) n≥1 ⊂ D converge to P ∈ D in the sup norm. As T P n and T P belong to D, in view of Problem 14.8 (c) in [10], pointwise convergence of T P n to T P will imply uniform convergence since T P is continuous.
To each P n and P we can associate functions H n and H, B n and B (see (4.35)), and the processes M n and M from Lemma 4.2.
Pointwise convergence of T P n to T P will follow immediately once we can show that for any continuous and bounded function g In view of the uniform bound on M n and M due to (4.36), the above convergence will hold if we can show that M n 1 converges to M 1 in Q-probability. In order to get the estimates to prove this convergence first note that, due to Lemma 4.2 for any stopping time, τ , bounded by 1, we have for some K independent of n. This shows that c t 0 W s H n y (s, σW s )dW s is a square integrable martingale on [0, 1] with where K is a constant independent of n. Let As H n are uniformly bounded and H n y ≤ C 1 √ 1−t for t < 1, if we can show that H n (1, y) → H(1, y) and H n y (t, y) → H y (t, y) for all y ∈ R and t ∈ [0, 1), the above will immediately imply that N n 1 converges to 0, Q-a.s.. Moreover, it will also imply convergence in L p (Q) for all p ∈ [1, ∞) in view of the bound obtained in (4.33 Together with N n 1 → 0, Q-a.s. the above implies M n 1 → M 1 , Q-probability. Thus, it remains to show that H n (1, y) → H(1, y) and H n y (t, y) → H y (t, y) for all y ∈ R and t ∈ [0, 1). Indeed, lim n→∞ H n (1, y) = lim n→∞ f (Φ −1 (P n (y))) = f (Φ −1 (P (y))) in view of the continuity of f • Φ −1 on (0, 1) and the fact that the sequence (P n (y)) converges to a limit P (y) ∈ (0, 1) for any y ∈ R due to the definition of D.
Next, observe that for t < 1 As H n and H are bounded by f ∞ , the convergence to 0 follows from the dominated convergence theorem and that H n (1, y) → H(1, y) as n → ∞. Thus, we have verified that T is continuous operator, D is a closed and convex subset of a Banach space and T D is precompact. Therefore, by Schauder's fixed point theorem, T has a fixed point P , i.e. T P = P . For this P , define H as the solution to (4.43) and Y as the corresponding unique solution to (3.19). Then (H, Y ) is the solution to the system of equations (3.18)- (3.20).
To finish the proof of the theorem, we need to show that the solution to (3.19) has a transition density with the required smoothness and positivity properties. This follows from the Lemma 4.3 once we observe that h(z) := f (Φ −1 (P (z))) satisfies the required conditions. It is obvious that h is bounded (since f is), non constant and nondecreasing (as f , Φ, and P are). Moreover, h ′ (y) = f ′ (Φ −1 (P (y)))(Φ −1 ) ′ (P (y))P ′ (y), is well defined for all y ∈ R as P (y) ∈ (0, 1) for all P ∈ D and y ∈ R, and therefore h is absolutely continuous. Let K ⊂ R be a compact, then since P is continuous, P (z) ∈ K 1 for all z ∈ K, where K 1 ⊂ (0, 1) is also a compact. As Φ −1 ∈ C 1 ((0, 1)), this implies that (Φ −1 ) ′ (P (y)) is bounded for all y ∈ K. Similarly, f ′ (Φ −1 (P (y))) is bounded for all y ∈ K. As boundedness of P ′ follows from the fact that P ∈ D, this yields that h ′ is bounded on compacts and therefore satisfies conditions of Lemma 4.3.

Construction of the equilibrium
Suppose H is the function determined in Theorem 4.1. As briefly discussed in Section 3, if we can identify an admissible strategy X such that: i) α is given by , and ii) X 1 satisfies (3.23), then (H, X) will be a candidate equilibrium in view of Proposition 3.1 once we show that U(G) is a true F M -martingale. The following theorem gives such an X.
Proof. We will first show that there exists a unique weak solution to (5.47) on [0, T ] for any T < 1. Then, Proposition IX.3.2 in [39] will imply the uniqueness of strong solutions since if Y 1 and Y 2 are two strong solutions, then Y 1 − Y 2 satisfies for some deterministic function b and, therefore, its local time process at level 0 is identically 0. The strong uniqueness combined with a weak solution will lead to the existence of a unique strong solution by a result due to Yamada and Watanabe (see Corollary 5.3.23 in [28]). To show the existence of a weak solution fix T < 1 and let N t := p(t, ζ t ; 1, H −1 (1, v)) for t ≤ T where v ∈ R and ζ is the unique strong solution of on [0, 1] under a probability measureP, where β is aP-Brownian motion as established in Theorem 4.1. The same theorem also gives p as the transition density of ζ. Then, (N t ) t∈[0,T ] is a strictly positive and bounded martingale with respect to the natural filtration of ζ as a consequence of Ito formula and the estimates on p obtained in (4.38). Thus, N T N 0 has expectation 1 underP and defines an equivalent change of measure on the σ-algebra F ζ T . Since then it follows from Girsanov's theorem that under the new measure, Q T , for some Q T -Brownian motion. Thus, ζ, as a solution of the above under Q T is a weak solution of (5.47) on [0, T ]. Moreover, the weak uniqueness holds since the distribution of ζ under Q T has a one-to-one correspondence with the distribution of ζ under the original measure via the the change of measure martingale p(t, ζ t ; 1, H −1 (1, v)). More precisely, for any bounded function F and points 0 = t 0 < . . . < t n = T E Q T [F (ζ t 1 , . . . , ζ tn )] = EP F (ζ t 1 , . . . , ζ tn ) p(T, ζ T ; 1, H −1 (1, v)) p(0, 0; 1, H −1 (1, v)) = R . . . R F (y 1 , . . . , y n ) p(0, 0; t 1 , y 1 ) . . . p(t n−1 , y n−1 ; t n , y n )p(T, y n ; 1, H −1 (1, v)) p(0, 0; 1, H −1 (1, v)) dy 1 . . . dy n .

(5.48)
Hence, we conclude the existence of a unique strong solution, Y T , of (5.47) over the interval [0, T ] under P v . Define Y by Y t = Y T t 1 t≤T and observe that due to the uniqueness of strong solutions Y is well-defined and is the unique process that solves (5.47).
Next we want to extend the process Y to time-1 by considering its limit. This limit exists in view of Theorem 2.4 in [15]. Moreover, the same result also yields Thus, it remains to show the claimed representation of Y under F M . Suppose that ξ is a solution of ξ s H y (s, ξ s ) ds.
The above theorem shows that the equilibrium demand process has a drift in its own filtration. This is in contrast with the other possible generalizations found in the literature (for the change in the pattern of private information arrival see [5], for a risk averse insider see [7], and for competition among insiders see [4]) of the original models of [30] and [2] lead to equilibria with total demand being a martingale in its own filtration.
Moreover, as H y > 0 the equilibrium total demand process is mean reverting. This suggests a theoretical explanation for the emergence of mean reversion in the specialists' inventories, which has strong empirical support (see, e.g., [12], [23], [25], and [35]): The mean reversion appears as a result of the insider's reaction to the market maker's demand for risk sharing. The speed of mean reversion is not constant and depends on the market makers' level of effective risk aversion, ρ N , as well as the level of informational asymmetry, σ, in a non-trivial way due to the definition of H y . This theoretical implication is in line with the empirical findings of [23] who observe that the speed of mean reversion depends on the inventory levels of the market makers in the London Stock Exchange.
Closely related to the observation that the total order has a drift, is the fact that the equilibrium price is no longer a martingale under the physical measure. Moreover, Y * and, therefore, H * (t, Y * t ) are mean-reverting processes. This meanreversion property of Y * also entails that Kyle's conclusion of constant market depth, which is the order size necessary to move the prices by one unit, does not hold in this model. Indeed, is not a martingale since H * is not linear. In particular, if H * is such that H * yy (1, y) = −H * yy (1, −y) with H * yy (1, y) ≤ 0 for y ≥ 0, then yH * yy (t, y) ≤ 0 and thus H * y (t, Y * t ) is a submartingale. Consequently, E[H * y (t, Y * t )] has an upward slope, i.e the executions costs increase in time in our model. This is consistent with the empirical findings of U-shaped patterns of execution costs on NYSE (see [34]).

Conclusion and further remarks
We have solved a long-standing open problem of existence of an equilibrium in a financial market with asymmetrically informed traders and risk averse market makers in a continuous-time version of a model first introduced by Kyle [30]. The equilibrium turns out to be the solution of a non-standard FBSDE. We've solved this FBSDE by transforming it into a forward-backward system of stochastic and partial differential equations and employing a novel application of Schauder's fixed point theorem.
Consistent with the empirical studies on the inventories of market makers we find that the risk aversion of market makers causes mean reversion in the equilibrium total demand (i.e. collective inventory of the market makers). This implies that the informed trader's strategy ceases to be inconspicuous and, therefore, provides the first example of an equilibrium in a Kyle-type model which does not satisfy inconspicuousness condition. The driving force behind this result is that the risk aversion of market makers makes them unwilling to bear risk. Instead of paying the extra compensation for the inventory risk, the informed trader chooses to absorb a part of large fluctuations in the market makers' inventories, i.e. participates in a risk sharing.
We also show that the sensitivity of prices to the total order, which is the reciprocal of the market depth, can be a submartingale for certain model parameters. This implies that the execution costs are, on average, increasing toward the end of a trading period, which is consistent with the empirical results obtained in [34].
Whereas, for general set of the parameters, the reciprocal of the market depth is not a submartingale, it is not a martingale either. This theoretical conclusion is in discord with the results obtained in [30], as well as in [7], who studies the effect of risk averse insider on the equilibrium, and in [5], who extend Kyle's model to the case when the informed trader receives a fluctuating signal over time. In fact, Kyle in [30] made a conjecture that: [...] neither increasing nor decreasing depth is consistent with behavior by the informed trader which is "stable" enough to sustain an equilibrium. If depth ever increases, the insider wants to destabilize prices (before the increase in depth) to generate unbounded profits. If depth ever decreases, the insider wants to incorporate all of his private information into the price immediately.
Thus, the results obtained from our model demonstrate that the necessity of risk sharing between the informed trader and the market makers makes exploitation of systematic movements in market depth unprofitable for the informed trader. Indeed, if the trader attempts to follow the strategy outlined by Kyle, i.e. acquiring a large position when depth is lower in order to liquidate at more favourable price when depth is higher, she would be moving the total order away from its mean, leaving the market makers exposed to the risk of large orders. Violation of risk sharing would cause the market makers to adjust the prices eliminating favourable liquidation opportunities for the informed trader. Thus, contrary to Kyle, such a strategy does not lead to unbounded profits.
Moreover, the appearance of systematic changes in market depth as a result of market makers' risk aversion demonstrates that competition of the informed traders, as in [4], is not the only possible mechanism that can make the strategy proposed by Kyle unprofitable, thus leading to a drift in the reciprocal of the market depth.
These observations show that a mere introduction of risk averse market makers to the setting of [30] changes the equilibrium outcome fundamentally.